Advanced Topics

in Quantum Mechanics

M. Fannes

Instituut voor Theoretische Fysica

K.U.Leuven

March 2013

Contents

1 Introduction 1

1.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Two qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Observables and states 9

2.1 Positive matrices . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Density matrices . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Pure and mixed states . . . . . . . . . . . . . . . . . . 16

2.2.4 The pure state space of Md . . . . . . . . . . . . . . . 17

2.3 State space of a qubit . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 A probability theory approach to states . . . . . . . . . . . . . 20

2.5 States of composite systems . . . . . . . . . . . . . . . . . . . 22

3 Quantum dynamics 25

3.1 Dynamics in discrete time . . . . . . . . . . . . . . . . . . . . 25

3.2 General quantum operations . . . . . . . . . . . . . . . . . . . 27

3.3 Examples of quantum operations . . . . . . . . . . . . . . . . 29

3.3.1 Unitary gates . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Dissipative operations . . . . . . . . . . . . . . . . . . 31

3.4 Dynamics in continuous time . . . . . . . . . . . . . . . . . . . 35

3.4.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2 Lindblad’s theorem . . . . . . . . . . . . . . . . . . . . 37

3.4.3 A semi-group of decoherent qubit maps . . . . . . . . . 39

3.4.4 Radiation loss to the vacuum . . . . . . . . . . . . . . 40

3.5 Thermalising maps . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.1 Reversible dynamics of the system . . . . . . . . . . . . 44

3.5.2 Equilibrium states of the bath . . . . . . . . . . . . . . 45

3.5.3 The weak-coupling limit . . . . . . . . . . . . . . . . . 47

3.5.4 Properties of the weak-coupling limit . . . . . . . . . . 47

3.5.5 Quantum detailed balance maps on Md . . . . . . . . 48

4 Entropy 50

4.1 Classical entropy . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Construction of the Shannon entropy . . . . . . . . . . . . . . 52

4.3 Quantum Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 55

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1 Introduction

Classical systems can be described at different levels of complexity. Thesimplest systems are registers that can be in a discrete number of stateslike an Ising spin that can be up or down. More complex systems appearin classical mechanics: point masses whose motion is governed by Newton’ssecond law. Still more complex are classical fields like e.g. the electromagneticfield.

A very similar situation holds in the quantum world. The simplest systemsare these that can only access a finite number of states, typically the groundstate and a few excited states. Here we just need finite dimensional vectorspaces and linear algebra to describe them. More complex are systems like thehydrogen atom where we need an infinite dimensional space of wave functions.The appropriate mathematical tools for such systems are Hilbert spaces andlinear operators on such spaces. Still more complex are quantum field theoriesor systems with an infinite number of degrees of freedom. There the Hilbertspace picture breaks down and one has to turn to operator algebras.

We will here mainly restrict our attention to the simplest, finite dimensionaldescription of quantum mechanics. This context is already wide enough todiscover a number of typical quantum phenomena. It is, moreover, also arealistic approach to small systems at low temperatures such as a few atoms,a molecule or a few photons.

To describe a quantum system with d accessible levels we will just need thecomplex d-dimensional orthogonal space. Fixing an orthonormal basis we canidentify the space with Cd equipped with its standard scalar product. Lineartransformations of the space can then simply be identified with matrices withcomplex entries.

1.1 General principles

The main postulates for describing an autonomous isolated quantum systemare:

Postulate 1: The maximal knowledge that can be obtained about anensemble of identically prepared isolated systems is encoded into thestate vector ψ of the system. This is a normalized vector in Cd.

1

Postulate 2: Every normalized vector in Cd is a possible state vectorof the system. This is equivalent to the superposition principle:if ψ1 and ψ2 are possible state vectors, then any normalized linearcombination of ψ1 and ψ2 also occurs.

Postulate 3: The observables of the system are the Hermitian lin-ear transformations of Cd. Let A be an observable and ej be theorthonormal basis of eigenvectors of A with corresponding eigenval-ues αj. The αj are the possible outcomes of a measurement of A.Measuring the observable A will return a random outcome and theprobability of the outcome αj for a system in the state ψ is |〈ej, ψ〉|2.This probability can be obtained by repeatedly observing the systemand establishing the relative frequencies of the different outcomes.

Postulate 4: The evolution of an autonomous isolated system is

governed by an Hermitian Hamiltonian matrix H . If ψ is thestate vector at time t1 then exp(i(t2 − t1)H)ψ is the state vector attime t2.

Postulate 1 supposes that we can produce an arbitrary number of systemsall identically and perfectly prepared. The state vector or wave functionassociated to such a preparation is not attached to any particular instance ofthe system but rather to the ensemble. We need such an ensemble to builda statistics of measurement outcomes using relative frequencies. This is themeaning of Postulate 3.

Postulate 2 also leads to the tensor product construction for dealing

with composite systems. If systems A and B are independent with statevectors ψA and ψB then a product vector ψA ⊗ ψB should describe the com-posite system. Superpositions of such vectors are also allowable states, wehave therefore to consider all linear combinations of such product vectorswhich leads to the tensor product construction.For systems composed of indistinguishable particles the Pauli principle ap-plies: the admissible state vectors for Fermions have to change sign underodd permutations of the particles while they remain invariant for Bosons.In such a situation we must consider the totally anti-symmetric or sym-

metric subspaces of a tensor product of copies of the single particle space.

Sending an instance of a system prepared in the state ψ through a measure-ment apparatus of an observable A will yield one of the outcomes αj . It is not

2

possible to predict which αj will be observed but the relative frequency

for obtaining αj will tend to |〈ej, ψ〉|2. Postulate 3 allows to compute theexpectation value of any function of an observable which is another way toobtain the measurement outcome statistics

〈f(A)〉ψ =∑

j

f(αj) |〈ej, ψ〉|2 = 〈ψ, f(A)ψ〉. (1)

It also follows from Postulate 3 that multiplying ψ by a phase doesn’t alterany expectation value. Therefore normalized vectors in Cd that differ onlyby a global phase yield the same outcome statistics for every observable andshould be identified. We arrive in this way at a more refined notion of statevector: a one-dimensional subspace of Cd also called a ray.

Postulate 4 tells us that unitary quantum dynamics is linear at the level ofstate vectors: superpositions evolve into the same superpositions of evolvedstate vectors: unitary quantum dynamics is a coherent evolution.

1.2 The qubit

This is the simplest non-trivial quantum system with state vectors in C2. Itis the quantum mechanical counterpart of an Ising spin which is a classicalsystem with only two configurations: spin up or down. Let us fix the notationfor the standard basis

|0〉 =

(10

)

and |1〉 =

(01

)

. (2)

In the context of quantum information theory this basis is often called thecomputational basis.

An arbitrary state vector ψ in C2 is of the form

ψ = α|0〉 + β|1〉 with α, β ∈ C and |α|2 + |β|2 = 1. (3)

We are, moreover, free to multiply ψ by a complex number of modulus one.This freedom can be used to make the coefficient of |0〉 positive. In this waywe arrive at a unique parametrisation of the rays of a qubit

ψ = cos(θ/2)|0〉 + sin(θ/2) eiφ|1〉 with 0 ≤ θ ≤ π and 0 ≤ φ < 2π. (4)

3

So we see, viewing (θ, φ) as spherical angular coordinates, that the space ofrays of a qubit is isomorphic to the unit sphere in R3. The state |0〉 is thenorth pole and |1〉 is the south pole. The unit sphere in R3, seen as the statespace of a qubit, is called the Bloch sphere.

Problem 1. Show that orthonormal bases of C2 correspond to antipodalpoints on the Bloch sphere.

Problem 2. Express the transition probability between two qubit statesin terms of a geometric property of the corresponding points on the Blochsphere. (The transition probability between ψ1 and ψ2 is |〈ψ1, ψ2〉|2.)

Let us observe the z-component of the spin which corresponds to the Pauliσz matrix. As σ2

z = 1, we only need to consider

〈σz〉ψ = 〈ψ, σzψ〉 = cos2(θ/2) − sin2(θ/2) = cos θ. (5)

Clearly, observing only σz does not provide enough information to recon-struct the state vector: there is no way to determine φ. The variance ofmeasurement outcomes is given by

∆2(σz) = 〈σ2z〉ψ − 〈σz〉2ψ = 1 − cos2 θ = sin2 θ. (6)

Hence we see that a perfectly accurate determination of σz is only possiblefor θ = 0 and θ = π or, in other words, when ψ is an eigenstate of theobservable.

Problem 3. Show that, up to a shift by a constant and a rescaling, any qubitobservable is unitarily equivalent to the Pauli σz matrix. (Two observablesA and B are unitarily equivalent if there exists a unitary matrix U such thatB = U AU †.)

Problem 4 (Uncertainty relation for a qubit). Show that the uncertaintiesin the x and y components of a qubit satisfy ∆2(σx) + ∆2(σz) ≥ 1.

The dynamics of an isolated autonomous qubit is quite simple. It is generatedby a Hamiltonian

H = ǫ1|f1〉〈f1| + ǫ2|f2〉〈f2| (7)

where ǫ1 and ǫ2 are the energies of the eigenstates f1 and f2 of H . BySchrodinger’s equation a state ψ evolves according to

ψ = α|f1〉 + β|f2〉 7→ ψt = αeitǫ1 |f1〉 + βeitǫ2 |f2〉. (8)

4

It is clear that the eigenstates |f1〉 and |f2〉 remain, up to a multiplicationby a global phase, unchanged. Therefore the dynamics leaves the points onthe Bloch sphere corresponding to |f1〉 and |f2〉 stationary.

Problem 5. Describe geometrically the dynamics of an isolated autonomousqubit on the Bloch sphere. Show in particular that it is a rotation of theBloch sphere around the axis through the eigenstates of the Hamiltonian ata constant angular velocity determined by ǫ2 − ǫ1, the Bohr frequency ofthe system.

1.3 Two qubits

We now turn to a composite system of two qubits. The components aresometime called parties and the composite system bipartite. The states ofsuch a system are the normalized vectors in C2 ⊗C2 = C

4 up to a factor ofmodulus one. The standard tensor basis is usually lexicographically ordered

|00〉 = |0〉⊗|0〉, |01〉 = |0〉⊗|1〉, |10〉 = |1〉⊗|0〉, and |11〉 = |1〉⊗|1〉. (9)

A general two-qubit state can then be written as

ψ = a00|00〉 + a01|01〉 + a10|10〉 + a11|11〉 with

|a00|2 + |a01|2 + |a10|2 + |a11|2 = 1.

As we can also fix a global phase there remain 6 real parameters.

Some two-qubit state vectors have a particular structure: they are elemen-

tary tensors

ψ = ϕ⊗ η, ϕ, η ∈ C2. (10)

This type of state vectors describes a composite system where both partiesare independent: the expectation of product observables A⊗B is simply theproduct of the expectations of these observables in the subsystems. Statesof this type are called non-entangled or separable and they are not veryuseful for many purposes.

Problem 6. Show that a two-qubit state vector is non-entangled iff a00a11 =a01a10. As this imposes two real constraints, we need 4 real parameters todescribe a generic non-entangled two-qubit states. This means that a generictwo-qubit state vector will be entangled.

5

Let us consider a very entangled state vector β00 = 1√2

(|00〉 + |11〉

)and let

us computed the expectation of an observable of the first subsystem

〈A⊗ 1〉β00 = 〈β00, A⊗ 1β00〉 = 12

(〈0|A 0〉 + 〈1|A 1〉

). (11)

It is easy to see that there is no qubit state vector φ such that

〈φ,Aφ〉 = 12

(〈0|A 0〉 + 〈1|A 1〉

). (12)

In fact we need an equal weight mixture of the expectations defined by thequbit state vectors |0〉 and |1〉 to reproduce such expectations. It meansthat we have to enlarge our description of expectation values to include alsomixtures of expectations defined by state vectors. This is the subject ofthe next section.

Problem 7. Show that there is no qubit state vector reproducing the ex-pectation

A 7→ 12

(〈0|A 0〉 + 〈1|A 1〉

). (13)

1.4 Teleportation

Suppose that we want to transmit a qubit state ψ from A(lice)’s lab B(ob)’slab using standard means of communication, that is to say, by transmittingnumerical data from A to B. The state ψ is just any possible unknownstate of a qubit and we are allowed a single use of the system. Measuringan observable in A will not be very helpful. Indeed, the outcome is randomand after the measurement the state has turned into one of the eigenstatesof the observable. Suppose, however, that beside transmitting numericalinformation A and B share a (maximally) entangled state two-qubit statelike β00 above. Sharing a two-qubit state means that A can act on thecomposite system consisting of the unknown qubit state ψ and the first C2

factor of the space C2 ⊗C2 to which β00 belongs. B on the other hand canact on the second factor in C2 ⊗C2.

