sissa - isas journal club

Measuring Entanglement and Renyi Entropies

Andrea Coser

SISSA - ISAS

Journal Club

11/02/2013

Outline Entanglement Cardy’s proposal Abanin-Demler’s proposal

Outline

1 Entanglement in Many-Body systemsEntanglement EntropyRenyi EntropiesMeasuring Entanglement Entropy

2 Cardy’s proposalPhys. Rev. Lett. 106, 150404 (2011)

3 Abanin-Demler’s proposalPhys. Rev. Lett. 109, 020504 (2012)The quantum switchExample of a two-dimensional case (n=2)

Measuring Entanglement and Renyi Entropies SISSA-ISAS 2 / 21


Entanglement in Many-Body systems

Entanglement:

bipartition of the Hilbert space: H = HA ⊗HB

non-entangled state

|1〉A ⊗ |0〉B

vs.

←→

entangled state

1√2

“|0〉A ⊗ |1〉B − |1〉A ⊗ |0〉B

”

Schroedinger, 1935:

I would not call that one, rather thecharacteristic trait of quantum mechanics,the one that enforces its entire departure fromclassical lines of thought.



Entanglement Entropy

We need a “measure” of entanglement of subsystem A with the rest B

Consider a system in its ground state:

ρ = |ΨG 〉〈ΨG |

Reduced density matrix of subsystem A:

ρA = TrB ρ

• zero entanglement: product state −→ |ΨG 〉 = |ψ〉A ⊗ |φ〉B

• low entanglement: few non-zero eigenvalues

• high entanglement: many non-zero eigenvalues




Entanglement entropy (EE):

von Neumann entropy of ρA: SA = −TrA ρA log ρA

Schmidt decomposition: |ΨG 〉 =X

k

λk |ψk〉A ⊗ |φk〉B

λ2k : non-zero eigenvalue of ρA

SA = −X

k

λ2k log λ2

k

Universal scaling at 1D critical points:

SA =c

3ln`

a+ c ′1

central charge

Area law:

SA = γ · Area(∂A)

ad−1+subleading terms




Entanglement Entropy −→ important tool in the study ofquantum many-body systems

• quantum computation

• quantum phase transition, topological quantum order

• entanglement spectrum

Concentrating partial entanglement by local operations

Charles H. BennettIBM Research Division, T. J. Watson Center, Yorktown Heights, New York 10598

Herbert J. BernsteinHampshire College and Institute for Science and Interdisciplinary Studies, Amherst, Massachusetts 01002

Sandu PopescuPhysics Department, Tel Aviv University, Tel Aviv, Israel

Benjamin SchumacherPhysics Department, Kenyon College, Gambier, Ohio 43022

~Received 9 August 1995!If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical

partly entangled pure states, one member of each pair being given to each observer, they can, by local actionsof each observer, concentrate this entanglement into a smaller number of maximally entangled pairs of par-ticles, for example, Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The con-centration process asymptotically conserves entropy of entanglement—the von Neumann entropy of the partialdensity matrix seen by either observer—with the yield of singlets approaching, for large n , the base-2 entropyof entanglement of the initial partly entangled pure state. Conversely, any pure or mixed entangled state of twosystems can be produced by two classically communicating separated observers, drawing on a supply ofsinglets as their sole source of entanglement.PACS number s : 03.65.Bz, 42.50.Dv, 89.70. c

Recent results in quantum information theory have shedlight on the channel resources needed for faithful transmis-sion of quantum states, and the extent to which these re-sources can be substituted for one another. The fundamentalunit of quantum information transmission is the quantum bit,or qubit 1 . A qubit is any two-state quantum system, suchas a spin- 12 particle or an arbitrary superposition of two Fockstates. If two orthogonal states of the system are used torepresent the classical Boolean values 0 and 1, then a qubitdiffers from a bit in that it can also exist in arbitrary complexsuperpositions of 0 and 1, and it can be entangled with otherqubits. Schumacher’s quantum data compression theorem1,2 characterizes the number of qubits, sent through thechannel from sender to receiver, that are asymptotically nec-essary and sufficient for faithfully transmitting unknownpure states drawn from an arbitrary known source ensemble.Quantum superdense coding 3 and quantum teleporta-

