Area Laws for Entanglement - PowerPoint Presentation

Slide1

Area Laws for Entanglement

Fernando

G.S.L.

Brand

ão

University College London

j

oint work with

Michal

Horodecki

arXiv:1206.2947

arXiv:1406.XXXX

Stanford University,

April 2014Slide2

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide3

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Ultimate limits to information transmission

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide4

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Entanglement as a resource

Ultimate limits to information transmission

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide5

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Entanglement as a resource

Ultimate limits to information transmission

Quantum computers

a

re digital

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide6

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Entanglement as a resource

Ultimate limits to information transmission

Quantum computers

a

re digital

Quantum algorithms with exponential speed-up

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide7

Quantum Information Theory

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

Entanglement as a resource

Ultimate limits to information transmission

Quantum computers

a

re digital

Quantum algorithms with exponential speed-upUltimate limits for efficient computation

Goal

: Lay down the theory for future quantum-based technology

(quantum computers, quantum cryptography, …)Slide8

QIT Connections

Quant. Comm.

Entanglement theory

Q. error

correc. + FTQuantum comp.Quantum complex. theo.

QITSlide9

Quant. Comm.

Entanglement theory

Q. error

c

orrec. + FTQuantum comp.Quantum complex. theo.

Strongly corr. systems

Topological order

Spin glasses

Condensed Matter

QIT

QIT Connections Slide10

Quant. Comm.

Entanglement theory

Q. error

c

orrec. + FTQuantum comp.

Quantum complex.

theo.

Strongly corr. systems

Topological order

Spin glasses

Condensed MatterQITStatMechQIT Connections EquilibrationThermodynamics@nano scaleSlide11

Quant. Comm.

Entanglement theory

Q. error

c

orrec

. + FT

Quantum comp.

Quantum complex.

theo.

Strongly corr. systems

Topological orderSpin glasses EquilibrationThermodynamics@nano scaleCondensed MatterQITStatMechHEP/GRTopolog. q. field theo.Black hole physicsHolographyQIT Connections Slide12

Quant. Comm.

Entanglement theory

Q. error

c

orrec

. + FT

Quantum comp.

Quantum complex.

theo.

Ion traps, linear optics, optical lattices,

cQED, superconduc. devices,many more Topolog. q. field theo.Black hole physicsHolographyStrongly corr. systemsTopological orderSpin glasses HEP/GRCondensed MatterQITStatMechExper. Phys.

QIT Connections

Equilibration

Thermodynamics

@

nano

scaleSlide13

This Talk

Goal:

give

an

example of these emerging connections: Connect behavior of correlation functions to entanglement Slide14

Entanglement

Entanglement

in quantum

i

nformation science is a resource (teleportation, quantum key distribution, metrology, …)Ex. EPR pairHow to quantify it?Bipartite Pure State Entanglement Given , its entropy of entanglement is

Reduced State:

Entropy: (

Renyi

Entropies: ) Slide15

Entanglement in Many-Body Systems

A quantum state

ψ

of

n qubits is a vector in ≅

For almost every state

ψ

,

S(X)

ψ

≈ |X| (for any X with |X| < n/2)|X| := ♯qubits in X Almost maximal entanglementExceptional SetSlide16

Area Law

X

X

c

ψ

Def

:

ψ

satisfies an area law if there is c > 0

s.t.

for every region X,

S(X) ≤ c Area(X)

Area(X)

Entanglement is HolographicSlide17

Area Law

X

X

c

ψ

Def

:

ψ

satisfies an area law if there is c > 0

s.t.

for every region X,

S(X) ≤ c Area(X)

Area(X)

When do we expect an area law?