We consider a general (α, β) superposition of the states |0〉 and |1〉 at the Alab

ψ = α|0〉 + β|1〉 (14)

6

and set up a procedure to generate the same (α, β) superposition of a set ofbasis states at lab B. We first introduce the four Bell states which form anorthonormal basis of C2 ⊗C2

β00 = 1√2

(|00〉 + |11〉

)β01 = 1√

2

(|01〉 + |10〉

)

β10 = 1√2

(|00〉 − |11〉

)β11 = 1√

2

(|01〉 − |10〉

).

(15)

The three-qubit state consisting of the generic ψ and β00 is then

ψ ⊗ β00 = 1√2

(α|0〉 + β|1〉) ⊗ (|00〉 + |11〉)= 1√

2

(α|000〉 + α|011〉 + β|100〉 + β|111〉

).

(16)

In the A lab a measurement of an observable that is diagonal in the Bellbasis is made and depending on the (random) outcome an instruction istransmitted to the B lab about what action should be taken. To do this weexpand the three-qubit state with respect to the Bell basis of the first twocomponents:

ψ ⊗ β00 =1

2β00 ⊗ (α|0〉 + β|1〉) +

1

2β01 ⊗ (α|1〉 + β|0〉)

+1

2β10 ⊗ (α|0〉 − β|1〉) +

1

2β11 ⊗ (α|1〉 − β|0〉).

(17)

The idea is now to perform a measurement in lab A in the Bell basis and totransmit the random outcome of this measurement to lab B where a suitableaction is undertaken so as to restore the original state in lab B. Suppose, e.g.that the measurement selected the Bell state β11, then we know that the partof the state that is accessible in lab B is α|1〉 − β|0〉 and we can reconstructthe original state by applying the unitary

(0 1−1 0

)

.

This procedure is called teleportation. It is actually not transporting physicalobjects from A to B but rather the structure of an arbitrary quantum stateat A to that of a state at B. Remark also that neither A nor B actuallymeasure the teleported state. The only effect of the whole procedure is thatan unknown (α, β) superposition of two states at lab A has been exactlyreconstructed at lab B.

Problem 8. Work out the actions that have to be taken in Bob’s lab torestore the original qubit for other measurement outcomes of Alice.

7

8

2 Observables and states

To describe the simplest possible classical system, such as a register, weneed a configuration space that labels the contents of the register. For anIsing spin this is just a set with two elements. More complex systems canhave continuous configuration spaces or, whenever motion is possible, a gen-eral phase space. There is no notion of configuration space or phase spacefor quantum systems. It is however possible to describe jointly classical andquantum systems by passing to the level of observables: real or complex func-tions on configuration or phase space for classical systems or Hermitian linearmaps on the space of wave functions for the quantum case. In both casesthe observables form a complex algebra, commutative in the classical andnon-commutative in the quantum. Algebra means that elements can be mul-tiplied by complex scalars, added and multiplied among themselves. Thesethree operations satisfy the usual axioms on associativity and distributivity.An important ingredient in the quantum is Hermitian conjugation A 7→ A†

which extends the notion of complex conjugation for the classical case whereone can actually do with real functions. Finally, and this is only relevant incontinuous or infinite dimensional situations, an adapted notion of continuityis needed.

2.1 Positive matrices

We shall mostly restrict ourselves to fully quantum systems with d accessiblelevels. For such systems the algebra of observables is Md, the algebra ofd× d complex matrices. Hermitian conjugation is transposition followed bycomplex conjugation:

(A†)

ij= Aji.

For classical systems with d states the algebra of observables Cd consists ofthe complex functions on the configuration space Ωd = 1, 2, . . . , d. It isnatural to identify a classical observable with a diagonal matrix

f ⇐⇒

f(1) 0 · · · 00 f(2) · · · 0...

.... . .

...0 0 · · · f(d)

.

9

So classical can always be naturally embedded in quantum: Cd → Md. TheHermitian conjugate of a diagonal matrix corresponding to a function f isthen the diagonal matrix corresponding to the conjugate function.

Hermitian and unitary matrices play an important role in quantum theory.A matrix A is Hermitian if A† = A, such matrices are general observablesof the system, their eigenvalues are real and they can be diagonalized byrepresenting them in a suitably chosen orthonormal basis. A matrix U isunitary if it satisfies U †U = 1, this automatically implies that also U U † = 1,hence U † is the inverse of U . Unitary matrices preserve the inner product:

〈U ϕ, U ψ〉 = 〈ϕ, ψ〉, ϕ, ψ ∈ Cd.

This is equivalent to stating that unitary matrices map orthonormal bases inorthonormal bases. Unitary matrices are important in describing symmetriesof a quantum system, think of unitary group representations, and reversibleevolution.

Orthogonal projection matrices, also called projectors, are particular Hermi-tian matrices:

P = P † = P 2.

It is easily seen that also 1−P is a projector and that P (1−P ) = (1−P )P =0. Any vector ϕ ∈ Cd can therefore be written as ϕ = P ϕ + (1 − P )ϕ =ϕ1 + ϕ2 with 〈ϕ1, ϕ2〉 = 0 and therefore ‖ϕ‖2 = ‖ϕ1‖2 + ‖ϕ2‖2. The vectorϕ1 is the vector in PCd that is as close as possible to ϕ, it is the orthogonalprojection of ϕ on P Cd. In this way there is a one to one correspondencebetween projectors and subspaces of Cd. Let P1 and P2 be projectors suchthat P1P2 = 0(= P2P1) then P1+P2 is again a projector. Conversely, P1+P2

with P1 and P2 projectors is again a projector only if P1P2 = 0. In terms ofthe associated spaces: P1P2 = 0 means that the subspaces P1C

d and P2Cd

are orthogonal and P1 + P2 is then the projector on the space spanned byP1C

d and P2Cd. Also, P1P2 is a projector if and only if P1 and P2 commute in

which case P1P2 projects on the intersection of P1Cd and P2C

d. For generalprojectors there are two interesting constructions: P1 ∨ P2 is the projectoron the space spanned by P1C

d and P2Cd and P1 ∧P2 is the projector on the

intersection of P1Cd and P1C

d. P1 ∨ P2 is called the join of P1 and P2 andP1 ∧ P2 their meet.

For Hermitian and for unitary matrices, in fact for normal matrices (A†A =AA†), one has an orthogonal decomposition of Cd in eigenspaces: to each

10

eigenvalues α of A there corresponds an eigenspace Vα and a correspondingprojector Pα. These projectors satisfy

PαPβ = δαβPα and∑

α

Pα = 1. (18)

The matrix A can then be decomposed as

A =∑

α

αPα. (19)

An important notion is that of positive observable: A = A† is non-negativeif all the eigenvalues of A are non-negative, sometimes one uses the termpositive semi-definite. So the measurement outcomes of a non-negative ob-servable are always non-negative. The terms positive and positive definiteare reserved for strict positivity. We will be sloppy and use positive for bothcases.

Proposition 1. The following conditions on A ∈ Md are equivalent

1. A is positive

2. A = B†B for some B ∈ Md

3. 〈ψ,Aψ〉 ≥ 0 for every ψ ∈ Cd.

Problem 9. Fill in the proof of the preceding proposition.

Problem 10. Find the necessary and sufficient conditions on a, b, c ∈ R tomake

a b cc a bb c a

positive.

We can now order pairs of Hermitian matrices: A ≤ B if and only A = A†,B = B† and B−A is positive. For projectors this ordering can be expressedas follows:: P1 ≤ P2 if and only if P1 = P2P1. The join and meet operationson projectors can be restated as follows: P1 ∨ P2 is the smallest projectorthat is larger or equal than P1 and P2 while P1 ∧ P2 is the largest projectorthat is dominated by both P1 and P2.

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2.2 States

We now come to the notion of state. This will encode all the statisticalinformation that we can obtain about measurements of observables for asystem prepared with infinite care following a given procedure. In this sensea state is a preparation procedure. So a state is an expectation functionalon the observables:

A 7→ 〈A〉 ∈ R. (20)

Up to now the term state vector was used for a quantum system. A statevector ψ defines a state 〈 〉ψ through

〈A〉ψ := 〈ψ,Aψ〉, A an observable.

It turns out that the following requirements on states fit very well the exper-imental observations and allow, moreover, for a probabilistic interpretationof the theory:

Definition 1. A state 〈 〉 on an algebra of observables A is a functional thatsatisfies the following requirements

1. A 7→ 〈A〉 ∈ C is a complex linear functional on the algebra of observ-ables

2. 〈1〉 = 1, this is a normalisation condition

3. 〈A〉 ≥ 0 whenever A is a positive observable.

The complex linearity is important and is asking strictly more than reallinearity on the (Hermitian) observables. Real linear theories have beendeveloped but they simply don’t work. An important property of the statespace of a system is its convexity: if 〈 〉1 and 〈 〉2 are states and 0 ≤ p ≤ 1then p〈 〉1 + (1 − p)〈 〉2 is also a state. This is an immediate consequence ofthe previous definition. States can thus be mixed in any proportion, we canin fact mix an arbitrary number of states.

2.2.1 Density matrices

We first characterise states on Cd:

12

Proposition 2. There is a one to one correspondence between states on Cdand probability vectors p of length d:

〈f〉 =

d∑

j=1

pjf(j). (21)

Here p = pj with pj ≥ 0 and∑

j pj = 1.

Problem 11. Prove the preceding proposition.

Next we characterise states on Md. To do so, we introduce the notion ofdensity matrix which is the quantum mechanical counterpart of a prob-ability vector. A density matrix of dimension d is a matrix ρ in Md thatsatisfies

ρ is positive and Tr ρ = 1.

It is easily seen that the set of d-dimensional density matrices is convex.We shall denote by Sd the space of d-dimensional density matrices. Thisnotation refers to the density matrices as the states on Md and is justifiedby the following proposition.

Proposition 3. There is a one to one affine correspondence between stateson Md and d-dimensional density matrices given by

〈A〉 = Tr ρA. (22)

Problem 12. Fill in the proof of the preceding proposition.

Let ψ ∈ Cd be a normalised vector, then

〈A〉ψ = 〈ψ,Aψ〉 = Tr(|ψ〉〈ψ|A

), A ∈ Md. (23)

Therefore the one-dimensional projector |ψ〉〈ψ| is the density matrix corre-sponding to the state 〈 〉ψ.

Canonical Gibbs matrices are another important example: given an inversetemperature β > 0 and a Hermitian Hamiltonian H , the canonical equilib-rium state has density matrix

ρβ =e−βH

Z with Z = Tr e−βH . (24)

The normalization factor Z is called the partition function and it yields thefree energy − log(Z)/β of the system.

13

2.2.2 Convex sets

There is a general theory for convex subsets of a real vector space that showshow a compact convex set can be reconstructed in terms of its extreme points.These results provide us with a number of useful notions about state spaces.

We consider for simplicity only subsets of finite dimensional real vectorspaces. Most results generalise to infinite dimensions modulo some addi-tional technical assumptions. A subset C ⊂ Rd is convex if px+(1−p)y ∈ Cwhenever x, y ∈ C and 0 ≤ p ≤ 1. If C ⊂ R

d is convex, x1, x2, . . . , xk ∈ C,pj ≥ 0 and

∑

j pj = 1 then also

k∑

j=1

pjxj ∈ C. (25)

This follows by recursion from the definition of convexity.

Given X ⊂ R

d the convex hull of X is the smallest convex set in Rd thatcontains X . It is easily seen that

Conv(X) =∑

j

pjxj

∣∣∣ xj ∈ X, pj ≥ 0,

∑

j

pj = 1

. (26)

The closed convex hull is the closure of Conv(X), it is the set obtained byadding all limits points.

Let C be a closed convex subset of Rd. A point c ∈ C is called an extreme

point of X if it cannot be written as a non-trivial mixture of points of C. Informula’s: x is extreme if x = px1 + (1 − p)x2 with 0 < p < 1 and x1, x2 ∈ Cimplies that x1 = x2. The set of extreme points of a closed convex set iscalled the extreme boundary of C and denoted by ∂ext(C).

Problem 13. Find the necessary and sufficient conditions on a, b, c to turnthe matrix of Problem 10 in a density matrix. Is the set of density matricesthat you obtain convex? If so, find its extreme points.