tion 4 consume a different quantum resource—namely, en-tanglement, in the form of maximally entangled pairs of par-ticles initially shared between sender and receiver—and useit to assist, respectively, in the performance of faithful clas-sical and quantum communication. Following Schumacher’sterminology, we define an ebit as the amount of entangle-ment between a maximally entangled pair of two-state sys-tems, such as two spin- 12 particles in the singlet state, and weinquire how many ebits are needed for various tasks. In 4 ,for example, it is shown that the consumption of one sharedebit, together with the transmission of a two-bit classicalmessage, can be substituted for the transmission of one qubit.An important concept in quantum data transmission is

fidelity, the probability that a channel output would pass atest for being the same as the input conducted by someonewho knows what the input was. If a pure state sent into aquantum channel emerges as the in general mixed staterepresented by density matrix W , the fidelity of transmissionis defined as F W . A quantum channel will be con-sidered faithful if in an appropriate limit the expected fidelityof transmission tends to unity. This means that the outputsare almost always either identical to the inputs or else soclose that the chance of distinguishing them from the inputsby any quantum meausrement tends to zero.Note that qubits are a directed channel resource, sent in a

particular direction from the sender to the receiver; by con-trast, ebits are an undirected resource shared between senderand receiver. For example, if you prepare two particles in asinglet state and give me one of them, the result is the sameas if I had prepared the particles and given you one of them.Ebits are a weaker resource than qubits, in the sense thattransmission of one qubit can, as just described, be used tocreate one ebit of entanglement; but the sharing of an ebit, ormany ebits, does not by itself suffice to transmit an arbitrarystate of a two-state quantum system, or qubit, in either direc-tion. To do that, the ebits must be supplemented by directedclassical bits, as in teleportation.One would naturally like to know whether, in order to be

useful for purposes such as teleportation, entanglement mustbe supplied in the form of maximally entangled pairs. Inparticular, could partly entangled pure states, such as pairs ofparticles in the state

cos A B sin A B 1be used instead, and, if so, how many such pairs would be

PHYSICAL REVIEW A APRIL 1996VOLUME 53, NUMBER 4

531050-2947/96/53 4 /2046 7 /$10.00 2046 © 1996 The American Physical Society








Topological Entanglement Entropy

Alexei Kitaev1,2 and John Preskill1

1Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA2Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA

(Received 13 October 2005; published 24 March 2006)

We formulate a universal characterization of the many-particle quantum entanglement in the ground

state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the

plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state,

by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator

for the degrees of freedom in the interior. The von Neumann entropy of , a measure of the entanglement

of the interior and exterior variables, has the form S Lÿ , where the ellipsis represents

terms that vanish in the limit L ! 1. We show that ÿ is a universal constant characterizing a global

feature of the entanglement in the ground state. Using topological quantum field theory methods, we

derive a formula for in terms of properties of the superselection sectors of the medium.

PRL 96, 110404 (2006)P H Y S I C A L R E V I E W L E T T E R S week ending

24 MARCH 2006








Entanglement Spectrum as a Generalization of Entanglement Entropy: Identificationof Topological Order in Non-Abelian Fractional Quantum Hall Effect States

Hui Li and F. D. M. Haldane

Physics Department, Princeton University, Princeton, New Jersey 08544, USA(Received 2 May 2008; published 3 July 2008)

We study the ‘‘entanglement spectrum’’ (a presentation of the Schmidt decomposition analogous to a

set of ‘‘energy levels’’) of a many-body state, and compare the Moore-Read model wave function for the

5=2 fractional quantum Hall state with a generic 5=2 state obtained by finite-size diagonalization of

the second-Landau-level-projected Coulomb interactions. Their spectra share a common ‘‘gapless’’

structure, related to conformal field theory. In the model state, these are the only levels, while in the

‘‘generic’’ case, they are separated from the rest of the spectrum by a clear ‘‘entanglement gap’’, which

appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement

spectrum can be used as a ‘‘fingerprint’’ to identify topological order.