Low-energy states of many-body local

models:

Entanglement is Holographic

H

ijSlide18

Area Law

X

X

c

ψ

Def

:

ψ

satisfies an area law if there is c > 0

s.t.

for every region X,

S(X) ≤ c Area(X)

Area(X)

(

Bombeli

et al

’

86)

massless free scalar field (connection to

B

ekenstein

-Hawking entropy

)

(Vidal

et al

‘03;

Plenio

et al

’

05, …)

XY model, quasi-free

bosonic

and

fermionic

models, …

(

Holzhey

et al ‘

94; Calabrese,

Cardy

‘04)

critical systems described by CFT (log correction)

(

Aharonov

et al

‘09;

Irani

‘10)

1D model with

volume

scaling of entanglement entropy!

When do we expect an area law?

Low-energy states of many-body local

models:

Entanglement is Holographic

H

ij

…Slide19

Why is Area Law Interesting?

Connection to

Holography

.

Interesting to study entanglement in physical states with an eye on quantum information processing.Area law appears to be connected to our ability to write-down simple

A

nsatzes

for the quantum state.

(e.g. tensor-network states: PEPS, MERA)

This is known rigorously in 1D:Slide20

Matrix Product States

(

Fannes

,

Nachtergaele

, Werner

’

92; Affleck, Kennedy,

Lieb

, Tasaki ‘87)

D : bond dimensionOnly nD2 parameters. Local expectation values computed in nD3 timeVariational class of states for powerful DMRG (White ‘)Generalization of product states (MPS with D=1)Slide21

MPS Area Law

X

Y

(Vidal

’

03;

Verstraete

,

Cirac

‘05) If ψ satisfies S(ρX) ≤ log(D) for all X, then it has a MPS description of bond dim. D

(obs: must use Renyi entropies)

For MPS, S(

ρ

X

) ≤ log(D) Slide22

Correlation Length

X

Z

Correlation Function

:

ψ

has correlation length

ξ

if for every regions X, Z:

cor

(X : Z)

ψ

≤ 2

-

dist

(X, Z) /

ξ

Correlation Length

:

ψSlide23

When there is a finite

correlation length?

(

Hastings ‘04)

In any dim at zero temperature for gapped models (for groundstates; ξ

= O(1/gap)

)

(Hastings

’

11;

Hamza et al ’12; …) In any dim for models with mobility gap (many-body localization)(Araki ‘69) In 1D at any finite temperature T (for ρ = e-H/T/Z; ξ = O(1/T))(Kliesch et al ‘13) In any dim at large enough T (Kastoryano et al ‘12) Steady-state of fast converging dissipative processes (e.g. gapped Liovillians) Slide24

Area Law from Correlation Length?

X

X

c

ψSlide25

Area Law from Correlation Length?

X

X

c

That’s incorrect!

Ex.

For almost every

n

qubit

state:

and for all

i

in

X

c

,

Entanglement can be

scrambled

, non-locally encoded

(e.g. QECC, Topological Order)

ψSlide26

Area Law from Correlation Length?

X

Y

Z

Suppose . Slide27

Area Law from Correlation Length?

X

Y

Z

Suppose .

Then

X is only entangled with Y

lSlide28

Area Law from Correlation Length?

X

Y

Z

Suppose

Then

X is only entangled with Y

What if merely ?

lSlide29

Area Law from Correlation Length?

X

Y

Z

Suppose

Then

lSlide30

Area Law from Correlation Length?

X

Y

Z

Suppose

Then

True (

Uhlmann’s

thm

)

. But

we need 1-norm (trace-distance):

l

In contrastSlide31

Data Hiding States

Well distinguishable globally, but poorly distinguishable locally

Ex. 1

Antisymmetric

Werner state ωAB = (I – F)/(d

2

-d

)

(

DiVincenzo

, Hayden, Leung, Terhal ’02) Slide32

Data Hiding States

Well distinguishable globally, but poorly distinguishable locally

Ex. 1

Antisymmetric

Werner state ωAB = (I – F)/(d

2

-d

)

Ex. 2

Random state

with |X|=|Z| and |Y|=l(DiVincenzo, Hayden, Leung, Terhal ’02) Slide33

Data Hiding States

Well distinguishable globally, but poorly distinguishable locally

Ex. 1

Antisymmetric

Werner state ωAB = (I – F)/(d

2

-d

)

Ex. 2

Random state

with |X|=|Z| and |Y|=lEx. 3 Quantum Expanders States: States with big entropy but s.t. for every regions X, Z far away from edge(DiVincenzo, Hayden, Leung, Terhal ’02) (Hastings ’07)Slide34

Area Law in 1D?