Theorem 1 (Minkowski). Let C be a closed, bounded, convex subset of Rd,then

C = Conv(∂ext(C)

). (27)

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In infinite dimensions closed and bounded has to be replaced by compactwhich is strictly stronger and one has to take the closed convex hull. Thegeneral theorem is known as the Krein-Milman theorem.

Theorem 2 (Caratheodory). Let C be a closed and bounded convex subsetof Rd, then every element in C is a convex combination of at most d + 1extreme points of C.

Closed, bounded, convex subsets of Rd with exactly d+ 1 extreme points arecalled simplices. Any point in a simplex can be decomposed in a uniqueway in a mixture of extreme points. The weights of this decompositionare sometimes called the convex or barycentric coordinates of the point.Simplices are basic building blocks for constructing more general sets, notnecessarily convex (‘triangulation’).

A basic tool in proving Minkowski’s theorem is the following separation prop-erty for convex sets: given a closed, convex subset C ofRd and a point x ∈ Rd

that does not belong to C then there exists a separating hyperplane inR

d in-between x and C. An hyperplane H in Rd is a d−1 dimensional planein Rd, it is uniquely determined by a unit vector n and a real number d:

H = x ∈ Rd | n · x = d. (28)

An hyperplane cuts Rd in a positive and a negative half-space

H+ = x ∈ Rd | n · x ≥ d and H− = x ∈ Rd | n · x ≤ d. (29)

The separation property claims that we can always find an hyperplane suchthat

C ⊂ H+ and x ∈ H−. (30)

Using this notion one can show that a closed convex set C ⊂ Rd different fromR

d is equal to the intersection of all the positive half-spaces that contain C.I.e. such a set can be characterised by a set of linear inequalities. In generalthere will be a lot of redundant inequalities and one can look for a minimalset, this is called ‘linear programming’.

Problem 14. Describe the unit disk in R2 by a set of linear inequalities.

15

2.2.3 Pure and mixed states

We now characterise the boundary and extreme boundary of Sd, the statespace of Md. To give a precise meaning to the topological boundary weconsider the state space as a subset of the plane of Hermitian d-dimensionalmatrices with trace 1. A basis a neighbourhoods of an element X of thisplane are open disks: Hermitian matrices of the form X + Y where Y isHermitian with trace 0 and norm less than a given ǫ.

Proposition 4. A density matrix ρ belongs to the (topological) boundary ofSd if and only if ρ is not invertible, equivalently if and only if at least one ofthe eigenvalues of ρ is equal to 0.

Problem 15. Work out the details of the proof of the proposition.

Proposition 5. A density matrix ρ belongs to the extreme boundary of Sd ifand only if ρ is a one-dimensional projector, equivalently if and only if thereis a normalized vector ψ ∈ Cd such that ρ = |ψ〉〈ψ|.

Problem 16. Work out the details of the proof of the proposition.

The vector states are also called pure states, they are the extreme pointsof the state space and correspond to density matrices that are projectors ofdimension one. A state that is not pure can be decomposed into a non-trivialconvex decomposition of pure states. Such states are called mixed and theycorrespond to density matrices that have at least two eigenvalues differentfrom zero.

Mixed states arise e.g. when describing quantum sources. Suppose that asource emits a number of pure states given by vectors ϕj which appear withprobabilities pj. Repeated measurements will eventually yield the densitymatrix

ρ =∑

j

pj|ϕj〉〈ϕj|. (31)

This ρ contains all the information that can be gained by measurements aboutthe particles emitted by the source. The details of the source (pj, ϕj) iscalled a quantum ensemble. Because the state space of a quantum systemis very different from a simplex different quantum ensembles can return thesame density matrix and no experiment can discern between such sources.

16

This is very different from the classical case where the state space is a simplexand where therefore a mixed state automatically defines a unique ensemble.

Let us count the number of real parameters needed to describe a d-dimensio-nal density matrix. For a general Hermitian matrix we need d+ d(d− 1) =d2. The positivity condition doesn’t change that number but normalisationremoves 1 degree of freedom, hence we need d2−1. The topological boundaryimposes one additional real condition and so we still need d2− 2. The pointsof the extreme boundary correspond to one-dimensional subspaces of Cd.Normalising a vector in such a subspace and multiplying it by a phase weremain with 2(d− 1) real parameters. So we see that the extreme boundaryis in general a much smaller set than the boundary except for a qubit (d =2). It turns out that in any dimension the pure state space is a very niceRiemannian manifold. The boundary of the state space is, however, verycomplicated and contains many ‘flat’ parts.

Suppose that x is a point on the topological boundary of a compact convexsubset C of Rd. Such a point need not be extreme but it defines a face of C

F(x) = y ∈ C | ∃z ∈ C such that x = λy + (1 − λ)z

with z ∈ C and 0 < λ < 1. (32)

Faces are compact and convex.

Let ρ be a density matrix on the boundary of the space of d-dimensionaldensity matrices, then ρ has at least one zero eigenvalue. Suppose that ρ hask strictly positive eigenvalues, taking possible multiplicities into account. Itis not hard to see that the face of ρ is actually the state space of a quantumsystem of dimension k.

Problem 17. Check the assertion of above about the faces of the state spaceof Md.

2.2.4 The pure state space of Md

Let ψ be a unit vector in Cd. The set

zψ | z ∈ C, |z| = 1 (33)

is called a ray in Cd. As multiplying a state vector by a complex numberof modulus one doesn’t affect the expectation value of any observable, all

17

vectors in a ray yield the same pure state on Md. It is easily seen that thecorrespondence between rays and pure states is actually one to one. Thespace of rays in Cd is sometimes called the complex projective Hilbert

space of dimension d: CP(d). We actually just said that

CP(d) = ψ ∈ Cd | ‖ψ‖ = 1/U(1). (34)

Here U(1) are the one-dimensional unitary matrices: the complex numbersof modulus one.

The usual norm distance in Cd induces a natural distance on CP(d). If wedenote by [ψ] the ray of ψ ∈ Cd, ‖ψ‖ = 1, then

dSF

([ψ1], [ψ2]

):= min

z∈U(1)‖ψ1 − zψ2‖. (35)

The distance dSF is called the Study-Fubini distance and it is easily com-puted

dSF

([ψ1], [ψ2]

)=√

2 − 2|〈ψ1, ψ2〉|. (36)

Let us consider a small perturbation dψ of the vectors zψ belonging to a raythat does not change the norm to first order

‖zψ + dψ‖2 = ‖ψ‖2 + z〈ψ, dψ〉 + z〈dψ, ψ〉 + o(‖dψ‖)

= 1 + z〈ψ, dψ〉 + z〈dψ, ψ〉 + o(‖dψ‖).(37)

Hence 〈ψ, dψ〉 = 0, i.e., the tangent space at [ψ] to CP(d) is the (d − 1)-dimensional subspace ψ⊥ of Cd.

For dψ ⊥ ψ we have‖ψ + dψ‖2 = 1 + ‖dψ‖2 (38)

and henceψ + dψ

‖ψ + dψ‖ = ψ + dψ − 12‖dψ‖2ψ + o(‖dψ‖2). (39)

We now compute the square ds2 of the Study-Fubini distance between [ψ]and ψ+dψ

‖ψ+dψ‖ :

ds2 = 2 − 2∣∣∣

⟨ψ,

ψ + dψ

‖ψ + dψ‖⟩∣∣∣

= 2 − 2∣∣1 − 1

2‖dψ‖2

∣∣+ o(‖dψ‖2)

= ‖dψ‖2. (40)

18

So we see that the Study-Fubini distance yields a Riemannian metric onCP(d) defined at [ψ] by the identity matrix on ψ⊥.

The length ℓ of a parametric curve t ∈ [0, 1] 7→ [ψ(t)] on CP(d) starting at[ψ1] and ending at [ψ2] is then

ℓ =

∫ 1

0

dt∥∥∥

dψ(t)

dt

∥∥∥. (41)

For two points [ψ1] and ψ2 we can always choose the phases so that 〈ψ1, ψ2〉 ≥0. The geodesic connecting [ψ1] and [ψ2] is then the circular arc in the(ψ1, ψ2)-plane connecting ψ1 and ψ2. Its length is

cos−1(|〈ψ1, ψ2〉|

). (42)

Problem 18. Verify the statements made in this section.

2.3 State space of a qubit

The state space of a qubit is particularly simple. We first parametrise ageneral qubit density matrix in terms of the Pauli matrices σ = (σ1, σ2, σ3)

σ1 =

(0 11 0

)

, σ2 =

(0 −ii 0

)

, and σ3 =

(1 00 −1

)

. (43)

The Pauli matrices together with 1 form a basis of Hermitian matrices ofM2. Therefore ρ = 1

2(a1 + x · σ) with a ∈ R and x ∈ R3. As Tr ρ = 1

we have a = 1 and in order to have ρ non-negative we still must imposedet(ρ) = 1

4(1 − ‖x‖2) ≥ 0 or ‖x‖ ≤ 1. So we see that the state space of a

qubit is affinely isomorphic to the unit ball in R3 also called the Bloch ball

ρ = 12

(1+ x · σ), x ∈ R3. (44)

The boundary and extreme boundary coincide in this case. The centre ofthe ball is the uniform state A 7→ 1

2TrA which is for instance obtained as

an equilibrium state at infinite temperature.

Problem 19. Check that the parametrisation of qubit vector states usedin (4) corresponds to the usual parametrisation of the unit sphere in termsof spherical angular coordinates.

19

To make a full tomography of a qubit state we can perform series of mea-surements to obtain reliable figures for the expectations of σ1, σ2, and σ3.Using the Bloch parametrisation (44) we have

xj = Tr ρσj = 〈σj〉ρ. (45)

From the positivity condition for a qubit density matrix we see that we musthave

〈σ1〉2ρ + 〈σ2〉2ρ + 〈σ3〉2ρ ≤ 1. (46)

The closer this expression comes to 1 the purer the state is. A full tomog-raphy of more complicated systems, like two qubits, requires many moremeasurements and is therefore a costly and lengthy operation.

2.4 A probability theory approach to states

In classical probability one usually starts with the notion of a universe Ω ofelementary events. For a dice this would be the faces: Ω = 1, 2, . . . , 6. Theevents to which a probability will be assigned are then the Borel subsets of Ω.For a finite set these are just the elements of the power set of Ω. In general, itis a Borel algebra B meaning that the set is closed under countable unions

and intersections and under taking complements. The empty set alsobelongs to B. A probability measure µ is now a function on B with values in[0, 1] such that µ(∅) = 0, µ(Ω) = 1, and

µ(⋃

j

Bj

)

=∑

j

µ(Bj)

for all countable collections Bj of disjoint Borel sets. This last property iscalled σ-additivity. The Borel sets are the subsets of Ω that can be given aprobability, they represent the events that can occur.

For a quantum system the projectors play the role of events: the correspond-ing measurement can only have two outcomes 0 or 1, true or false. Theprojectors (on closed subspaces of a Hilbert space) form a lattice. Recallthat a projector P1 is smaller than P2 if P2 − P1 is positive definite and thisis equivalent to P1 = P2P1. Being a lattice means that for any two projectorsP1 and P2 there is a projector that dominates both, e.g. P1 ∨ P2. There isalso a projector that is dominated by both such as P1 ∧ P2. Actually, the

20

lattice is finite because there is a smallest element, the zero operator, and alargest one, the identity. The lattice is also orthocomplemented: the join ofP and 1− P is 1 and their meet is 0. One can also show that the lattice isclosed under countable joins and meets. A quantum probability measure cannow be characterized as in the classical case: a function 〈 〉 from the latticeof projectors to [0, 1] such that 〈0〉 = 0, 〈1〉 = 1, and

⟨∑

j

Pj

⟩

=∑

j

〈Pj〉

for every countable collection Pj of pairwise orthogonal projectors.

