Monogamy of Quantum Entanglement

Fernando G.S.L. Brandão ETH Zürich

Princeton EE Department, March 2013

Simulang quantum is hard

More than 25% of DoE supercomputer power is devoted to simulang quantum physics

Can we get a beer handle on this simulaon problem? Simulang quantum is hard, secrets are hard to conceal

More than 25% of DoE All current cryptography is supercomputer power is devoted based on unproven hardness to simulang quantum physics assumpons

Can we get a beer handle on this Can we have beer security simulaon problem? guarantees for our secrets? Quantum Informaon Science…

… gives a path for solving both problems. Physics CS But it’s a long journey

Engineering QIS is at the crossover of computer science, engineering and physics The Two Holy Grails of QIS

Quantum Computaon: Quantum Cryptography:

Use of engineered quantum Use of engineered quantum systems for performing systems for secret key computaon distribuon

Exponenal speed-ups over Uncondional security based classical compung solely on the correctness of

E.g. Factoring (RSA) quantum mechanics Simulang quantum systems The Two Holy Grails of QIS

Quantum Computaon: Quantum Cryptography:

Use of well controlled State-of-the-art: quantum Use of engineered quantum systems+-5 qubits for performing computer systems for secret key computaons . distribuon Can prove (with high Exponenal speed-ups probability) that 15 = 3 x 5 over Uncondional security based classical compung solely on the correctness of quantum mechanics The Two Holy Grails of QIS

Quantum Computaon: Quantum Cryptography:

Use of well controlled State-of-the-art: quantum Use of well controlled State-of-the-art: quantum systems+-5 qubits for performing computer systems for secret key +-100 km computaons . distribuon Can prove (with high Sll many technological Exponenal speed-ups probability) that 15 = 3 x 5 over Uncondional securitychallenges based classical compung solely on the correctness of quantum mechanics While waing for a quantum computer …how can a theorist help?

Answer 1: figuring out what to do with a quantum computer and the fastest way of building one

(quantum communicaon, quantum algorithms, quantum fault tolerance, quantum simulators, …)

While waing for a quantum computer …how can a theorist help?

Answer 1: figuring out what to do with a quantum computer and the fastest way of building one

(quantum communicaon, quantum algorithms, quantum fault tolerance, quantum simulators, …)

Answer 2: finding applicaons of QIS independent of building a quantum computer While waing for a quantum computer …how can a theorist help?

Answer 1: figuring out what to do with a quantum computer and the fastest way of building one

(quantum communicaon, quantum algorithms, quantum fault tolerance, quantum simulators, …)

Answer 2: finding applicaons of QIS independent of building a quantum computer

This talk Quantum Mechanics I probability theory based on L2 norm instead of L1

States: norm-one vector in Cd: T ψ := (ψ1,...,ψd )

ψ = ψ ψ =1 2

Dynamics: L2 norm preserving linear maps, unitary matrices: UUT = I

Observaon: To any experiment with d outcomes we

associate d PSD matrices {Mk} such that ΣkMk=I

Born’s rule: Pr(k) = ψ M k ψ

Quantum Mechanics II probability theory based on PSD matrices

Mixed States: PSD matrix of unit trace:

ρ ≥ 0,tr(ρ) =1 ρ = pi ψi ψi ∑ i

Dirac notaon reminder: * * ψ := (ψ1 ,...,ψd )

Dynamics: linear operaons mapping density matrices to density matrices (unitaries + adding ancilla + ignoring subsystems)

Born’s rule: Pr(k) = tr(ρM k )

Quantum Entanglement

d l Pure States: C C ψ AB ∈ ⊗

If , it’s separable ψ AB = φ A ⊗ ϕ B

otherwise, it’s entangled.

d l Mixed States: ρAB ∈ D(C ⊗ C )

If , it’s separable ρ = ∑ pi ψi ψi ⊗ φi φi i otherwise, it’s entangled.

