Fernando G.S.L. Brand ão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info

Information-Theoretic Techniques in Many-Body Physics

Day 1

Fernando G.S.L. BrandãoUCL

Based on joint work with A. Harrow and M. Horodecki

New Mathematical Directions for Quantum Info

Quantum Many-Body Systems

iHdC

n

Quantum Hamiltonian

Interested in computing properties such as minimum energy, correlations functions at zero and finite temperature, dynamical properties, …

Constraint Satisfaction Problems vs Local Hamiltonians

k-arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

Constraint Satisfaction Problems vs Local Hamiltonians

k-arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

k-local Hamiltonian H:

n qudits in

Constraints:

qUnsat := E0 : min eigenvalue

H1

qudit

Classical vs Quantum Optimal Assignments

Finding optimal assignment of CSPs is usually hard(NP-hard)

Finding optimal assignment of quantum CSPs (groundstates) seems even harder (QMA-hard; See Daniel Nagaj’s talk)

Main difference: Optimal Assignment can be a highly entangled state (unit vector in )

The Plan

Today: Product-State Approximations to Groundstates

- de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas., …)

Tomorrow: Groundstates in 1D

- area laws and matrix product states - information theory approach (decoupling, state merging, single-shot protocols, …)

Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H):

Small total error:

Small extensive error:

today

Mean-Field……consists in approximating the groundstate by a product state

is a CSP


is a CSP


is a CSP

Successful heuristic in Quantum Chemistry (Hartree-Fock) Condensed matter (e.g. BCS theory)

Intuition: Mean-Field good when Many-particle interactions Low entanglement in state

It’s a mapping from quantum Hamiltonians to CSPs

Hamiltonian on the Complete Graph

Consider a Hamiltonian on the complete graph G of size n

Hij

The Hamiltonian is permutation symmetric:

with

Quantum de Finetti Theorems

(Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a permutation-symmetric state. Then

(Stormer ’69, Hudson, Moody ’76) Infinite Quantum de Finetti Theorem(Raggio, Werner ’89) Connection of Infinite Quantum de Finetti with Mean-Field (Caves, Fuchs, Sachs ’01) Proof infinite de Finetti using info-complete measurements(Koenig, Renner ’05) Finite Quantum de Finneti Theorem

(remember Graeme Mitchison’s talk)


Product-States Approximation and de Finetti Theorem


By de Finetti:



By de Finetti:

So

Product-states achieve error 2d2/n for mean-energy


The Role of Permutation Symmetry

To apply quantum de Finetti we need a permutation-invariant Hamiltonian.

Can we relax this assumption?

Can we show product states do a good job for models not on the complete graph?

Product-State Approximation without Symmetry

(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.

Let {Xi} be a partition of the sites with each Xi having m sites.

Ei : expectation over Xi

Deg : degree of GS(Xi) : entropy of groundstate in Xi

X1

X2size m

Product-State Approximation without Symmetry

(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.

Let {Xi} be a partition of the sites with each Xi having m sites. Then there are states ψi in Xi s.t.

Ei : expectation over Xi

Deg : degree of GS(Xi) : entropy of groundstate in Xi

X1

X2size m

Approximation in terms of degree

Implications to the quantum PCP problem (whether to compute is QMA-hard ):

Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP”


Implications to the quantum PCP problem (whether to compute is QMA-hard ):

Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP”

Bound: ΦG < ½ - Ω(1/deg) implies product states work well on highly expanding graphs (ΦG -> ½) Obs: Restricted to 2-local models (Aharonov, Lior ‘13) k-local, commuting models


1-D

2-D

3-D

∞-D…shows mean field becomes exact in high dim

See (Cirac, Kraus, Lewenstein) for rotationally invariant systems

Approximation in terms of average entanglement

Product-states do a good job if entanglement of groundstate satisfies a subvolume law:

X1

X3X2

m < O(log(n))


If , Pinsker’s inequality shows product states give error


If , Pinsker’s inequality shows product states give error

In constrast, if merely , the theorem shows product states give error

When does it fail?

Expander graph G(V, E) with expansion ΦG

E.g.

Intuition: Monogamy of EntanglementQuantum correlations are non-shareable(see Aram Harrow’s and Thomas Vidick’s talks)

Cannot be highly entangled with too many neighbors

S(Xi) quantifies how much entangled Xi can be with the rest

Intuition: Monogamy of EntanglementQuantum correlations are non-shareable

Cannot be highly entangled with too many neighbors

S(Xi) quantifies how much entangled Xi can be with the rest

(see Aram Harrow’s and Thomas Vidick’s talks)

Proof uses information-theoretic techniques to make this intuition precise

Inspired by classical information-theoretic ideas for bounding convergence of Sum-Of-Squares hierarchy for CSPs (Tan, Raghavendra ’10; Barak, Raghavendra, Steurer ‘10)