Gleason showed that for Hilbert spaces of dimension larger than two everyquantum probability measure corresponds to a density matrix ρ:

〈P 〉 = Tr ρP, P a projector. (47)

A basic difference between classical and quantum probability is that condi-tioning doesn’t work in the quantum case. Let X and Y be two Borel subsetsof Ω and let Prob(Y ) > 0. The conditional probability of X given Y isdefined to be

Prob(X|Y ) :=Prob(X ∩ Y )

Prob(Y ). (48)

Suppose that Yj is a partition of Ω in at most a countably infinite numberof Borel subsets such that for every j Prob(Yj) > 0. We can then write foran arbitrary Borel subset X of Ω

Prob(X) =∑

j

Prob(Yj) Prob(X|Yj). (49)

This is often called Bayes’s law. In the quantum context this doesn’t work,the reason being that for a countable partition Pj of the identity, i.e. fora countably infinite family of projectors Pj such that

∑

j Pj = 1 and for ageneral projector Q one usually has the strict inequality

Q ≥∑

j

Q ∧ Pj, (50)

while in the classical case, for a partition Yj of Ω

X =⋃

j

X ∩ Yj. (51)

21

2.5 States of composite systems

Often one has only access to local information contained in a state of acomposite system. Assume for simplicity a system composed of two partiesA and B that are sufficiently distant so that one can not reasonably observequantities that correlate both parties. In such a case one will not so much beinterested in the full joint density matrix ρAB but rather in the marginals

of this state

X 7→ Tr ρAB(X ⊗ 1B) and Y 7→ Tr ρAB(1A ⊗ Y ). (52)

It is clear that these defines states on A and B. So there is a density matrixρA for the first party such that

Tr ρAX = Tr ρAB(X ⊗ 1B), X observable of first party. (53)

ρA is called a reduced density matrix of ρAB and it is obtained by com-puting a partial trace

ρA = TrB ρAB or 〈j|ρA|k〉 =∑

ℓ

〈jℓ|ρAB|kℓ〉. (54)

Problem 20. Show that the partial trace does not depend on the choice ofbasis in the space over which the partial trace is taken.

Problem 21. Verify (54).

Given local density matrices ρA and ρB then there is always an extension tothe composite system: ρA ⊗ ρB. This corresponds to independence betweenboth parties. In most cases there will be more possibilities compatible withthe local restrictions. Indeed, for a general bipartite state we have d2Ad

2B − 1

real freedoms while specifying the reduced density matrices consumes onlyd2A + d2B − 2 parameters.

How strong quantum systems are correlated is partly captured in the notionof entanglement. A state ρAB of a bipartite system is called separable ifit is a convex combination of product states

ρAB =∑

α

pα σAα ⊗ τB,α. (55)

22

A state that is not separable is called entangled. By construction, the sep-arable states form a convex set. Deciding whether a given bipartite state isseparable or entangled turns out to be a very difficult problem. Moreover,a more refined distinction between classes of states seems to be necessaryin order to understand their usefulness for various tasks. These questions avery difficult and go far beyond the scope of this course. Restricting to purebipartite states is, however, quite simple: a pure state is separable if andonly if it is a product of pure states.

Problem 22. Show that the extreme points of the set of separable statesare the pure product states.

There is a simple quantitative characterisation of the degree of entanglementof a pure state based on the following proposition:

Proposition 6. For any ψAB ∈ CdA ⊗CdB of a composite system there existorthonormal families ej in CdA and fj in CdB and positive numbers cjsuch that

ψAB =∑

j

cjej ⊗ fj . (56)

This decomposition is called the Schmidt decomposition and the numberof necessary terms is the Schmidt number or Schmidt rank of ψAB.

Problem 23. Fill out the details of the proposition.

Problem 24. Find the Schmidt decomposition of the two qubit state

1√3

(|00〉 + |01〉 + |10〉).

Clearly a pure state is separable if and only if its Schmidt number is equalto 1. It is also clear that the Schmidt number of a pure state is quite discon-tinuous as it takes values in N0. A very useful consequence of the Schmidtdecomposition of bipartite pure states is:

Proposition 7. The eigenvalues, with their multiplicities, of both marginalsof a bipartite pure state are equal up to zero’s.

Problem 25. Let ψ be a state vector in Cd1 ⊗ Cd2 . What is the maximalnumber of non-zero eigenvalues of the reduced density matrices?

23

Problem 26. Could one extend the Schmidt decomposition to more thantwo parties?

Closely related to the Schmidt decomposition is the purification of a generalstate. Let ρ be a density matrix on Cd, then we can find an orthonormalbasis ej and a probability vector rj such that

ρ =∑

j

rj|ej〉〈ej|. (57)

In fact we can limit the sum to the j with rj > 0. Suppose there are d′ suchj, the number d′ is the rank of ρ, it is the dimension of the range space ofρ. Pick now an orthonormal basis fj in Cd′ and construct the normalisedvector

Ω =∑

jrj>0

√rj ej ⊗ fj (58)

in Cd⊗Cd′ . One then checks that the reduction of |Ω〉〈Ω| to the first systemis precisely equal to ρ. The pure state on the composite system defined byΩ is called a purification of ρ and Cd′ is called an ancillary space.

Problem 27. Verify the statements above. Can you parametrise all possiblepurifications of ρ?

24

3 Quantum dynamics

We suppose that we can start the dynamics of a system at some initial timet0 with an arbitrary initial state ρ0 of the system. At some later time t thestate of the system is ρ(t; t0, ρ0). The following general requirements on anevolution appear to agree well with observations

1. The map ρ0 7→ ρ(t; t0, ρ0) is affine: it preserves convex mixtures. Thisallows us to introduce a linear evolution map Γ(t, t0) on Mk such that

ρ(t; t0, ρ0) = Γ(t, t0) ρ0, t ≥ t0. (59)

Clearly the map Γ is positive because it maps any density matrix inanother density matrix and it preserves normalisation, in other wordsΓ(t, t0) is trace-preserving.

2. The maps Γ(t, t0) depend continuously on t and Γ(t0, t0) = id.

3. It should be possible to trivially extend the dynamics to an enlargedsetting system + environment: for any dimension d of the environment

σ ⊗ ρ 7→ σ ⊗(Γ(t, t0) ρ

)(60)

should extend to an affine transformation of the global state space ofenvironment + system. Here σ is an arbitrary d-dimensional densitymatrix and ρ an arbitrary density matrix of the system. This conditionis called complete positivity of Γ and it is a strong requirement aswe shall see later on.

3.1 Dynamics in discrete time

In this section we concentrate our attention mostly on the complete positivityof dynamical maps. More precisely, we consider a single map Γ from thestate space of Mk to that of Mn, i.e. a linear map that sends k-dimensionaldensity matrices to n-dimensional density matrices. Clearly, this is equivalentto imposing that Γ is positive and trace-preserving. Here positive meansmapping positive matrices to positive matrices. As we think of Γ as actingon states we say that Γ is a dynamical map in Schrodinger picture.

25

There is of course also an Heisenberg version Γ∗ of Γ obtained throughduality

Tr Γ(ρ)X = Tr ρΓ∗(X), ρ k-dimensional density matrix,

X n-dimensional observable.(61)

Note that Γ∗ : Mn → Mk. It is easily seen that Γ∗ is unity-preserving,Γ∗(1n) = 1k, whenever Γ is trace-preserving.

Example 1. Transposition with respect to some given basis is a positivetransformation of M2 because it preserves hermiticity and the spectrum.However, if we trivially extend transposition to a composite system of twoqubits we loose positivity. Consider the projector P onto 1√

2

(|00〉 + |11〉

):

P =1

2

1 0 0 10 0 0 00 0 0 01 0 0 1

=1

2

(1 00 0

) (0 10 0

)

(0 01 0

) (0 00 1

)

.

We now transform the density matrix P by acting with the transpositiononly on the second qubit:

(id2⊗T)(P ) =1

2

1 0 0 00 0 1 00 1 0 00 0 0 1

∼ 1

2

(1 00 1

)

⊕(

0 11 0

)

which is not positive because of the last term. Hence, we already loosepositivity if we extend the transposition to a single additional qubit.

Let d ∈ N0. A C-linear map Γ : Mk → Mn can be extended to a C-linearmap idd⊗Γ from Md⊗Mk to Md⊗Mn. Any element X ∈ Md⊗Mk canbe written as a d× d matrix with entries in Mk:

X =

X11 · · · X1d...

......

Xd1 · · · Xdd

, Xij ∈ Mk. (62)

We then put

(idd⊗Γ

)(X) =

Γ(X11) · · · Γ(X1d)...

......

Γ(Xd1) · · · Γ(Xdd)

. (63)

26

Problem 28. Show that (63) is equivalent to putting for A ∈ Md andB ∈ Mk (

idd⊗Γ)(A⊗ B) := A⊗ Γ(B).

A C-linear map Γ : Mk → Mn is d-positive if idd⊗Γ is positive. A C-linearmap Γ : Mk → Mn is completely positive if it is d-positive for d = 1, 2, . . .

It can be shown that there exist for any d maps that are d-positive but not(d+ 1)-positive. This is, however, not easy.

Problem 29. Show that Γ is completely positive iff Γ∗ is completely positive.

The d-positive maps from Mk to Mn form a convex cone: if Γ1 and Γ2 ared-positive and if a and b are non-negative numbers then aΓ1 + bΓ2 too isd-positive. A composition of two d-positive maps with matching dimensionsis again d-positive.

3.2 General quantum operations

Sometimes the term super-operator is used to denote linear transforma-tions of Mk, or linear maps from Mk to Mn. This is just to warn you thatwe are not working on the level of the space of wave functions but ratheron that of states or observables considered as elements of the linear space oftransformations of vector states.

There is a standard way of encoding such a super-operator called the Choi

encoding. We start by introducing the standard matrix units eij := |i〉〈j|,these are the matrices with all entries equal to 0 except for a 1 on row i andcolumn j. Obviously these matrix units form a basis of Mk considered asvector space and we have

Mk ∋ A =∑

ij

Aijeij. (64)

It is clear that a super-operator Γ : Mk → Mn is completely specified if weknow its action on each of the eij . The Choi encoding does this in a globalway:

C(Γ) :=∑

ij

eij ⊗ Γ(eij). (65)

27

In this way one associates to the super-operator Γ a kn-dimensional matrixand vice versa. It is easily seen that C(αΓ1 + Γ2) = αC(Γ1) + C(Γ2) but it isNOT TRUE that C(Γ1Γ2) = C(Γ1)C(Γ2).

A super-operator Γ is called positive if it maps positive definite matrices inpositive definite matrices. Note that this is a completely different notion thanpositive definiteness for which you need an inner product space. A super-operator Γ is trace-preserving if Tr Γ(A) = TrA for all A, it is calledunity-preserving if Γ(1) = 1. Suppose that Γ is both positive and trace-preserving, then it maps by definition density matrices to density matrices.

We provide now a basic example of a completely positive super-operator thatis generally neither trace nor unity-preserving but that will prove very usefulin describing quantum operations

Example 2. Pick a linear map V : Ck → C

n, then

Γ : X 7→ V XV † is completely positive. (66)

Indeed:

(idd⊗Γ

)(A⊗ B) = A⊗ Γ(B) = A⊗ V BV † = (1d ⊗ V )(A⊗ B)(1d ⊗ V )†.

Now, if C ∈ Md ⊗Mk is positive then it is of the form D†D and

(idd⊗Γ

)(C) = (1d ⊗ V )D†D(1d ⊗ V )†

=D(1d ⊗ V †)

†D(1d ⊗ V †)

≥ 0.

The following theorem characterises the completely positive maps from Mk

to Mn:

Theorem 3 (Choi - Kraus - Jamio lkowski). Γ : Mk → Mn is completelypositive if and only if C(Γ) is a positive semi-definite matrix.

A quantum operation in Schrodinger picture is a completely positive trace-preserving map from Mk to Mn. Quantum operations transform densitymatrices into density matrices and can, moreover, be trivially extended tocomposite systems without loosing this property. In the dual Heisenbergpicture trace-preserving should be replaced by unity-preserving.

28

Corollary 1. Every quantum operation Γ from a k-level system to a n-levelsystem is of the form

Γ(ρ) =∑

j

VjρV†j with

∑

j

V †j Vj = 1. (67)

There are at most kn terms needed in the summation.

The Vj appearing in (67) are called Kraus operators and this way ofwriting Γ is a Kraus decomposition. There are in general many waysof writing such a decomposition, i.e. the Kraus operators are not uniquelydetermined by the map Γ. Actually one can show that two different sets ofKraus operators Vj and Wα of a same map are connected by an isometrictransformation u

Wα =∑

j

uαjVj.

Problem 30. Fill out the details of the proof of the theorem and the corol-lary.

Problem 31. Express the trace-preserving and unity-preserving conditionson a quantum operation in terms of its Choi matrix.

3.3 Examples of quantum operations

We now consider a number of examples of quantum operations.

3.3.1 Unitary gates

Example 3 (Unitary gate).

Γ(ρ) = UρU † with U †U = 1(= UU †). (68)

Unitary gates are single-shot unitaries that are applied to qubit systemsextending operations on bits to qubits or introducing operations that haveno classical counterpart. These gates are intended to represent basic logicaloperations performed on systems of qubits.