A Physical Definion of Entanglement

LOCC: Local quantum Operaons and Classical Communicaon

Separable states can be created by LOCC:

ρ = ∑ pi ψi ψi ⊗ φi φi i Entangled states cannot be created by LOCC:

non-classical correlaons

Quantum Features Superposion principle uncertainty principle

interference non-locality Monogamy of Entanglement Monogamy of Entanglement

00 11 / 2 Maximally Entangled State: φ AB = ( + )

Correlaons of A and B in φ φ AB cannot be shared:

If ρABE is is s.t. ρAB = , then φ φ AB ρABE = φ φ AB ⊗ ρE

Only knowing that the quantum correlaons of A and B are very strong one can infer A and B are not correlated with anything else. Purely quantum mechanical phenomenon! Quantum Key Distribuon… … as an applicaon of entanglement monogamy

(Benne, Brassard 84, Ekert 91)

- Using insecure channel Alice and Bob share entangled state ρAB - Applying LOCC they disll 00 11 / 2 φ AB = ( + )

Quantum Key Distribuon… … as an applicaon of entanglement monogamy

(Benne, Brassard 84, Ekert 91)

- Using insecure channel Alice and Bob share entangled state ρAB - Applying LOCC they disll 00 11 / 2 φ AB = ( + )

Let πABE be the state of Alice, Bob and the Eavesdropper. Since π = , π = . Measuring A and B AB φ φ AB AB φ φ AB ⊗ π E in the computaonal basis give 1 bit of secret key. Outline Entanglement Monogamy

How general is the concept? Can we make it quantave? Are there other applicaons/implicaons?

1. Detecng Entanglement and Sum-Of-Squares Hierarchy

2. Accuracy of Mean-Field Approximaon

3. Other Applicaons Quadrac vs Biquadrac Opmizaon

Problem 1: For M in H(Cd) (d x d matrix) compute

T * max x 1 x Mx = max x 1 Mij xi x j = = ∑ i, j Very Easy!

Problem 2: For M in H(Cd Cl), compute max (x y)T M(x y) max M x x*y y* x = y =1 ⊗ ⊗ = x = y =1 ∑ ij;kl i j k l

ijkl

Next: Best known algorithm (and best hardness result) using ideas from Quantum Informaon Theory

Quadrac vs Biquadrac Opmizaon

Problem 1: For M in H(Cd) (d x d matrix) compute

T * max x 1 x Mx = max x 1 Mij xi x j = = ∑ i, j Very Easy!

Problem 2: For M in H(Cd Cl), compute max (x y)T M(x y) max M x x*y y* x = y =1 ⊗ ⊗ = x = y =1 ∑ ij;kl i j k l

ijkl

Next: Best known algorithm using monogamy of entanglement

The Separability Problem

d l • Given ρAB ∈ D(C ⊗ C ) is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

SEP ε D

The Separability Problem

d l • Given ρAB ∈ D(C ⊗ C ) is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

Frobenius norm T 1/2 X = tr X X 2 ( ) SEP ε D

The Separability Problem

d l • Given ρAB ∈ D(C ⊗ C ) is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

Frobenius norm T 1/2 X = tr X X 2 ( ) ε D SEP LOCC norm

ρ −σ = LOCC max tr(M(ρ −σ )) M ∈LOCC

The Separability Problem

d l • Given ρAB ∈ D(C ⊗ C ) is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

• Dual Problem: Opmizaon over separable states

T hSEP (M ) := max tr(Mσ ) = max (x ⊗ y) M(x ⊗ y) σ ∈SEP x = y =1

The Separability Problem

d l • Given ρAB ∈ D(C ⊗ C ) is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

• Dual Problem: Opmizaon over separable states

T hSEP (M ) := max tr(Mσ ) = max (x ⊗ y) M(x ⊗ y) σ ∈SEP x = y =1 • Relevance: Entanglement is a resource in quantum cryptography, quantum communicaon, etc…

Previous Work

When is ρ entangled? AB

- Decide if ρAB is separable or ε-away from separable

Beauful theory behind it (PPT, entanglement witnesses, etc)

Horribly expensive algorithms dim of A State-of-the-art: 2O(|A|log|B|) me complexity