4 JULY 2008




Entanglement Entropy −→ studied in different fields

• quantum quenches

• AdS/CFT correspondance

Evolution of entanglement entropy inone-dimensional systems

Pasquale Calabrese1 and John Cardy1,2

1 Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road,Oxford OX1 3NP, UK3

2 All Souls College, Oxford, UKE-mail: [email protected]

Received 21 March 2005Accepted 6 April 2005Published 21 April 2005

Online at stacks.iop.org/JSTAT/2005/P04010doi:10.1088/1742-5468/2005/04/P04010

Abstract. We study the unitary time evolution of the entropy of entanglementof a one-dimensional system between the degrees of freedom in an interval oflength ℓ and its complement, starting from a pure state which is not an eigenstateof the Hamiltonian. We use path integral methods of quantum field theory as wellas explicit computations for the transverse Ising spin chain. In both cases, thereis a maximum speed v of propagation of signals. In general the entanglemententropy increases linearly with time t up to t = ℓ/2v, after which it saturatesat a value proportional to ℓ, the coefficient depending on the initial state. Thisbehaviour may be understood as a consequence of causality.

t

2t 2t

l

t

2t > l

2t < l

A

B B

A

BB




Entanglement Entropy −→ studied in different fields

• quantum quenches

• AdS/CFT correspondance

Holographic Derivation of Entanglement Entropy from the anti–de SitterSpace/Conformal Field Theory Correspondence

Shinsei Ryu and Tadashi Takayanagi

Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA(Received 8 March 2006; published 9 May 2006)

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed

from anti–de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement

entropy in d 1 dimensional conformal field theories can be obtained from the area of d dimensional

minimal surfaces in AdSd2, analogous to the Bekenstein-Hawking formula for black hole entropy. We

show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS3.

We also compare the entropy computed in AdS5S5 with that of the free N 4 super Yang-Mills theory.


12 MAY 2006

z

x1

(a) (b) xi>1

l



Renyi Entropies

Renyi Entropies: S(n)A =

1

1− nln Tr ρn

A

• analytical continuation SA = limn→1

S(n)A

• 1 + 1 CFT computation for integer n, replica trick:

(i) density matrix in Eucl. time, path integral formalism

(ii) trace over degrees of freedom in B

(iii) single factor ρA: Riemann sphere with cuts incorrespondence of subsystem A

(iv) n Riemann spheres are glued together to forma complicated Riemann surface



Renyi Entropies


1

1− nln Tr ρn

A


S(n)A



Ψ(φ−) =

Z φ(τ=0−,x)=φ−

τ=−∞Dφ e−S(φ) ρφ−,φ+ = Ψ(φ−)Ψ(φ+)






Renyi Entropies


1

1− nln Tr ρn

A


S(n)A



Ψ(φ−) =

Z φ(τ=0−,x)=φ−τ=−∞

Dφ e−S(φ)ρφ−,φ+

= Ψ(φ−)Ψ(φ+)


ρAφ+,φ− =1

Z1

ZDφ e−S(φ)

Yx∈A

δ(φ(0+, x)−φ+)·δ(φ(0−, x)−φ−)





Renyi Entropies


1

1− nln Tr ρn

A


S(n)A



Ψ(φ−) =

Z φ(τ=0−,x)=φ−τ=−∞

Dφ e−S(φ)ρφ−,φ+

= Ψ(φ−)Ψ(φ+)


ρAφ+,φ−=

1

Z1

ZDφ e−S(φ) Y

x∈A

δ(φ(0+, x)−φ+)·δ(φ(0−, x)−φ−)





Renyi Entropies

Tr ρnA: partition function

on a complicated Riemann surface ZRn

↓move the complicated topology

from the world sheet to the target space

n copies of the original model φj(x , τ)(x,τ)∈C

with “boundary conditions”: φj(x , 0+) = φj+1(x , 0

−), x ∈ A

Twist Fields:

ZRn ∝ 〈Tn(u, 0)Tn(v , 0)〉CTn : j → j + 1 mod n

Tn : j + 1→ j mod n



Measuring Entanglement Entropy

Measuring Entanglement Entropy:

Entanglement entropy is a very important theoretical and numerical tool, but

why is it so diffcult to measure?