Mutual Information1. Mutual Information

2. Pinsker’s inequality

3. Conditional MI

4. Chain Rule

5. Upper bound

4+5 for some t ≤ k

Quantum Mutual Information1. Mutual Information

2. Pinsker’s inequality

3. Conditional MI

4. Chain Rule

5. Upper bound

4+5 for some t ≤ k

But… …conditioning on quantum is problematic

For X, Y, Z random variables

No similar interpretation is known for I(X:Y|Z) with quantum Z

Conditioning DecouplesIdea that almost works. Suppose we have a distribution p(z1,…,zn)

1. Choose i, j1, …, jk at random from {1, …, n}.Then there exists t<k such that

Define

So

i

j1

j2

jk

Conditioning Decouples2. Conditioning on subsystems j1, …, jt causes, on average, error <k/n and leaves a distribution q for which

, and so

By Pinsker:

j1

jt

j2

Choosing k = εn

Informationally Complete Measurements

There exists a POVM M(ρ) = Σk tr(Mkρ) |k><k| s.t. for all k and ρ1…k, σ1…k in D((Cd)k)

(Lacien, Winter ‘12, Montanaro ‘12)

Proof Overview1. Measure εn qudits with M and condition on outcomes.

Incur error ε.

2. Most pairs of other qudits would have mutual information ≤ log(d) / ε deg(G) if measured.

3. Thus their state is within distance d3(log(d) / ε deg(G))1/2 of product.

4. Witness is a global product state. Total error isε + d6(log(d) / ε deg(G))1/2.Choose ε to balance these terms.

5. General case follows by coarse graining sites (can replace log(d) by Ei H(Xi))

Proof OverviewLet

…

previous argument

q: probability distribution obtained conditioning on zj1, …, zjt

Proof Overview

(σ: state obtained by measuring M on j1, …, jt and conditioning on the outcome). Choosing k = εn

info complete measurement

Other Applications 1: New Classical Algorithms for Q. Hamiltonians

Following same approach one obtains polynomial time algorithms for approximating the groundstate energy of

1. Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07)2. Dense Hamiltonians, improving on (Gharibian, Kempe ‘10)3. Hamiltonians on graphs with low threshold rank, building on

(Barak, Raghavendra, Steurer ‘10)

In all cases we prove that a product state does a good job and use efficient algorithms for CSPs.

Other Applications 2: New de Finetti Theorems

- Classical de Finetti without symmetry: For p(x1,…,xn)

with

- Q. version using info-complete measurement

- Q. version using locality constrained norms (see Aram’s talk)

- Version replacing uniform randomness by Santa-Vazirani source (Ramanathan et al ‘13)

Thank you!

Information-Theoretic Techniques in Many-Body Physics

Day 2

Fernando G.S.L. BrandãoUCL

Based on joint work with A. Harrow and M. Horodecki

New Mathematical Directions for Quantum Info

The Plan

Yesterday: Product-State Approximations to Groundstates

- de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas.)

Today: Groundstates in 1D

- matrix product states - area law and exponential decay of correlations - information theory approach (decoupling, state merging, single-shot protocols)

(see Nilanjana Datta’s talk)

Quantum Many-Body Systems

iHdC

n

Quantum Hamiltonian

Interested in computing properties such as minimum energy, correlations functions, etc…

Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H):

Small total error:

Small extensive error:

today

Matrix Product States(Fannes, Nachtergaele, Werner ‘92)

D : bond dimension

• Only nD2 parameters. • Local expectation values computed in poly(D, n) time• Variational class of states for powerful DMRG • Generalization of product states (MPS with D=1)

Area Law in 1DLet be a n-qubit quantum state

Entanglement Entropy:

Area Law: For all partitions of the chain (X, Y)

X Y

(Bekenstein ‘73, ….…, Eisert, Cramer, Plenio ’10)

MPS Area Law

X Y

For MPS,

MPS Area Law

X Y

If is s.t.then it has a MPS description of bound dim. D(Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06)

For MPS,

MPS Area Law

X Y

If is s.t.then it has a MPS description of bound dim. D(Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06)

(Approx. version) If is s.t.then it can be approximated by a MPS of bound dim. D up to error ε

For MPS,

Def:

Exponential Decay of CorrelationsLet be a n-qubit quantum state

Correlation Function:

Exponential Decay of Correlations: There is ξ > 0 s.t. for all cuts X, Y, Z with |Y| = l

X ZY

l

MPS EDC≈Let

Define (w.l.o.g. )

and let λj be the second largest eingenvalue of and λ := max |λj|

If λ is independent of n we say is a gMPS

has (1/log(1/|λ|))-EDC

How good are MPS?