Gapped Ham

Finite Correlation

Length

Area Law

MPS

Representation

???

(Hastings

’

04)(Vidal ’03)Slide35

Area Law in 1D

Gapped Ham

Finite Correlation

Length

Area Law

MPS

Representation

???

(Hastings

’

07)(Hastings ’04)thm (Hastings ‘07) For H with spectral gap Δ and unique groundstate Ψ0, for every region X, S(X)ψ ≤ exp(c / Δ)

X

(Arad,

Kitaev

, Landau,

V

azirani

‘12)

S

(X)ψ ≤

c / Δ

(Vidal ’03)Slide36

Area Law in 1D

Gapped Ham

Finite Correlation

Length

Area Law

MPS

Representation

???

(Hastings

’

07)(Hastings ’04)(Rev. Mod. Phys. 82, 277 (2010)) “Interestingly, states that are defined by quantum expanders can have exponentially decaying correlations and still have large entanglement, as has been proven in (…)”(Vidal ’03)Slide37

Correlation Length

vs

Entanglement

t

hm 1 (B., Horodecki

‘12)

Let be a quantum state in

1D

with correlation length

ξ

. Then for every X, X

The statement is only about quantum states, no Hamiltonian involved.Applies to

degenerate

groundstates

,

and

gapless models with finite correlation length (e.g. systems with mobility gap; many-body localization) Slide38

Summing Up

Area law

always holds in

1D

whenever there is a finite correlation length: Groundstates (unique or degenerate) of gapped modelsEigenstates of models with mobility gap (many-body localization)

Thermal states at any non-zero temperature

Steady-state of gapped dissipative dynamics

Implies that in all such cases the state has an

efficient classical

parametrization

as a MPS (Useful for numerics – e.g. DMRG. Limitations for quantum information processing e.g. no-go for adiabatic quantum computing in 1D) Slide39

Proof Idea

X

We want to bound the entropy of X

using the fact the correlation length of the state is finite.

Need to

relate entropy to correlations

. Slide40

Random S

tates

H

ave Big Correl.

: Drawn from

Haar

measure

X

Z

Y

Let size

(XY)

< size

(Z)

.

W.h.p

. ,

X

is

decoupled from Y. Extensive entropy, but also large correlations: Maximally entangled state between XZ1. Cor(X:Z) ≥

Cor(X:Z1) = 1/4 >> 2-Ω(n)

: long-range correlations

Slide41

Entanglement Distillation

C

onsists of extracting EPR pairs from bipartite entangled states by

L

ocal Operations and Classical Communication (LOCC)Central task in quantum information processing for distributing entanglement over large distances (e.g. entanglement repeater)

(Pan

et al

’

03)

LOCCSlide42

Entanglement

Distillation Protocol

We apply

entanglement distillation

to show large entropy implies large correlationsEntanglement distillation: Given Alice can distill

-S(A|B) = S(B) – S(AB) EPR pairs with Bob by making

a

measurement with N≈ 2

I(A:E)

elements, with

I(A:E) := S(A) + S(E) – S(AE), and communicating the outcome to Bob. (Devetak, Winter ‘04) ABESlide43

l

X

Y

Z

B

E

A

Distillation BoundSlide44

Distillation Bound

l

X

Y

Z

S(X) – S(XZ) > 0

(EPR pair distillation rate)

Prob. of getting one of the 2

I(X:Y)

outcomes

B

E

ASlide45

Area Law from “Subvolume

Law”

l

X

Y

ZSlide46

Area Law from “Subvolume

Law”

l

X

Y

ZSlide47

Area Law from “Subvolume

Law”

l

X

Y

Z

Suppose

S(Y) < l/(4ξ)

