typical entanglement of small quantum systems in a d-dimensional hilbert space h ’cd. a system s=...

Universita degli Studi di Bari Aldo Moro

FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Magistrale in Fisica

Typical Entanglementof Small Quantum Systems

Relatori:

Prof. Paolo FacchiProf. Saverio Pascazio

Laureando:

Cunden Fabio Deelan

Anno Accademico 2010/2011

I

Abstract

Let a pure state vector ∣ ⟩ be chosen randomly in an NM -dimensional Hilbert space, and

consider the reduced density matrix �A of an N -dimensional subsystem. What are the

typical properties of �A? This work is an attempt to provide a partial answer to this

question. For unbiased states, with no a priori information, “random” refers to the unitarily

invariant Haar measure. We have investigated both unbiased states and fixed-purity states.

In this work we have followed a general approach for studying the statistics of bipartite

entanglement, relying on the techniques of classical statistical mechanics. In order to study

these statistical properties, our first aim has been to find the most probable eigenvalues of

the reduced density matrix �A by means of a saddle point method. More precisely, our task

has been to investigate in detail how much mixed the subsystem is, and if and how much

residual entanglement in the subsystem can be found.

Contents

Introduction 1

1 Quantum States and Entanglement 3

1.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Measures of entanglement . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Entanglement everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Pair of qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Photon: two degrees of freedom . . . . . . . . . . . . . . . . . . . . . 15

2 Random Matrices and Random States 17

2.1 Topological groups and Haar measures: an overview . . . . . . . . . . . . . 18

2.1.1 Groups of matrices and Haar measure . . . . . . . . . . . . . . . . . 18

2.2 Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Random unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Random pure states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Induced measure in the space of mixed quantum states . . . . . . . . 21

2.3.2 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Joint distribution of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Measure on the Bloch ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Most probable distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Numerical investigation 31

3.1 Numerical simulations: random sampling . . . . . . . . . . . . . . . . . . . 32

3.2 Numerical simulations II: entanglement within the pair . . . . . . . . . . . . 34

IV CONTENTS

4 Typical entanglement 37

4.1 Saddle point equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Heuristic discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Simplified saddle point equations . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 The typical entanglement spectrum . . . . . . . . . . . . . . . . . . . . . . . 42

4.6 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 A single qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Four-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8.1 Behavior of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.8.2 Measures of entanglement . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Isopurity typical states 57

5.1 Isopurity manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 The single qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Small deviation from typical purity . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Balanced bipartition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 High purity regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.6 Four-level system: numerical solutions . . . . . . . . . . . . . . . . . . . . . 65

5.6.1 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6.2 Elementary symmetric invariants . . . . . . . . . . . . . . . . . . . . 69

6 Entanglement in mixed states 77

6.1 Separability criteria for mixed states . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Separability of the reduced density matrix . . . . . . . . . . . . . . . . . . . 78

6.3 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 An integral over the unitary group . . . . . . . . . . . . . . . . . . . . . . . 80

Conclusions 87

A Probability measure on the ensemble of Hermitian matrices 91

B Vieta’s formula 95

C Stieltjes trick 97

CONTENTS V

D Orthogonal polynomials 99

References 99

Introduction

Entangled states are inherently quantum in the sense that there is no classical analog for

them. Classically, if we put together a system encoding n bits of information with another

one that stores m bits of information, the total capacity of the global system is n+m bits.

In other words, the number of degrees of freedom of a “sum” of classical systems, is the

sum of the degrees of freedom of their parts. For quantum systems, the sum of their parts

has nm degrees of freedom. This fundamental property of quantum systems arises from

the tensor product structure of the Hilbert space of a composite system, whereas classical

phase space is the Cartesian product of the component phase spaces.

Some peculiar techniques of quantum information (such as dense coding, teleportation,

cryptography, etc.) involve processes which make use of entangled states, and for this reason

entanglement is considered a fundamental resource, and it appears increasingly important

to quantify it.

The purpose of this work is to tackle the following problem: let a random pure state

vector ∣ ⟩ be chosen in an NM -dimensional Hilbert space, and consider the reduced density

matrix �A of an N -dimensional subsystem. For unbiased states, with no a priori information,

“random” refers to the unitarily invariant Haar measure. The reduced density matrix �A

will usually be close to the totally unpolarized mixed state if the bipartition is strongly

unbalanced (if M is large). The operation of tracing out generates mixedness. In all realistic

systems, the mixedness of the density matrix describing the system is inevitably increased

by the correlation with its surrounding environment.

We are interested in typical properties of �A. In this work we will use a general approach

for studying the statistics of bipartite entanglement, relying on techniques of classical

statistical mechanics. In classical statisical mechanics a system may be in any of the

admissible states but one is not interested in which particular configuration the system is.

2 Introduction

We will consider an ensemble of density matrices each of which is a particular state of the

subsystem. As in orthodox statistical mechanics, there is a overwhelming probability that

the properties of a member of this large supply will be correctly described by an ensemble

average. Only an exponentially small number of members will deviate appreciably from the

ensemble average.

The ensemble of reduced states �A is correctly described by a probability measure. It

turns out that, due to the spectral decomposition, the appropriate probability density

function factorizes in a product measure; one is responsible for the choice of the eigenvectors,

the other for the choice of the eigenvalues. Many of the interesting properties of a mixed

state are encoded in its spectrum.

Then our aim will be to find the most probable spectrum for the reduced density matrix

of the subsystem by means of a saddle point method. More precisely, our task will be

to investigate in detail how much the subsystem is mixed, and if and how much residual

entanglement in the subsystem can be found.

The outline of this thesis is as follows: the first chapter is devoted to a quick overview

of the quantum physical framework; in Chapter 2, we review the basic tools of Random

Matrix Theory which are necessary to an unambiguous formulation of our problem. The last

chapters are entirely devoted to its solution: in Chapter 3 we will present numerical results

which have guided our research; the topic of Chapters 4 and 5 are the solutions (exact and

approximated) of the saddle point equations, which give the most probable spectrum of the

reduced density matrix of a subsystem, when the sampling is done according to various

measures. Finally, in Chapter 6 we consider a four-level system as a two-qubit system, and

we try to compute the residual entanglement within the pair.

Some technical issues and some extensively used tools are discussed in the appendices.

The Bibliography contains a list of all reference material. Some references are not explicitly

cited in the text, nevertheless they have been indispensable to a better understanding of

the subject.

Chapter 1

Quantum States and Entanglement

Nowadays, entanglement is viewed as a fundamental resource in quantum applications

and quantum information. Though in mathematical terms entanglement has a simple

definition, its consequences are among the most puzzling traits of quantum mechanics. The

primary goal of this chapter is to present an overview of basic mathematical concepts and

tools of quantum mechanics and quantum information. In the last section we will present

some familiar physical systems that exhibit entanglement in a natural way.

1.1 Mathematical Preliminaries

This overview will serve to set our notation and highlight the main concepts that we

will need. In particular we will define and briefly discuss entanglement and how to quantify

it. Let us give some basic definitions.

1.1.1 Quantum States

Definition 1 (Qudit and System of qudits). We define qudit a quantum system living

in a d-dimensional Hilbert space ℋ ≃ ℂd. A system S = {1, . . . , l} of l qudits is a quantum

system with a dl-dimensional Hilbert space ℋS =⊗

1≤i≤lℋi, ℋi ≃ ℂd. Its pure states are

normalized vectors ∣ ⟩ ∈ ℋS , ⟨ ∣ ⟩ = 1.

As an example (the most popular one), a quantum system with a two-dimensional

Hilbert space is called a qubit. Qubits, or quantum bits, are the basic building blocks that

4 Quantum States and Entanglement

provide a mathematically simple framework in which to introduce the basic concepts of

quantum physics. Qubits are those physical systems that we can describe well enough as

two-states quantum systems. The most popular examples among physicists, are provided

by the polarization of a photon, the spin of a 1/2-spin particle, or the atomic orbitals (in

the situations where only two energy levels are relevant).

The concept of entanglement naturally arises in the context of composite quantum

systems. A composite quantum system is a tensor product space. The dimension of the

global system is the product of the dimensions of its factor spaces.

Definition 2 (Bipartition). A bipartition of S is a pair (A,B) with ∣A∣ = n and ∣B∣ = m

s.t. 0 < n ≤ m < l and n + m = l. Given a bipartition (A,B) the total Hilbert space

of a system of l qudits is partitioned into ℋS = ℋA ⊗ℋB, with ℋA =⊗

1≤i≤nℋi and

ℋB =⊗

1≤i≤mℋi:

L = dimℋS = dl = dn+m = dndm = dimℋA dimℋB = NM .

Definition 3 (Entanglement). A state ∣ ⟩ ∈ ℋS is said to be separable with respect to

the bipartition (A,B) if it can be expressed as a tensor product ∣ ⟩ = ∣�⟩ ⊗ ∣�⟩ for some

∣�⟩ ∈ ℋA and ∣�⟩ ∈ ℋB. A state that is not separable is called entangled.

Remark. Note that what has really meaning is entanglement with respect to a given

(bi)partition. Entanglement is a remarkable property which naturally arises when one

endeavours to answer to questions like: “given a pure quantum state ∣ ⟩ ∈ ℂN ⊗ ℂM , can

it be written as a product ∣�⟩ ⊗ ∣�⟩?”.

For pure states there is a very simple necessary and sufficient criterion for separability:

the Schmidt decomposition.

Lemma 1.1.1 (Schmidt decomposition). Given a bipartition (A,B) every state ∣ ⟩ ∈ℋS can be written as

∣ ⟩ =

N∑k=1

√�k ∣uk⟩ ⊗ ∣vk⟩ (1.1)

with �k ≥ 0,∑N

k=1 �k = 1, and {∣uk⟩} {∣vk⟩} orthonormal sets of ℋA and ℋB, respectively.

The number r of nonzero Schmidt coefficients is called Schmidt number. A pure state ∣ ⟩is separable iff r = 1.

1.1 Mathematical Preliminaries 5

Remark. The Schmidt coefficients {�k} are uniquely determined, up to their ordering. This

useful decomposition is possible, in general, only for bipartite systems.

Definition 4 (Density operator). A density operator � is a nonnegative (self-adjoint)

linear operator having trace equal to one.

The set of all bounded linear operators B : ℋ 7→ ℋ is denoted ℒ(ℋ). We will write

D(ℋ) ={� ∈ ℒ(ℋ) : � = �† , � ≥ 0 , Tr� = 1

}to refer to the collection of all density operators on ℋ. A density matrix � ∈ D(ℋ) is said

to be pure if it has rank 1, which is equivalent to it taking the form ∣ ⟩ ⟨ ∣ . A quantum

state is a density operator acting on the Hilbert space associated with the quantum system.

An ensemble interpretation of density matrices arises by writing the spectral decomposition

of �:

� =L∑i=1

�i ∣ i⟩ ⟨ i∣ . (1.2)

The eigenvalues of � are nonnegative and sum up to one. Physicists like to consider

(�1, . . . , �L) as a probability vector which weights the ensemble of states {∣ i⟩} (statistical

interpretation of �).

Remark. D(ℋ) is a convex subset of the real linear space ℳ(ℋ) of Hermitian operators,

whose extremal points are the pure states.

Definition 5 (Reduced density matrix). If ∣ ⟩ ∈ ℋS is a pure state of system S, the

reduced density matrix of the subsystem A ⊂ S is

�A = TrB(∣ ⟩ ⟨ ∣) , (1.3)

with B = S∖A.

It is easy to verify that �A is a self-adjoint, nonnegative, unit-trace matrix N × N .

Given a bipartition (A,B), a state ∣ ⟩ ∈ ℋS is separable iff its reduced density operator

can be written as �A = ∣'⟩ ⟨'∣ for some normalized ∣'⟩ ∈ ℋA; the state is said maximally

entangled iff �A = 1/N , N being the dimension of ℋA.


Definition 6 (Purification). Any mixed state �A ∈ D(ℋA), acting on ℋA, can be purified

by finding an ancillar space ℋB and a pure state ∣ ⟩ ⟨ ∣ ∈ D(ℋA ⊗ℋB) in the composite

Hilbert space, such that �A is given by the partial trace over the auxiliary subsystem,

TrB (∣ ⟩ ⟨ ∣) = �A . (1.4)

Remark. Notice that a purification of �A ∈ D(ℋA) is a pure state on a tensor product space

whose ℋA is a factor, that yields �A when traced out. A necessary and sufficient condition

for a purification of �A to exist in ℋA ⊗ℋB is that dimℋB ≥ rank �A.

Many interesting and useful norms can be defined on the linear space of bounded

operators ℒ(ℋ).

Definition 7 (Schatten norms). For any A ∈ ℒ(ℋ) and any p ≥ 1, we define the

Schatten p-norms of A as

∥A∥p =[Tr(A†A)p/2

]1/p. (1.5)

Remark. The p-norms satisfy many properties (for more details, see [23]), e.g. ∀p ≥ 1, ∥⋅∥pis unitarily invariant. The Schatten 1-norm is commonly known as the trace norm:

∥A∥1 = Tr(√A†A) . (1.6)

This norm is linked to some commonly used quantities such as fidelity and concurrence.

The usual Euclidean norm of ordinary vectors is reproduced by the Schatten-2 norm, called

also Hilbert-Schmidt (or Frobenius) norm:

∥A∥2 =[Tr(A†A)

]1/2. (1.7)

For p =∞ one recover the usual operator norm:

∥A∥∞ = sup {∥A ∣ ⟩∥ : ⟨ ∣ ⟩ = 1} . (1.8)

There are several way to quantify the degree of similarity of two states. One of them is

the fidelity function, that is closely related with the notion of purification.

Definition 8 (Fidelity). Given two nonnegative operators A,B ∈ ℒ(ℋ), we define their

fidelity as

F (A,B) =∥∥∥√A√B∥∥∥

1= Tr

(√√AB√A). (1.9)


Fidelity is a symmetric function and multiplicative with respect to tensor product.

When restricted to pure quantum states, fidelity is equal to the overlap of vector states, i.e.

the transition amplitude:

F (∣ ⟩ ⟨ ∣ , ∣�⟩ ⟨�∣) = ∣ ⟨ ∣ �⟩ ∣ . (1.10)

Moreover, for mixed states �, � ∈ D(ℋ), it turns out that

F (�, �) = max{∣ ⟨ ∣ �⟩ ∣ : ∣ ⟩ , ∣�⟩ purifications of �, � respectively

}. (1.11)

It is not difficult to prove that 0 ≤ F (�, �) ≤ 1; F = 1 iff � = �, and F = 0 for density

operators with mutually orthogonal ranges. Fidelity is also jointly convex (convex in both

arguments).

Definition 9 (Separable density operators). A density operator � ∈ D(ℋA′ ⊗ℋA′′) is

said separable iff there exist a positive integer p and density operators

�′1, . . . , �′p ∈ D(ℋA′) , �′′1, . . . , �

′′p ∈ D(ℋA′′) ,

and a p-dim probability vector {wi}1≤i≤p, s.t.

� =

p∑i=1

wi(�′i ⊗ �′′i ) . (1.12)

We write S(ℋA′ ⊗ℋA′′) to denote the collection of all such operators.

Remark. A mixed state � is said entangled if it cannot be written as a convex combination

of product states. The maximum integer p in the previous decomposition is dimℋA′ +

dimℋA′′ + 2.

Despite the simplicity of the above definition, it is in general quite hard to check whether

a nonpure state is separable or not, with respect to a given bipartition. This has led to the

introduction of some criteria in order to properly check the separability of states. There is

a simple algebraic test, due to Peres [16], which is a necessary condition for the existence of

the decomposition eq. (1.12). Preliminary, let us define superoperators, that play a key

role in quantum mechanics.

Definition 10 (Linear superoperators). A superoperator from ℋ into K (ℋ, K Hilbert

spaces) is a linear mapping of the form Φ : ℒ(ℋ)→ ℒ(K). The set of all such mappings is

denoted T(ℋ,K).


Definition 11 (Positive Partial Transpose test). A separable density operator �A ∈D(ℋA′ ⊗ℋA′′) has positive partial transpose (PPT):

�A ∈ S(ℋA′ ⊗ℋA′′)⇒ (TA′)�A ≥ 0 , (1.13)

where the superoperator TA′ = (TA′ ⊗ 1A′′) is named partial transpose superoperator.

Explicitly:

�A =

p∑i=1

wi(�′i ⊗ �′′i ) ∈ S(ℋA1 ⊗ℋA2)⇒ � =

p∑i=1

wi((�′i)T ⊗ �′′i ) ≥ 0 . (1.14)

Remark. For systems of dimensions 2 × 2 and 2 × 3 the partial transposition condition

is also sufficient, and thus the set of separable states is completely characterized by this

condition.

1.1.2 Measures of entanglement

The problem of quantifying entanglement is a difficult issue, and a number of entangle-

ment measures have been proposed. However, for pure quantum states, measures of bipartite

entanglement, that is the entanglement of a subset of the qudits with the complementary

subset, involve essentially the Schmidt coefficients. One could rely on the Schimdt number

r, but soon one realizes that this is not very satisfactory, in the sense that it does not

capture all aspect of entanglement. Bipartite pure states represent one of the few cases for

which the problem of revealing and also quantifying quantum correlations has an exhaustive

answer. Bipartite entanglement is essentially measured by the von Neumann entropy of

the reduced density matrix (this statement will be justified later). At least, one can rely

on the linearized version of it, that turns out to be a more convenient measure from a

computational point of view.

We will use three quantities that measure the degree of mixedness of a state: von

Neumann entropy, purity and visibility. All these quantities are locally invariant, i.e.

invariant under local unitary transformations, since they depend only on the spectrum {�i}of the state �A. The eigenvalues of the reduced density matrix form a probability vector

and the degree of mixedness of a state is linked to how “nearly equal” the eigenvalues are.

To this end, it is useful to have in mind the basic ideas of majorization. Majorization theory

provides a way to capture the idea that a probability distribution is more “uniform” or more

“mixed” than another one (see Fig. 1.1). In the following we will refer to finite-dimensional


æ

æ

ææ æ æ æ æ æ æ

à à à à à à à à à à

1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

xi¯

yi¯

æ

æ

ææ

æ æ æ æ æ æ

à

à

à

à

à

à

à

à

à

à

1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

âi=1

k

xi¯

âi=1

k

yi¯

Figure 1.1: Majorization: two probability vectors to explain eq. (1.15). The blue vector (squares)

is majorized by the the red one (cirles). Figures show the components of the ordered vectors and

cumulative probabilities.

probability distributions (probability vectors) as real vectors belonging to the positive

hyperoctant ℝN+ of ℝN whose components sum up to one. Let us introduce the following

notation.

Definition 12 (Descending ordering). For every vector x ∈ ℝN+ , we write x↓ to denote

the vector obtained by sorting the entries of x from largest to smallest x↓i ≥ x↓i+1.