1 we are considering systems with a large number of particles

⇒ detecting entanglement is much more challenging

2 measuring the full reduced density matrix is clearly impossible• realistic measurement processes cannot simultaneously access

all degrees of freedom

• the size grows exponentially with the subsystem size

3 what makes EE so important, also makes it difficult to measure• universality

• EE is defined without reference to the observables of the system



Measuring Entanglement Entropy

Quick review of some previous proposals:

(i) Horodecki, Ekert, 2002:

detect entanglement in a state of several coupled qubits measuring theminimal eigenvalue of some positive maps on ρ→ finite dimensional Hilbert space, complexity grows with system size

(ii) Klich, Rafael, Silva, 2006:

relate EE to the measurement entropy of suitably chosen observables→ system-specific

(iii) Klich, Levitov, 2009:

relation between EE and cumulants of transmitted charge statistic in aQuantum Point Contact→ valid for free fermions

(iv) Song, Rachel, Le Hur, 2010:

comparison between EE and particle number fluctuations in simple models



Cardy, Phys. Rev. Lett. 106, 150404 (2011)

Cardy’s proposalPhys. Rev. Lett. 106, 150404 (2011)

Question:

Is entanglement entropy even measurable?

→ many proposals, but:

• often system-specific

• the complexity increases with system size

Cardy:

• focus on 1-dim systems, close to a critical point

• it is easier to measure Renyi entropies S(n)A

• n copies of a (1-dim) systems are “swapped” at time t = 0

• Renyi entropies are related to the fidelity P0 = |˙0′ | 0

¸|2

• P0 is related to P(E): probability of finding the system in low-lying states→ turns out to be universal




Observation:

Permutation operators Πn are twist operators

Πn =Y

i

Tn(ui )

Examples:• Renyi entropy between left and right part of a system 〈0| Tn(0) |0〉• Renyi entropy between a single interval and the rest 〈0| Tn(u)Tn(v) |0〉• . . .

After the quench → energy radiated in non-universal manner

→ P(E) is related to Renyi entropies (E cutoff)ZP(E) e−EτdE = 〈0| e−(H′−H)τ |0〉

= 〈0|“ Y

i

Tn(ui , τ)†”“ Y

i

Tn(ui , 0)”|0〉




Last step: relate P(E) and P0

critical point → for p twists we need to know the 2p-point function of twist fields

(i) single twist:DTn(0, τ)

†Tn(0, 0)E∼ bnτ

−2 xn

scalingdimension of Tn

(ii) A = interval of length `:

• E `−1 → results for single, independent twists

• E `−1 → P(E) = P0

Xx

dn(x) (`E)4x−1 + . . .

CFT scalingdimensions

computablefactor

calculablecorrections

Pros:

• Renyi entropies are in principlemeasurable

• the swap should be robust in gappedsystems if imperfections are ∆

Cons:

• P(E) may be difficult to measure

• difficulty increases with system size




Last step: relate P(E) and P0

critical point → for p twists we need to know the 2p-point function of twist fields

(i) single twist:DTn(0, τ)

†Tn(0, 0)E∼ bnτ

−2 xnscaling

dimension of Tn

(ii) A = interval of length `:

• E `−1 → results for single, independent twists

• E `−1 → P(E) = P0

Xx

dn(x) (`E)4x−1 + . . .