29

A unitary gate maps pure states into pure states

U |ψ〉〈ψ|U † = |Uψ〉〈Uψ|. (69)

Because the action of the gate can be expressed at the level of state vectors,we also have

U(αψ1 + βψ2) = αUψ1 + βUψ2. (70)

Hence, unitary gates not only preserve mixtures but also coherences.

Denoting by ǫ a classical bit ∈ 0, 1 we denote by ǫ1 ⊕ ǫ2 addition modulo2. A few common classical gates are

NOT : ǫ −→ 1 ⊕ ǫ (71)

AND :ǫ1ǫ2

−→ ǫ1ǫ2 (72)

CNOT :ǫ1ǫ2

−→ ǫ1ǫ1 ⊕ ǫ2

(73)

FAN OUT : ǫ −→ ǫǫ

(74)

NAND :ǫ1ǫ2

−→ 1 ⊕ ǫ1ǫ2 (75)

Problem 32 (Universality of NAND). Write all the mentioned gates in termsof NAND’s.

One can get quantum analogues of such gates by associating ǫ to |ǫ〉. Asquantum gates should be unitary and therefore reversible one can only applythis recipe to reversible classical gates, i.e. to classical gates where the inputcan be uniquely reconstructed in terms of the output. The gates NOT andCNOT are examples of reversible gates, this is not the case for AND andalso not for FAN OUT as not every output is reached. There exist howeverclassical reversible gates that allow to realise AND and FAN OUT modulointroducing additional bits and setting some of these to a particular value.A nice example is the Toffoli gate

ǫ1ǫ2ǫ3

−→ǫ1ǫ2

ǫ3 ⊕ ǫ1ǫ2

(76)

30

The AND gate can be realised by putting ǫ3 = 0, feeding ǫ1 and ǫ2 to thefirst two leads and only retaining the last lead as output. We can get FANOUT by feeding ǫ to the first lead, setting the second lead equal to 1 andthe third to 0. We can then get two copies of ǫ on the first and third outputlead.

The single qubit gate that implements the NOT on the computational basisis the unitary U determined by

U : |0〉 7→ |1〉 and U : |1〉 7→ |0〉.

It is easily seen that this unitary is the Pauli σ1 matrix

[0 11 0

]

. A general

superposition of the computational basis states is then mapped into the samesuperposition of the ‘negated’ basis states. Such extensions of classical gatesto the quantum setting don’t really fulfil the expectations one could have.For example the quantum NOT defined above does not map every qubit stateinto its orthogonal complement. FAN OUT would map, modulo neglectingsome outputs, |0〉 to |00〉 and |1〉 to |11〉 and is therefore a classical copier.A general qubit state would then be mapped as follows

α|0〉 + β|1〉 7→ α|00〉 + β|11〉 6= (α|0〉 + β|1〉) ⊗ (α|0〉 + β|1〉)for general α and β. Hence this quantum extension of the classical copieris not a quantum copier. In fact, one can prove that no unitary gate canever make two copies a general (unknown) pure quantum state, this is theno-cloning theorem.

It should also be remarked that many quantum gates have no classical coun-terpart. A very useful single qubit gate in this class is the Hadamard gate

H =1√2

[1 11 −1

]

.

There is a well-understood theory on how to approximately and efficientlygenerate an arbitrary gate by concatenating gates from a finite basic set.This goes, however, beyond the scope of these lectures.

3.3.2 Dissipative operations

In contrast with unitary gates, a general quantum operation might send apure state into a mixed one or inversely. They should be compared to classical

31

stochastic maps that cannot be seen as flows or maps on the configurationspace but that rather send a point in configuration space with a certainprobability to another one.

Example 4 (von Neumann measurement without selection). Let ejbe an orthonormal basis corresponding to an observable. The correspondingprojectors pj := |ej〉〈ej| are mutually orthogonal and form a resolution of theidentity

pjpk = δjkpj and∑

j

pj = 1. (77)

If a system described by a state vector ϕ is sent through the measuringdevice corresponding to pj then we obtain the read-out j with probability|〈ej, ϕ〉|2 and the incoming state collapses to the corresponding eigenstate|ej〉〈ej|. If we don’t filter out any particular set of outcomes we obtain theoutgoing state

∑

j

|〈ej, ϕ〉|2|ej〉〈ej| =∑

j

pj |ϕ〉〈ϕ|pj. (78)

Sending in an arbitrary state ρ will produce the outcome

ρ 7→∑

j

pjρpj . (79)

It is obvious that a pure state will be transformed in a mixed state by thisprocedure and that this is also an irreversible process: there is no possibilityfor reconstructing the incoming state.

Example 5 (Coupling to an external system). This is an example of amap between states on Mk, the system, and on Mk⊗Mℓ, environment plussystem. Let ω be a fixed density matrix of the environment then

Γ(ρ) = ρ⊗ ω, ρ density matrix in Mk. (80)

The extension to an auxiliary d-level system is then

(idd⊗Γ

)(X) = X ⊗ ω, X ∈ Md ⊗Mk (81)

and this is positive.

32

Example 6 (Reduction to a subsystem). We are now considering thepartial trace

ρ 7→ Trℓ ρ ∈ Mk (82)

where ρ is a density matrix on Mk ⊗Mℓ. We write

Trℓ ρ =

k∑

ij=1

⟨i∣∣(Trℓ ρ

)j⟩|i〉〈j| =

k∑

ij=1

ℓ∑

a=1

〈ia|ρja〉|i〉〈j|

=

ℓ∑

a=1

( k∑

i=1

|i〉〈ia|)

ρ( k∑

j=1

|ja〉〈j|)

=

ℓ∑

a=1

VaρV†a

(83)

whereVa : Ck ⊗Cℓ → C

k : |ib〉 7→ δab|i〉.In this way we obtain a Kraus form for the partial trace which proves that itis completely positive. Combining this example with the three previous oneswe can realise a generalised measurement set-up using a pointer system.

Example 7 (A generalised measurement). A more complicated andmore realistic set-up uses an auxiliary ‘pointer’ system C

k: the incomingstate is composed with an initial state of the pointer system. This compos-ite system then unitarily evolves by passing through a unitary gate. Thena von Neumann measurement, without filtering any particular outcome, isapplied to the pointer part of the system and we are finally left with thereduction of the resulting state to the system we observe. We compute thetransition from initial to final state using some simplifying features but itturns out that the overall result is still a completely general quantum oper-ation.

Let fj denote the measurement basis in our pointer system C

k and supposethat the initial state of the pointer system is f1. It is useful to write theunitary gate U in the given basis of the pointer system

U =[Uij]

ij(84)

where the Uij are d× d matrices that satisfy the unitarity relations

∑

j

(U †)

ijUjℓ =

∑

j

(Uji)†Ujℓ = δiℓ1d. (85)

33

We now compute what happens to an incoming pure state ϕ ∈ Cd. It is firstcoupled to the initial state of the pointer system and so we have f1 ⊗ ϕ atour disposal. This evolves through the unitary gate U into

∑

j fj ⊗ Uj1ϕ.Sending this through the measuring apparatus that observes the pointer partof the system we obtain the state

∑

j |fj〉〈fj| ⊗ |Uj1ϕ〉〈Uj1ϕ|. Finally tracing

over the pointer system we obtain∑

j |Uj1ϕ〉〈Uj1ϕ| =∑

j Uj1|ϕ〉〈ϕ|U †j1. The

global quantum operation on a general mixed state can therefore be writtenas

ρ 7→∑

j

Uj1ρ(Uj1)†

=∑

j

VjρV†j . (86)

Because of the unitarity condition the matrices Vj := Uj1 satisfy

∑

j

V †j Vj = 1. (87)

Example 8 (Qubit decoherence).

Γ(ρ) =

(ρ11 γρ12γρ21 ρ22

)

with γ ∈ C. (88)

In order to find out under which condition on γ this is a quantum operationwe write down the Choi matrix

C(Γ) =

1 0 0 γ0 0 0 00 0 0 0γ 0 0 1

. (89)

This matrix is positive iff |γ| ≤ 1. Repeated applications of this map, with|γ| < 1 will map any initially pure state with state vector α|0〉 + β|1〉 to themixed state |α|2|0〉〈0|+ |β|2|1〉〈1|. This means that the off-diagonal terms inthe original pure state, αβ|1〉〈0|+αβ|0〉〈1|, are eventually sent to zero. Thisis loss of coherence.

Example 9 (Qubit depolarisation).

Γ(ρ) = λρ + (1 − λ)1

2, λ ∈ R. (90)

In order to find out under which condition on λ this is a quantum operationwe write down the corresponding Choi matrix. We first rewrite the map in

34

such a way that the affinity in ρ is manifest:

Γ(ρ) = λρ+ (1 − λ)1

2Tr ρ. (91)

This allows us to compute the Choi matrix

C(Γ) =

1+λ2

0 0 λ0 1−λ

20 0

0 0 1−λ2

0λ 0 0 1+λ

2

. (92)

This matrix is positive iff −13≤ λ ≤ 1. The extreme value λ = −1

3corre-

sponds to σ 7→ −σ/3. This is the most we can do reversing the sign of allcomponents of angular momentum without destroying complete positivity.

3.4 Dynamics in continuous time

We now return to evolution maps Γ(t, t0) | t ≥ t0 as introduced at thebeginning of this chapter (59).

3.4.1 Generators

We assume the additional simplifying property of divisibility also calledMarkovianity in time: if we know the state of a system at a given timethen we know it for all later times, put differently, the evolution of a systemis independent of its history. This can be stated as

Γ(t2, t0) = Γ(t2, t1) Γ(t1, t0), t0 ≤ t1 ≤ t2. (93)

Even simpler are autonomous evolutions. Here Γ(t, t0) only depends on t−t0.Slightly abusing notation we write

Γ(t, t0) = Γ(t− t0). (94)

In this case we obtain a one-parameter semi-group of trace-preserving com-pletely positive maps

Γ(t) | t ∈ R+, Γ(t1 + t2) = Γ(t1) Γ(t2). (95)

35

Even more restrictive are reversible autonomous dynamics where one has aone-parameter group of trace-preserving completely positive maps

Γ(t) | t ∈ R, Γ(t1 + t2) = Γ(t1) Γ(t2). (96)

In this case it turns out that the dynamics is unitary:

Γ(t)(ρ) = U(t)ρU(t)† with U(t) | t ∈ R a unitary group. (97)

Markovian dynamics are characterised by their generator, which is in gen-eral time-dependent. Let us define this time-dependent generator as

Λ(t) := limǫ↓0

Γ(t+ ǫ, t) − id

ǫ= lim

ǫ↓0

Γ(t, t− ǫ) − id

ǫ. (98)

Using the Markovian property, we can write

Γ(t+ ǫ, t0) − Γ(t, t0) =(Γ(t+ ǫ, t) − id

)Γ(t, t0). (99)

Dividing by ǫ and taking the limit ǫ ↓ 0 we obtain a differential equationwith initial condition for Γ

d

dtΓ(t, t0) = Λ(t) Γ(t, t0), t > t0 and Γ(t0, t0) = id . (100)

This equation rightly deserves the name Schrodinger equation.

Conversely, we can solve the linear differential equation (100) by applyingPicard iteration in a time-interval [t0, tmax] for which Λ depends continuouslyon time

Γ(t, t0) = Texp(∫ t

t0

dsΛ(s))

= id +

∫

t0≤s1≤tds1 Λ(s1) +

∫∫

t0≤s2≤s1≤tds1ds2 Λ(s1)Λ(s2) + · · ·

(101)

The series that defines the solution is called a time-ordered exponential.

By the unicity of the solution of (100) we obtain Markovianity in time. Lin-blad’s theorem gives the general form of a generator t 7→ Λ(t) of a Marko-vian dynamics that leads to completely positive dynamical maps.

36

3.4.2 Lindblad’s theorem

Theorem 4. A family of super-operators t 7→ Λ on the d × d matrices isa generator of a divisible family of quantum operations if and only if it is ofthe form

Λ(ρ) = i[H(t), ρ] +∑

k

(Vk(t)ρV

†k (t) − 1

2V †

k (t)Vk(t), ρ). (102)

Here t 7→ H(t) is a continuous family of Hermitian d-dimensional matricesand the t 7→ Vk(t) are continuous families of transformations of Md.