Definition 13 (Majorization). Consider two vectors x and y in ℝN+ . We say that x is

majorized by y:

x ≺ ydef⇔

⎧⎨⎩∑k

i=1 x↓i ≤

∑ki=1 y

↓i , ∀k ∈ {1, . . . , N} ,

∑Ni=1 xi =

∑Ni=1 yi = 1 .

(1.15)

Remark. The relation ≺ provides a partial ordering in the set of probability vectors of fixed

dimension. An equivalent characterization of majorization is as follows: we say that x is

majorized by y iff there exist a double stochastic linear operator B s.t. x = By. Thus, when

x ≺ y we can imagine that y is the input probability vector to a noisy channel described by

the doubly stochastic matrix B, inducing a more disordered output probability vector, x.

Roughly speaking, the statement x ≺ y means that y is “more ordered” than x, since

we can reach x from y by a stochastic operator. We state two important facts: for every

probability vector x: (1

N, . . . ,

1

N

)≺ x ≺ (1, 0, . . . , 0) ,


and (1

N, . . . ,

1

N

)≺(

1

N − 1, . . . ,

1

N − 1, 0

)≺ ⋅ ⋅ ⋅ ≺

(1

2,1

2, 0 . . . , 0

)≺ (1, 0 . . . , 0) .

Definition 14 (Schur-convexity). Real-valued functions which preserve majorization

order are called Schur convex :

x ≺ y =⇒ f(x) ≤ f(y) . (1.16)

A function f(x) is said Schur concave if −f(x) is Schur convex.

A useful characterization for Schur-convexity is provide by the following result.

Theorem 1.1.2 (Schur). Let f : ℝN+ → ℝ be a real-valued function with continous first

partial derivatives ∂f∂xi

. Then

x ≺ y implies f(x) ≤ f(y)

iff f(x1, . . . , xN ) is permutation invariant and

(xi − xj)(∂f

∂xi− ∂f

∂xj

)≥ 0 , ∀x . (1.17)

Definition 15 (Visibility). A very quick way to get an idea of the degree of entanglement

of a state � is to compute the visibility, that is

v(�) =�max − �min

�max + �min, (1.18)

where �max = max {�1, . . . , �N} = �↓1 and �min = min {�1, . . . , �N} = �↓N . It is easy to

check that 0 ≤ v(�) ≤ 1; the visibility attains its minimum value 0 iff the state is maximally

entangled, and reaches its maximum v(�) = 1 iff the state is pure. Visibility is a nice

quantity since the range of v(�) is a compact interval which does not depend on dimℋ.

Remark. The normalization condition constraints the extreme eigenvalues: 1/N ≤ �max ≤ 1

and 0 ≤ �min ≤ 1/N .


Definition 16 (Purity). A good measure of entanglement is purity

�AB = TrA(�2A) = TrB(�2

B) =∑i

�2i , 1/N ≤ �AB ≤ 1 . (1.19)

Purity has the Schur-convexity property; �AB = 1 and �AB = 1/N iff ∣ ⟩ is, respectively,

separable and maximally entangled with respect to the given bipartition.

Remark. The inverse of purity R = 1Tr�2A

is known as partecipation ratio. It ranges from 1

(separable states) to N (fully entangled), and it has an interpretation as the effective rank

of �A, i.e. the effective number of orthogonal pure states in the mixture.

Definition 17 (von Neumann entropy). The von Neumann entropy of a state � is

S(�) = −Tr(� log �

)= −

N∑i=1

�i log �i .(1.20)

Logs are understood to have base 2 unless otherwise specified. Entropy is a nonnegative,

Schur-concave quantity. It has also an upper bound: entropy is maximized when the

quantum state is maximally entangled S(� = 1/N) = logN . The von Neumann entropy

has many other nice properties which are not relevant for our purposes.

Schur-concave functions of eigenvalues include the elementary symmetric polynomials

(they are permutation invariant and eq. (1.17) holds for all N -uples).

Definition 18 (Elementary invariants). Let the elementary invariant sj be the quantity

formed by adding together all distinct products of the N eigenvalues of a matrix A, i.e. the

elementary symmetric polynomial ej (see appendix B) evaluated on the eigenvalues �i:

s1(A) = e1(�1, . . . , �N ) =∑j

�j

s2(A) = e2(�1, . . . , �N ) =∑j1<j2

�j1�j2

s3(A) = e3(�1, . . . , �N ) =∑

j1<j2<j3

�j1�j2�j3

...

sN (A) = en(�1, . . . , �N ) = �1�2 . . . �N .

(1.21)


These functions provide information on how much a state � is separable or mixed. It is

easy to verify that

0 ≤ sk(�) = ek(�1, . . . , �N ) ≤ 1

Nk

(N

k

), (1.22)

since each probability vector � = (�1, . . . , �N ) majorizes the uniform population, � ≻(1/N, . . . , 1/N).

Measures of entanglement are quantities that try to answer two subtle questions. The

first one is: “how much does it cost to create a certain state �?”, the building blocks being

Bell pairs and the allowed operation being LOCC. The acronym LOCC stands for “Local

(quantum) Operations and Classical Communcations”. The second question is essentially

the opposite: “how much can we extract out from a state?”; again the “unit measure” are

intended to be Bell pairs. Entanglement cost and distillable entanglement are fundamental

quantities if one looks at entanglement as a resource. From this point of view the question

“how much a state is entangled” acquires sense.

Definition 19 (Ebit). We define unit of entanglement, or simply ebit, the pure state

∣�+⟩ ∈ ℂ2 ⊗ ℂ2 expressed in Dirac notation as∣∣�+⟩

=1√2

(∣00⟩+ ∣11⟩) . (1.23)

Other commonly used variations are singlet state, Bell pair or EPR state. The ebit ∣�+⟩ is

a pure maximally entangled state (its reduced density matrix is 1/2).

Definition 20 (Entanglement cost). Given a state � ∈ D(ℋ), its entanglement cost is

defined as the the minimal number EC(�) of Bell pairs (ebits) needed to produce � by

LOCC operations.

Definition 21 (Distillable entanglement). The distillable entanglement from � ∈ D(ℋ)

is the maximal number ED(�) of Bell pairs (ebits) can be produced from � by means LOCC

operations.

Remark. The above definitions must be understood in a asymptotic sense. Given n copies

of �, we can extract at most nED(�) singlet states from it, in the n→ +∞ limit. Similarly,


we can construct n states � with at least nEC(�) singlets, in the n→ +∞ limit. In these

building/distillating processes, the only allowed protocols are LOCC, and the monitoring

quantity is usually the fidelity, eq. (1.9).

The following theorem justifies the choice of von Neumann entropy S as the proper

measure of bipartite entanglement.

Theorem 1.1.3. For all pure states ∣ ⟩ ∈ ℋA ⊗ℋB:

EC(∣ ⟩) = ED(∣ ⟩) = S(TrA ∣ ⟩ ⟨ ∣) = S(TrB ∣ ⟩ ⟨ ∣). (1.24)

Definition 22 (Convex roof). Consider a continous real function g on the space of pure

states ∣ ⟩ ∈ ℋ. The real function gext : D(ℋ) 7→ ℝ is a convex-roof extension of g if it

coincides with g on pure states, and

gext(�) = min∑i

pig(∣ i⟩ ⟨ i∣) (1.25)

in the space of states D(ℋ), where the minimum is taken over all decompositions of � as

convex combinations.

For a four-level system regarded as a bipartite system of two qubits, one can avoid the

convex roof construction and use a more manageable quantity.

Definition 23 (Concurrence). For a two-qubit state � ∈ D(ℂ2 ⊗ ℂ2), let us define the

spin-flipped matrix

� = (�y ⊗ �y) �∗ (�y ⊗ �y) , (1.26)

where symbols ∗ and �y denote, respectively, conjugation and the second Pauli matrix in

the computational basis. Then, the concurrence is defined as

C = max{

0, c↓1 − c↓2 − c

↓3 − c

↓4

}, (1.27)

where ci are the eigenvalues of the matrix

R =√√

��√� . (1.28)


It has been proved [25] that concurrence is a measure of entanglement. C(�) = 0 for

separable states and C(�) = 1 for the fully unpolarized state. Concurrence has the handicap

of being a measure of entanglement just for two-qubit states.

Remark. Note that TrR is the fidelity F (�, �) between the state � and the spin-flipped

matrix � ≥ 0.

1.2 Entanglement everywhere

Most people look at entanglement as something exotic and mysterious. In fact, quantum

mechanical phenomena are far from an intuitive comprehension, and entanglement is one of

the properties of quantum states that makes quantum theory “strange” from a classical

perspective. However, entanglement is everywhere. Schrodinger once remarked that entan-

glement is the characteristic trait of quantum mechanics [19]. Let us end this introductory

chapter with a few examples. Entanglement is nothing but a natural consequence of the

superposition principle in quantum mechanics. When applied to composite systems, the

superposition principle is responsible for the existence of nonseparable states. Therefore,

any (composite) quantum system can exhibit peculiarly quantum correlations, that is

entanglement.

1.2.1 Pair of qubits

The simplest possible scenario that allows entanglement is a two-qubit system. A pair

of 1/2-spin particle in a singlet state

∣∣ −⟩ =1√2

(∣↓↑⟩ − ∣↑↓⟩)

is perhaps the most popular example of pure nonseparable states. In fact, ∣ −⟩ is one

member of a famous basis of mutually orthogonal maximally entangled states of ℂ2 ⊗ ℂ2,

called Bell states:

∣∣�±⟩ =1√2

(∣00⟩ ± ∣11⟩) ,∣∣ ±⟩ =1√2

(∣01⟩ ± ∣10⟩) .(1.29)

1.2 Entanglement everywhere 15

We can go from one Bell state to another one with simple local operations, but local

operations (that is, operations on single qubit) cannot change the degree of entanglement.

1.2.2 Photon: two degrees of freedom

Many people think that entanglement is connected to a description of multiparticle

systems. In fact, this is not correct. Even a single particle can be a “composite” system.

A simple example of entangled state is produced when a photon passes through a

Polarizing Beam Splitter (PBS). Let us clarify what happens. The photon has two relevant

degrees of freedom: its polarization, and the direction of the path (the ray) that it follows.

These two degrees of freedom can be treated as if they were two constituents of a composite

system. The Hilbert space describing the state of the photon is the direct product of a

polarization space ℋA ≃ ℂ2 and a direction (momentum) space ℋB ≃ L2(ℝ3). Let ∣H⟩

and ∣V ⟩ (∈ ℂ2) denote the two orthogonal linear polarization states defined by the PBS

orientation, and let ∣x⟩ and ∣y⟩ (∈ L2

(ℝ3)) denote the directions of the transmitted and

reflected rays, respectively. A complete basis for photon states may thus be the product

base: {∣H⟩ ⊗ ∣x⟩ , ∣V ⟩ ⊗ ∣x⟩ , ∣H⟩ ⊗ ∣y⟩ , ∣V ⟩ ⊗ ∣y⟩}. For example, if we say that a photon

state is ∣H⟩ ⊗ ∣x⟩, this means that we can predict that if the photon is subjected to a test

(a photodetection apparatus) for horizontal polarization ∣H⟩, and if that test is located in

the direction of ray ∣x⟩, the photon will certainly pass the test. Moreover, that photon will

not excite a detector located in the orthogonal direction ∣y⟩ (for any polarization) and it

will not pass a test for linear polarization ∣V ⟩ (at any location). Suppose now that the

initial state of the photon, before passing through the beam splitter, is (a ∣H⟩+ b ∣V ⟩)⊗ ∣x⟩,where ∣a∣2 + ∣b∣2 = 1. The polarization state of the photon is a linear combination of linear

horizontal and vertical states, and ∣x⟩ denotes the direction of the incident ray. This state

is a direct product since the two degrees of freedom are factorized: the photon can be found

only in the incident direction. According to quantum theory, the state of the photon, after

passing through the PBS, can be written as

a ∣H⟩ ⊗ ∣x⟩+ b ∣V ⟩ ⊗ ∣y⟩ . (1.30)

This is a very simple operative manner to create an entangled state (with Schmidt number

r = 2). The PBS acts as a (global) unitary transformation that sends a separable state into


Figure 1.2: A Polarizing Beam Splitter is a unitary transformation (namely a CNOT transforma-

tion) able to send separable states in entangled ones.

an entangled one (see Fig. 1.2). Such a “nonlocal” unitary is commonly known in quantum

computation as CNOT gate.

Chapter 2

Random Matrices and Random

States

Random matrix theory (RMT) was introduced as a subject of intensive study with

the work by Wigner on nuclear physics about scattering resonances for neutrons off heavy

nuclei. Wigner was the first to propose that the local statistical behavior of the resonance

levels could be modeled by the local statistical behavior of the eigenvalues of a large random

matrix. The seminal work of Wigner was soon elaborated and developed by many authors,

both physicists and mathematicians. The idea underlying Wigner’s work is the following: a

random matrix can model the Hamiltonian operator of a not-understood quantum system.

We briefly explain why. Any physical information about the energy levels of a system can

be deduced, in principle, if one is able to solve the Schrodinger equation:

H ∣ i⟩ = Ei ∣ i⟩ .

Unfortunately, in nuclear physics one faces two problems. First, the nuclear interaction is

still somewhat unclear. Thus we do not known the Hamiltonian and, second, even if we did,

the corresponding equation will turn out to be a too complicate problem. Wigner’s idea was

to use random matrices to mimic the eigenvalues of the unknown Hamiltonian of heavy nuclei.

Lacking any particular information, the strategy is to consider an ensemble of Hamiltonians

defined by the appropriate symmetries of the system. Typically, one is interested in the

discrete part of the energy spectrum. Then we can represent the Hamiltonians as random

Hermitian matrices whose distributions are restricted by the imposed symmetry properties.

18 Random Matrices and Random States

Since its inception, methods and main results of RMT have been extensively used in

disparate areas of physics and mathematics. In this chapter we will be concerned with

probability distributions on the space D(ℂN ) of N ×N density matrices. In order to clarify

the meaning of ensembles of matrices, the first sections will be devoted to an informally

introduction on Random Matrix Theory. We will deal almost exclusively with a few families

of matrices: unitary, hermitian and density matrices. From a technical point of view, such

ensembles are the more intensively studied and, probably, the easiest to understand.

2.1 Topological groups and Haar measures: an overview

Random matrix theory is strongly related with topological group theory. This is linked

to the possibility of defining integrals over these groups. In the following we will give some

basic notions of topological groups and Haar measures; our paradigm will be the unitary

group.

Definition 24 (Topological groups). A topological group is a tern (G, ⋅ , T) s.t.

(a) (G, ⋅) is a group, ⋅ being the composition law;

(b) G is a topological space with respect to the topology T;

(c) the internal law � : (g, ℎ) ∈ G×G 7→ g ⋅ ℎ−1 ∈ G is continuous.

Note that commutativity is not required. A topological group has the familiar group

structure with a topology on such that the group’s binary composition law and the group’s

inverse function are continuous functions with respect to the topology.

2.1.1 Groups of matrices and Haar measure

A very important class of topological groups is that of groups of matrices such as the

linear general group:

GL (n,ℂ) := {A ∈Mn(ℂ) ∣detA ∕= 0} ,

with the standard product of matrices and the topology induced by Mn(ℂ) ≃ ℂn2. Since

the entry of a product AB is a polynomial in the entries of A and B, the composition law

of the group is continuous.

2.2 Random matrices 19

Remark. Recall thatGL (n,ℂ) is an open set ofMn(ℂ) (its complement, {A ∈Mn(ℂ) ∣detA = 0} ,

is the set of the zeros of a continuous function and therefore is closed). Then, the topology

of the linear general group follows naturally from the one of Mn(ℂ). The linear general

group is non commutative, and locally compact (as ℂn2).

Definition 25 (Haar measure). A left-Haar measure on a locally compact topological

group G is a nonzero regular Borel measure � on G which is left invariant by the group, i.e.

�(gH) = �(H) , ∀Borel setH ⊂ G , ∀ g ∈ G . (2.1)

Equivalently, for all f ∈ L1(G):∫Gd�(H)f(gH) =

∫Gd�(H)f(H) , ∀ g ∈ G . (2.2)

Every locally compact topological group has a left-Haar measure. A left-Haar measure

which is also right invariant is called a Haar measure.

Theorem 2.1.1 (Haar). Every compact topological group G has an essentially unique Haar

measure. Since the group is compact, this (finite) measure can be made into a probability

measure.

2.2 Random matrices

A matrix-valued random variable or, informally, a “random matrix”, is a random variable

which take values in a space Mm,n(ℝ) or Mm,n(ℂ) of m×n real or complex-valued matrices,

equipped with a Borel �-algebra, where m,n are integers. Note that the “shape” (size)

m×n of the matrix is deterministic (although it can be very fascinating to consider matrices

whose shapes are themselves random variables). In the following we will consider the square

case (m = n). We can view a random matrix A = (aij)1≤i,j≤n as the joint random variable

of its scalar components aij . One can apply all the usual matrix operations (e.g. sum,

product, determinant, trace, inverse, etc.) on random matrices to obtain new interesting

random variables with appropriate ranges.

Then, for our purposes, an ensemble of random matrices is defined by a matrix space

(e.g. a subset of Mn(ℂ)) equipped with a Borel �-algebra, and a probability measure on


it. The choice of the measure is more delicate. The measure, in all the relevant cases, is

an invariant (under a suitable operation) measure. If there is a underlying compact group

structure, this measure can be deduced from the Haar measure on the symmetry group.

We now turn our attention to the construction of the ensembles used in this thesis.

2.2.1 Random unitaries

We will focus our attention on the unitary group:

U (N) :={U ∈MN (ℂ)

∣∣∣UU † = I}.

This is the symmetry group of the hermitian product on ℂN . U (N) is a compact group:

it is closed since UU † − I is continuous, and it is bounded because every unitary matrix

U = (u)ij is unimodular (∣detU ∣ = 1) and then, ∣uij ∣ ≤ 1.

Unitary condition UU † = I forces N2 terms. Therefore an N -dimensional unitary

matrix has 2N2 −N2 = N2 real degrees of freedom. The Haar measure normalized to one

is a natural choice for a probability measure on the unitary group because, being invariant

under group multiplication, any region of U(N) carries the same weight in a group average.

It is the analogue of the uniform density on a finite interval.

In order to understand this point consider the simplest example: U(1), i.e the set{e{�}

of unimodular complex numbers. U(1) has the topology of the unit circle S1. The group

point of view reveals that, since in this case matrix multiplication is simply addition of

the phase mod2�, U(1) is isomorphic to the group of translations on S1. A probability

density function (pdf) that equally weighs any part of the unit circle is the constant density

f(�) = 1/2�. This pdf correspond to a measure which is invariant under translations: this

is the unique Haar measure on U(1). Note that it is not possible to define an unbiased

probability measure on a non-compact manifold. For example, it makes no sense try to

write a constant pdf on the whole real line ℝ.