CFT scalingdimensions

computablefactor

calculablecorrections

Pros:

• Renyi entropies are in principlemeasurable

• the swap should be robust in gappedsystems if imperfections are ∆

Cons:

• P(E) may be difficult to measure

• difficulty increases with system size



Abanin, Demler, Phys. Rev. Lett. 109, 020504 (2012)

Abanin-Demler’s proposalPhys. Rev. Lett. 109, 020504 (2012)

Main idea:

same as Cardy’s, but using a quantum switch → 2-level system |↑〉, |↓〉

The quantum switch has two functions:

(1.) swaps the subsystems Ai → Ai+1

(2.) allows for a direct measurement of P0

Consider for example d = 1, n = 2

• initially all Aj systems are coupled to all Bj systems

• |↑〉 → tunneling is blocked between

(A1 ↔ B2

A2 ↔ B1

• |↓〉 → tunneling is blocked between

(A1 ↔ B1

A2 ↔ B2




1. the two states in the quantum switch are decoupled

2. introduce weak tunneling between |↑〉 ↔ |↓〉

Spectrum:

thermodynamic gap or

finite size gap ∆ ∼ ~v/L

→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸

coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”

T ∆ ⇒effective low-energy

HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”T = T

˙0 | 0′

¸Rabi oscillations: PGS − PGS’ = cos

`Ω t

´

Ω = T/~




1. the two states in the quantum switch are decoupled2. introduce weak tunneling between |↑〉 ↔ |↓〉

Spectrum:



→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸ coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”


HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”T = T

˙0 | 0′


`Ω t

´

Ω = T/~





Spectrum:



→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸ coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”


HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”

T = T˙0 | 0′


`Ω t

´

Ω = T/~





Spectrum:



→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸ coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”


HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”T = T

˙0 | 0′

¸

Rabi oscillations: PGS − PGS’ = cos`Ω t

´

Ω = T/~





Spectrum:



→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸ coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”


HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”T = T

˙0 | 0′


`Ω t

´

Ω = T/~





Spectrum:



→ two sectors:

(|GS〉 = |↑〉 ⊗ |0〉

|GS’〉 = |↓〉 ⊗˛0′

¸ coupling: Ht = T“|↑〉〈↓|+ |↓〉〈↑|

”


HamiltonianHeff = T

“|GS〉〈GS’|+ |GS’〉〈GS|

”T = T

˙0 | 0′


`Ω t

´Ω = T/~



The quantum switch

Quantum switch: n dipolar molecule in a 2n-well potential

Requirements:

(i) negligible tunneling between the wells

→ the ground state is doubly degenerate

(ii) strong repulsive interactions dipolar molecule ↔ chain

→ the presence of the molecule blocks tunneling between half chains

(iii) strong repulsive interactions dipolar molecule ↔ dipolar molecule

→ neglect excited states



Example of a two-dimensional case (n=2)

Focus on one-dimensional systems, for two-dimensional systems:

• conceptually the same

• experimentally more challenging

Case n = 2:

(i) tunneling between A1 and B1 or between A1 and B2

(ii) quantum switch: two dipolar molecules trapped in a wave guide

around the boundary of A in the (x , y) plane

|↑〉: both molecules localized between the planes

→ hopping between the two planes is forbidden

|↓〉: superposition of states localized in the two planes

→ hopping between region A and B of the same plane is forbidden


Conclusions References

Conclusions

Cardy - quantum quench:

• showed that in principle Renyi entropies are measurable

• distribution of energy fluctuations following the quench

• measuring energy fluctuations is challenging

Abanin, Demler - quantum switch:

• conceptually very similar to Cardy’s approach

• use a quantum switch and Rabi oscillations→ one needs only to measure the population of the two states of the switch

• in principle may be generalized to d > 1, n > 1,but it may turn to be experimentally impracticable


Conclusions References

References

J. Cardy, Phys. Rev. Lett. 106 150404 (2011)

D. A. Abanin and E. Demler, Phys. Rev. Lett. 109 020504 (2012)

P. Horodecki and A. Ekert, Phys. Rev. Lett. 89 127902 (2002)

I. Klich, G. Refael and A. Silva, Phys. Rev. A. 74 032306 (2006)

I. Klich and L. Levitov, Phys. Rev. Lett. 102 100502 (2009)

H. F. Song, C. Flindt, S. Rachel, I. Klich and K. Le Hur, Phys. Rev. B 83161408(R) (2011)

H. F. Song, S. Rachel and K. Le Hur, Phys. Rev. B 82 012405 (2010)


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