Same for esmang h (no beer than exausve search!) SEP

Hardness Results

When is ρ entangled? AB

- Decide if ρAB is separable or ε-away from separable

(Gurvits ’02, Gharibian ’08, …) NP-hard with ε=1/poly(|A||B|)

(Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|)) me algorithm with ε=Ω(1) and any ν+μ>0 for unless ETH fails

ETH (Exponenal Time Hypothesis): SAT cannot be solved in 2o(n) me (Impagliazzo&Paru ’99) Quasipolynomial-me Algorithm

(B., Christandl, Yard ’11) There is an algorithm with run me -2 exp(O(ε log|A|log|B|)) for WSEP(||*||, ε) Quasipolynomial-me Algorithm

(B., Christandl, Yard ’11) There is an algorithm with run me -2 exp(O(ε log|A|log|B|)) for WSEP(||*||, ε)

Corollary: Solving WSEP(||*||, ε) is not NP-hard for ε = 1/polylog(|A||B|), unless ETH fails

Contrast with:

(Gurvits ’02, Gharibian ‘08) Solving WSEP(||*||, ε) NP-hard for ε = 1/poly(|A||B|)

Quasipolynomial-me Algorithm

(B., Christandl, Yard ’11) For M in 1-LOCC, can compute hSEP(M) within addive error ε in me exp(O(ε-2log|A|log|B|))

M in 1-LOCC: M = ∑Ak ⊗ Bk, Ak, Bk ≥ 0, Bk ≤ I,∑Ak ≤ I k k Contrast with

(Harrow, Montanaro ’10) No exp(O(log1-ν|A|log1-μ|B|)) algorithm for hSEP(M) with ε=Ω(1), for separable M

M in SEP: M = ∑Ak ⊗ Bk, Ak, Bk ≥ 0, M ≤ I k Algorithm: SoS Hierarchy h (M ) max M x x*y y* SEP = x = y =1 ∑ ij;kl i j k l ijkl Polynomial opmizaon over hypersphere

Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)

- Round k takes me dim(M)O(k)

- Converge to hSEP(M) when k -> ∞

SoS is the strongest SDP hierarchy known for polynomial opmizaon (connecons with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)

We’ll derive SoS hierarchy by a quantum argument

Algorithm: SoS Hierarchy h (M ) max M x x*y y* SEP = x = y =1 ∑ ij;kl i j k l ijkl Polynomial opmizaon over hypersphere

Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)

- Round k SDP has size dim(M)O(k)

- Converge to hSEP(M) when k -> ∞

Algorithm: SoS Hierarchy h (M ) max M x x*y y* SEP = x = y =1 ∑ ij;kl i j k l ijkl Polynomial opmizaon over hypersphere

Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)

- Round k SDP has size dim(M)O(k)

- Converge to hSEP(M) when k -> ∞

SoS is the strongest SDP hierarchy known for polynomial opmizaon (connecons with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)

We’ll derive SoS hierarchy exploring ent. monogamy

Classical Correlaons are Shareable

Given separable state σ AB = ∑ pj ψ j ψ j ⊗ ϕ j ϕ j j

Consider the symmetric extension B1 B2 B ⊗k 3 σ = p ψ ψ ⊗ ϕ ϕ B4 AB1,...,Bk ∑ j j j j j A … j Bk

Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all j in [k], tr\ Bj (ρAB1…Bk) = ρAB 2

Measure Esq ED KD EC EF ER ER EN normalisation y y y y y y y y faithfulness y n ? y y y y n LOCC monotonicity y y y y y y y y asymptotic continuity y ? ? ? y y y n convexity y ? ? ? y y y n strong superadditivity y y y ? n n ? ? subadditivity y ? ? y y y y y monogamy y ? ? n n n n ?

TABLE I: If no citation is given, the property either follows directly from the definition or was derived by 3 the authors of the main reference. Many recent results listed in this table have significance beyond the study of entanglement measures, such as Hastings’ counterexample to the additivity conjecture of the minimum output entropy [76] which implies that entanglement of formation is not strongly superadditive [79].