2.3 Random pure states

The idea of randomly taking a quantum state is equivalent to assuming minimal a

priori knowledge about the system. So, the point is what statistical ensemble corresponds

to minimal prior knowledge about a quantum system. In the space of pure states of an

2.3 Random pure states 21

L-dimensional Hilbert space ℋS , isomorphic with the complex projective space ℂPL−1,

there is a natural answer to the question. In particular, identifying minimal knowledge

with maximal symmetry, it is natural to require that the ensemble be invariant under the

full group of unitary transformations (so that there is no preferred measurement basis

for extracting information). Thus the sampling criterion corresponds to a unique natural

measure, PL(∣ ⟩), induced by the Haar measure d�H(U) on the unitary group U(L). In

other words, a random pure state, defined by the action of a random unitary matrix on an

arbitrary reference state, ∣ ⟩ = U ∣ 0⟩, can be represented, in an arbitrary basis, as a given

column of the random unitary matrix U . Such an ensemble may be identified as the “most

random” ensemble of possible states of the system.

However, there does not appear to be a natural generalization of the above ensemble

when the restriction of pure states is removed. Indeed, if general states described by density

operators are allowed, the requirement of unitary invariance only implies that the probability

measure over the set of possible states be a function of the density operator eigenvalue

spectrum alone.

2.3.1 Induced measure in the space of mixed quantum states

Consider a bipartite N ×M composite quantum system. Pure states ∣ ⟩ of this system

may be represented by a L-dimensional normalized vector, where L = MN is the dimension

of the composite Hilbert space ℋS = ℋA ⊗ℋB. Without loss of generality we will suppose

M ≥ N . In view of the operations of partial tracing it is convenient to work in a product

basis {∣i⟩ ⊗ ∣�⟩}, where {∣i⟩}1≤i≤N and {∣�⟩}1≤�≤M are orthonormal bases of ℋA and ℋB,

respectively. A pure state ∣ ⟩ ∈ ℋ is then represented by Ψi�, i.e. a rectangular complex

matrix N ×M . The normalization condition, ⟨ ∣ ⟩ = TrΨΨ† = 1, is the only constraint

imposed on this matrix. The density matrix � = ∣ ⟩ ⟨ ∣, a 1-rank projector acting on the

composite Hilbert space ℋS , is represented in the chosen basis by a matrix labelled by four

indices �i�,j� = Ψi�Ψ∗j� . The partial tracing with respect to the M -dimensional subspace

ℋB gives a reduced density matrix of size N

�A = TrB� , where (�A)ij =M∑�=1

Ψi�Ψj�∗ .

In what follows, we will show how the natural measure in the space of L = NM -dimensional

pure states induces a measure PN,M (�A) on the space of the reduced density matrices of size


N . In words, PN,M (�A) d�A is the probability of drawing an N -dimensional density matrix

from the volume element d�A around �A by means of the following operations: i) sampling

∣ ⟩ according to a given measure d�(∣ ⟩) over the space of pure states ∣ ⟩ ∈ ℋS = ℋA⊗ℋB,

and ii) partial tracing over ℋB.

First, we calculate the distribution of matrix elements:

PN,M (�A) ∝∫

[dΨ] �(�A −ΨΨ†

)�(

TrΨΨ† − 1), (2.3)

where [dΨ] =∏i� dΨi�dΨ∗i�. In the second delta function we can directly substitute

TrΨΨ† with Tr�A. Since �A is generically positive definite (see 2.3.2), we can make the

transformation

Ψ =√�AΨ , [dΨ] = det (�A)M

[dΨ]

The transformation of the differentials follows from the fact that [dΨ] contains both real

and imaginary parts. The matrix delta function becomes:

�(�A −ΨΨ†

)= �

(√�A

(1− ΨΨ†

)√�A

)= det (�A)−N�

(1− ΨΨ†

).

Then, we obtain:

PN,M (�A) ∝ � (�A) det (�A)M−N� (Tr�A − 1) . (2.4)

Now we have to deal with the measure �(�A) on the space of N ×N Hermitian matrices.

2.3.2 Hermitian matrices

Recall that a density matrix � ∈ D(N) ⊂ℳ(N) has N2 − 1 independent parameters.

The natural parametrization is given by the matrix elements: {�ii}1≤i≤N , {Re�ij}i<j and

{Im�ij}i<j . It is not easy to write the measure d�(�) in terms of these coordinates. However,

a standard result of linear algebra provides a convenient set of coordinates.

Every N × N self-adjoint matrix A has a spectral representation A = UΛU †, where

U ∈ U(N) is a unitary matrix, and Λ = diag (�1, . . . , �N ) is the diagonal matrix of eigen-

values of A. It is not difficult to prove that, typically, the spectrum of � is simple, i.e. the

set ℳ(N) of hermitian matrices A with simple spectra is open and dense in ℝN2. Another

natural question is: “given A and Λ, how many (real) degrees of freedom does U have?”.

Although the answer is intuitive, from a detailed derivation we can obtain a precious result.

2.3 Random pure states 23

Let us suppose:

∃U1, U2 s.t. A = U1ΛU †1 = U2ΛU †2 .

Then

(U †2U1)Λ = Λ(U †2U1) =⇒[(U †2U1),Λ

]= 0 ,

V =(U †2U1

)= (vjk)1≤j,k≤n commutes with the eigenvalues diagonal matrix Λ = diag (�1, . . . , �n):

⎛⎜⎜⎜⎜⎜⎝v11 v12 . . . v1n

v21 v22 . . . v2n

......

. . ....

vn1 vn2 . . . vnn

⎞⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝�1 0 . . . 0

0 �2 . . . 0...

.... . .

...

0 0 . . . �n

⎞⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝�1 0 . . . 0

0 �2 . . . 0...

.... . .

...

0 0 . . . �n

⎞⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝v11 v12 . . . v1n

v21 v22 . . . v2n

......

. . ....

vn1 vn2 . . . vnn

⎞⎟⎟⎟⎟⎟⎠ ,

i.e. vjk�k = �jvjk. So, if A has a simple spectrum (distinct eigenvalues), vjk = 0 for j ∕= k:

V is a diagonal unitary matrix. Then, a matrix U that diagonalizes A belongs to U(N)/TN ,

a (N2 −N)-dimensional set (we denote by TN the N -dimensional torus).

What happens if A has at least one degenerate eigenvalue, say �1 = �2? Then, V has

the block form ⎛⎜⎝⎞⎟⎠v11 v12

0v21 v22

0 Diag,

and belongs to a subgroup of U(N). The matrices U ’s that diagonalizes A belongs to

U(N)/ (set of matrices of type V), of dimension N2 − (N + 2). The diagonal matrix Λ has

now (N − 1) independent real parameters. Then the submanifold of ℳ(N) of A’s with

at least one repeated eigenvalue has dimension N2 − (N − 2) + (N − 1) = N2 − 3, i.e.

codimension 3. Observe that, naively, one would expect a codimension 1 set of Hermitian

matrices with two equal eigenvalues. Indeed, the unexpectedly higher codimension of the

non-simple matrices space suggests a repulsion phenomenon: since it is rare for eigenvalues

to be equal, there must be some force that repels the eigenvalues of Hermitian matrices

hindering them from getting too close to each other. Note also that the above discussion

holds for every ensemble of normal matrices.


We will consider the spectral decomposition for a typical (simple-spectrum) Hermitian

matrix as a change of variables ':

�'7→(Λ , U modTN

), (2.5)

where the N -torus TN takes into account the above mentioned N undetermined phases of

the unitary U . Spectral variables give rise to a convenient, smooth local parameterization

of the manifold ℳ(N) of hermitian matrices (with simple spectra) in terms of N2 = N + k

real variables {�1, . . . , �N ; �1, . . . �k}, where k = N2 −N is the dimension of the manifold

U(N)/TN .

By standard methods of random matrix theory (see Appendix A) one can show that

the measure �(�) of density matrices factorizes into a product measure

�(�) = Δ(Λ)× �(U) , (2.6)

where the first factor defines a measure in the (N − 1)-dimensional simplex of eigenvalues

defined by the unit-trace condition. The measure � on the space of unitary matrices U(N)

is responsible for the choice of the eigenvectors of �. A unitarily invariant measure over

the pure states of a composite system induces a measure �A over the eigenvectors of the

reduced density matrix �A which is still rotationally invariant, i.e. a Haar measure. Observe

that the space of Hermitian matrices ℳ(N) is not compact, while the space D(N) of states

is the product of two compact spaces, a simplex and a sphere.

2.4 Joint distribution of the eigenvalues

For random pure states sampled from the ensemble

({U ∣ 0⟩ ⟨ 0∣U †

}, d�H(U)

),

the eigenvalues of the reduced N−dimensional density matrix

�A = TrB(U ∣ 0⟩ ⟨ 0∣U †)

are distribuited according to the following joint distribution (see Appendix A):

f(N,M)(�1, . . . , �N ) = CN,M∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

), (2.7)

2.4 Joint distribution of the eigenvalues 25

where � is the Dirac delta function and CN,M is the normalization factor

CN,M =(NM − 1)!∏

0≤j≤N−1 (M − j − 1)!(N − j)!. (2.8)

Let us summarize some known results about the joint pdf eq. (2.7). Since the eigenvalues

joint pdf is permutation invariant and∑

i �i = 1, one immediately obtains:

1 =

⟨∑i

�i

⟩=∑i

⟨�i⟩ = N ⟨�i⟩ =⇒ ⟨�i⟩ =1

N, ∀1 ≤ i ≤ N . (2.9)

Calculation of the second moment⟨(�i − 1

N

)2⟩needs more work. Lubkin [11] calculated

that:

�rms =

√√√⎷⟨(�i − 1

N

)2⟩

=

(1− 1/N2

MN + 1

)1/2

. (2.10)

Denoting � s.t. �N = M −N , in the large N limit, with M −N finite and fixed, the width

of the distribution becomes:

�rms =1

N√

1 + �. (2.11)

Various aspects of the entanglement properties of random pure states have been studied

in previous works. Lubkin in the same work found (incidentally):

⟨Tr(�2

A)⟩

= ⟨�AB⟩ =N +M

MN + 1. (2.12)

For M −N finite and fixed, and large N one has ([7]):

⟨�AB⟩ =1

N

2 + �

1 + �. (2.13)

A balanced (� = 0) bipartite large system has ⟨�AB⟩ = 2/N typical purity. Bipartite


entanglement of pure states is appropriately measured by von Neumann entropy Tr� log �.

In a pioneering work, Page [14] conjectured from numerical evidence that the average von

Neumann entropy for finite N,M should read:

⟨S⟩ =

NM∑k=M+1

1

k− N − 1

2M. (2.14)

This conjecture has been confirmed later by other authors [20].

The elementary symmetric polynomial eN (�1, . . . , �N ) = �1 ⋅ ⋅ ⋅�N is a mixedness mea-

sure. In fact, this is nothing but the determinant of the density matrix det �A. It is a

bounded Schur-concave function of the spectrum (and then unitarily invariant) of the density

matrix. If the state is sampled according to the pdf (2.7), the moments of eN = det �A can

be calculated straightforwardly. The probability density for eN is:

PeN (y) = P (y∣y = �1 ⋅ ⋅ ⋅�N )

= CN,M

∫dN�

∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

)⋅ �(y −

∏k

�k

);

(2.15)

expectation values of ekN readily follow:

⟨det �k

⟩N,M

= CN,M

∫dy

∫dN� yk

∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

)⋅ �(y −

∏k

�k

)= CN,M

∫dN�

∏i<j

(�i − �j)2∏l

�M−N+kl

∏i

�(�i)�(

1−∑k

�k

)=

CN,MCN,M+k

.

(2.16)

It is not difficult to show that, in the limit of large environment, M → +∞:⟨det �A

k⟩−→(

1N

)kN.

2.5 Measure on the Bloch ball 27

2.5 Measure on the Bloch ball

A well known fact is the one-to-one correspondence between single qubit density ma-

trices � ∈ D(ℂ2) and points of the unit-ball, named Bloch ball. This nice geometrical

interpretation of two-dimensional states follows from the general representation of a N

dimensional density matrix as

� =1

N(1 + � ⋅ r) , (2.17)

where �’s are the N2 − 1 generalized Pauli matrices of size N .

In the two-dim case, �’s are the well-known 2 × 2 three Pauli matrices �i, and the

representation shows explicitly that the set of single qubit states is homeomorphic to the

ball of radius r = ∣r∣ = 1. In deference to the theory of polarization of light, r is named

polarization vector (by analogy with Stoke’s parameters and light coherence).

It is easy to write the joint pdf eq. (2.7) in terms of coordinates on the Bloch ball.

First, note that, since local unitary transformation in D(ℂ2) correspond to rotations about

the origin, unitarily equivalent states are equally distant points from the origin (equally

mixed states lie on the same spherical surface). Then, the only relevant coordinate must

be the distance r from the origin. From det � = 14(1−r2), and then �1/2 = 1

2±r2 , one obtains:

P2,M (r) = P2,M (r, �, �) =CN,M

4�r2

(1− r2

4

)M−2

. (2.18)

For a balanced bipartition (M = 2), the pdf describes a uniformly populated ball (3D

ball). For higher dimensions the geometry of quantum states (of density matrices) is not so

simple!

2.6 Most probable distribution

We are looking for most probable eigenvalues and then, we have to find the points that

maximizes the probability density function f(N,M)(�1, . . . , �N ) = eFN,M ({�i}):

f(N,M)(�1, . . . , �N ) = CN,M∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

),


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0M = 4

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0M = 5

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0M = 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0M = 50

0.0 0.2 0.4 0.6 0.8 1.0

-8

-6

-4

-2

0

Λ1

Log

f 1HΛ 1

L

M = 4

0.0 0.2 0.4 0.6 0.8 1.0

-8

-6

-4

-2

0

Λ1

Log

f 1HΛ 1

L

M = 5

0.0 0.2 0.4 0.6 0.8 1.0

-20

-15

-10

-5

0

Λ1

Log

f 1HΛ 1

L

M = 10

0.0 0.2 0.4 0.6 0.8 1.0

-150

-100

-50

0

Λ1

Log

f 1HΛ 1

L

M = 50

Figure 2.1: Contour plots of the function ln f2 for a single qubit (N=2) for different dimensions

of the environment M ; the domain is the full square �1, �2 ∈ [0, 1]× [0, 1]. Below, the projections

of ln f2 on the simplex (i.e. the diagonal of the square from (1, 0) to (0, 1) ). When M > N ,

two maxima appear in the interior of the simplex. The intersection of the simplex with the Weyl

chambers is the segment (0, 1)(1/2, 1/2). As M increases, the repulsion between the eigenvalues

becomes dominated by a “uniformization” effect of the environment.

or, equivalently, of

FN,M (�) = ln f(N,M)(�) = 2∑i<j

log ∣�i − �j ∣+ (M −N)∑l

log �l , (2.19)

with the constraint∑

i �i − 1 = 0. This is a constrained optimization problem. By

introducing a Lagrange multiplier, we have to maximize the (N + 1)-variable function:

F (�, �M ) = 2∑i<j

log ∣�i − �j ∣+ (M −N)∑l

log �l − �M (1−∑i

�i) (2.20)

The function ln f(N,M) is continous and derivable in the subset of the N -dimensional

simplex defined by the inequalities ∣�i − �j ∣ > 0 and �i > 0 if M > N . The existence of

a maximum is not guaranteed since the domain is not compact. In fact, since the pfd is

permutation invariant (f(�) = f(Π�), ∀permutation Π ∈ SN ), one can consider the Weyl

chamber only, i.e. the set {(�i, . . . , �N ) : �1 ≤ ⋅ ⋅ ⋅ ≤ �N}. To establish a convention on the

ordering of the eigenvalues, our region of interest (ROI) will be:

2.6 Most probable distribution 29

{(�i, . . . , �N ) ∈ ℝN+ : �1 ≤ ⋅ ⋅ ⋅ ≤ �N ,

∑i

�i = 1}. (2.21)

In the Weyl chamber, for the balanced bipartition (M = N) the unique maximum point

is (0, 0, 0, 0, . . . , 1) which belongs to the boundary and then cannot be detected by the

Lagrange multiplier method (see Fig. 2.1). This is not a serious problem; we are interested

to the peaks of the distribution since we expect that the average will be located near the

most probable value. Maximum values which correspond to separable states are, obviously,

not close to the mean spectrum ⟨�i⟩ ∼ 1/N, ∀i. Then, we will say simply that the saddle

point method does not hold for balanced bipartitions. In the unbalanced situation, ln f(N,M)

is concave in the restriction of its domain to the Weyl chamber (it follows from the concavity

of logarithms) and it tends to minus infinity on the boundary. Then, a (unique) maximum

in the interior must exist.

Chapter 3

Numerical investigation

The joint distribution (2.7) of the Schmidt eigenvalues is a very hard object to manipulate.

Most of the analytic difficulties are due to the unit-trace constraint, that is, the joint pdf is

defined on the N -dim simplex. In many cases, one is interested in the study of quantities

that do not contain products of different eigenvalues, such as purity or von Neumann

entropy. The averages of these can be evaluated using a single integral over one-body

density, that is the marginal pdf defined as:

f(1)(N,M)(�1) =

∫ +∞

0. . .

∫ +∞

0d�2 ⋅ ⋅ ⋅ d�N f(N,M)(�1, . . . , �N )

= CN,M

∫ +∞

0. . .

∫ +∞

0d�2 ⋅ ⋅ ⋅ d�N×

×∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

).

(3.1)

The above integral over the simplex is very hard to perform. We have tried to do it

analytically, but Mathematica R⃝ fails to evaluate it, because of the interdependence of the

integration limits in each dimension (since it’s over a simplex). The case N = 2 is trivial

(we need to integrate just a delta function); for small dimensions N = 3, 4, eq. (3.1) can be

evaluated, but the calculation time grows exponentially in N and M . In this scenario, we

decided to tackle the problem numerically.

32 Numerical investigation

3.1 Numerical simulations: random sampling

In order to have a picture of what happens when one considers a random state of a

bipartite system, it is very useful to implement numerical simulations. A very nice resource

is provided by the Mathematica package QI.m for the analysis of quantum states and

operations [1]. What has been done is a canonical simulation procedure: we generated a

large number of random pure states in ℂNM , and then, at fixed bipartition (N ≤M), we

found the Schmidt coefficients, that are the N square roots of the eigenvalues of the reduced

density matrices. In our problem, we are interested to strongly unbalanced bipartitions: N

is a small number, typically 4, while M is a large number (we want to tackle the problem

of a small quantum system in a quantum environment).