The differential equation

d

dtΓ(t, t0) = Λ(t) Γ(t, t0), t > t0 and Γ(t0, t0) = id . (103)

with the explicit form (102) for the generator is called Lindblad’s equation.It is the generalisation of Schrodinger’s equation to general non-reversibleMarkovian dynamics. The differential equation for a state ρ with initialcondition ρ0 reads

dρ

dt= i[H(t), ρ]+

∑

k

(Vk(t)ρV

†k (t)− 1

2V †

k (t)Vk(t), ρ)

and ρ(t0) = ρ0. (104)

To derive this result it suffices to consider the autonomous case and we needthe general characterisation of positive semi-definiteness for a 2 × 2 blockmatrix.

Proposition 8. The block matrix

(A CC† B

)

acting on Cd1 ⊕Cd2 is positive

semi-definite if and only if

1. A and B are positive semi-definite and

2. there exists a U : Cd2 → C

d1 with ‖U‖ ≤ 1 such that C =√AU

√B.

Problem 33. Prove Proposition 8.

Problem 34 (Schur complement). Suppose that B in Proposition 8 isinvertible. Show then that the positivity conditions for the block matrix canbe written as

A,B ≥ 0 and CB−1C† ≤ A. (105)

37

Proof of Theorem 4. Because we are dealing with a semi-group of quantumoperations we have

exp(tΛ) = exp(tΛ/n)n. (106)

For ǫ sufficiently small we may write

exp(ǫΛ) = id +ǫΛ + o(ǫ) (107)

Because of Choi’s theorem we then have that

C(id +ǫΛ) + o(ǫ) is positive semi-definite. (108)

An easy computation shows that

C(id) =(∑

i

|ii〉)(∑

j

〈jj|)

. (109)

We now split the space Cd ⊗Cd into

C

d ⊗Cd = C

(∑

i

|ii〉)

⊕(∑

i

|ii〉)⊥

(110)

which leads to the block matrix decomposition

C(id +ǫΛ) + o(ǫ) =

(d+ ǫC(Λ)11 ǫC(Λ)12ǫC(Λ)21 ǫC(Λ)22

)

. (111)

By Proposition 8 and taking the limit ǫ ↓ 0 we remain with the condition

C(Λ)22 ≥ 0 and C(Λ)12 ∈ Ran(C(Λ)22

). (112)

Now we can essentially repeat the proof of the Choi-Jamio lkowski-Kraus the-orem. The main difference is that C(Λ)22 acts on the orthogonal complementof the maximally entangled vector

∑

i |ii〉. Combining this with preservationof the trace leads to (102).

All these arguments can be reversed to show that a super-operator of theform (102) is the generator of a semi-group of quantum operations.

Problem 35. Fill out the details of the proof of Theorem 4

Before turning to applications we mention the quite useful two-positivity

inequality:

38

Proposition 9. Let Γ∗ be a quantum operation in Heisenberg picture, then

Γ∗(X†)Γ∗(X) ≤ Γ∗(X†X) for any X. (113)

Let Λ∗ be a generator in Heisenberg picture of a continuous one-parametersemi-group of 2-positive unity-preserving maps, then

X†Λ∗(X) + Λ∗(X†)X ≤ Λ∗(X†X) for any X. (114)

To prove this inequality we consider the positive 2 × 2 block matrix

(X†

1

)(X 1

)=

(X†X X†

X 1

)

. (115)

As Γ∗ is 2-positive we have also

0 ≤ (id⊗Γ∗)

(X†X X†

X 1

)

=

(Γ∗(X†X) Γ∗(X†)

Γ∗(X) 1

)

. (116)

This is the case iff

Γ∗(X†) =(Γ∗(X)

)†and Γ∗(X†)Γ∗(X) ≤ Γ∗(X†X). (117)

The inequality (114) follows from (113) by expanding exp(tΛ∗) for smallpositive t.

3.4.3 A semi-group of decoherent qubit maps

We propose a simple form for a time-independent generator of a completelypositive trace-preserving semi-group of quantum operations acting on a singlequbit

Λ

(ρ11 ρ12ρ21 ρ22

)

=

(−aρ11 + bρ22 cρ12

c ρ21 aρ11 − bρ22

)

(118)

and determine the necessary and sufficient conditions on the parameters a,b, and c that ensure that Λ is a Lindblad generator. We certainly need thatΛ annihilates the trace, Tr Λρ = 0, because the exponential of tΛ has to betrace-preserving. This is already incorporated in the form of Λ.

39

From the proof of Lindblad’s theorem we need to pick a, b, and c in such away that

C(Λ) =

−a 0 0 c0 a 0 00 0 b 0c 0 0 −b

≥ 0 on Ω⊥ (119)

where Ω = 1√2

(1 0 0 1

)T. This leads to

a ≥ 0, b ≥ 0, and a+ b ≤ −c− c. (120)

From (118) we see that the off-diagonal matrix elements are eigenvectors ofΛ and that Λ mixes the diagonal entries of a density matrix. This makes theexponentiation of Λ rather straightforward:

etΛ(ρ11 ρ12ρ21 ρ22

)

=1

a + b

(

b+ (aρ11 − bρ22)e−t(a+b) etc ρ12

etc ρ21 a− (aρ11 − bρ22)e−t(a+b)

)

.

(121)

The conditions (120) on the parameters of the generator are equivalent with

∣∣etc∣∣2 ≤ e−t(a+b). (122)

Inspecting (121) we see that every initial density matrix evolves toward theequilibrium state ρ∞ when t→ ∞

ρ∞ =1

a+ b

(b 00 a

)

. (123)

The convergence rate of the diagonal elements to this limit is determined bya+b. The off-diagonal elements decay even faster to zero, this is the meaningof (122).

Problem 36. Check that the details of the example above.

3.4.4 Radiation loss to the vacuum

Suppose that we consider a single mode of a radiation field, say to describea laser. We are also not interested in polarisation but just in the number

40

of photons. The photon field is described in terms of a creation and aannihilation operator that satisfy the canonical commutation relations

[a, a†] = 1. (124)

The vector state with zero photons is called the vacuum, we represent it byΩ or |0〉. The defining feature of Ω is

aΩ = 0. (125)

Using this and (124) we can compute that

am(a†)n Ω = 0 for n > m and an(a†)n Ω = n! Ω. (126)

In this way we introduce an orthonormal family |n〉 | n ∈ N of vectors

|n〉 :=1√n!

(a†)n Ω. (127)

|n〉 is the n photon state and |n〉 is called the particle number basis. Itis easy to express a and a† in this basis

a†|n〉 =√n+ 1 |n+ 1〉 and a|n〉 =

√n |n− 1〉. (128)

The number operator N := a†a counts the number of photons in a state

N |n〉 = n|n〉. (129)

We will need the following:

Lemma 1. Let A and B be linear transformations such that [A, [A,B]] =[B, [A,B]] = 0, then

eA+B = e−12[A,B] eA eB. (130)

Problem 37. Prove the lemma by showing that

t 7→ F (t) := e−t2

2[A,B] etA etB (131)

satisfies the differential equation

dF

dt= (A+B)F (t) and F (0) = 1. (132)

You need to use [A, [A,B]] = [B, [A,B]] = 0.

41

We can apply this to exponentials of linear combinations of the creation andannihilation operator. E.g.:

eza†−za = e−

12|z|2 eza

†

e−za, z ∈ C. (133)

Remark that the left hand side is the exponential of a skew Hermitian trans-formation and that it is therefore unitary.

It turns out that the radiation of a laser is well-described by coherent states

eza†−za Ω (134)

depending on a complex parameter z. Using (133) and aΩ = 0 we expandsuch a coherent state in the particle number basis

eza†−za Ω = e−

12|z|2 eza

†

Ω = e−12|z|2

∞∑

n=0

zn√n!

|n〉. (135)

This means that the statistics of the number operator is Poisson:

ProbN = n = e−|z|2 |z|2nn!

. (136)

|z|2 fixes the average number of photons, i.e. the intensity of the beam.

A phenomenological description of radiation loss to the vacuum can be ob-tained through a weak-coupling type limit. It leads to a Lindblad generator

Λ(ρ) = aρa† − 12a†a, ρ. (137)

Computing the action on the matrix units in the number basis yields

Λ(|n〉〈m|) = a |n〉〈m| a† − 12a†a |n〉〈m| − 1

2|n〉〈m| a†a (138)

=√mn |n− 1〉〈m− 1| − 1

2(n+m) |n〉〈m|. (139)

This means that

etΛ(|n〉〈m|) = e−(n+m)t/2 |n〉〈m| +m∧n∑

k=1

ck(t) |n− k〉〈m− k|. (140)

Hence, for any initial density matrix ρ we have

limt→∞

etΛ(ρ) = |0〉〈0|. (141)

42

It is possible to compute explicitly the time evolution of functions of thenumber operator and the evolution of coherent states. Deriving the generalresults is a bit too lengthy in this context and we limit ourselves to anexample.

Λ∗(N) = a†Na− 12a†a,N = −N. (142)

HenceetΛ

∗

(N) = e−tN. (143)

Λ∗(N2) = a†N2a− 12a†a,N2 = −2N2 +N. (144)

HenceetΛ

∗

(N2) = e−2tN2 + e−t(1 − e−t)N. (145)

Computing the intensity of the beam at time t leads to exponential decay

〈N〉t = e−t 〈N〉0. (146)

Computing the long-time behaviour of the variance we get

〈∆N〉t ≈ e−t/2√

〈N〉0. (147)

Problem 38. Verify the computations above.

Problem 39. For those who like more challenging computations, prove thefollowing

Λ∗(N(N − 1) · · · (N − k1))

= −(k + 1)N(N − 1) · · · (N − k1) (148)

etΛ∗(

e−sN)

= exp(

N log(1 + e−s−t − e−t

))

. (149)

Here k is a natural number and s and t can be taken non-negative in orderto avoid convergence problems.

3.5 Thermalising maps

We present here very schematically a simple black box dynamics that de-scribes how an environment in thermal equilibrium that is weakly coupled

43

to a small system drives the system towards equilibrium at the temperatureof the environment. This is an approximate description that is only validin a situation where the system is weakly interacting with the thermal bathand where it consequently takes a long time to equilibrate the small system.This explains the name weak-coupling limit and the need to consider longtimes.

Let us denote by S the system and by B the thermal bath or environment.We assume that S has d accessible levels and that B is an infinite system.In particular, the energy spectrum of S is discrete while B has a continuousspectrum. Although we will not enter in the technical details, this is animportant feature. We will briefly describe the outcome of the weak-couplinglimit but in order to prove its existence one needs some technical propertiesthat can not hold for systems with a discrete energy spectrum.

3.5.1 Reversible dynamics of the system

Let HS denote the Hamiltonian of the system, i.e. a d-dimensional Hermitianmatrix. We then know that there is an orthonormal basis |j〉 of eigenvectorsof HS: HS|j〉 = Ej |j〉. For a generic Hamiltonian we can always assume thatthe eigenvalues are non-degenerate. The dynamics, in Heisenberg picture, isgiven by

A 7→ eitHS

A e−itHS

= exp(it[HS, ·])(A), A ∈ Md. (150)

In this formula [HS, ·] is the short-hand notation for the super-operator

[HS, ·](A) := [HS, A], A ∈ Md. (151)

It is not difficult to compute the eigenvalues and eigenvectors of the super-operator [HS, ·]:

[HS, |k〉〈ℓ|] = (Ek − Eℓ)|k〉〈ℓ| = ωℓk |k〉〈ℓ|, k, ℓ = 1, 2, . . . , d. (152)

The energy gaps ωℓk between the levels of HS are called the Bohr fre-quencies of the evolution. Generically, each ωℓk is non-degenerate except for0 = ωkk which has a degeneracy d. An arbitrary observable A can be ex-panded in the matrix units |k〉〈ℓ| and this yields the following expressionfor the dynamics

eitHS

A e−itHS

=d∑

k,ℓ=1

Akℓeitωℓk |k〉〈ℓ|. (153)

44

It is clear from this formula that, for a generic Hamiltonian, the constantsof the motion are precisely the linear combinations of the spectral projectors|k〉〈k| of HS. In other words, the constants of the motion are just thealgebra of diagonal matrices. In the sequel we use the notation dia(ϕ) withϕ ∈ Cd to denote the diagonal matrix whose entries on the diagonal are thecomponents of ϕ. It is also clear from (153) that the evolution of any systemobservable is essentially periodic in time. It is not truly periodic becausethe Bohr frequencies are in general not integer multiples of a fundamentalfrequency, nevertheless after a sufficiently long time the system repeats itselfup to an arbitrary small error. Such a behaviour is called quasi or almostperiodic. It is in particular impossible for systems with this type of energyspectrum to converge in the long run to some equilibrium situation. Conver-gence or return to equilibrium can only happen in infinite systems and sucha behaviour is needed for the weak-coupling limit.