I. INTRODUCTION 1 00 1 2 2 Monogamy Defines Entanglement2 AB 1 R| | 0000⇥ ⇥AB =(Stormer ’69, Hudson & Moody ⇥ ’76, Raggio & Werner ’89) 0000 ρAB separable iff ρAB is k-extendible for all k 1 ⌃⇧⌅⌥1 ⇧ 0011 ⌃ 2 all quantumAB2 states ⇧ | | ⌃ 2-extendible ⇤ min( A , B ⌅ | | | | ii jj i,j | ⌃⇧ | 3-extendibleAB 3 ⇤ | | > 0 00 log A | ⌃ 2 2 O 2| | O( log A log B ) n = A B = e | | | | log A 137-extendible| | | | ⇣ ⌘ O | | 2 ⇥ O( 2 log A log B ) separable states= poly(n)=e | | | | extendible ⌅ Squashed entanglement is the quantum analogue of the intrinsic information, which is defined as 1000 searchI(X;Y Z for) :=inf a 2-extension,I(X; Y Z¯), 3-extension...... ⇤ PZ¯ Z | | 0000 How close to separable is ⇥ AB =if a k-extension AB = is found? ⇥ ⇤ 0000 for a triple of random variablesHow longX, Y, Zdoes[16]. Theit take minimisation to check extends if| a over⌃⇧k-extension| all conditional exists? prob- ability distributions mapping Z to Z¯. It has been shown that the minimisation can be⇧ restricted0000⌃ to random variables Z¯ with size Z¯ = Z [17]. This implies that the minimum is achieved⇧ and in ⌃ | | | | ⇤ ⌅

1000 0000 ⇥AB = ⇥ ⇧⇥ 0000 ⇧ 0000⌃ ⇧ ⌃ ⇤ ⌅

1 1 2 00 2 0000 ⇥AB = AB = ⇥ ⇥ | ⌃⇧ | 0000 ⇧ 1 00 1 ⌃ ⇧ 2 2 ⌃ ⇤ ⌅

1 1 2 00 2 0000 ⇥AB = ⇥ ⌅ 0000 ⇧ 1 00 1 ⌃ ⇧ 2 2 ⌃ 1 ⇤ ⌅ 2 000 > 0 1 002 0 ⇥AB = 1 ⇥ 0 2 00 ⇥ ⇧ 0001 ⌃ ⇧ 2 ⌃ ⌅ ⇤ ⌅

1 1 ⇤ 3 006 1 1 2 000 0 00 1 6 002 0 ⇥AB = 1 ⇥ ⇥AB = 1 ⇥ 0 2 00 00 0 ⇧ 0001 ⌃ 6 ⇧ 2 ⌃ ⇧ 1 001 ⌃ ⇤ ⌅ ⇧ 6 3 ⌃ ⇤ ⌅

2 1 3 00 3 0000 ⇥AB = ⇥ 0000 ⇧ 1 00 1 ⌃ ⇧ 3 3 ⌃ ⇤ ⌅

3 2 8 008 1 0 8 00 ⇥AB = 1 ⇥ 008 0 ⇧ 2 003 ⌃ ⇧ 8 8 ⌃ ⇤ ⌅

i i i i ⇥AB1B2 Bk = pi⇥A ⇥B ⇥B ⇥B ··· 1 2 ··· k ⌥i

= 0 0 | ⌃AB | ⌃A | ⌃B SoS as opmizaon over k-extensions

(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to opmizaon over k-extendible states (plus PPT (posive paral transpose) test):

maxtr(Mπ ) : ∃ σ ≥ 0, tr(σ ) =1, σ = π ∀j AB1...Bk ABj

How close to separable are k-extendible states? (Koenig, Renner ‘04) De Fine Bound % 2 ( ' B * If ρ is k-extendible min ρAB −σ AB ≤ Ω AB σ ∈SEP ' k * & )

Improved de Fine Bound (B., Christandl, Yard ’11) 1/2 $ 4ln2log A ' If ρ is k-extendible: min ρAB −σ AB ≤ & ) AB SEP σ ∈ % k ( How close to separable are k-extendible states? (Koenig, Renner ‘04) De Fine Bound % 2 ( ' B * If ρ is k-extendible min ρAB −σ AB ≤ Ω AB σ ∈SEP ' k * & )