To generate an NM -dimensional pure state, i.e. a density matrix that is a projection

� = ∣ ⟩ ⟨ ∣, means to generate an NM -dimensional vector. In order to sample a pure state

according to the Haar measure, we have to extract a vector whose coordinates (given basis)

are “uniformly” distribuited in a way that satisfies the normalization condition of vector

states. The package QI.m provides the function RandomKet. RandomKet[NM] uniformly

extracts a vector of nonnegative numbers in the (NM − 1)-dimensional simplex. These

numbers will be the squared moduli ∣ai∣2 . Then, the function extracts NM random phases

�i i.i.d. uniformly in [0, 2� [ and outputs the vector

∣ ⟩ =

NM∑i=1

aie{�i ∣i⟩ (3.2)

in the computational basis. Given the random state ∣ ⟩, one obtains the Schmidt decom-

position by VectorSchmidtDecomposition[∣ ⟩ , N,M ] (or alternatively by constructing a

singular values decomposition of the matrix ∣ ⟩ ⟨ ∣); VectorSchmidtDecomposition sorts

the Schmidt coefficients√�i in decreasing order. This is a simple procedure to generate the

N eigenvalues �i of the reduced density matrix obtained by partially tracing a � = ∣ ⟩ ⟨ ∣pure sampled according to the Haar measure. Once the eigenvalues are obtained, one can

proceed and calculate every kind of quantity (averages). As an example, fig.3.1 shows

the numerical distributions of the radial coordinate in the Bloch sphere for a single qubit,

compared with the analytic curves eq. (2.18). The histograms in Fig. 3.2 show the

distribution of a single eigenvalue corresponding to f4,M(1)(�) of eq. (3.1). Numerical

experiments make clear that the single-eigenvalue’s population is concentrated around its

3.1 Numerical simulations: random sampling 33

M=2M=10

M=100

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

r

PHrL

Figure 3.1: Numerical simulations: radial distribution in the Bloch ball for a single qubit � =

(1 + � ⋅ r)/2. Solid lines are the analytic curves P2,M (r) = C2,Mr2(

1−r24

)M−2.

mean value ⟨�⟩ = 1/4, with a dispersion that is decreasing with M . More precisely, our

numerics confirms Lubkin’s result �(�) ∼ 1/√M , eq. (2.10). For a crude analysis just look

at the support of histograms. As an example, the first numerical distributions in Fig. 3.2

correspond to M1 = 80 and M2 = 300 ≃ 4M1; therefore, the dispersion should change by

a factor√

4 = 2: indeed the support of the second histogram (M = 300, Supp ∼ 0.12) is

two times smaller than the support of the first (M = 80, Supp ∼ 0.24). Moreover, the

profile of the histograms suggest that f4,M(1)(�) is a 4-peaks pdf. Similar investigations for

qudit states, with d = 2, 3, 5, exhibit the same shape: the distribution of single-eigenvalue

fN,M(1)(�) has N peaks close to the mean value; in all cases, except single-qubit states

d = 2, these maxima are not symmetric with respect to the mean 1/N . These and many

other numerical observations, suggest the following guess for the marginal pdf fN,M(1) as a

polynomial in [0, 1]:

f(1)M,N (�1) =

∫ΔN−1

d�2 . . . d�Nf(N,M)(�1, . . . , �N )

= (1− �1)MN−(M+N)�M−N1 P2(N−1)(�1) ,

(3.3)

where P2(N−1)(x) is a polynomial of degree 2(N − 1) with integer coefficients.


0.10 0.15 0.20 0.25 0.30 0.35 0.400.000

0.001

0.002

0.003

0.004

0.005

0.006

Λ

M = 80

0.10 0.15 0.20 0.25 0.30 0.35 0.400.000

0.002

0.004

0.006

0.008

Λ

M = 300

0.10 0.15 0.20 0.25 0.30 0.35 0.400.000

0.002

0.004

0.006

0.008

0.010

0.012

Λ

M = 700

0.250 0.255 0.260 0.265 0.270 0.275 0.2800.00

0.01

0.02

0.03

0.04

0.05

0.06

ΠAB

M = 80

ΠAB= H0.261668 ± 0.00421929L

0.250 0.255 0.260 0.265 0.270 0.275 0.2800.00

0.05

0.10

0.15

0.20

ΠAB

M = 300

ΠAB= H0.25314 ± 0.0011351L

0.250 0.255 0.260 0.265 0.270 0.275 0.2800.0

0.1

0.2

0.3

0.4

ΠAB

M = 700

ΠAB= H0.25131 ± 0.000463454L

Figure 3.2: Numerical simulations, four-level system (N = 4): distributions of a single eigenvalue

(above) and distributions of purity (below) for typical states, eq. (2.7), for different values of

M = dimℋB. We have sampled k = 2000 random pure states (4M -dim normalized vectors) for

each M . Observe asymmetry of peaks with respect to the mean value ⟨�⟩ = 1/4.

This pdf is determined up to the 2(N − 1) coefficients of P2(N−1)(x) (they obviously depend

on M). Recently (2009), some authors [2] derived the exact analytical formula of the one-

body distrbution function f(1)M,N (�1), and our guess (suggested by numerical investigations)

agrees with this result. The exact analytical formula has been found from very hard

mathematics and appears very cumbersome.

3.2 Numerical simulations II: entanglement within the pair

What happens if the four-level system A is regarded as two qubits A1, A2? The saddle

point method returns the most probable spectrum of the reduced density matrix �A. The

state of A is determined by (N2 − 1) independent numbers, but we have only (N − 1) of

them. That is, we know the most probable spectrum, but the eigenbasis of �A is completely

unknown. Strictly speaking, it is meaningless to ask if the two qubits A1 and A2 are

entangled in the state �A. However, one can pose other kinds of questions. The state of

system A is �A = U ΛU † where Λ is the most probable spectrum. Given the spectrum,

the state �A is determined modulo a unitary transformation. What is the distribution of

3.2 Numerical simulations II: entanglement within the pair 35

M Starting purity �AB Concurrence C(�A) Entropy �A1 Purity Tr�2A1

5 0.437500 0.0800686 ±0.0739789 0.887681 ±0.074908 0.575058±0.0485846 0.400000 0.0311272 ±0.0424090 0.910376 ±0.059916 0.560127±0.0400307 0.375000 0.0087741 ±0.0191714 0.925644 ±0.050374 0.550233 ±0.0328738 0.357143 0.0012072 ±0.0054863 0.937633 ±0.042617 0.542639±0.0285069 0.343750 0.0000335 ±0.0005970 0.945079 ±0.037702 0.537711±0.02530610 0.333333 0 0.950452 ±0.034003 0.533512±0.02270464 0.261905 0 0.993085 ±0.004827 0.504764±0.003335

1000 0.250751 0 0.999566 ±0.000304 0.500300±0.000210

Table 3.1: Numerical study: for each average spectrum Δ we have constructed k = 10000 four-level

quantum states by conjugation with random unitaries. The table shows the average quantities

computed for such states. Note that concurrence is always zero for M ≥ 10. Every run, i.e. every

procedure of constructing and manipulating k = 10000 random states of fixed spectrum, requires

less than 5min on a personal computer.

matrices �A with fixed spectrum, when U varies according to the Haar measure? What is

the expectation value ⟨�A⟩H?

In a more formal language, we are interested to the coadjoint orbit of �A

O�A ={U�AU

† ∈ D(ℂ4) : U ∈ U(ℂ4)}

(we will denote by �A one of the diagonal forms of �A). Since entanglement does not change

under local unitaries, the relevant coadjoint orbit is the one generates by U(ℂ4)/U(ℂ2)×U(ℂ2) (16 − 4 = 12 free real parameters). Again, a quick way to have an idea of what

happens is to perform numerical simulations. We have randomly generated a large number

of density matrix with fixed spectrum �A = U ΛU † by randomly sampling the unitaries

U ’s with function RandomUnitary[], which works with the Hurwitz parametrization [28].

Briefly, we can sketch our approach as:

∣ ⟩ ∈ Pℂ4M Saddle Point Method−→ Δ ∈ Diag(ℂ4)Random Unitary−→ �A = U ΛU †

Average (Haar)−→ ⟨�A⟩H .

(3.4)

Numerical simulations give empirical averages. Here Δ is the average spectrum and not

the most probable one. Given the average density matrix ⟨�A⟩H , one obtains the reduced


density matrix for a single qubit, since the partial trace is linear:

⟨TrA2(�A)⟩H = ⟨�A1⟩H =

∫d�H TrA2(U ΛU †) = TrA2

(∫d�H U ΛU †

)= TrA2(⟨�A⟩H) .

(3.5)

On the other hand, if one desires to manipulate nonlinear quantities (such as all the

entanglement measures), the average on the unitary group has to be the last step in

the computation. This means that, given a function g over the states �A, generally

⟨g(�A)⟩H ∕= g(⟨�A⟩H). The table summarizes the numerical study for some dimensions M

of the environment. Simulations reproduce the subadditivity property of von Neumann

entropy

S(�A) ≤ S(�A1) + S(�A2) .

Incidentally, the above mentioned package QI.m contains an error in the definition of

function Concurrence4[], because it uses the square roots of matrix of eq. (1.28); this

error is not too serious and it can be easily corrected. In the following chapters we will see

how all these numerical experiments have guided our research.

Chapter 4

Typical entanglement

In this chapter we start to tackle our main problem: the statistical properties of the

entanglement of a random pure state. We will start from unbiased states, i.e. states sampled

according to the unitarily invariant Haar measure. In particular, we are concerned with the

Schmidt coefficients with respect to a given bipartition. These coefficients give information

about the degree of mixedness of the reduced density matrices of the subparts of the global

system. We are interested in the typical entanglement of a small subsystems of a large

random pure state. Our approach will rely on the saddle point method: given the joint

distribution of the eigenvalues of the reduced density matrix �A ∈ D(ℋA), we will search

the most probable spectrum, that is, the density matrix (up to local unitaries U ∈ U(ℋA))

that maximizes the probability.

The saddle point problem for typical states can be reduced to the problem of finding the

equilibrium configurations of a system of identical charges on a line (electrostatic models).

These problems are elegantly connected with the theory of orthogonal polynomials, as

Stieltjes first showed.

For unbiased states, we are able to fully solve the problem. The complete solution of

the saddle point method will be provided for all possible (unbalanced) bipartitions. This

result will be the starting point to compute all quantities of interest. For some well-known

quantities, such as purity or elementary symmetric invariants, we are able to give compact

and usable analytic expressions, for all N and M . Moreover, we will present typical

entanglement properties for the quasi-balanced balanced bipartition in the large size limit

(N,M → +∞, with M −N ≪ N,M).

38 Typical entanglement

In the last sections we will specialize the solution to strongly unbalanced bipartitions.

We will also present the complete scenario for the four-level system (N = 4).

4.1 Saddle point equations

To get a clearer insight of the joint pdf of the eigenvalues, eq. (2.7), physicists invoke

statistical physics intuition for a “Coulomb Gas” of N repelling electrical charges on a line.

In fact, the joint pdf can be written (see section 2.6) as the exponential of

F (�, �M ) = 2∑i<j

log ∣�i − �j ∣+ (M −N)∑l

log �l − �M (1−∑i

�i) ,

that can be interpreted as the potential energy of a gas of N point charges at positions �i’s.

The potential energy is given by the mutual interaction of these charges,

Vmutual =∑j ∕=i

log ∣�i − �j ∣ , (4.1)

and a part given by the external field

'ext(x) = −� log x+ �x , (4.2)

where we have denoted � = (M −N), and we have dropped the subscript in the Lagrange

multiplier for simplicity, � ≡ �M . In other words, the total energy is

Etot = Vmutual (�) +N∑k=1

'ext(�k) . (4.3)

Sometimes, we will refer to the potential (4.3) with the expression “landscape”. We are

interested on the stationary points of this landscape. By deriving F (�, �) with respect to

both the �i’s and �, we get N + 1 saddle point equations:⎧⎨⎩2∑

j ∕=i1

�i−�j + M−N�i− � = 0 , ∀i ∈ {1, . . . , N}

∑i �i = 1 .

(4.4)

4.2 Heuristic discussion 39

In the framework of the electrostatic model, the saddle point equations are nothing but

static equations of balance of the forces (the derivatives of the potential energy, ∇Etot = 0),

with the additional constraint that the charges average position be equal to 1/N .

4.2 Heuristic discussion

Before trying to write down a solution for the saddle point equation, it is useful to

give a picture of what will happen. Our N charges interact via a logarithmic repulsion.

Therefore, in absence of an external field, the charges will try to stay as far apart as possible.

Indeed, ther are two external forces acting on them. The first one is due to a charge �,

with � = M −N > 0, at position x = 0 that repels the unit charges (the eigenvalues) via a

logarithmic potential. This repulsion constrains the charges on the positive half-line. The

second field is a constant force (the gradient of the linear potential −�x). In order to ensure

the existence of an equilibrium configuration, the external potential must have a minimum.

Then, � must be positive (necessary condition to have a concave potential). The minimum

is located at xc = �� . Since we expect that, in the typical case, all eigenvalues be located

near the maximally mixed value, this critical point has to be close to 1/N . By setting the

trial value xc = 1/N , we can immediately guess that � ∼ NM in the large-M limit. To

check that the stationary points of the landscape Etot(�; �) are minima, one can study the

Hessian matrix:

H(Etot) =

(∂2

∂�i∂�jEtot(�; �)

). (4.5)

We can readily write:

Hii = 2∑j ∕=i

1

(�i − �j)2+M −N�2i

, Hij = − 1

(�i − �j)2, 1 ≤ i, j ≤ N . (4.6)

The Hessian H is (obviously) everywhere real and symmetric, stricly diagonally dominant

and with a positive diagonal. Therefore, it is positive definite everywhere (in our ROI), and

so, every stationary point of eq. (4.3) is a local minimum.


4.3 Simplified saddle point equations

As suggested by numerical investigations (see previous chapter), when M (the dimension

of the environment) is large, the eigenvalues �i fluctuate around 1/N . We try a solution of

the form:

�i =1

N+ Δi =

1

N+

�iM�

.

The saddle point equations in terms of �i read⎧⎨⎩2M�

∑j ∕=i

1�i−�j + M−N

1N

+�iM�− �M = 0 , ∀i ∈ {1, . . . , N}

∑i �i = 0 .

In the limit of large M , the first N equations become

2∑j ∕=i

1

�i − �j+NM1−� −N2M1−2��i − �MM−� = 0 , ∀i ∈ {1, . . . , N} ,

Since the variance of the random variable � in the typical case is⟨(1/N − �)2

⟩= 1−1/N2

MN−1

(eq. (2.10)), we expect � = 1/2. Moreover, the previous heuristic discussion (section 4.2)

justifies the position � = limM→+∞�MM . The set of simplified equations is then:⎧⎨⎩

2∑

j ∕=i1

�i−�j +N√M −N2�i − �

√M = 0 , ∀i ∈ {1, . . . , N}

∑i �i = 0.

(4.7)

The saddle point equations can be tackled by using an ingenious method due to Stielt-

jes (1885), which deals with the electrostatic interpretation of the zeros of some families

of orthogonal polynomials. The first N equations are equivalent to a single polynomial

equation. Indeed, one can write:

2∑j ∕=i

1

�i − �j=g′′(�i)

g′(�i)

where g(x) is a polynomial whose zeros �i have multiplicity one: g(x) =∏i (x− �i) (see

Appendix C).

4.4 The solution 41

Then, the first N equations of eq. (4.7) can be rewritten as

g′′(�i)− (N2�i −N√M + �

√M)g′(�i) = 0 . (4.8)

This conditions means that the N -degree polynomial eq. (4.8) has its zeros at �1, . . . , �N ,

and then it is a multiple of g(x). Comparing the coefficient of xN one finds:

g′′(x)− (N2x−N√M + �

√M)g′(x) +N3g(x) = 0 . (4.9)

The polynomial solution of this differential equation is the Hermite polynomial of degree N

[21]

g(x) = HN

(−√MN − �

√M√

2N+Nx√

2

). (4.10)

By imposing that the solution be even, one finds � = N and is left with:

HN

(Nx√

2

)= N !

[N/2]∑k=0

(−1)k(Nx/√

2)N−2k

k!(N − 2k)!. (4.11)

The zeros of this polynomial (all real, see appendix D) give the most probable eigenvalues

for the N -dimensional subsystem, namely

�i =1

N+

�i√M, with HN

(N�i√

2

)= 0 . (4.12)

Since the polynomial HN (x) is even, the condition∑

i �i = 0 is automatically fulfilled.

4.4 The solution

We can use Stieltjes’s trick to handle the complete saddle point equations (4.4). Starting

from the first N equations

2∑j ∕=i

1

�i − �j+M −N�i

− � = 0 ,


and following the previous procedure, we arrive at the differential equation

xg′′(x)− (M −N − �x)g′(x) + �Ng(x) = 0 , (4.13)

where g(x) is a polynomial whose simple zeros are the unknown �i’s. The polynomial

solution of the previous ordinary differential equation is an associated Laguerre polynomial:

L(M−N−1)N (�x) =

N∑�=0

(M − 1

N − �

)(−�x)�

�!. (4.14)

In order to fix the Lagrange multiplier �, we have to equalize (up to the sign) the two leading

coefficients aN to −aN−1: this is the condition under which the roots of a polynomial

equation sum up to 1. It is easy to find:

� =(M − 1)!

(M − 2)!

N !

(N − 1)!= N(M − 1) , (4.15)

in agreement with the heuristic guess � ≃ NM of section 4.2.

4.5 The typical entanglement spectrum

Using some general results of elementary algebra (see Appendix B), we can extract

plenty of information from the polynomial equation

L(M−N−1)N (N(M − 1)x) =

∑k

ckxk = 0 , (4.16)

without solving it. As an example, in order to compute the purity, eq. (1.19), note that(∑i �i

)2=∑

i �i2 + 2

∑i≤j �i�j , that is

�AB(N,M) = 1− 2cN−2(M)

cN (M)= 1− (N − 1)(M − 2)

N(M − 1), ∀N < M . (4.17)

Similarly, we can readily compute the values of the elementary symmetric invariants (1.21):

4.6 Thermodynamic limit 43

sk(�A) = ek(�1, . . . , �N ) =cN−kcN

=N !(M − 1)!

k!(N − k)!(M − k − 1)![N(M − 1)]−k . (4.18)

As an example, the most probable determinant is:

sN (�A) = det (�A) = eN (�1, . . . , �N ) =(M − 1)!

(M −N − 1)![N(M − 1)]−N . (4.19)

Remark. The most probable value is typical and close to the average if the pdf is sharply

peaked. Numerical simulations have suggested that the width of the peak decreases with M .