(“

subvolume

law” assumption)Slide48

Area Law from “Subvolume

Law”

l

X

Y

Z

Suppose

S(Y) < l/(4ξ)

(“

subvolume

law” assumption)

Since

I(X:Y) < 2S(Y)

<

l

/

(2ξ)

, a correlation length

ξ

implies

Cor

(X:Z) < 2

-l/

ξ

< 2

-I(X:Y)

Thus: S(X) < S(Y) Slide49

Actual Proof

We apply the bound from entanglement distillation to prove

finite correlation length -> Area Law

in

3 steps:c. Get area law from finite correlation length under assumption there is a region with “

subvolume

law”

b

.

Get region with “

subvolume law” from finite corr. length and assumption there is a region of “small mutual information”a. Show there is always a region of “small mutual info”Each step uses the assumption of finite correlation length. Obs: Must use single-shotinfo theory (Renner et al) Slide50

Area Law in Higher Dim?

Wide open…

Known proofs in 1D (for groundstates gapped models):1. Hastings ‘07. Analytical (Lieb

-Robinson bounds,

F

ourier analysis,…)

2.

Arad,

Kitaev, Landau, Vazirani ‘13. Combinatorial (Chebyshev polynomial, …)3. B., Horodecki ‘12 (this talk). Information-Theoretical All fail in higher dimensions….Slide51

Area Law in Higher Dim?

New Approach:

“Conditional

Correlation length”:

X

Z

ψ

Measurement on site

a

k

: post-measured state after measurement on sites (a

1

,…,

a

k

) with outcomes (i

1

, …,

i

k

)

in {0, 1}

k

lSlide52

Area Law from Finite Conditional

Correlation length

t

hm

(B. ‘14) In any dim, if ψ

has conditional correlation length

ξ

, then

S(X)

ψ ≤ 4ξ Area(X) Which states have a finite conditional correlation length?Conjecture1: Any groundstate of gapped local Hamiltonian..Conjecture 2: Any state with a finite correlation length.Obs: Can prove it for 1D models (finite CL -> area law -> MPS -> finite CCL)

X

X

c

ψSlide53

Area Law from Finite Conditional

Correlation length

t

hm

(B. ‘14) In any dim, if ψ

has conditional correlation length

ξ

, then

S(X)

ψ ≤ 4ξ Area(X) Proof by quantum information theory:

X

X

c

ψ

c

onditional corr. lengthSlide54

Application to Systems with

Robust

Ga

p

thm (B. ‘14) In any dim, if

ψ

has conditional correlation length

ξ

, then

S(X)ψ ≤ 4ξ Area(X) (Verstraete ‘14) Groundstates of Hamiltonians with local topological order have finite conditional correlation length.LTQO : Closely related to “robust gap”, i.e. is gapped for ε small enough and all Vk.cor Every groundstate of a system with local topological order fulfills area law Slide55

thm (B. ‘14) In any dim, if ψ

has conditional correlation length

ξ

, then S(X)ψ ≤ 4ξ Area(X)

(

Verstraete

‘14)

Groundstates

of Hamiltonians

with local topological order have finite conditional correlation length.LTQO : Closely related to “robust gap”, i.e. is gapped for ε small enough and all Vk.cor Every groundstate of a system with local topological order fulfills area law Improves on (Michalakis, Pytel ‘11) who proved S(X) ≤ Area(X)log(vol(X)). Obs: Strict area law is important, as it allows us to define the concept of topological entanglement entropy (Kitaev, Preskill ’05, Levin, Wen ‘05) Application to Systems with Robust GapSlide56

Summary

Finite correlation length gives an area law for entanglement in 1D. We don’t know what happens in higher dimensions.

More generally, thinking

about entanglement from the perspective of quantum information theory is

useful. Growing body of connections between concepts/techniques in quantum information science and other areas of physics.

Thanks!