Improved de Fine Bound (B., Christandl, Yard ’11) 1/2 $ 4ln2log A ' If ρ is k-extendible: min ρAB −σ AB ≤ & ) AB SEP σ ∈ % k ( How close to separable are k-extendible states? Improved de Fine Bound 1/2 $ 4ln2log A ' min ρAB −σ AB ≤ & ) If ρAB is k-extendible: σ ∈SEP % k (

k=2ln(2)ε-2log|A| rounds of SoS

solves WSEP(ε) with a SDP of size

k -2 SEP ε |A||B| = exp(O(ε log|A| log|B|)) D Proving… Improved de Fine Bound 1/2 $ 4ln2log A ' min ρAB −σ AB ≤ & ) If ρAB is k-extendible: σ ∈SEP % k (

Proof is informaon-theorec

Mutual Informaon:

I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))

Proving… Improved de Fine Bound 1/2 $ 4ln2log A ' min ρAB −σ AB ≤ & ) If ρAB is k-extendible: σ ∈SEP % k (

Proof is informaon-theorec

Mutual Informaon:

I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))

Let ρAB1…Bk be k-extension of ρAB

log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) (chain rule)

For some l

Proving… Improved de Fine Bound 1/2 $ 4ln2log A ' min ρAB −σ AB ≤ & ) If ρAB is k-extendible: σ ∈SEP % k (

Proof is informaon-theorec

Mutual Informaon:

I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))

Let ρAB1…Bk be k-extension of ρAB

log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) (chain rule) What does it imply? For some l

Quantum Informaon?

Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. Quantum Informaon?

Nature isn't classical, dammit, and if you want to make a simulation of informaon theory Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.

Good news Bad news • I(A:B|C) still defined • Only definition I(A:B|C)=H(AC) +H(BC)-H(ABC)-H(C) • Chain rule, etc. still hold • Can’t condition on quantum • I(A:B|C) =0 implies ρ is ρ information. separable (Hayden, Jozsa, Petz, Winter‘03) • I(A:B|C)ρ ≈ 0 doesn’t imply ρ is approximately separable in 1-norm (Ibinson, Linden, Winter ’08) Proving the Bound

Thm (B., Christandl, Yard ’11) 1 2 I(A : B | C) ≥ min ρAB −σ AB 2ln2 σ ∈SEP

Obs: I(A:B|E)≥0 is strong subaddivity inequality (Lieb, Ruskai ‘73)

Chain rule:

2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)

k 2 For ρAB k-extendible: 2log A ≥ min ρAB −σ AB 2ln2 σ ∈SEP Condional Mutual Informaon Bound

1 2 I(A : B | C) ≥ min ρAB −σ AB 2ln2 σ ∈SEP

• Coding Theory Strong subaddivity of von Neumann entropy as state redistribuon rate (Devetak, Yard ‘06)

• Large Deviaon Theory Hypothesis tesng for entanglement (B., Plenio ‘08)

∞ ∞ 1 2 I(A : B | C)ρ ≥ER (ρA:BE )− ER (ρA:E )≥D1−LOCC (ρA:B )≥ min ρA:B −σ ( 2ln2 σ ∈SEP 1−LOCC ) hSEP equivalent to

1. Injecve norm of 3-index tensors

2. Minimum output entropy quantum channel, Opmal acceptance probability in QMA(2), ….

P := max x 1 Px 4. 2->q norms of projectors: 2→q 2 = q

P = h (M ), M = P k k P ⊗ P k k P 2→4 SEP ∑( ) ( ) k

(||*||2->q for q > 2 are hypercontracve norms: useful in Markov Chain Monte Carlo (rapidly mixing condion), etc) Quantum Bound on SoS Implies (Barak, B., Harrow, Kelner, Steurer, Zhou ‘12) 1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t. 4 4 2 2 A ≤ x ≤ A +ε A A 2→4 2→4 2→2 2→∞

2. nO(ε) -level SoS solves ε-Small Set Expansion Problem (closely related to unique games). Alternave algorithm to (Arora, Barak, Steurer ’10)