Let us verify it. The width of the peak is encoded in the Hessian Hij(�) of the landscape

eq. (4.3) evaluated at the minima. In the gaussian approximation, the proper values ℎi’s of

Hij give the widths (�−2i = ℎi) of the pdf around its maximum. In order to extract the

scaling, we can consider just the trace of the Hessian, obtaining:

TrHij ≃ N3(M −N) + 2N(N − 1)M , (4.20)

where we use the solution of the simplified saddle point equation, eq. (4.12).

4.6 Thermodynamic limit

In this section we want to use our solution to investigate the large size limit. We

introduce the parameter � s.t. �N = M −N ; the most probable purity reads:

�AB = 1− (N − 1)(M − 2)

N(M − 1)=M +N − 2

NM − 1=

N(2 + �)− 2

N2(1 + �)− 1, (4.21)

to be compared with the average value computed by Lubkin eq. (2.12). When N,M →∞with �N fixed and finite, we obtain:

�AB =1

N

(2 + �)

(1 + �), (4.22)

a value derived in the literature by more sophisticated methods [7]. Recall that the saddle

point method and its results hold for unbalanced bipartitions. However, one can (somewhat

arbitrarily) extend this results, and consider the quasi-balanced case M = N + 1, that


becomes the balanced bipartition in the N → +∞ limit. Then, one recovers some classical

results such as �AB = 2/N , for an equally bipartite large system.

Similarly, one can compute the quantities Tr�kA, and then derive Renyi’s entropies. A

very inexpensive method is to write∑

i �ki in terms of elementary symmetric polynomials,

and then use eq. (4.18). The above procedure can be easily implemented by the most

common symbolic manipulating softwares (Mathematica R⃝ provide the suitable function

SymmetricReduction[] to expand any symmetric polynomial). We give a list of the first

seven traces in the thermodinamic limit, N,M → +∞ with M −N = �N fixed:

Tr�A2 =

(2 + �)

(1 + �)

1

N+O

(1

N

)2

,

Tr�A3 =

5 + 5�+ �2

(1 + �)2

1

N2+O

(1

N

)3

,

Tr�A4 =

14 + 21�+ 9�2 + �3

(1 + �)3

1

N3+O

(1

N

)4

,

Tr�A5 =

42 + 84�+ 56�2 + 14�3 + �4

(1 + �)4

1

N4+O

(1

N

)5

,

Tr�A6 =

132 + 330�+ 300�2 + 120�3 + 20�4 + �5

(1 + �)5

1

N5+O

(1

N

)6

,

Tr�A7 =

429 + 1287�+ 1485�2 + 825�3 + 225�4 + 27�5 + �6

(1 + �)6

1

N6+O

(1

N

)7

.

(4.24)

The same can be done for the N -degree elementary invariant, that is det �A. For

example, for the quasi-balanced bipartition M = N + 1:

det �A =N !

N2N≃ e−N

NN. (4.25)

Let us conclude this section by extract the scaling of the width eq. (4.20) in the large

4.7 A single qubit 45

sizes limit. For �N = M −N fixed, for N → +∞, it is easy to find

�i ≃1

2N√

1 + �, (4.26)

in agreement with eq. (2.11).

4.7 A single qubit

We will begin our study with a simple solvable prototype: the binary quantum system.

For a single qubit, N = 2, the system of 2 + 1 equations

⎧⎨⎩2

�1−�2 + M−2�1− �M = 0

2�2−�1 + M−2

�2− �M = 0

�1 + �2 = 1 ,

(4.27)

is satisfied by the zeros of L(M−3)2 (�x), (� = 2M − 2):

�1/2 =1

2

(1± 1√

M − 1

). (4.28)

4.8 Four-level system

Let us discuss in more details what happens for a four-level system, N = 4. The solution

of the 4 + 1 simplified equations


⎧⎨⎩

2�1−�2 + 2

�1−�3 + 2�1−�4 − 16�1 + 4

√M − �

√M = 0

2�2−�1 + 2

�2−�3 + 2�2−�4 − 16�2 + 4

√M − �

√M = 0

2�3−�1 + 2

�3−�2 + 2�3−�4 − 16�3 + 4

√M − �

√M = 0

2�4−�1 + 2

�4−�2 + 2�4−�3 − 16�4 + 4

√M − �

√M = 0

�1 + �2 + �3 + �4 = 0 ,

(4.29)

can be directly written down using eq. (4.11). First, the asymptotic Lagrange multiplier is

set to be � = 4M . The most probable eigenvalues are given by the four zeros of

H4

(4x√

2

)∝ x4 − 3

8x2 +

3

256, (4.30)

i.e. �j = 14 +

�j√M

(j = 1, 2, 3, 4) with:

�1 = −1

4

√3 +√

6

�2 = −1

4

√3−√

6

�3 = +1

4

√3−√

6

�4 = +1

4

√3 +√

6 .

(4.31)

In order to recover the asymmetry of eigenvalues distribution showed by numerical

simulations (fig.3.2), we need to see how the complete solution works. Using eqs. (4.14)(4.15),

one can fix the Lagrange multiplier to � = 4(M − 1), and find the most probable �i’s as the

roots of the degree-four polynomial equation L(M−5)4 ((4M − 4)x) = 0, that is:

c4(M)�4 − c3(M)�3 + c2(M)�2 − c1(M)�+ c0(M) = 0 , (4.32)

4.8 Four-level system 47

where the coefficients ci are polynomial functions of M :⎧⎨⎩

c4(M) = 256(M − 1)3

c3(M) = c4(M)

c2(M) = 96(M − 2)(M − 1)2

c1(M) = 16(M − 3)(M − 2)(M − 1)

c0(M) = (M − 4)(M − 3)(M − 2) .

(4.33)

A remarkable fact is that the dimension M of the environment ℋB is prevented to be a zero

of the polynomial (M − 4)(M − 3)(M − 2). The four roots of this equation (all nonnegative,

see appendix D) and the Lagrange multiplier � = 4(M − 1) give the stationary points of

the landscape (2.20). Obviously, there are 4! stationary points due to the invariance of the

system under permutations of the �i’s.

As previously explained (section 4.5), we are able to compute the typical purity �AB

without solving the equation for eigenvalues �i’s:

�AB = 1− 2c2(M)

c4(M)= 1− 3(M − 2)

4(M − 1)=

1

4+

3

4M+O

( 1

M2

). (4.34)

This equation states that, in thermodynamic limit M → +∞ (when system ℋB becomes a

reservoir), large deviations from the maximally mixed state 1/4 are forbidden.

Similarly, we can readily compute the values of Tr�q for q = 3, 4, just using the general

formulas (Tk = Tr�k):

s3 = −T3

3+T1T2

2− T1

3

6,

s4 = −T4

4+T1T3

3− T1

2T2

4+T2

8− T1

4

24,

(4.35)


obtaining

Tr�3 =1

16+

9

16(M − 1)+

3

8(M − 1)2,

Tr�4 =1

64+

9

32(M − 1)+

39

64(M − 1)2+

3

32(M − 1)3.

(4.36)

Remark. As expected, when M → +∞, Tr�k = N−(k+1). The next powers Tr�k, k > 4, are

not independent quantities since every N -symmetric polynomial can be written in terms

of (as a polynomial in) the elementary N -symmetric polynomial ek(x1, . . . , xN ). So, the

first N powers enable us to compute all the next ones. For example, one easily obtains

Tr�5 = T5 = −T 51

24 +5T 3

1 T212 − 5T1T 2

28 − 5T 2

1 T36 + 5T2T3

6 + 5T1T44 .

The so-called Renyi’s entropies Sq(M) = 11−q log2

∑i �i

q for q = 2, 3, 4 follow immedi-

ately:

S2(M) ≃ 2− 1

log 2

3

(M − 1),

S3(M) ≃ 2− 1

log 4

[9

(M − 1)+

6

(M − 1)2

],

S4(M) ≃ 2− 1

log 8

[18

(M − 1)+

39

(M − 1)2+ +

6

(M − 1)3

].

(4.37)

4.8.1 Behavior of solutions

In the limit M → +∞, eq. (4.32) reduces to

�4 − �3 +3

8�2 − 1

16�+

1

256= 0 ,

which has the solution � = 14 with multiplicity four: the larger the system HB, the more

mixed the state �A.

Since for large M the roots of eq. (4.32) are near the limiting point 14 , it is useful to

solve equation L(M−5)4 ((4M − 4)�) = 0 in terms of Δi = (�i − 1/4) in the limit of large M :


256(M − 1)3Δ4 − 96(M − 1)2Δ2 − 32(M − 1)Δ + 3(M − 3) = 0 , (4.38a)

Δ4 − 3

8(M − 1)Δ2 − 1

8(M − 1)2Δ +

3(M − 3)

256(M − 1)3= 0 , (4.38b)

Δ4 − 3

8MΔ2 − 1

8M2Δ +

3

256M2= 0 . (4.38c)

The above equation tells us that, as we expect, the quantities Δi sum to zero; since their

product is equal to 3/256M2 > 0 we expect two negative and two positive solutions. The

same argument yields Δi = O(1/√M). Neglecting the third term of eq. (4.38c) which is of

-0.3 -0.2 -0.1 0.1 0.2 0.3

-50

50

100

150

-0.2 -0.1 0.1 0.2

-100

100

200

300

400

-0.05 0.05

1000

2000

3000

4000

5000

6000

-0.02 -0.01 0.01 0.02

10 000

20 000

30 000

40 000

50 000

60 000

Figure 4.1: The polynomial g(Δ;M) of eq. (4.38a), for M = 6, 12, 100, 1000; as M increases the

roots of g(Δ;M) become closer and more symmetric with respect to the origin.

order O(1/M5/2), we recover the Hermite polynomial equation (4.30):

Δ4 − 3

8MΔ2 +

3

256M2= 0 . (4.39)

The solutions Δ1,4 = ±√

3+√

64M ≃ ±0.583/

√M and Δ2,3 = ±

√3−√

64M ≃ ±0.185/

√M exhibit


0.0 0.2 0.4 0.6 0.8 1.0

4

6

8

10

12

14

16

18

Λ

E8Λ i

<M = 5

0.0 0.2 0.4 0.6 0.8 1.0

20

25

30

35

40

45

50

Λ

E8Λ i

<

M = 10

0.0 0.2 0.4 0.6 0.8 1.0

120

140

160

180

200

220

240

260

Λ

E8Λ i

<

M = 50

0.0 0.2 0.4 0.6 0.8 1.0

250

300

350

400

450

500

Λ

E8Λ i

<

M = 100

Figure 4.2: Equilibrium configuration of the four charges in the external field 'ext(x) = −� log x+

�x. The charges repel each other electrostatically, via logarithmic Coulomb potential. The graphs

show the positions � of the charges when the dimension of the environment M = dimℋB (and then

the charge � in the origin) changes.

a symmetry of the deviations from the mean ⟨�⟩ = 1/4 at the order 1/M1/2. In order

to recover the asymmetry showed by numerical data, one has to solve eq. (4.38) exactly.

Quartics are the highest degree polynomial equation that can be solved by radicals. the

solutions can be easily obtained by using symbolic manipulation software packages like

Mathematica. We avoid writing the explicit solutions whose expressions are not transparent.

In order to fully extract the scaling of the solutions Δi(M), the suggested strategy is to write

them in terms of m = 1/M and to expand about m = 0. The first terms of the expansion are:


æ

æ

æ

æ

ææ

ææ æ æ æ æ æææææ æ ææææ

1 5 10 50 100 500 1000-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

M

D1

æ

æ

æ

ææ æ

æ æ æ æ æ æ æææææ æ ææææ

1 5 10 50 100 500 1000-0.20

-0.15

-0.10

-0.05

0.00

M

D2

æ

æ

æ

ææ æ

æ æ æ æ æ æ æææææ æ æææ æ

1 5 10 50 100 500 10000.0

0.1

0.2

0.3

0.4

0.5

M

D4

æ

æ

æ

æ

æ æ

ææ æ æ æ æ æææææ æ æææ æ

1 5 10 50 100 500 10000.00

0.02

0.04

0.06

0.08

M

D3

Figure 4.3: The four solutions of g(Δ,M) = 0 (solid lines), eq. (4.38a), give a good agreement

with the numerical averages of simulations (k = 2000 samples) when M is large. Note that the

first numerical value for Δi appreciably differs from the solution, because it refers to M = 4 and

equation g(Δ,M) = 0 holds when the environment ℋB has dimension greater than 4.

⎧⎨⎩

Δ1(M) = − 1

M1/2a1/2 +

1

Mb1 +

1

M3/2a3/2 +

1

M2b2 +

1

M3/2a3/2 +

1

M3b3 + o

( 1

M7/2

)Δ2(M) = − 1

M1/2a′1/2 −

1

Mb1 −

1

M3/2a′3/2 −

1

M2b2 −

1

M3/2a′3/2 −

1

M3b3 + o

( 1

M7/2

)Δ3(M) = +

1

M1/2a′1/2 −

1

Mb1 +

1

M3/2a′3/2 −

1

M2b2 +

1

M3/2a′3/2 −

1

M3b3 + o

( 1

M7/2

)Δ4(M) = +

1

M1/2a1/2 +

1

Mb1 −

1

M3/2a3/2 +

1

M2b2 −

1

M3/2a3/2 +

1

M3b3 + o

( 1

M7/2

),

(4.40)

where the values of coefficients ai/2’s and bi’s are listed in Tab. 4.1.

4.8.2 Measures of entanglement

We will now analyze three quantities: von Neumann entropy, purity and visibility. In the

following computations, we will retain only the first two terms in the asymptotic solutions in

eqs. (4.40) since they are sufficient to describe the deviation from the maximally entangled

state (the terms O(1/M1/2)) and the asymmetry of the distribution around 1/4 (the terms

O(1/M)):


Coefficients Numerical values

a1/214

√3 +√6 ≃ 0.583

a′1/214

√3−√6 ≃ 0.185

a3/2124

√332+ 37√

6≃ 0.234

a′3/2124

√332− 37√

6≃ 0.049

a5/2

√16857+6877

√6

1728≃ 0.206

a′5/2365

576√

50571+20631√6≃ 0.002

b11

2√6≃ 0.204

b27

12√6≃ 0.238

b3313

432√6≃ 0.295

Table 4.1: Coefficients of the expansion eq. (4.40). Note that the leading coefficients a1/2 and a′1/2give the approximated solutions eq. (4.31).

⎧⎨⎩

�1(M) =1

4− 1

M1/2a1/2 +

1

Mb1 + o

( 1

M3/2

)�2(M) =

1

4− 1

M1/2a′1/2 −

1

Mb1 + o

( 1

M3/2

)�3(M) =

1

4+

1

M1/2a′1/2 −

1

Mb1 + o

( 1

M3/2

)�4(M) =

1

4+

1

M1/2a1/2 +

1

Mb1 + o

( 1

M3/2

).

(4.41)

Visibility The visibility of a typical pure state of a bipartite system ℋA ⊗ℋB, with

dimℋA = 4 ,dimℋB = M is

v(M) =�max − �min

�max + �min

=�4 − �1

�4 + �1

=4a√M

4b+M+O

(1

M

)=

(√3 +√

6)

√M

+O

(1

M

).

(4.42)

The trend of visibility is showed in Fig. 4.4.


æ

æ

æ

æ

æ

æ

æ

ææ

æ æ æ æ ææææ æ æææ æ

5 10 50 100 500 10000.0

0.2

0.4

0.6

0.8

1.0

M

vHM

L

Figure 4.4: Behavior of the visibility v in eq. (4.42) as a function of M = dimℋB . The analytic

solutions for the most probable values (full line) are compared with empirical averages obtained via

simulations.

Purity The purity of a typical pure state of a bipartite system ℋA⊗ℋB, with dimℋA =

4 ,dimℋB = M has been previously calculated (eq. (4.34)):

�AB(M) =4∑i=1

�2i

=1

4+

3

4M+O

( 1

M2

).

(4.43)

Purity �AB tends to its minimum faster (as 1/M) than the visibility v. Fig. 4.5 shows the

behavior of the typical purity as function of M .


æ

æ

æ

ææ æ æ æ æ æ æ æ æ ææææ æ æææ æ

5 10 50 100 500 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

M

ΠHM

L

Figure 4.5: Behavior of the typical purity �AB as a function of M = dimℋB. Analytic solution

(blue solid line), eq. (4.17), for the most probable values are compared with empirical averages

obtained via simulations. The most probable values of purity are also compared with the average

value computed by Lubkin (red solid line), eq. (2.12). The most probable values is (correctly) lower

than the averages.

von Neumann entropy The von Neumann entropy of a typical pure state of a bipartite

system ℋA ⊗ℋB, with dimℋA = 4 ,dimℋB = M is

S(M) = −4∑i=1

�i log2 �i

∼ log2 4− 1

log 2

(16b1

2

M2− 8

M

(a1/2

2 + a′1/22))

= 2− 3

log 2

1

M− 2

3 log 2

1

M2.

(4.44)

See Fig. 4.6. With figure 4.7 we summarize the behaviors of the three quantities just

calculated.

Remark. The saddle point method returns the most probable eigenvalues, i.e. the peaks of

the joint density distribution fM,4(�1, . . . �4). If we plug in the solutions �1, . . . , �1 of the

saddle point equations into a four-dimensional vector �, the most probable joint distribution


æ

æ

æ

ææ æ

æ æ æ æ æ æ æ æ æææ æ æ ææ æ

10 20 50 100 200 500 1000 20001.0

1.2

1.4

1.6

1.8

2.0

M

SHM

L

Figure 4.6: Behavior of Von Neumann entropy eq. (4.44) as a function of M = dimℋB. In the

figure, the most probable values (blue solid line), the empirical averages obtained via simulations

and the mean values, conjectured by Page eq. (2.14) (red line) are compared. Again, the most

probable value is always lower than the average one.

can be written as

fM,4(�1, . . . �4) =1

4!

∑�

�(4)(�− � ⋅ �)

=1

4!

∑�

∏1≤i≤4

�(�i − �ij �j) ,(4.45)

where �’s are the 4-dimensional permutation matrices.

The saddle point method has to be used with care: the solution of the saddle point

equations tell us what is the most probable vector � of eigenvalues (modulo a permutation)

when � is sampled according to the invariant measure (2.7). In general we cannot infer

anything about the most probable values for a single eigenvalue, i.e the maxima of the

marginal one-body pdf f(1)M,4(�1), eq. (3.1). What really happens is showed in fig. 4.8: the

peaks of the joint pdf fM,4(�1, . . . �4) are no longer maxima when projected on the �1-axis.


æ

æ

ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

æ

æ

æ

æ

ææ

ææ

æ æ æ æ æ æ æ æ æ æ æ æ æ æ

æ

æ

æ

ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

5 10 50 100 500 10000.0

0.5

1.0

1.5

2.0

M

SHM

L,v

HML,

ΠHM

L

Figure 4.7: Graphics of eqs. (4.42), (4.43), (4.44): most probable visibility, purity and von Neumann

entropy of a typical pure state. For M −→ +∞ these quantities reach their asympotic values, 0,

1/4 and log2 4, respectively,.