Negative results:

(Aharonov, Gottesman, Irani, Kempe ‘07)1D Hamiltonians can be QMA-hard(see Daniel Nagaj’s talk)

(Irani ’09; Gottesman, Hastings ‘09)1D Hamiltonians with volume scaling of entanglement

(Irani, Gottesman ‘09)1D Hamiltonians with translational-invariance still hard

… is there hope?

1D gapped models

iHdC

n

Given

Let

Then Hn is gapped if

(remember Toby Cubbit’s talk)

1D gapped models

Area Law gMPS

EDC

(Hastings ’07Arad, Kitaev,

Landau, Vazirani ’12)

GroundstateGapped model

(Hastings ’05)

(Landau, Vidick, Vazirani ‘12)

gMPSEDC

Area Law

E.g. there are gapless Hamiltonians with a mobility gap/dynamical localization, which imply EDC in the groundstate(Hastings ’10; Hamza, Sims, Stolz ‘11)

Do we need the gap?

Do we need the gap?

gMPSEDC

Area Law

(B., Horodecki ’12)

Do we need the gap?

gMPSEDC

Area Law

(B., Horodecki ’12)

Area Law vs. Decay of CorrelationsExponential Decay of Correlations suggests Area Law

Area Law vs. Decay of CorrelationsExponential Decay of Correlations suggests Area Law:

X ZY

l = O(ξ)

ξ-EDC implies


X ZY

ξ-EDC implies which implies

(by Uhlmann’s theorem)

X is only entangled with Y!

l = O(ξ)


X ZY



X is only entangled with Y! Alas, the argument is wrong…

Uhlmann’s thm require 1-norm:

l = O(ξ)


X ZY



X is only entangled with Y! Alas, the argument is wrong…

Uhlmann’s thm require 1-norm:

l = O(ξ)

Data Hiding StatesWell distinguishable globally, but poorly distinguishable locally

Ex. 1 Antisymmetric Werner state ωAB = (I – F)/(d2-d)

Ex. 2 Random state with |X|=|Z| and |Y|=l

(DiVincenzo, Leung, Terhal ’02)

X ZY

What data hiding implies?

1. Intuitive explanation is flawed



2. No-Go for area law from exponential decaying correlations?







3. We fixed a partition; EDC gives us more…



2. No-Go for area law from exponential decaying correlations? 3. We fixed a partition; EDC gives us more…

4. It’s an interesting quantum information problem: How strong is data hiding in quantum states?

Exponential Decaying Correlations Imply Area Law

Thm 1 (B., Horodecki ‘12) If has ξ-EDC then for every X and m,

X Xc



Obs1: Implies

Obs2: Only valid in 1D…

Obs3: Reproduces bound of Hastings for GS 1D gapped Ham., using EDC in such states

X Xc



Obs1: Implies

Obs2: Only valid in 1D…

Obs3: Reproduces bound of Hastings for GS 1D gapped Ham., using EDC in such states

X Xc



Obs4: Implies stronger form of EDC: For l > exp(O(ξlogξ)) and split ABC with |B|=l

X Xc

EDC gMPS

(Cor. Thm 1) If has ξ-EDC then for every ε>0 there is gMPS with poly(n, 1/ε) bound dim. s.t.

X Xc

Random States Have EDC?

: Drawn from Haar measure

X ZY

l

w.h.p, if size(X) ≈ size(Z):

and

Small correlations in a fixed partition do not imply area law.

Random States Have EDC?

: Drawn from Haar measure

X ZY

l

w.h.p, if size(X) ≈ size(Z):

and

Small correlations in a fixed partition do not imply area law.

But we can move the partition freely...

Random States Have Big Correl.: Drawn from Haar measure

X ZY

l

Let size(XY) < size(Z). W.h.p. ,

X is decoupled from Y.


X ZY

l



Extensive entropy, but also large correlations:


X ZY

l




Maximally entangled state between XZ1.

(Uhlmann’s theorem)


X ZY

l




Maximally entangled state between XZ1. Cor(X:Z) ≥ Cor(X:Z1) = Ω(1) >> 2-Ω(n) : long-range correlations!



X ZY

l




Maximally entangled state between XZ1. Cor(X:Z) ≥ Cor(X:Z1) = Ω(1) >> 2-Ω(n) : long-range correlations!