3.5.2 Equilibrium states of the bath

The equilibrium state at inverse temperature β = 1/kT for a N -particlesystem is given by the canonical Gibbs matrix

ρβ =e−βH

N

Z =e−βH

N

Tr e−βHN(154)

where HN is the N -particle Hamiltonian. This expression makes sense forfinite systems but does not survive the thermodynamic limit N → ∞. Wecan, however, use a purification of the state to get

• a Hilbert space HN

• a normalized vector ΩN in HN that generates the expectations of ob-servables

• an algebra AN of observables of the N -particle system that acts on HN .For finite systems, HN has a tensor product structure and the observ-ables act only on the first factor. Expectations of bath observables aregiven in the usual way

〈X〉β = 〈ΩN , X ΩN 〉, X ∈ AN .

45

• a unitary dynamics on the observables AN

There is, moreover, a deep connection between the dynamics and the thermalexpectations. For any two observables X and Y of the system

〈XY 〉β =Tr e−βH

N

XY

Z =Tr e−βH

N

Y e−βHN

X eβHN

Z= 〈Y, αiβ(X)〉β.

(155)

Hereαt(X) := eitH

N

X e−itHN

(156)

is the reversible dynamics of the N -particle system.

This set-up survives the thermodynamic limit at a price: the infinite systemHilbert space H has no simple tensor decomposition such that the bath ob-servables act only on the first factor. The infinite system can be describedas follows:

• there is an infinite dimensional separable Hilbert space H,

• a normalized vector Ω in H that generates the expectations of observ-ables, and

• an algebra A of bath observables that acts on H.

The dynamics is generated by a bath Hamiltonian HB that acts on H andthat enjoys the following properties:

• HB Ω = 0 which expresses the time-independence of the expectations.

• There is a global dynamics of the observables: if A ∈ A then alsoeitH

B

A e−itHB

is an observable.

• The dynamics and thermal expectations are still connected as in (155)

〈Ω, XY Ω〉 = 〈Ω, Y e−βHB

X eβHB

)〉, X, Y ∈ A. (157)

Because of the infinite dimensions, we can now have a very different behaviourof the dynamics, such as return to equilibrium

limt→∞

〈Ω, X(eitH

B

Y e−itHB)Z Ω〉 = 〈Ω, XZ Ω〉 〈Ω, Y Ω〉. (158)

Such a behaviour is excluded for finite systems as in that situation all time-dependent expectations are quasi-periodic.

46

3.5.3 The weak-coupling limit

The weak-coupling limit describes the simplified situation that arises whenwe couple a d-level system weakly with a thermal bath at inverse temperatureβ. Because of the weak interaction we need to wait a long time before thecoupling affects the expectation values of system observables. It turns out,however, that after a proper rescaling of coupling strength and time, we endup with a simple dissipative dynamics of the small system: all memory effectsdisappear and we are left with a semi-group of completely positive maps ofthe system that drives any initial state toward the equilibrium state of thesystem at the inverse temperature β imposed by the thermal bath.

So, we consider a total Hamiltonian of system + bath

Htot = HS ⊗ 1 + 1⊗HB + λH int = H free + λH int. (159)

The interaction part H int is assumed to be generic. The proper scaling forthe limit is to let λ→ 0 and t→ ∞ such that λ2t→ τ . This causes, however,very fast oscillations that have to be corrected for using the free evolution(λ = 0). Modulo technical assumptions on the bath one can show for τ ≥ 0,for any A ∈ Md, and for any d-dimensional density matrix ρ the existenceof the following limit

Tr ρΓ∗τ (X)

= limλ→0, t→∞λ2t→τ

Tr ρ⊗ |Ω〉〈Ω|(

e−itHfree

eitHtot

X ⊗ 1 e−itHtot

eitHfree

)

. (160)

3.5.4 Properties of the weak-coupling limit

The weak-coupling limit provides us with a set of super-operators Γ∗t | t ≥ 0

in Heisenberg picture or with their dual Schrodinger version Γt | t ≥ 0.These maps are completely positive and unity- or trace-preserving accordingto the chosen picture.

• A first remarkable property is that they are actually semi-groups:

Γt1 Γt2 = Γt1+t2 , t1, t2 ≥ 0. (161)

Physically, it means that all memory in the dynamics at finite coupling disap-pear in the limit. As a consequence there exists a time-independent generator

47

Λ∗ of Lindblad type such that

Γ∗t = exp(tΛ∗), t ≥ 0. (162)

(The notation Λ is only for very local use and intended to make the letter Λavailable.)

• The canonical equilibrium state of the system has density matrix

ρβ =e−βH

S

Z . (163)

This state is not only invariant under each Γt but any initial state of thesystem will eventually be driven to ρβ

limt→∞

Γt(ρ) = ρβ , ρ arbitrary density matrix in Md. (164)

• The Lindblad generator has a very special form:

Λ∗(A) = i[HS, A] + Λ∗(A), A ∈ Md. (165)

Here Λ is the dissipative part of Λ and

• The super-operators Λ∗ and [HS, ·] commute, i.e. the reversible dynam-ics of the system commutes with the dissipative part.

• Λ∗ satisfies quantum detailed balance which is expressed as Λ∗ beingHermitian with respect to the scalar product induced by ρβ on theobservables of the system

Tr ρβXΛ∗(Y ) = Tr ρβΛ∗(X)Y, X, Y ∈ Md. (166)

3.5.5 Quantum detailed balance maps on Md

In this last section we try to find out the structure of a quantum detailedbalance semi-group on Md in a generic case. We start out with the systemHamiltonian HS such as in Section 3.5.1 and assume that the Bohr frequen-cies of HS are non-degenerate, except for 0 that has a degeneracy d.

We first express that the dissipative and reversible parts of the dynamicscommute. On diagonal matrices this leads to

[HS,Λ∗(dia(ϕ))] = Λ∗([HS, dia(ϕ)])

= 0, ϕ ∈ Cd. (167)

48

This means that Λ∗ sends constants of the motion into constants of themotion. Because of linearity it implies that there is a d-dimensional matrixL∗ such that

Λ∗(dia(ϕ)) = dia(L∗ϕ), ϕ ∈ Cd. (168)

For off-diagonal matrix units |k〉〈ℓ|, k 6= ℓ, we have

[HS,Λ∗(|k〉〈ℓ|)] = Λ∗([HS, |k〉〈ℓ|])

= ωℓk Λ∗(|k〉〈ℓ|). (169)

This means that either Λ∗(|k〉〈ℓ|) is zero or that it is an eigenvector of [H∗, ·]with eigenvalue ωℓk. As the non-zero Bohr frequencies are assumed to benon-degenerate we conclude that there exists ckℓ ∈ C such that

Λ∗(|k〉〈ℓ|) = ckℓ |k〉〈ℓ|, k 6= ℓ. (170)

We can put the c’s in a square table and fill it up with zeroes on the diagonalto get a second d-dimensional matrix C. So, commutation of the dissipativeand reversible parts of the dynamics leads us to specify Λ∗ in terms of onlytwo d-dimensional matrices L∗ and C.

Λ∗ is the generator of a semi-group of unity-preserving completely positivemaps if and only if Λ∗(1) = 0 and the Choi encoding C(Λ∗) is positive on theorthogonal complement of the maximally entangled state 1√

d

∑

k |kk〉. Theseconditions are equivalent with

1. L∗ is a generator of a semi-group of stochastic matrices preserving theconstant function 1 and

2. the matrix

ℓ11 c12 · · · c1dc21 ℓ22 · · · c2d...

.... . .

...cd1 cd2 · · · ℓdd

is positive semi-definite, where L∗ =

[ℓkℓ]kℓ.

It remains to find out the consequences of quantum detailed balance. Astraightforward application of (166) yields that L∗ has to be the generatorof a classical detailed balance process with invariant measure given by thediagonal elements of ρβ and moreover the matrix C has to be real symmetric.

Problem 40. Fill out the details in Section 3.5.5.

49

4 Entropy

4.1 Classical entropy

A stationary classical source emits letters belonging to an alphabet X =x1, x2, . . . , xd at discrete times. After N emissions we obtain a messagexi0xi1 · · ·xiN−1

. We now assume that we can empirically establish the statis-

tics of the source by counting relative frequencies. This is actually themeaning of stationarity. So the probability distribution of single letters isgiven by

µ(xj) = limN→∞

1

NN(xj) (171)

where N(xj) is the number of times that the letter xj appears up to timeN − 1. We can also look for relative frequencies of two letter words such asxj0xj1

µ(xj0xj1

)= lim

N→∞

1

NN(xj0xj1

). (172)

Here N(xj0xj1

)is the number of times that the letters xj0 and xj1 consecu-

tively appear in the message. It is easily seen from the construction that

µ(xj0)

=d∑

j1=1

µ(xj0xj1

). (173)

This reflects the compatibility between the probability distributions on oneand two letter words. Another relation that we obtain is

µ(xj1)

=d∑

j0=1

µ(xj0xj1

). (174)

This expresses the stationarity of the source or equivalently shift-invarianceof our probability measure on two letter words: the probability of all twoletter words with a given first letter is the same as that of two letter wordswith the same letter on the second place. This procedure can be extended towords of arbitrary length and computing relative frequencies for all possiblefinite words we obtain a shift-invariant probability measure µ on XN. It is inthese terms that a stationary source is described. Depending on additionalproperties of µ one distinguishes particular classes of sources.

50

A source is memoryless if µ is a product measure:

µ(xj0xj1 · · ·xjk

)= µ

(xj0)µ(xj1)· · ·µ

(xjk). (175)

This means that the letters sent out by the source are completely uncorre-lated. Correlations of time length 1 yield Markov measures:

µ(xj0xj1 · · ·xjk

)=µ(xj0xj1

)µ(xj1xj2

)· · ·µ

(xjk−1

xjk)

µ(xj1)µ(xj2)· · ·µ

(xjk−1

)

= µ(xj0)Tj0j1Tj1j2 · · ·Tjk−1jk

(176)

with

Tjk :=µ(xjxk)

µ(xj). (177)

The matrix T = [Tjk] is called the transition matrix. It has non-negativeentries and its row sums are equal to 1. This matrix describes the probabilityof jumping from j to k. Such matrices are also called stochastic. In orderto obtain a shift-invariant Markov measure one has to impose

d∑

k=1

µ(xjxk) =

d∑

k=1

µ(xkxj) (178)

Obviously there exist much more complicated measures.

An important informational task is to compress long messages before sendingthem through a transmission channel and then to restore the original mes-sage by a suitable decompression. Such a procedure is very useful whenevertransmission or storing of information is costly or space limited. Instead ofrequiring a perfect reconstruction of the original unknown message one mayrequire that most messages emitted by the source are restored in the end,this can allow for big savings whenever some words appear more frequentlythan others. The extreme example is that of a source emitting only a singleletter. The main idea is compression is to assign short code words to partsof message words that often appear.

Let us consider a memoryless source and determine the probability of oc-currence of a word xj0xj1 · · ·xjN−1

of length N . Suppose that the letter xjoccurs Nj times. We find therefore the probability

d∏

j=1

µ(xj)Nj . (179)

51

Now there are N !/(N1!N2! · · ·Nd!) such words. Therefore the probabilitydistribution over the possible words is multinomial. For large N such adistribution is sharply peaked around its maximum that is attained at Nj =µ(xj)N . This follows from Stirling’s formula. It also follows that almostthe full weight of the probability distribution is concentrated on a subset ofwords of size exp(N(h + δ)) where

h := −d∑

j=1

µ(xj) logµ(xj). (180)

The quantity h is called the Shannon entropy of the source. Shannon’snoiseless (memoryless) compression theorem states that for such a source thespace of relevant messages of length N cannot be further compressed thanexp(Nh).

Problem 41. Fill in the details of the computation of above.

4.2 Construction of the Shannon entropy

Define the function

η(x) := −x log x for 0 < x ≤ 1 and η(0) := 0. (181)

It is not hard to check that η is non-negative, strictly concave, and smoothon ]0, 1]. Moreover it vanishes only at x = 0 and x = 1.

Let X be a finite set equipped with a probability measure µ. The Shannon

entropy of µ is defined as

H(X) :=∑

x∈Xη(µ(x)). (182)

The notation H(X) is traditional and not quite optimal, H(µ) would fit better.H takes values in [0, log|X|]. The extreme value 0 is attained for degeneratemeasures (giving weight 1 to one of the points in X) while log|X| is attainedfor the uniform measure µ(x) = 1/|X|. Furthermore H is strictly concave:the entropy of a non-trivial mixture of two measures is strictly larger thanthe corresponding mixture of entropies.