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

M=12 , N=4

Figure 4.8: For small M one can compute the marginal one-body pdf f(1)M,4(�1), eq. (3.1). The

figure shows the graph (blue line) of the derivative of f(1)12,4(�1) with respect to �1 computed exactly,

and the polynomial (purple line) of eq. (4.32) whose roots are the most probable eigenvalue of the

joint pdf f12,4(�1, . . . �4). One has to look at the points where the derivative df(1)12,4(�1)/d�1 vanishes

with a negative slope (maximum points of f(1)12,4(�1)); the zeros of interest do not coincide with the

maxima of the joint pdf.

Chapter 5

Isopurity typical states

So far, we have dealt with unbiased pure states. We now look for the most probable eigen-

values sampled on isopurity manifolds, i.e. the most probable value of f(N,M)(�1, . . . , �N ),

eq. (2.7), on the the submanifolds:

ℳ(�AB) ={

(�i, . . . , �N ) ∈ ℝN+ : �1 ≤ ⋅ ⋅ ⋅ ≤ �N ,∑i

�i = 1 ,∑i

�i2 = �AB

}. (5.1)

This optimization problem can be converted in a “Coulomb gas” electrostatic problem

but, unfortunately, it does not seem to have a simple solution as for unbiased states. Our

strategy will be to find the analytic behavior for small deviations from typicality, and for

high purity regions. These analytic results have been used as starting points for a numerical

approach to the saddle point equations. Finally, from numerical solutions we will be able

extract information about atypical random states.

5.1 Isopurity manifolds

We look for the most probable eigenvalues sampled on isopurity manifolds, i.e. the most

probable spectrum when sampling is done according to the following joint distribution

f(N,M)(�1, . . . , �N ) = CN,M∏i<j

(�i − �j)2∏l

�M−Nl

∏i

�(�i)�(

1−∑k

�k

)�(�AB−

∑k

�k2).

(5.2)

58 Isopurity typical states

By introducing a second Lagrange multiplier � to take into account the new constraint, the

saddle point equation, eq. (4.4), modifies into:⎧⎨⎩

−2�M�i + 2∑

j ∕=i1

�i−�j + M−N�i− �M = 0 , ∀i ∈ {1, . . . , N}

∑i �i = 1

∑i �i

2 = �AB .

(5.3)

The new constraint,∑

i �i2 = �AB, enables us to compute immediately the second

moment of f(N,M)(�1, . . . , �N ), that is, the width of the pdf restricted on the isopurity

manifolds. From the permutation invariance of fN,M :

�AB =

⟨∑i

�i2

⟩=∑i

⟨�i

2⟩

= N⟨�i

2⟩

=⇒⟨�i

2⟩

=�ABN

, ∀1 ≤ i ≤ N , (5.4)

and then, the variance is:⟨⟨�i

2⟩⟩

=⟨(⟨�i⟩ − �i

)2⟩=⟨�i

2⟩− ⟨�i⟩2 =

1

N

(�AB −

1

N

), ∀1 ≤ i ≤ N , (5.5)

where, in the last equality the first constraint was used.

As pointed out, finding the most probable spectrum is analogous to the problem of finding

the equilibrium positions of N interacting charges in an external field. The charges repel

electrostatically (2D-Coulomb potential) in an external field that now includes a new term

due to � ∕= 0:

'ext(x) = −� log x+ �x+ �x2 , (5.6)

where � = M − N ≥ 0. In the origin there is a charge � that repels the unit charges.

Moreover, the unit charges experience a constant force whose direction is given by the sign

of �, and the Lagrange multiplier � plays the role of the elastic constant of a harmonic

potential �x2. We can look at � as an inverse (fictitious) temperature which fixes the energy

�AB. As discussed in the previous chapter, when � = 0 the Lagrange multiplier has to be

positive in order to have an energy minimum in (0, 1). The minimum is at xc = �� . Since

we expect that this critical point is close to 1/N we can immediately guess � ∼ NM in

5.2 The single qubit 59

the large M limit. When � ∕= 0 the scenario becomes more interesting. Again the N unit

charges will arrange themselves in the external potential in a configuration of minimum

energy. In the following analysis, we will consider the unbalanced situation � = M −N > 0.

The critical points of the external potential are:

xc± =−� ±

√8�� + �2

4�. (5.7)

We want to find some relations among �, � and � such that at least one of the two critical

points falls in the admissible region xc ∈ (0, 1). First of all, we notice that the discriminant

Δ = 8�� + �2 (5.8)

has to be nonnegative (� ≥√−8��).

Positive temperature, � > 0. When � > 0, the only significant critical point is x+.

This minimum point falls in (0, 1) iff � ≤ 2� + �.

Negative temperature, � < 0. Fixed � < 0, the two critical points x± depend on �.

For all � > 0, the critical point x+ belongs to the admissible interval and it is a minimum

point. When � ≤ 2�+ �, x− (≤ 1) is a maximum point. If � < 0 the potential is everywhere

repulsive.

5.2 The single qubit

Let us briefly discuss the prototype system, that is the quantum bit. For a single qubit,

the most probable pairs of eigenvalue obey the equations:⎧⎨⎩

−2�M�1 + 2�1−�2 + M−2

�1− �M = 0

−2�M�2 + 2�2−�1 + M−2

�2− �M = 0

�1 + �2 = 1 .

�12 + �2

2 = �AB

(5.9)


where we have rescaled the Lagrange multiplier � = �M (for large M).

For a single qubit the above system of equations is immediately solvable. With the

additional condition �AB ∕= 1/2, 1, the solution is:

∙ the Lagrange multipliers are

� = −2(1− 3�AB +M(2�AB − 1))

(�AB − 1)(2�AB − 1)M

� =4M + (2�AB − 3)� − 6

2M

(5.10)

∙ the two eigenvalues, in terms of purity, are �1/2 = 12

(1±√

2�AB − 1).

One can invert the relation �(�AB) since the level surfaces of the Lagrange multiplier

are isopurity manifolds. Once the dependence �AB(�;M) is established, by taking the

limit M → +∞, one finds the behavior of the eigenvalues with respect to the fictitious

temperature:

⎧⎨⎩

�AB(� ∕= 0) =2+3�−∣2+�∣

4�

�AB(� = 0) = 12

�1/2 = 12

(1±

√2�AB(�)− 1

).

(5.11)

At the value � = −2 the symmetry S2 (permutations of the two eigenvalues) suddenly breaks.

One of the two eigenvalue, increases (while the other one decreases) and tends towards the

limiting value �max = 1 (�min = 0). Figure 5.1 displays what happens. This symmetry

breaking does not surprise: a not completely mixed 2× 2 density matrix necessarily has

different eigenvalues.

5.3 Small deviation from typical purity 61

-20 -15 -10 -5 0 5 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Β�M

ΠA

B

-20 -15 -10 -5 00.0

0.2

0.4

0.6

0.8

Β�M

Λ1,

2

Figure 5.1: Most probable purity and eigenvalues of states belonging to the level surfaces of �. At

� = −2 the symmetry S2 breaks, and purity increases.

5.3 Small deviation from typical purity

Following the approach adopted for typical states, we try a simplified equation valid

for large M and �M close to 0. Our guess is that, for � ∼ 0, the eigenvalues configuration

will resemble the typical one, �i = 1/N + �i/M�. An educated guess for the purity is

�AB = 1/N + p/M2�. Then, the saddle point equations read:

⎧⎨⎩

−2 �MNM� − 2�M

�iM + 2

∑j ∕=i

1�i−�j +NM1−� −N2M1−2��i − � = 0 , ∀i ∈ {1, . . . , N}

∑i �i = 0

∑i �i

2 = p .

(5.12)

With � = 1/2 (see equations (4.7)), �M ∝M and then, introducing � = limM→+∞ �M/M ,

we can write the first N equations as

−2�√M

N− 2��i + 2

∑j ∕=i

1

�i − �j+N√M −N2�i − � = 0 .

The simplified equations previously derived, eqs. (5.12), can be tackled by means the


Stieltjes trick (appendix C). The corresponding differential equation is

g′′(x) +

(−2�

√M

N− 2�x−N2x+N

√M − �

)g′(x) +N(N2 + 2�)g(x) = 0 . (5.13)

Again, the polynomial solution is provided by a Hermite polynomial of degree N :

g(x) = HN

⎛⎝−−2�√M +

√MN2 −N�

√2N

√2� +N2

+

√2� +N2x√

2

⎞⎠ , (5.14)

and we impose that the polynomial be even. This requirement yields the unit-trace Lagrange

multiplier

� =−2�√M +

√MN2

N⇒ � = MN − 2

�

N, (5.15)

and the solution reads

g(x) = HN

⎛⎝√

2� +N2

√2

x

⎞⎠ . (5.16)

For example, specializing the solution to the four-level case,⎧⎨⎩

− �√M

2 − 2��i + 2∑

j ∕=i1

�i−�j + 4√M − 16�i − � = 0 , ∀i ∈ {1, . . . , 4}

∑i �i = 0.

∑i �i

2 = p .

(5.17)

we get:

H4

⎛⎝√

2� + 16√

2x

⎞⎠ ∝ x4 − 6

16 + 2�x2 +

3

(16 + 2�)2, (5.18)

whose zeros �i give the equilibrium configuration of the four charges:

�1,2,3,4 =1

4±

√3±√

6

16 + 2�

1√M

, (5.19)

�AB(M) =1

4+

6

M(8 + �), (5.20)

5.4 Balanced bipartition 63

When � → 0 we recover the typical case eqs. (4.31). The value � = −8 seems to be a

critical value for �i’s and purity �AB . The convergence radius of this approximate solution,

can be estimated as follow. The simplified equations were obtained by linearization of the

logarithmic repulsion due to the charge � = M −N around the mean value ⟨�⟩ = 1/N . The

linearization ceases to hold approximatively when �i ≃√M/N . For N = 4, this translates

into the condition on the fictious temperature∣∣∣�∣∣∣ <∼ ∣∣∣2(3+

√6)

M − 8∣∣∣.

5.4 Balanced bipartition

The balanced case, M = N , is exactly solvable in the context of orthogonal polynomial.

Although we are not really interested in the balanced situation, we will see that the solution

will give us some precious information. If we add a constraint on the purity by means of

the Lagrange multiplier �, in the balanced case M = N , we have to solve the following

linear differential equation:

g′′(x)− (2�x− �)g′(x) + 2�Ng(x) = 0 , (5.21)

whose polynomial solution is the Hermite polynomial HN ( �2√�

+√�x). As done previously,

we fix the Lagrange multiplier � = − 2N � in order to satisfy the unit-trace condition; for

N = 4 one finds � = −�2 . The family of polynomial solution for the four-level system is

parametrized by �:

P�(x) = HN

(√�

(x− 1

4

))= x4−x3 +

(3

8− 3

�

)x2−

(3

2�+

1

16

)x+

3

4�2− 3

16�+

1

256.

(5.22)

This polynomial correctly describes a four-levels spectrum if the coefficients satisfy the

inequalities of eqs. (5.37) and, therefore, the zeros of P�(x) are meaningful when 0 ≤3

4b2− 3

16b + 1256 ≤ 1/256, i.e. iff � ≥ 8

(3 +√

6)

= 43.5959. The purity �AB in terms of � is

given by:

�AB(�) = 1− 2

(3

8− 3

�

)=

1

4+

6

�; (5.23)

when � varies from 8(3 +√

6)

to +∞, the purity �AB spans all values from 0.387 to the

totally mixed value 0.25. The lower bound on � arises because in the balanced bipartition


(� = 0) there is no logarithmic repulsion in the external field of eq. (5.6) that constraints

the rank of the reduced density matrix �A to be the maximal one, namely N . Equivalently,

we say that in the balanced case, the joint pdf fN,M (�1, . . . , �N ) does not vanishes when

one of its arguments become zero. However, the main point of this section is the behavior

� = − 2N � (for � = 0).

5.5 High purity regions

It is not easy explore regions of high purity. We tried the following strategy. The

condition �AB ≃ 1 means that the greatest eigenvalue �max = max {�1, . . . , �N} is near the

maximum allowed value � = 1 and, consequentely, the others are pushed very close to 0 by

the unit-trace condition, that is

�max = �, �i ∼1− �N − 1

, 1 ≤ i ≤ N − 1 , (5.24)

where the last condition follows by considering the baricenter of the smallest N−1 eigenvalues

(∑N−1

i �i = 1− �). Observe that, due to the unit-trace constraint, the extreme eigenvalues

can vary in 1/N ≤ �max ≤ 1 and 0 ≤ �min ≤ 1/N , �min = min {�1, . . . , �N}. For typical

states �>∼ 1/N . Now we will treat the max eigenvalue as a parameter which controls the

distribution of the remaining N − 1. We hope that this approach can give reasonable results

at least when � ∼ 1. We suppose that, given the larger �max = �, the remaining eigenvalues

will fluctuate around their centre of mass imposed by normalization condition, with a

deviation which again decreases with the squared root of the dimension of environment:

�i = 1−�N−1 + �i√

M, 1 ≤ i ≤ N − 1. Later, it will be clear that to fix �max is quasi-equivalent

to fixing purity �AB.

The saddle point equation for the three charges (now indices i, j run from 1 to N − 1)

with the constraint �(�− �max) can be written as:

⎧⎨⎩2∑

j ∕=i1

�i−�j −(

2(N−1)2

(1−N�)2M+ (N−1)2

(1−�)2

)�i + (N−1)

√M

1−� + 2(N−1)

(1−N�)√M− �√

M= 0 , ∀i ∈ {1, . . . , N − 1}

∑N−1i=1 �i = 0 .

(5.25)

5.6 Four-level system: numerical solutions 65

The tool of choice is again the Stieltjes trick. The (N − 1) equilibrium points are the zeros

of a Hermite polynomial of degree (N − 1). The resolving polynomial must be even and

this condition yields the multiplier �, which in the large M limit is

� =(N − 1)M

1− �, (5.26)

and we recover the typical value for � ≃ 1/N .

We write down the solution for a four-level system, N = 4. The Hermite polynomial

that solve eq. (5.25), with � correctly adjusted, is

54√

2(2(1− �)2 +M(1− 4�)2

)3/2M3/2(1− �)3(1− 4�)3

x3 −18√

2√

2(1− �)2 +M(1− 4�)2

√M(1− �)x(1− 4�)

, (5.27)

whose solutions, in the large M limit, are

�i =1− �√

3

ri√1 + 2(1−�)2

M(1−4�)2

∼ ri1− �√

3

(1− (1− �)2

M(1− 4�)2

), (5.28)

where ri = −1, 0,+1 respectively for i = 1, 2, 3, with a discrete agreement with the typical

case. Now it is easy to compute the most probable value of the purity (using Vieta’s

formulas, see appendix B):

�AB = �2 +(1− �)2

3+s2(e1, e2, e3)

M∼ 1

3(4�2 − 2�+ 1) +

2

3M(�− 1)2 . (5.29)

In the limit M → +∞, we can invert the above relation and get:

�max =1

4

(1 +√

12�AB − 3)

�i =1

4−√

4�AB − 1

4

( 1√3− ri√

M

),

(5.30)

with ri = −1, 0, 1 respectively for i = 1, 2, 3. The above trends are displayed in Fig. 5.2.

5.6 Four-level system: numerical solutions

The previous result could be a starting point for a numerical approach to the saddle

point equations eqs. (5.9). Recall that our goal is to find the most probable spectrum,


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Π

Μ,

<Λ

i>

=H1-

ΜL

3

Figure 5.2: Behavior of � = �max and ⟨�i⟩ versus purity, eq. (5.30).

when the density matrix is sampled according to eq. (5.2), i.e. �i(�AB;M). We expect

that eqs. (5.30) hold in the M → +∞ limit. Therefore, the idea is to use a standard

root-finding algorithm, such as the Newton-Raphson method, with starting test values

close to the analytic approximated solutions eqs. (5.30). Figure 5.5 shows the comparison

of the analytic solution and the numerical ones for some values of the dimension M of

the environment. Morever, Fig. 5.6 displays the behavior of the eigenvalues �i’s when the

fictitious temperature � = �/M varies (figure refers to M = dimℋB = 50).

5.6.1 Lagrange multipliers

Numerical solutions provide new information about the trend of the Lagrange multipliers

� and � versus purity �AB (see Figg. 5.6). First, numerical solutions are in agreement

with our heuristic discussions about the cuts on the Lagrange multipliers space (section

5.1). Both � and � cannot be negative at the same time, because such a choice prevents

the existence of an attractive well (see eq. 5.6). For the same reason, regions � ≤√−8��

(� ≤ 0) and � ≤ 2� + � (� ≥ 0) are forbidden. For � ≥ 0 we expect that the charges

will stay close to 1/N . We may crudely impose that the minimum of 'ext(x) is located at

x+ ∼ 1/N and find, for large positive �’s:


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0.0 0.2 0.4 0.6 0.8 1.0

-40

-20

0

20

40

ΠAB

Ξ�M

,Β

�M

M = 5

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0.0 0.2 0.4 0.6 0.8 1.0

-40

-20

0

20

40

60

ΠAB

Ξ�M

,Β

�M

M = 10

Figure 5.3: Trend of the rescaled Lagrange multipliers � = �/M (red) and � = �/M (blue).

By thinking at � as an inverse temperature that controls energy �AB, we find that far from the

maximally entangled state and the separable ones the “heat capacity” ∂�AB

∂� is large (while purity

�AB increases the fictious temperature � slowly varies).

�(� ≫ 0) = �N − 2�

N. (5.31)

In terms of the rescaled Lagrange multipliers � = �/M and � = �/M we obtain, in the

large M limit:

�(� ≫ 0)M→+∞−→ N − 2�

N. (5.32)

Moreover, when � = 0, � must be stricly positive (and it is). The charges are close to the

maximally entangled configuration �i ∼ 1/N , and then, we are allowed to write:

�(� = 0) = �N2

2, (5.33)

and in the large M limit:

limM→+∞

�(� = 0) =N2

2, (5.34)

When � is negative, a maximum point x− appears in the admissible region. The presence

of this maximum makes the region x− ≤ � ≤ 1 unphysical. Notice that, if we fix the purity

�AB =∑

i �2i , the eigenvalues �i’s cannot exceed the upper bound

√�AB. By imposing


ææ

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-40 -30 -20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

Β�M

Λi

M =50

Figure 5.4: Trend of the eigenvalues versus �/M .

x− =−�−√

8��+�2

4� =√�AB, we find �(� ≪ 0) = �−2��AB√

�AB, in agreement with the limit

value � = �− 2� for pure states. All these facts can be read in Fig. 5.6.