3. Open Problem: Improvement in bound (addive -> mulplicave error) would solve SSE in exp(O(log2(n))) me.

Can formulate it as a quantum informaon-theorec problem

Quantum Bound on SoS Implies (Barak, B., Harrow, Kelner, Steurer, Zhou ‘12) 1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t. 4 4 2 2 A ≤ x ≤ A +ε A A 2→4 2→4 2→2 2→∞

2. nO(ε) -level SoS solves ε-Small Set Expansion Problem (closely related to unique games). Alternave algorithm to (Arora, Barak, Steurer ’10)

3. Open Problem: Improvement in bound (addive -> mulplicave error) would solve SSE in exp(O(log2(n))) me.

Can formulate it as a quantum informaon-theorec problem

Quantum Bound on SoS Implies (Barak, B., Harrow, Kelner, Steurer, Zhou ‘12) 1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t. 4 4 2 2 A ≤ x ≤ A +ε A A 2→4 2→4 2→2 2→∞

2. nO(ε) -level SoS solves ε-Small Set Expansion Problem (closely related to unique games). Alternave algorithm to (Arora, Barak, Steurer ’10)

3. Open Problem: Improvement in bound (addive -> mulplicave error) would solve SSE in exp(O(log2(n))) me.

Can formulate it as a quantum informaon-theorec problem

Outline Entanglement Monogamy

How general is the concept? Can we make it quantave? Are there other applicaons/implicaons?

1. Detecng Entanglement and Sum-Of-Squares Hierarchy

2. Accuracy of Mean Field Approximaon

3. Other applicaons Constraint Sasfacon Problems vs Local Hamiltonians

k-arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints: c : Σk → {0,1} j Assignment: σ :[n] → Σ

Unsat := min cj (σ (x j ),...,σ (x j )) σ ∑ 1 k j Constraint Sasfacon Problems vs Local Hamiltonians qudit

H1

k-arity CSP: k-local Hamiltonian H:

d ⊗n Variables {x1, …, xn}, alphabet Σ n qudits in C ( ) k d ⊗k H ∈ Her C Constraints: cj : Σ → {0,1} Constraints: j ( ) ( ) Assignment: σ :[n] → Σ " % qUnsat := E0 $ H j ' $∑ ' # j & Unsat := min cj (σ (x j ),...,σ (x j )) σ ∑ 1 k E : min eigenvalue j 0 C. vs Q. Opmal Assignments

Finding opmal assignment of CSPs can be hard C. vs Q. Opmal Assignments

Finding opmal assignment of CSPs can be hard

Finding opmal assignment of quantum CSPs can be even harder

(BCS, Laughlin states for FQHE,…)

Main difference: Opmal Assignment can be a ⊗n highly entangled state (unit vector in ) (C d ) Opmal Assignments: Entangled States

Non-entangled state: a1 0 + b1 1 ⊗ ...⊗ an 0 + bn 1 ( ) ( ) e.g. ↑ ⊗ ...⊗ ↑

c i ,...,i Entangled states: ∑ i ,...,i 1 n 1 n i ,...,i 1 n e.g. (↑ ↓ − ↓ ↑ ) / 2

To describe a general entangled state of n spins requires exp(O(n)) bits How Entangled?

Given biparte entangled state ψ ∈ C n ⊗ C m AB

the reduced state on A is mixed: ρA ≥ 0, tr(ρA ) =1

The more mixed ρA, the more entangled ψAB:

Quantavely: E(ψAB) := S(ρA) = -tr(ρA log ρA)

Is there a relaon between the amount of entanglement in the ground-state and the computaonal complexity of the model?

Mean-Field… …consists in approximang groundstate

by a product state ψ1 ⊗…⊗ ψn

max ψ1,…,ψn H ψ1,…,ψn is a CSP ψ1,…,ψn

Successful heurisc in Quantum Chemistry (e.g. Hartree-Fock) Condensed maer (e.g. BCS theory)

Folklore: Mean-Field good when Many-parcle interacons Low entanglement in state Mean-Field Good When (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interacon graph G(V, E) and |E| local terms.