Problem 42. Verify the statements just made.

52

Consider a composite system X ×Y equipped with a probability measure µ.The marginals µX and µY of this measure define probabilities on X and Y :

µX(x) :=∑

y∈Yµ(xy). (183)

The term joint entropy is used for

H(XY ) := H(X × Y ). (184)

We have the following relations

H(X) ≤ H(XY ) monotonicity and (185)

H(XY ) ≤ H(X) + H(Y ) subadditivity. (186)

Moreover, the subadditivity inequality is saturated iff µ is a product measure.The mutual information is the degree of non-saturation of subadditivity

H(X : Y ) := H(X) + H(Y ) − H(XY ). (187)

H(X : Y ) = 0 iff the joint measure is the product of its marginals, this meansthat there is independence between the two subsystems and so certainlyno strong correlations that would allow to identify with high probabilityelements of X with elements of Y . The maximal value that H(X : Y ) canattain is H(X). This happens when there is a bijective map f connecting theelements with positive probability in X with those of positive probability inY such that

µ(xf(y)) > 0 and µ(xy) = 0 for y 6= f(x). (188)

An other important property is strong subadditivity

H(XY Z) + H(Y ) ≤ H(XY ) + H(Y Z). (189)

Markov measures saturate this inequality, where Markov means

µXY Z(x, y, z) =µXY (x, y)µY Z(y, z)

µY (y). (190)

The subscripts to µ indicate the marginals of the joint measure.

53

Problem 43. Verify the statements just made.

Theorem 5. Let µ be a shift-invariant probability measure on XN then

h(µ) := limn→∞

1

nH(×nX) (191)

exists. It is called the entropy of the source µ.

Proof. The proof is based on subadditivity. In fact one shows that

h(µ) = infn

1

nH(×nX). (192)

Problem 44. Compute the entropy of a Markov measure.

Actually, this result can be refined. Using the short notation Hn = H(×nX)

Theorem 6. For n = 1, 2, . . .

0 ≤ Hn+2 − Hn+1 ≤ Hn+1 − Hn (193)

andh = lim

n→∞

(Hn+1 − Hn

). (194)

Theorem 6 is much stronger than Theorem 5: it tells that h is the entropyentropy produced each time unit while the former just states that h is theaverage entropy produced over a large time.

In order to compute the probability of all the words with given first letter xkand third letter xℓ we have to compute

d∑

i=1

µ(xkxixℓ). (195)

In a similar way we can obtain probabilities of general configurations fixedat a number of times. A measure is mixing if probabilities of finite wordspulled far apart factorise in probabilities of the words. For two single letterwords

limn→∞

∑

i1,i2,...,in

µ(xkxi1xi2 · · ·xinxℓ) = µ(xk)µ(xℓ). (196)

We can now state the general Shannon compression theorem:

54

Theorem 7. Let µ be a shift-invariant, mixing probability measure on XN

with entropy h. For any ǫ, δ > 0 and n sufficiently large there exist coding anddecoding maps Cn and Dn to and from a set Rn with at most exp(n(h + δ))elements such that

µ(x ∈ ×nX | Dn Cn(x) = x

)≥ 1 − ǫ. (197)

If |Rn| ≤ exp(n(h− δ)) then no such maps exist.

4.3 Quantum Entropy

We now deal with a memoryless source emitting pure states ϕi with prob-abilities pi. This defines a quantum ensemble (pi, ϕi) of pure states. Wecan in the same way consider a quantum ensemble (pi, ρi) of mixed states,this can be thought to be a source emitting pure composite states states ofwhich only one party is accessible. Observing the source by repeated mea-surements, we see an average state

ρ =∑

i

pi|ϕi〉〈ϕi| or ρ =∑

i

piρi. (198)

Moreover, observations between consecutive times will be uncorrelated

A1 ⊗A2 7→ Tr ρ⊗ ρA1 ⊗ A2. (199)

Note that, in contrast to classical systems, we cannot interpret ρ as an en-semble. Differently built sources may generate the same ρ because a quantumstate space is very far from a simplex.

As before, we have an entropy S that measures the uncertainty in the statespit out by the source, this is the von Neumann entropy

S(ρ) := Tr η(ρ) = −Tr ρ log ρ. (200)

It is not hard to see that S(ρ) = 0 iff all ϕ coincide (or all ρi are pure andcoincide).

For more complicated sources (not memoryless) one has a description as inthe classical case. The multi-time correlated expectations are given by re-

55

duced density matrices ρ(n) on Md ⊗ · · · ⊗Md︸ ︷︷ ︸

n times

and we have the relations

Tr ρ(n+1) (An ⊗ 1) = Tr ρ(n)An, compatibility (201)

Tr ρ(n+1) (1⊗ An) = Tr ρ(n)An, shift-invariance. (202)

Here An is an arbitrary observable in Md ⊗ · · · ⊗Md︸ ︷︷ ︸

n times

. The collection of (re-

duced) density matrices ρ(n) now defines a shift-invariant state on ⊗NMd.

The von Neumann entropy of a density matrix ρ on Md has properties similarto those of the Shannon entropy. It is easily seen that its range is [0, log d],S(ρ) = 0 iff ρ is pure and the maximal value log d is attained iff ρ is theuniform state 1/d. It is also a concave function which follows from Lemma 2.

Lemma 2. Let f : [a, b] → R be concave, then A 7→ Tr f(A) is concave onthe set of Hermitian matrices in Md whose eigenvalues are a subset of [a, b].

Problem 45. Fill out the details of the proof.

Let ρ12 be a density matrix of a bipartite system Md1 ⊗Md2 with reduceddensity matrices ρ1 and ρ2, then

S(ρ12) ≤ S(ρ1) + S(ρ2), subadditivity and (203)

S(ρ2) ≤ S(ρ12) + S(ρ1). (204)

The term S(ρ1) in the right hand side of (204) is needed because mono-

tonicity does not hold in the quantum case, e.g., ρ12 can be pure inwhich case ρ1 and ρ2 have the same entropy, generally strictly positive. Inthis case (204) is saturated. A more symmetric version of (204) reads

|S(ρ1) − S(ρ2)| ≤ S(ρ12). (205)

The proof of subadditivity relies on Klein’s inequality while that of (204)uses strong subadditivity, see later.

Lemma 3. Let x ∈ R → A(x) be a continuously differentiable functiontaking values in the Hermitian matrices and let f : R → R be continuouslydifferentiable, then

d

dxTr f(A(x)) = Tr

(

f ′(A(x))dA

dx(x))

. (206)

56

Problem 46. Fill out the details of the proof.

Lemma 4. A continuously differentiable function f :]a, b[→ R is concave iff

f(y) ≤ f(x) + (y − x)f(x) for all x, y ∈]a, b[. (207)

Problem 47. Fill out the details of the proof.

Lemma 5 (Klein’s inequality). Let f :]a, b[→ R be continuously dif-ferentiable and concave and let A, B be Hermitian matrices in Md whoseeigenvalues belong to ]a, b[, then

Tr f(B) ≤ Tr(f(A) + (B −A)f ′(A)

). (208)

Problem 48. Fill out the details of the proof.

Lemma 6 (Positivity of relative entropy). Let ρ and σ be density ma-trices and assume that σ is strictly positive, then

S(ρ|σ) := Tr ρ(log ρ− log σ) ≥ 0. (209)

Relative entropy is an important quantity. In statistics it is related to hy-pothesis testing. In statistical mechanics it appears in the context of thevariational principle for the free energy. Suppose that σ is a canonicalequilibrium state: σ = exp(−βH) where the appropriate normalisation con-stant, the log of the partition function, has been absorbed in the Hamiltonian,then

S(ρ|σ) = β〈H〉ρ − S(ρ) (210)

is the expected internal energy of the system in the state ρ minus the entropyof ρ. Thermodynamic equilibrium is attained when this quantity reaches itsminimal value.

Problem 49. Fill out the details of the proof.

Subadditivity now follows from

0 ≤ S(ρ12 | ρ1 ⊗ ρ2) = S(ρ1) + S(ρ2) − S(ρ12). (211)

A really profound result, the proof of which goes beyond the scope of theselectures, is quantum strong subadditivity

S(ρ123) + S(ρ2) ≤ S(ρ12) + S(ρ23). (212)

57

Let ρ be a density matrix. Consider then an orthonormal basis ej ofeigenvectors of ρ

ρ ej = rjej , and so ρ =∑

j

rj|ej〉〈ej|. (213)

Suppose that there are k eigenvalues of ρ strictly positive. We then introducea vector ϕ ∈ Cd ⊗Ck

ϕ :=∑

j

√rj ej ⊗ fj . (214)

Here fj is an orthonormal basis of Ck. It is then easily checked that ρ isthe partial trace of |ϕ〉〈ϕ| over Ck. The state vector ϕ is called a purificationof ρ. There are obviously many purification because we can freely pick theorthonormal basis fj.

Suppose now that we are given a two party state ρ12. We can always purifythis to a pure state ρ123. Because marginals of a bipartite pure state have thesame non-zero eigenvalues, and hence the same entropy, we have S(ρ23) =S(ρ1). Strong subadditivity then yields

S(ρ2) ≤ S(ρ12) + S(ρ1) (215)

which is exactly (204).

Existence of entropy of a shift-invariance state on ⊗NMd can now be proved,more or less as in the classical case on the basis of subadditivity

s = limn→∞

1

nS(ρ(n)). (216)

Using strong subadditivity one can show that for shift-invariant states theentropy is monotonically increasing in the volume and that

0 ≤ S(ρ(n+2)) − S(ρ(n+1)) ≤ S(ρ(n+1)) − S(ρ(n)). (217)

This suffices to prove that also in the quantum case

s = limn→∞

(S(ρ(n+1)) − S(ρ(n))

). (218)

We conclude this short series of lectures with Schumacher’s noiseless com-

pression theorem for sources that emit pure quantum states. After encod-ing and decoding a sequence ϕi1 ⊗ ϕi2 ⊗ · · · ⊗ ϕin of states emitted by the

58

source we should end up with a state that is close to the original state fora subset of states that appears with high probability. In order to comparestates, fidelity can be used. For two vector states ϕ and ψ fidelity is theordinary transition probability:

Fid(ϕ, ψ) := |〈ϕ , ψ〉|2. (219)

A fidelity close to 1 means that the vectors are also close. This notion can beextended to arbitrary mixed states. The idea is to jointly purify both densitymatrices and then to use the transition probability for these purification.More explicitly: let ϕ1 and ϕ2 be joint purifications of ρ1 and ρ2:

ρj = TrC

k |ϕj〉〈ϕj|, j = 1, 2 (220)

thenFid(ρ1, ρ2) := max |〈ϕ1 , ϕ2〉|2 (221)

where the maximum is taken over all joint purifications of ρ1 and ρ2. Uhlmannobtained an explicit expression for the mixed state fidelity

Fid(ρ1, ρ2) =(

Tr√√

ρ1ρ2√ρ1

)2

. (222)

Problem 50. Estimate ‖ϕ− ψ‖ in terms of the fidelity.

Problem 51. Compute the right hand side of (222) for the case were ρ1 ispure.

Problem 52. Show for the case that ρ1 is pure that the maximal transitionprobability between all joint purifications of ρ1 and ρ2 indeed agrees with theexpression obtained in the previous problem.

We now formulate a simple case of Schumacher’s theorem

Theorem 8. Suppose that a memoryless sources emits pure states ϕj ∈ Cd

with probability pj, let ρ =∑

j pj |ϕj〉〈ϕj| and s = S(ρ). For any givenǫ, δ > 0 and n sufficiently large there exist quantum operations (coding maps)Cn : L(⊗n

C

d) → L(Kn) and decoding maps Dn : L(Kn) → L(⊗nC

d) suchthat

∑

j1 j2 ...jn

pj1pj2 · · · pjn Fid(|ϕj1〉〈ϕj1| ⊗ · · · ⊗ |ϕjn〉〈ϕjn| ,

Dn Cn(|ϕj1〉〈ϕj1| ⊗ · · · ⊗ |ϕjn〉〈ϕjn|))≥ 1 − ǫ.

(223)

59

Here Kn is a Hilbert space of dimension not exceeding exp(n(s+δ)) and L(V)denotes the linear transformations of the complex vector space V. Moreover,there are no reliable coding and decoding maps in the sense of (223) if for alln the dimension of the code space Kn is not exceeding exp(n(s− δ)).

60