This reasoning hints at the idea of the general proof:

We’ll show large entropy leads to large correlations by choosing a random measurement that decouples A and B

Entanglement Distillation by Decoupling

We apply the state merging protocol to show large entropy implies large correlations

State merging protocol: Given Alice can distill -S(A|B) = S(B) – S(AB) EPR pairs with Bob by making a random measurement with N≈ 2I(A:E) elements, with I(A:E) := S(A) + S(E) – S(AE), and communicating the outcome to Bob. (Horodecki, Oppenheim, Winter ‘05)

A B E

We apply the state merging protocol to show large entropy implies large correlations

State merging protocol: Given Alice can distill -S(A|B) = S(B) – S(AB) EPR pairs with Bob by making a random measurement with N≈ 2I(A:E) elements, with I(A:E) := S(A) + S(E) – S(AE), and communicating the outcome to Bob. (Horodecki, Oppenheim, Winter ‘05)

A B EDisclaimer: only works for

Let’s cheat for a while and pretend it works for a single copy, and later deal with this issue

Entanglement Distillation by Decoupling

State merging protocol works by applying a random measurement {Pk} to A in order to decouple it from E:

log( # of Pk’s )

# EPR pairs:

A B E

Optimal Decoupling

l

X Y Z

A E B

Distillation Bound

Distillation Boundl

X Y Z

S(Z) – S(XZ) > 0 (EPR pair distillation by random measurement)

Prob. of getting one of the 2I(X:Y) outcomes in random

measurement

A E B

Proof Strategy

We apply previous result to prove EDC -> Area Law in 3 steps:

1. Get area law from EDC under assumption there is a region with “subvolume” law

2. Get region with “subvolume” law from assumption there is a region of “small mutual information”

3. Show there is always a region of “small mutual info”

1. Area Law from Subvolume Lawl

X Y Z


X Y Z


X Y Z

Suppose S(Y) < l/(4ξ) (“subvolume law” assumption)


X Y Z

Suppose S(Y) < l/(4ξ) (“subvolume law” assumption) Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2-l/ξ < 2-I(X:Y)


X Y Z

Suppose S(Y) < l/(4ξ) (“subvolume law” assumption) Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2-l/ξ < 2-I(X:Y)

Thus: S(Z) < S(Y)

2. Subvolume Law from Small Mutual Info

YL YC YR

l l/2l/2

YL YC YR

R := all except YLYCYR :

RR


l l/2l/2

Suppose I(YC: YLYR) < l/(4ξ) (small mutual information assump.)

YL YC YR


RR


l l/2l/2


ξ-EDC implies Cor(YC : R) < exp(-l/(2ξ)) < exp(-I(YC:YLYR))

YL YC YR


RR




From distillation bound H(YLYCYR) = H(R) < H(YLYR)

l l/2l/2

YL YC YR


RR




From distillation bound H(YLYCYR) = H(R) < H(YLYR)

Finally H(YC) ≤ H(YC) + H(YLYR) – H(YLYCYR) = I(YC:YLYR) ≤ l/(4ξ)

l l/2l/2

Getting Area Law

Z

To prove area law for Z it suffices to find a not-so-far and not-so-large region YLYCYR with small mutual information

We show it with not-so-far = not-so-large = exp(O(ξ))

Getting Area LawYL YC YR

Since I(YC: YLYR) < l/(4ξ) with l = exp(O(ξ))

by part 2, H(YC) < l/(4ξ)

by part 1, H(Z’) < l/(4ξ)

by subadditivity and Araki-Lieb: H(X) < exp(O(ξ))

Z’

Z

< 2O(1/ε) < 2O(1/ε)

3. Getting Small Mutual Info.

X

Lemma (Saturation Mutual Info.) Given a site s, for all ε > 0 there is a region Y2l := YL,l/2YC,lYR,l/2 of size 2l with 1 < l < 2O(1/ε) at a distance < 2O(1/ε) from s s.t.

I(YC,l:YL,l/2YR,l/2) < εl

Proof: Easy adaptation of result used by Hastings in his area law proof for gapped Hamiltonians (based on successive applications of subadditivity)

sYL YC YR

Making it WorkSo far we have cheated, since merging only works for many copies of the state. To make the argument rigorous, we use single-shot information theory (see Nilanjana Datta’s talk)

Single-Shot State Merging (Dupuis, Berta, Wullschleger, Renner ‘10) + New bound on correlations by random measurements

Saturation max- Mutual Info. Proof much more involved; based on - Quantum substate theorem, - Quantum equipartition property, - Min- and Max-Entropies Calculus - EDC Assumption

State Merging

Saturation Mutual Info.

Overview• Condensed Matter (CM) community always knew EDC implies

area law


area law

• Quantum information (QI) community gave a counterexample (hiding states)


area law


• QI community sorted out the trouble they gave themselves (this talk)


area law


• QI community sorted out the trouble they gave themselves (this talk)

• CM community didn’t notice either of these minor perturbations

”EDC implies Area Law” stays true!

Conclusions and Open problems

1. Can we improve the dependency of entropy with correlation length?

2. Can we prove area law for 2D systems? HARD!3. Can we decide if EDC alone is enough for 2D area law?4. See arxiv:1206.2947 for more open questions

• EDC implies Area Law and MPS parametrization in 1D.

• Proof uses state merging protocol and single-shot information theory: Tools from QIT useful to address

problem in quantum many-body physics.

Documents

Fernando G.S.L. Brand ão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info