We can compare these numerical solution for the four-level system, N = 4, with the

analytic result we have obtained for the single quibt N = 2. From eqns. (5.10) one may

extract the behavior of � as a function of �:

�(�) =2(M − 1)− 3�M +

√4− 4(3� + 2)M + (� + 2)2M2

2M. (5.35)

Taking the limit M → +∞ one obtains:

�(M → +∞) =1

2

(2− 3� +

∣∣∣2 + �∣∣∣)

=

⎧⎨⎩ 2− �

2, � ≥ −N

2

2= 2

−2�, � ≤ −N2

2= 2 .

(5.36)

For N = 2, we get that our heuristic analysis is full confirmed. In the M → +∞ limit, the

critical point is �c = −2; eq. (5.36) means � = N − 2N � = 2 − �, for � ≥ �c = −2, and


� = 1− 2�, for � ≥ �c = −2. The points �(� = 0) = N = 2 and �(� = 0) = N2/2 = 2 are

fully confirmed.

5.6.2 Elementary symmetric invariants

We have already said that the eigenvalues of a density matrix form a probability vector

(�i ≥ 0 and∑

i �i = 1). For a four-level system we have:

s1(�A) = e1(�1, . . . , �4) =∑i

�i = 1 ,

s2(�A) = e2(�1, . . . , �4) =∑j1<j2

�j1�j2 ≤3

8,

s3(�A) = e3(�1, . . . , �4) =∑

j1<j2<j3

�j1�j2�j3 ≤1

16,

s4(�A) = e4(�1, . . . , �4) = �1�2�3�4 ≤1

256,

(5.37)

where the inequalities follow from the Schur-concavity of ei’s. When rank� = 3, i.e. one

eigenvalue vanishes, we have

e4 = 0; 0 < e3 ≤1

27; 0 < e2 ≤

1

3.

If rank� = 2:

e4 = 0; e3 = 0; 0 < e2 ≤1

4.

while, for a pure state, e2,3,4 = 0.

Elementary symmetric polynomials seem to provide a more manageble tool to investigate

the transitions of �, since they have definite boundary values for different regions (rank of

�). Given the values of ei’s, the eigenvalues have to be solutions of a quartic of the form:

x4 − x3 + c2x2 − c1x+ c0 = 0 , (5.38)

where the coefficients ci are all positive and equal to e4−i. For a rank-1 density matrix, i.e.

when 3 eigenvalues vanish, the unique admissible spectrum is the solution of x4 − x3 = 0.

When two eigenvalues are greater than zero, the state ceases to be pure, and the admissible

spectrum is the solution of x4−x3 + c2x2 = 0, that is �1,2 = 0 and �3,4 = 1

2

(1±√

1− 4c2

);


this set of solution is meaningful for all 0 < c2 ≤ 14 . Things become more complicated when

a third eigenvalue becomes nonzero: we have to study the set of solutions of x4 − x3 +

c2x2 − c1x = 0. Such an equation has nonnegative solutions for all 0 < c1 ≤ 1

27 and c2 ≤ 13 ,

but the parameter space (0, c1, c2) is not totally accessible, since we must consider only

equations with real solutions. It is not easy to understand which region is admissible in the

parameter space, and difficulties increase when equation (5.38) is complete. The quantities

ci’s satisfy some inequalities which restrict the parameters space. In particular, from the

inequalities (see [12])[s2(�)

s2(1/N)

]2

≥[

s3(�)

s3(1/N)

]3

≥ ⋅ ⋅ ⋅ ≥[

sN (�)

sN (1/N)

]N, (5.39)

we have, for the four-level system:

8

3c2 ≥ 16c1 ≥ 256c0

4 c01/4 ≤ (16c1)1/3

4 c01/4 ≤

(8

3c2

)1/2

.

(5.40)

To get the admissible subset in the ci’s space, we recall Ungar characterization [22].

Given a real quartic

x4 + a3x3 + a2x

2 + a1x+ a0 = 0 ,

let

Q = 12a0 + a22 − 3a1a3 , R = 27a2

1 − 72a0a2 + 2a32 − 9a1a2a3 + 27a0a

23 ,

T = 3a23 − 8a2 + 8Re

[R+

√R2 − 4Q3

2

]1/3

;(5.41)

the four roots of the quartic are all real iff(R2 − 4Q3

)≤ 0 and T ≥ 0 for all the three

possible cubic roots. These constraints leave the connected subset Ω as admissible parameters

domains.

In our problem, we focus our attention on the full rank situation, since the form of the

joint pdf, eq. (2.7), prevents the eigenvalues to be zero. Then, both c2, c3 and c4 are strictly

positive. A constrained value of purity means that the four eigenvalues are the zeros of the


Coefficients Numerical values

a1 −0.250032a3/2 0.192487

b1 −0.0313681b3/2 0.0484017

b2 −0.0210097

Table 5.1: Coefficients of the expansion fits eq. (5.45). Note that the signs of coefficients agree

with the general forms of eq. (4.35).

polynomial:

x4 − x3 +1

2(1− �AB)x2 + c1x− c0 = 0 . (5.42)

From our numerical solution, we can compute the most probable c1 and c0 as functions

of �AB (see Fig. 5.8), an then try to fit these data. The suggested strategy in order to

interpolate the ci’s is to try an expression of the form

c1(�AB;M) =1

16+

k1∑m

am/2

(�AB −

1

4

)m/2

c0(�AB;M) =1

256+

k0∑m

bm/2

(�AB −

1

4

)m/2,

(5.43)

with the additional conditions

c1(�AB = 1;M) =1

16+

k1∑m

am/2

(3

4

)m/2= 0

c0(�AB = 1;M) =1

256+

k0∑m

bm/2

(3

4

)m/2= 0 .

(5.44)

To find the numbers k1 and k2 of such expansions, one may refer to the general expressions

of eq. (4.35), and conclude that k1 = 3, and k0 = 4. Equivalently, we can justify this

choice by dimensional analysis arguments (assuming that ci(�AB)’s are analytic functions).

Eq. (4.35) says also that, since Tr�A is not a degree of freedom, there will not be a �1/2AB


dependence. We have used a standard algorithm for searching for values of parameters that

yield the best least-squares fits to data. For M = 100000 numerical data we have found the

following best fits:

c1(�AB;M = 100000) =1

16+ a1

(�AB −

1

4

)+ a3/2

(�AB −

1

4

)3/2

c0(�AB;M = 100000) =1

256+ b1

(�AB −

1

4

)+ b3/2

(�AB −

1

4

)3/2+ b2

(�AB −

1

4

)2.

(5.45)

The values of parameters ai’s and bi’s that yield the best fits, are listed in table Tab. 5.1.


ææ

æ

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Figure 5.5: Most probable eigenvalues versus purity, numerical and analytic (dashed blue and

red lines, eq. (5.30)) solutions. For small values of the purity �AB , �4 and �3 increase with purity

and the other two descrease (our convention is �4 ≥ �3 ≥ �2 ≥ �1). At a critical value �cAB, also

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typical eigenvalues, eqs. (4.17) and (4.32).


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-100 -50 0 50

0

50

100

150

200

Β�M

Ξ�M

M = 1000

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-100 -50 0 50

0

50

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150

200

Β�M

Ξ�M

M = 10000

Figure 5.6: Plots of � = �/M versus � = �/M (blue points). The blue region is the admissible

region, that is the domain corresponding the existence of a minimum of the external field in the

admissible region [0, 1]. The red line are the asymptotical (rescaled) trends � = �N − 2�N (large

positive �’s, towards the totally mixed state) and � = � − 2� (large negative �’s towards the

separable state), and the lower bound � =√−8�� (positive discriminant Δ, eq. (5.8)). Data

intersect the �-axis at the correct value �(� = 0) = N(M − 1)/M , eq. (4.15).


0.0 0.2 0.4 0.6 0.8 1.0

0

500

1000

1500

2000

2500

3000

Λ

E8Λ i

<

M = 50

ΠAB = 0.251

0.0 0.2 0.4 0.6 0.8 1.0

100

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150

160

Λ

E8Λ i

<

M = 50

ΠAB = 0.3

0.0 0.2 0.4 0.6 0.8 1.0

140

150

160

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190

200

210

Λ

E8Λ i

<

M = 50

ΠAB = 0.4

0.0 0.2 0.4 0.6 0.8 1.0

160

180

200

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260

280

Λ

E8Λ i

<

M = 50

ΠAB = 0.5

0.0 0.2 0.4 0.6 0.8 1.0

200

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Λ

E8Λ i

<

M = 50

ΠAB = 0.7

0.0 0.2 0.4 0.6 0.8 1.0

400

600

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1000

1200

Λ

E8Λ i

<

M = 50

ΠAB = 0.9

Figure 5.7: Equilibrium configuration of the four charges in the external field 'ext(x) = −� log x+

�x+ �x2. The charges repel electrostatically via a logarithmic Coulomb potential. Graphics refer to

M = 50 and show the behavior of the charges at different values of purity �AB . Equilibrium positions

of the charges and parameters � an � (the Lagrange multipiers) are those found by numerical solution

of the saddle point eq. (5.9).





M=5

M=10M=1000

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

ΠAB

s 3

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M=5

M=10

M=1000

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.000

0.001

0.002

0.003

ΠAB

s 4

Figure 5.8: Numerical solutions: trend of s3(�1, . . . , �4) =∑j1<j2<j3

�j1�j2�j3 and

s4(�1, . . . , �4) = �1�2�3�4 = det �A versus purity �AB. The figures show the trend for some

values of dimℋℬ, M = 5, 10, 1000. When �AB = 1, s3 = s4 = 0; invariants s3 and s3 reach their

maximum values for the totally mixed state. Red solid lines show the behavior of the invariants s3

and s4 for the typical case � = 0 (when M increase), see eq. (4.18).

Chapter 6

Entanglement in mixed states

Once we have found the most probable spectrum of the reduced density matrix �A =

TrB ∣ ⟩ ⟨ ∣, a natural question is how much entanglement is still in �A, regarded as a

composite system. To this end, we will recall a separability criterion for mixed states due

to Zyczkowski et al. [10].

6.1 Separability criteria for mixed states

Consider a bipartite system ℋA = ℋA1 ⊗ℋA2 and its states (generally not pure). Then

all states � sufficiently close to the maximally mixed one, with

Tr�2 ≤ 1

dim �− 1, (6.1)

have the PPT property [10]. Thus, all N -dimensional density matrices whose purity is less

than �PPT = 1/(N − 1) have positive partial transpose.

Remark. A simple geometric interpretation of the critical value is as follows. Density

matrices are positive operators forming a convex set. Then, density matrices with one

null eigenvalue belong to the boundary of this set. It is easy to show that, the critical

value �(PPT )AB = 1/(N − 1) identifies a ball (with respect to the Frobenius norm, eq. (1.7))

inscribed in the convex set of density matrices, centered on the maximally mixed state. All

states belonging to this ball are not entangled.

78 Entanglement in mixed states

6.2 Separability of the reduced density matrix

Let us consider N -dimensional mixed states �A ∈ D(ℋA) obtained by tracing a random

pure states ∣ ⟩ of size MN over the M -dim space ℋB. The Haar measure on the unitary

group U(MN) induces the measure (2.7) in the space of N -dimensional mixed states. For

typical states, we have shown (see eq. (4.17)) that the most probable purity is

�AB = 1− (N − 1)(M − 2)

N(M − 1),

and numerical investigations show that purity is concentrated around this most probable

value. Given a large supply of copies of our randomly drawn state ∣ ⟩, most of the reduced

density matrices have purity “very close” to �AB. Let us compare it with the critical value

�PPTAB :

�AB(M) = 1− (N − 1)(M − 2)

N(M − 1)≤ 1

N − 1= �PPTAB ⇒M ≥ N2 − 2N + 2 = MPPT . (6.2)

Typical N -dimensional states obtained by tracing from a big environment are separable.

Starting from a typical pure vector of length L, if we restict ourselves at a portion of the

system of size N ≤ L1/3, trying to find residual entanglement is a hopeless task. When the

size of the subsystem increases beyond N = L1/3, the situation becomes manageable again.

Looking at the N = 4 case, eq. (6.2) gives the critical value MPPT = 10. For a 2× 2

state, the PPT property is sufficient to guarantee separability, which can be detected by

computing concurrence. The limit value for the size of ℋB in order for the pair of qubits to

be entangled, is confirmed by numerical experiments (see Tab. 3.1). As a result of the so

called monogamy of entanglement, large purity, i.e. low entanglement with the environment,

is required in order to exhibit large concurrence, i.e. entanglement within the pair.

6.3 Concurrence

In the following we will refer to the four-level system. We started with a typical pure

state ∣ ⟩ ⟨ ∣, i.e. a projector in ℋAB. The partial trace over B returns a mixed state �A.

The most probable �A’s belong to the coadjoint orbit OΛ of the most probable spectrum Λ

�A ∈{U ΛU † ∈ D(ℂ4) : U ∈ U(ℂ4)

}, (6.3)

6.3 Concurrence 79

à

à

à

à

à

à à à

æ

æ

ææ æ æ æ æ

4 6 8 10 120.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

M

C

Figure 6.1: Typical states: behavior of concurrence (entanglement between the pair of qubits)

versus the dimension M of subsystem HB . The spectrum of �A is fixed to be the most probable one.

Blue points refer to the concurrence of states like eq. (6.4), that are maximal. Reds are numerical

averages (with error) over the unitary group. For M ≥ 10 (for a system of qubits: HB ≃ 3−4 qubits)

the two qubits described by �A are almost surely unentangled.

that is, the set of matrices unitarly equivalent to Λ. The exact value of the purity is given

by �AB(M) = 1− 3(M−2)4(M−1) which is greater than 1/3 for small values of M . It was shown

[24] that, given the spectrum �, the general form of a density matrix that is maximal, i.e.

that maximize the concurrence (1.27), is (up to local unitary transformations):

�max =

⎛⎜⎜⎜⎜⎝x+ r

2 0 0 r2

0 y 0 0

0 0 z 0r2 0 0 x+ r

2

⎞⎟⎟⎟⎟⎠ (6.4)

with x = (�↓1 + �↓3)/2, y = �↓2, z = �↓4 and r/2 = �↓1 − �↓3. We have computed ex-

plicitly �max(�1, . . . , �4) and its concurrence. Fig. 6.1 show both the higher concurrence

C(�max) for some dimensions M = dimℋB, and empirical average over the unitary group⟨C(UΔU †)

⟩U(4)

. Numerical computations show that, even when the environment is suffi-

ciently small to have residual entanglement within the pair of qubits, concurrence attains a





M=5

M=10

M=1000

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

ΠAB

C

Figure 6.2: Concurrence vs Purity. Using numerical solutions (see Fig. 5.5) for the atypical case

(� ∕= 0), its easy to build up a graphic of the most probable concurrence when the purity of �A is

fixed. The more mixed the state, the lower the entanglement. The threshold value of the purity

above which entanglement within the pair can exist is �PPTAB = 1/3. Blue, green and red lines refer

respectively to M = 5, 10, 1000.

rather small value. This means that a state of two qubits, extracted from a typical pure

state, is characterized by low entanglement. We conclude with the numerical results for

concurrence of isopurity typical states. In section 5.6, we have shown the numerical solution

of the saddle point equation when the sampling is done over isopurity manifolds. Proceeding

as before, from the most probable spectrum (which now is a function of �AB), we can find

the upper bound of entanglement by computing C(�max(�AB)). As a result, we obtain (see

fig. 6.2) the locus of maximum concurrence of a pair of qubits whose state has purity �AB.

6.4 An integral over the unitary group

In the previous chapters we studied the statistical properties of random pure states

of a bipartite system. The subparts of the whole system are described by the reduced

density matrices, generally mixed. In chapter 4, we have studied the typical properties

of pure states sampled according to the unitarly invariant measure on U(L). Successively,

we focused the attention on typical states of fixed purity (chapter 5). In both cases, the

6.4 An integral over the unitary group 81

used tool has been the saddle point method, which gives us the most probable spectrum

of the reduced density matrices of the substystems. Neverthless, pure states are special

quantum states (the extremal points of D(L)); the state of a quantum system is most

frequently better described as a mixture of pure states, that is a mixed state. However, a

generalization of the above approach in order to perform a statistical analysis for mixed

states is not so straightforward [3].

So far, we tackled the problem of the most probable spectrum for random states picked

out from isopurity manifolds (submanifolds of D(ℋS)). Now we ask the following. Let us

start from a fixed-purity set of typical states. This means that the state of system A is

described by a density matrix:

�A ={UΛU †

}U∈U(N)

, (6.5)

where Tr�2A = TrΛ2 = �AB ∈ [1/N, 1] is fixed. Let us introduce a bipartition of the

subsystem in two smaller parts ℋA = ℋA1 ⊗ℋA2 , dimℋA1 = N1, dimℋA2 = N2 with

N1N2 = N . What is the average purity of the reduced density matrices TrA2�A = �A1 ,

if in the coadjoint orbit eq. (6.5) the U ’s are Haar distribuited? Note that for mixed states �A

Tr�2A1

= TrA1(TrA2�A)2 ∕= TrA2(TrA1�A)2 = Tr�2A2

,

and we will speak in terms of A1-local purity and A2-local purity.

In the following, we will use the convention

⎧⎨⎩Capital indices: 1 ≤ R ≤ N

Latin indices: 1 ≤ j ≤ N1

Greek indices: 1 ≤ � ≤ N2 .

Our mixed states have the form:


�A = U( N∑R=1

�R ∣ R⟩ ⟨ R∣)U †

=N∑R=1

�R

(U ∣ R⟩ ⟨ R∣U †

)

=

N∑R=1

�RP(R) ∈ D(ℋA) .