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

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

X3 X2 X1

m < O(log(n)) Mean-Field Good When (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interacon graph G(V, E) and |E| local terms.

Let {Xi} be a paron of the sites with each Xi having m sites. Then there are products states ψi in Xi s.t. 1/8 # & 1 6 1 S(X i ) ψ1,...,ψm H ψ1,...,ψm ≤ e0 (H) +Ω%d Ei ( | E | $ deg(G) m ' deg(G) : degree of G E : expectaon over X i i X3 S(X ) : entropy of X2 i X1 groundstate in Xi

m < O(log(n)) Approximaon in terms of degree …shows mean field becomes exact in high dim

1-D ∞-D

2-D

3-D Approximaon in terms of degree # &1/8 1 6 1 S(X i ) ψ1,...,ψm H ψ1,...,ψm ≤ e0 (H) +Ω%d Ei ( | E | $ deg(G) m ' Relevant to the quantum PCP problem:

(Aharonov, Arad, Landau, Vazirani ’08; Freedman, Hasngs ‘12; Bravyi, DiVincenzo, Loss, Terhal ’08, Aaronson ‘08)

Is approximang e0(H) to constant accuracy quantum NP-hard?

It shows that quantum generalizaons of PCP theorem and parallel repeon cannot both be true (assuming “quantum NP”= QMA ≠ NP)

Approximaon in terms of average entanglement # &1/8 1 6 1 S(X i ) ψ1,...,ψm H ψ1,...,ψm ≤ e0 (H) +Ω%d Ei ( | E | $ deg(G) m '

Mean-field works well if entanglement S(X i ) of groundstate sasfies a Ei = o(1) subvolume law: m m < O(log(n))

Connecon of amount of entanglement in groundstate X3 X2 and computaonal X1 complexity of the model

Approximaon in terms of average entanglement # &1/8 1 6 1 S(X i ) ψ1,...,ψm H ψ1,...,ψm ≤ e0 (H) +Ω%d Ei ( | E | $ deg(G) m '

Systems with low entanglement are expected to be simpler

So far only precise in 1D:

Area law for entanglement -> Succinct classical descripon (MPS) Here: Good: arbitrary lace, only subvolume law

Bad: Only mean energy approximated well New Algorithms for Quantum Hamiltonians

Following same approach we also obtain polynomial me algorithms for approximang 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.

Proof Idea: Monogamy of Entanglement

Cannot be highly entangled with too many neighbors

Entropy S(Xi) quanfies how Xi entangled it can be

Proof uses informaon-theorec techniques (chain rule of condional mutual informaon, informaonally complete measurements, etc) to make this intuion precise

Inspired by classical informaon-theorec ideas for bounding convergence of SoS hierarchy for CSPs (Tan, Raghavendra ’10; Barak, Raghavendra, Steurer ‘10) Outline Entanglement Monogamy

How general is the concept? Can we make it quantave? Are there other applicaons/implicaons?

1. Detecng Entanglement and Sum-Of-Squares Hierarchy

2. Accuracy of Mean Field Approximaon

3. Other applicaons Other Applicaons

(B., Horodecki ‘12) Proving entanglement area law from exponenal decay of correlaons for 1D states

(Almheiri, Marolf, Polchinski, Sully ’12) Black hole firewall controversy

(Vidick, Vazirani ’12; Barte, Colbeck, Kent ’12; …) Device independent key distribuon

(Vidick and Ito ‘12) Entangled mulple proof systems: NEXP in MIP* Conclusions Entanglement Monogamy is a disnguishing feature of quantum correlaons

(quantum engineering) It’s at the heart of quantum cryptography

(convex opmizaon) Can be used to detect entanglement efficiently and understand Sum-Of-Squares hierarchy beer

(computaonal physics) Can be used to bound efficiency of mean-field approximaon

(quantum many-body physics) Can be used to prove area law from exponenal decay of correlaons

QIT useful to bound efficiency of mean-field theory

- Cornering quantum PCP - Poly-me algorithms for planar and dense Hamiltonians

Thank you!