(6.6)

In eq. (6.6),∑N

R=1 �2R = �AB is fixed; ∣ R⟩’s and P (R)’s are pure states of ℋA, and

U ∈ U(ℋA). In the product basis {∣k⟩ ⊗ ∣�⟩}, 1 ≤ k ≤ N1, 1 ≤ � ≤ N2, we write

P (R) =(P

(R)k�,l�

)1≤k,l≤N1

1≤�,�≤N2

, ∀1 ≤ R ≤ N . (6.7)

The reduced density matrix of system A1:

�A1 = TrA2 �A =N∑R=1

�RTrA2P(R)

=N∑R=1

�R

N2∑�=1

P(R)k�,l�

=N∑R=1

�R�(R)A1∈ D(ℋA1) ,

(6.8)

is a convex combination of N density matrices (not pure) all belonging to D(ℋA1). The

next step is to compute the square of �A1 , which reads

�2A1

=( N∑R=1

�R�(R)A1

)2=

N∑R=1

�2R(�

(R)A1

)2 +∑R ∕=S

�R�S�(R)A1�

(S)A1

. (6.9)


And finally, from the linearity of the partial trace, the A1-local purity turns out to be:

�A1 = Tr�2A1

=

N∑R=1

�2RTr(�

(R)A1

)2 + 2∑S<R

�R�STr(�

(R)A1�

(S)A1

). (6.10)

Our task is to compute the average on the unitary group of the local purity �A1 . We

desire to underline an important fact. The �(R)A1

’s under the summation symbols are density

matrices obtained by partial tracing a pure state of ℋA = ℋA1 ⊗ℋA1 . Thus, for these

states, one can use all the results concerning the statistical properties of pure states. It is

not difficult to see that:

⟨�A1⟩Haar =N∑R=1

�2R

⟨�

(R)A1A2

⟩+ 2

∑S<R

�R�S

⟨�

(S<R)A1A2

⟩, (6.11)

with self-explaining meaning of symbols. The averages are understood to be with respect

to the Haar measure on U(N) = U(N1N2). The calculation proceeds smoothly:

⟨�A1⟩Haar =

⟨�

(R)A1A2

N∑R=1

�2R +

⟨�

(S<R)A1A2

⟩2∑S≤R

�R�S

⟩

=⟨�

(R)A1A2

⟩�AB +

⟨�

(S<R)A1A2

⟩(1− �AB)

=N1 +N2

N1N2 + 1�AB +

⟨�

(S<R)A1A2

⟩(1− �AB) .

(6.12)

In the last step we have invoked Lubkin result, eq. (2.12). We have obtained that, the

average purity of a subpart of a mixed state depends upon the spectrum of the larger system

only through its purity.

We see that, the average local purity of the subpart is an affine function of the purity of

the bigger system. In order to obtain the desired result, we should compute the average on

the unitary group:

⟨�

(S<R)A1A2

⟩=⟨

Tr(�

(R)A1�

(S)A1

)⟩. (6.13)


Since eq. (6.12) predicts an affine trend, we could avoid the calculation of eq. (6.13) with a

little trick. We will try to fix two suitable points in the (�AB, �A1)-plane, and then we will

write the linear trend through these two points.

First, we know that, the image of the maximally mixed density matrices under partial

trace operation is still a fully entangled density matrices. Thus, we know that the straight

line will pass through the point (1/N, 1/N1) in the (�AB, �A1)-plane.

A second suitable point can be find by the following consideration. If we start from a

composite system

ℋS = ℋA ⊗ℋB = (ℋA1 ⊗ℋA2)⊗ℋB ,

tracing over system B, leaving A and then performing a partial trace on A2 leaving the

subpart A1 is equivalent to trace out all except A1 in a unique operation, that is:

TrA2 ∘ TrB = TrB1 , B1 = S∖A1 . (6.14)

Thus, taking a purification of �A in a L = NM -dimensional space, it readily follows that

another good point for our interpolation is:

(�AB, ⟨�A1⟩) =( N +M

NM + 1,N1 +MN/N1

MN/N1 + 1

), (6.15)

where we have used again the average purity computed by Lubkin for pure random states.

The straight line in the (�AB, �A1)-plane that passes through the points (1/N, 1/N1) and

eq. (6.15) is:

�A1 =N(N2

1 − 1)

N1 (N2 − 1)�AB +

N2 −N21

N1 (N2 − 1)

=N2

(N2

1 − 1)

N21N

22 − 1

�AB +N1(N2

2 − 1)

N21N

22 − 1

,

(6.16)

in agreement with [3]. The same result can be obtained using the point

(�AB, ⟨�A1⟩) =(

1,N1 +N2

N1N2 + 1

),


æ

æ

ææ

æ

æ

æ

æ

æ

æ

æ

æ

ææ

æ

æN=4 , N1=2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.5

0.6

0.7

0.8

0.9

1.0

ΠAB

<Π

A1

>

Figure 6.3: Average on the unitary group of local purity �A1for N = 4 and N1 = 2 (a mixed state

of two qubits). Analytic trend (red solid line) of eq. (6.16), and numerical values from simulations.

We have generated k = 10000 states with fixed spectrum (fixed purity �AB) by conjugation with

random unitaries. Then, the averages of local purity ⟨�A1⟩ has been computed for all values of fixed

purity of the larger system.

because we know how the average works for pure states. By comparing eq. (6.16) with eq.

(6.12), we finally obtain

⟨�

(S<R)A1A2

⟩=⟨

Tr(�

(R)A1�

(S)A1

)⟩=N1(N2

2 − 1)

N21N

22 − 1

. (6.17)

This average has the following meaning. Let ∣ ⟩ and ∣�⟩ two mutually orthogonal N -dim

pure states sampled according to the Haar measure. We can consider the overlap between

the reduced N1-dimensional density matrices �(R) = Tr ∣ ⟩ ⟨ ∣ and �(S) = Tr ∣�⟩ ⟨�∣, using

the common Hilbert-Schmidt inner product⟨�(R), �(S)

⟩. The average over the unitary group

of this overlap is exactly the quantity eq. (6.17). The values of local purity summarized

in Tab. 3.1 are in agreement with eq. (6.16). In Fig. 6.3 we have plotted the function

⟨�A1(�AB)⟩ and some values from numerical simulations.

Conclusions

In this thesis, we have discussed the behavior of the entanglement of a bipartite quantum

system. We now proceed to formulate our conclusions.

In the first two chapters, we gave a quick overview on quantum mechanics and random

matrices theory. In particular, chapter 1 was devoted to a summary of the concepts of

quantum mechanics used in this thesis; the most important section is the one pertinent to

separability criteria and measures of entanglement (section 1.1.2).

In chapter 2 we discussed some aspects of random matrices theory, including a small

discussion about the relevant ensembles useful for our topics; the main point of the chapter

is the unambiguous definition of what we mean by “random state” and the construction of

the probability measure on the reduced density matrices ensemble.

Some numerical evidence about random quantum states is presented in chapter 3. Most

of the numerics has been used as a starting point for our investigation.

In chapter 4 we began to tackle our problem of investigating the statistical properties of

a random pure state of a bipartite system. Random refers to the unique unbiased measure

over unitaries, and states sampled according to this measure are named “unbiased states”.

We have seen that given a bipartite system, the state of its small subparts is highly mixed

with overwhelming probability. Our approach relies on the saddle point method applied on

the joint distribution of the eigenvalues of the reduced density matrix. Namely, we have

found the most probable spectrum of the reduced density matrices, for every unbalanced

bipartition of a global system. In order to solve the saddle point equations, we have

invoked the physical interpretation of the so called “Coulomb gas”, showing the strong

link of our problem about eigenvalues with 2D-electrostatic models. It turns out that the

most probable eigenvalues of the density matrix coincide with the equilibrium positions of

movable charges on a line when the interaction forces arise from a logarithmic potential. It

88 Conclusions

is very fascinating to discover that this equilibrium configuration (the N -uple of eigenvalues

that maximizes probability) is reached at the zeros of a class of orthogonal polynomial

(associated Laguerre). Then, the spectrum of a typical state is completely known to be

formed by the zeros of a certain polynomial. Even for high-degree polynomials, we have

obtained some useful analytic results (in compact and computable forms) for the typical

purity, and typical elementary symmetric invariants. All these results agree with known

results about the statistical averages of typical states obtained with other methods. In

particular, we have focused our attention on typical four-level quantum systems. Moreover,

our method enables one to recover the thermodynamic limit results, generally obtained by

others with methods based on statistical mechanics or random matrix theory. Our analysis

has the advantage of being clear and of not involving very cumbersome mathematical

methods or unclear physical hypotheses.

The question introduced in chapter 5 represents a new kind of problem. We have tried

to explore typicality on isopurity submanifolds of random pure states again through the

saddle point method. An interpretation in term of Coulomb gas is suggestive and provides

some hints about the behavior of the eigenvalues. However, an analytic solution similar to

the one written explicitly for Haar-distributed states seems to be not so immediate to be

found. The adopted strategy is the following. With suitable techniques, we have studied

two extreme situations: quasi-typicality and high purity regions. For these extremes we

have been able to write down reasonable analytic solutions. Then, we have specialized these

solutions for the four-level system to see in more details how they work. The next step has

been to use our analytic solutions as starting points for numerical root-finding algorithms

applied to the exact saddle point equations at fixed purity. As we vary the correlation of

the system with the heat bath, and as we move on region of fixed purity, our numerics

show a complex scenario for the eigenvalues. The numeric curves give full agreement with

exact results for typical states. The analysis of the behavior of the Lagrange multipliers

that take into account the unit-trace condition and the constraint on purity, agrees with

our preliminary heuristic discussions. The chapter ends with a best fit for the polynomial

whose roots are the most probable eigenvalues of a fixed-purity four-level state.

The thesis ends (chapter 6) with a last question. In chapters 4 and 5 we have found a

mixture of analytic and numeric solutions for the most probable spectrum of the four-level

state (typical and non-typical). Then, we ask: what about the four-level system regarded

as a composite systems of two qubits? We have shown that typical states are close to the

Conclusions 89

maximally mixed state, but are typically not-entangled (the entanglement within the pair

being measured by concurrence). As we move in the increasing-purity direction, the residual

entanglement of the two qubits increases. In particular, there exist a lower bound in purity,

below which entanglement within the pair cannot be found. We have underlined that this

is a consequence of the so-called monogamy of entanglement. As a conclusive example, we

have computed the average purity of a mixed state. Although this kind of averages over

the unitary group have been recently performed by other authors, we have showed a very

quick and simple method to obtain the same result avoiding hard mathematical techniques,

and with a direct physical interpretation.

Summarizing, in this work we have followed a broad variety of approaches in order

to explore the typical properties of random quantum states. Our main task has been to

study how entanglement is distributed among two subparts of a large quantum system.

The picture for typical states is subject of interest among scientists by long time, and many

results have been achieved. In our work, simple techniques have been chosen while solving

these problems, and these tools will be useful in further investigations. All the approaches

used in our investigation are very transparent in the final goals and in the results obtained.

The mathematical techniques are easy to follow, and always guided by the physical insight.

The new topic addressed in this thesis has been the study of the fixed-purity ensemble.

This ensemble seems to do not allow a simple solution as the one found for typical states.

However, we have been able to give some partial answers to many questions by mixing

approximate analytic solutions and numerical experiments. Also the other topic, that is

the residual entanglement in the substystems has been a formidable problem. However, we

have showed that our results about typical pure states agree with some separability criteria

for mixed states. We have also performed another numerical analysis for the two-qubits

system, and we have showed a simple method to obtain some kinds of averages on the

unitary group.

We end by reviewing open problems arising directly as a result of this research. Firstly,

it would be interesting to find an analytic solution of the saddle point equation for the

fixed-trace ensemble. Besides searching for the most probable quantities (spectrum, purity,

entropy, etc.), a second direction which one can look, is an extension of our mathematical

tools to give some information about the dispersion of these quantities about the most

probable ones.

Appendix A

Probability measure on the

ensemble of Hermitian matrices

Recall that a Hermitian matrix A ∈ ℳ(N) can be parametrized in terms of spectral

coordinates:

A = A(�i; ��)

by means the tranformation

A'7→(Λ , U modTN

)where �i’s are the N real eigenvalues of A, and �� are the N(N − 1) “angles” of the (family

of) unitaries U ∈ U(N)modTN that provide the spectral form A = UΛU †. In terms of this

coordinates, the measure transforms as

[dA] =( N∏i=1

[dAii])(∏

i<j

[dReAij ] [dImAij ])

=

∣∣∣∣∂ (Aii; ReAij ; ImAij)

∂ (�i; ��)

∣∣∣∣ N∏i=1

[d�i]

N(N−1)∏�=1

[d��] .

(A.1)

We will compute the Jacobian determinant

92 Probability measure on the ensemble of Hermitian matrices

J (') =

∣∣∣∣( ∂A∂�1, . . .

∂A

∂�N;∂A

∂�1, . . . ,

∂A

∂�N(N−1)

)∣∣∣∣=


∂ (�i; ��)

∣∣∣∣(A.2)

Differentiating A(�i; ��) we obtain:

∂A

∂��=

∂

∂��UΛU † =

∂U

∂��ΛU † + UΛ

∂U †

∂��, 1 ≤ � ≤ N(N − 1) (A.3)

From unitarity, UU † = I =⇒ ∂U∂��

U † + ∂U†

∂��U = 0, the N(N − 1) matrices S� = U † ∂U∂�� are

antihermitian. Hence:

U †∂A

∂��U = S�Λ− ΛS� = [S�,Λ] , 1 ≤ � ≤ N(N − 1) (A.4)

Similarly, one finds:

U †∂A

∂�iU =

∂Λ

∂�i, 1 ≤ i ≤ N (A.5)

Let us consider now the real Hilbert space ℳ(N) endowed with the Hilbert-Schmidt inner

product:

⟨A,B⟩ℋ = Tr(AB). (A.6)

The linear map induced by a unitary U (conjugation)

VU : A 7→ U †AU (A.7)

is clearly orthogonal with respect to the product ⟨ , ⟩ℋ, and hence, detVU = ±1. We can

write eqs. (A.4) and (A.5) as

VU

(∂A

∂�1, . . .

∂A

∂�N;∂A

∂�1, . . . ,

∂A

∂�N(N−1)

)=

(∂Λ

∂�1, . . .

∂Λ

∂�N; [S1,Λ] , . . . ,

[SN(N−1),Λ

]).

93

Then,

J(') =


∂ (�i; ��)

∣∣∣∣=

∣∣∣∣( ∂A∂�1, . . .

∂A

∂�N;∂A

∂�1, . . . ,

∂A

∂�N(N−1)

)∣∣∣∣=

∣∣∣∣( ∂Λ

∂�1, . . .

∂Λ

∂�N; [S1,Λ] , . . . ,

[SN(N−1),Λ

])∣∣∣∣ .

(A.8)

From (∂Λ

∂�i

)jk

=∂�j∂�i

�jk = �ij�jk and [S�,Λ]jk = (S�) (�j − �k), (A.9)

we obtain:

det(J(')) = det

(1N 0 0

0 �R �I

)

= det

(1N 0 0

0 (SR� )jk(�j − �k) (SI�)jk(�j − �k)

)

= det (S�)��∏i<j

(�i − Λj)2.

(A.10)

The N2×N2 Jacobian matrix J(') decomposes into a sum 1N⊕B; 1N is the N -dimensional

indentity and B is N(N −1)×N(N −1). The (S�)’s are N(N −1) (equals to the number of

independent “angles” ��); since S†� = −S� ∀�, each S� has N(N − 1) independent entries,

in particular N(N − 1)/2 real numbers (the entries of (SR� )) and N(N − 1) imaginary pure

numbers (entries of (SI�)). Thus, in our notation, �R and �I have N(N − 1) rows and

N(N − 1)/2 columns. The dimensions of the remaining block matrix, denoted generically

as 0, follow immediately.

Appendix B

Vieta’s formula

Given a polynomial of degree n

Pn(X) = anXn + an−1X

n−1 + ⋅ ⋅ ⋅+ a1X + a0

Vieta’s formula relates the coefficients ak to sums and products of its roots xi.

Let si be the quantity formed by adding together all distinct products of the n roots of the

polynomial equation P (X) = 0, i.e. the elementary symmetric polynomial evaluated on the

roots xi:

s1 = e1(x1, . . . , xn) =∑j

xj

s2 = e2(x1, . . . , xn) =∑j1<j2

xj1xj2

s3 = e3(x1, . . . , xn) =∑

j1<j2<j3

xj1xj2xj3

...

sn = en(x1, . . . , xn) = x1x2 ⋅ ⋅ ⋅xn.

Vieta’s formula states that

sk = (−1)kan−kan

. (B.1)

for 1 ≤ k ≤ n. The proof can be done inductively (the base case n = 1 is trivial), and

follows immediately from the factorized form Pn(x) =∑n

k=1 akxk = an

∏nk=1 (x− xk). This

formula is a very useful tool to discuss polynomial equations. As an example, from �A ≥ 0

96 Vieta’s formula

and TrA�A = 1, one knows that the eigenvalues �i’s of the four-level state �A are the zeros

of a polynomial of the form

x4 − x3 + c2x2 − c1x+ c0 = 0, (B.2)

with all coefficients ci’s nonnegative.

Appendix C

Stieltjes trick

The problem is to determine the equilibrium positions x = (x1, . . . , xN ) of N charges,

in an external field 'ext, when the interaction forces arise from a logarithmic potential:

Etot = 'ext (x)− 2∑i<j

log ∣xi − xj ∣. (C.1)

This continuous function yields the equilibrium points by setting ∇xEtot = 0, namely

∂'ext

∂xi− 2

∑j ∕=i

1

xi − xj= 0 , (C.2)

and solving for x1, . . . xN . To solve this system of N nonlinear equations in N unknowns,

Stieltjes introduced the polynomial

g(x) =N∏i=1

(x− xi) = (x− x1)(x− x2) ⋅ ⋅ ⋅ (x− xN ) , (C.3)

whose zeros are the unknown equilibrium points x1, . . . xN . Logarithmic differentiation gives:

∑j ∕=i

1

x− xj=g′(x)

g(x)− 1

x− xi=g′(x)(x− xi)− g(x)

g(x)(x− xi),

and then,∑j ∕=i

1

xi − xj= lim

x→xi

g′(x)(x− xi)− g(x)

g(x)(x− xi)= lim

x→xi

g′′(x)(x− xi)g′(x)(x− xi) + g(x)

,

98 Stieltjes trick

from L’Hospital’s rule. Finally we get:

2∑j ∕=i

1

xi − xj=g′′(xi)

g′(xi). (C.4)

Appendix D

Orthogonal polynomials

We list some facts about classes of orthogonal polynomials.

A family of real polynomials {pn(x)}n∈ℕ is said to be a set of orthogonal polynomials

on the interval [a, b] with respect to the continous weight function w(x) ≥ 0 on ]a, b[ if

⟨pn(x), pm(x)⟩w =

∫ b

adxw(x)pn(x)pm(x) =

1

ℎn�n,m (D.1)

with ℎn > 0, ∀n.

Several properties of the zeros of real orthogonal polynomials are well known for a long

time:

- All the zeros of pn, ∀n are real, simple an lie in the interval ]a, b[ (more generally, in

the support of the weight function and so may be ordered

a < x1,n < x2,n < ⋅ ⋅ ⋅ < xn−1,n < xn,n < b ;

- The zeros for succesive polynomials have the interlacing property, i.e. between every

pair of consecutive zeros of pn+1 there is one and only one zero of pn, ∀n

a < x1,n+1x1,n < x2,n+1 < ⋅ ⋅ ⋅ < xk,n+1xk,n < xk+1,n+1 ⋅ ⋅ ⋅ < xn,n+1 < xn,n < xn+1,n+1 < b .

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