Norm-One Vector in Cd

Home , LOCC

The Complexity of Quantum Entanglement

Fernando G.S.L. Brandão ETH Zürich

Based on joint work with M. Christandl and J. Yard

Journees Deferaon de Reserche en Mathemaques de Paris Centre/GT Informaque Quanque Paris, 09/05/2012

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

This talk: 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

This talk: 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

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

Outline • The Problem Quantum States Quantum Entanglement

• The Algorithm Parrilo-Lasserre Relaxaon Monogamy of Entanglement Quantum de Fine Theorem

• Applicaons A new characterizaon of Quantum NP Small Set Expansion

• Proof Ideas

Quantum States

• Pure States: norm-one vector in Cd:

T ψ := (ψ1,...,ψd )

• Mixed States: posive semideﬁnite matrix of unit trace:

ρ = ∑ pi ψi ψi i

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

Quantum Measurements

• To any experiment with d outcomes we associate d

posive matrices {Mk} such that ∑M k = I k

and calculate probabilies as Pr(k) = tr(M k ρ)

E.g. For pure states, Pr(k) = ψ M k ψ

Quantum Entanglement

d l • Pure States: C C ψ AB ∈ ⊗

If , it’s separable ψ AB = φ A ⊗ ϕ B

otherwise, it’s entangled.

• Mixed States: d l ρAB ∈ M(C ) ⊗ M(C )

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

Quantum Entanglement

d l • Pure States: C C ψ AB ∈ ⊗

If , it’s separable ψ AB = φ A ⊗ ϕ B

otherwise, it’s entangled.

• Mixed States: d l 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

The Separability Problem

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

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

SEP D

The Problem (for experimentalists)

one run of experiment A Measurement Source Measurement result B 1st run 1st result Is the 2nd run 2nd result statistical state analysis entangled ? nth run nth result The Problem (for experimentalists)

one run of experiment A Measurement Source Measurement result B 1st run 1st result Is the 2nd run 2nd result statistical state analysis entangled ? nth run nth result Relevance

• Quantum Cryptography Security only if state is entangled

• Quantum Communicaon Advantage over classical (e.g. teleportaon, dense coding) only if state is entangled

• Computaonal Physics Entanglement responsible for diﬃculty of simulaon of quantum systems

Deciding Entanglement

The problem of deciding whether a state is entangled

• has been considered since the early days of the ﬁeld of quantum informaon theory

• is regarded as a computaonally diﬃcult problem

In this talk I’ll discuss the fastest known algorithm for this problem

The Separability Problem (again)

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

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

SEP D

The Separability Problem (again)

d l • Given ρAB ∈ D(C ⊗ C ) Which norm should we use? is it separable or entangled?

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

SEP D

Norms on Quantum States

How to quanfy the distance in Weak-Membership?

• Euclidean Norm (Hilber-Schmidt):

||X|| = tr(XTX)1/2 2 • Trace Norm

T 1/2 ||X||1 = tr((X X) )

-1/2 Obs: ||X||1 ≥||X||2≥d ||X||1 The LOCC Norm

• Operaonal interpretaon trace norm:

||ρ – σ||1 = 2 max 0

• For ρAB, σAB deﬁne

||ρ – σ||LOCC = 2 max 0

LOCC: Local quantum Operaons and Classical Communicaon The LOCC Norm

• Operaonal interpretaon trace norm:

||ρ – σ||1 = 2 max 0

• For ρAB, σAB deﬁne the LOCC norm

||ρ – σ||LOCC = 2 max 0

Opmal bias of disnguishing two states by LOCC measurements

E.g. (one-way LOCC) M = ∑Ak ⊗ Bk, ∑Ak ≤ I, 0 ≤ Bk ≤ I k k Opmizaon Over Separable States

(Best Separable State BSS(ε)) Given M ∈ H(C d ⊗ Cl )

esmate

max tr(Mσ ) = max , ψ,ϕ M ψ,ϕ σ ∈SEP ψ φ

T max T T (x y) M(x y) = x x=y y=1 ⊗ ⊗

to addive error ε

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

State-of-the-art: 2O(|A|log|B|log (1/ε)) me complexity for either ||*|| or ||*|| norms 2 1 (Doherty, Parrilo, Spedalieri ‘04)

Hardness Results

When is ρ entangled? AB

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

(Gurvits ‘02) NP-hard with ε=1/exp(|A||B|)

(Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly((|A||B|)1/2)

(Beigi&Shor ‘08) Favorite separability tests fail

(Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ)) me algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails

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

When is ρ entangled? AB

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

(Gurvits ‘02) NP-hard with ε=1/exp(|A||B|)

(Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|)

(Beigi&Shor ‘08) Favorite separability tests fail

(Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ)) me algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails

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

When is ρ entangled? AB

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

(Gurvits ‘02) NP-hard with ε=1/exp(|A||B|)

(Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|)

(Beigi, Shor ‘08) Favorite separability tests fail

(Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ)) me algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails

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

When is ρ entangled? AB

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

(Gurvits ‘02) NP-hard with ε=1/exp(|A||B|)

(Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|)

(Beigi, Shor ‘08) Favorite separability tests fail

(Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|)) me algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP, and any ν+μ>0, unless ETH fails

ETH (Exponenal Time Hypothesis): SAT cannot be solved in 2o(n) me (Impagliazzo&Paru ’99) Algorithms for BSS

Esmate with addive error max tr(Mσ ) ε σ ∈SEP

State-of-the-art: 2O((|A|+|B|)log (1/ε)) me complexity

Exhausve search over ε-nets on A and B!

Hardness Results for BSS

Esmate with addive error max tr(Mσ ) ε σ ∈SEP

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

(Harrow, Montanaro ’10, built on Aaronson et al ‘08) No exp(O(log1-ν|A| 1-μ log |B|||M||∞)) me algorithm for any ν+μ>0 and constant ε, unless ETH fails

Main Result 1: Weak Membership

(B., Christandl, Yard ‘10) There is a exp(O(ε-2log|A|log|

B|)) me algorithm for WSEP(||*||, ε) (in ||*||2 or ||*|LOCC) Main Result 1: Weak Membership

(B., Christandl, Yard ‘10) There is a exp(O(ε-2log|A|log|

B|)) me algorithm for WSEP(||*||, ε) (in ||*||2 or ||*|LOCC)

Remind: NP-hard for ε = 1/poly(|A||B|) in ||*||2 (Gurvits ‘02, Gharibian ‘08, Beigi ‘08)

Corollary: the problem in ||*||2 is not NP-hard for ε = 1/polylog(|A||B|), unless ETH fails

Main Result 2: Best Separable State

(BCY ‘10) -2 2 1. There is a exp(O(ε log|A|log|B|(||M||2) )) me algorithm for BSS(ε) 2. For M in LOCC, there is a exp(O(ε-2log|A|log|B|)) me algorithm for BSS(ε) Main Result 2: Best Separable State

(BCY ‘10) -2 2 1. There is a exp(O(ε log|A|log|B|(||M||2) )) me algorithm for BSS(ε) 2. For M in LOCC, there is a exp(O(ε-2log|A|log|B|)) me algorithm for BSS(ε)

Contrast with:

1-ν 1-μ (Harrow, Montanaro ‘10) No exp(O(log |A|log |B|||M||∞)) me algorithm for any ν+μ>0 and constant ε, unless ETH fails, even for

separable M: . M = Ak ⊗ Bk, 0 ≤ M ≤ I ∑ k Remember: Part 2 works for M = ∑Ak ⊗ Bk, ∑Ak ≤ I, 0 ≤ Bk ≤ I k k Main Result 2: Best Separable State

(BCY ‘10) Quantum Info Remark: -2 2 1. The diﬃculty to show opmality of the algorithm is the There is a exp(O(ε log|A|log|B|(||M||2) )) me existence of separable measurements that are algorithm for BSS(ε) not LOCC, 2. a well studied phenomena in quantum informaon (e.g. For M in LOCC, there is a exp(O(ε-2log|A|log|B|)) me Benne et al ‘98algorithm for BSS(). Here we have a new computaonal-ε) complexity movaon for further studying the problem!

Contrast with:

1-ν 1-μ (Harrow, Montanaro ‘10) No exp(O(log |A|log |B|||M||∞)) me algorithm for any ν+μ>0 and constant ε, unless ETH fails, even for

separable M: . M = Ak ⊗ Bk, 0 ≤ M ≤ I ∑ k Remember: Part 2 works for M = ∑Ak ⊗ Bk, ∑Ak ≤ I, 0 ≤ Bk ≤ I k k The Algorithm

• We consider the a Parrilo-Lasserre hierarchy of SDP relaxaons to the problem introduced in (Doherty, Parrilo and Spedalieri ’01)

• We prove it converges to a good approximate soluon in a O(log|B|) number of rounds. Previously convergence only in Ω(|B|) rounds was known. Opmizaon Over Separable States (again)

(Best Separable State BSS(ε)) Given M ∈ H(C d ⊗ Cl ) esmate

max tr(Mσ ) = max , ψ,ϕ M ψ,ϕ σ ∈SEP ψ φ T max T T (x y) M(x y) = x x=y y=1 ⊗ ⊗ to addive error ε

This is a polynomial opmizaon problem. One can calculate a sequence of SDP approximaons to it following the approach of (Parrilo ‘00, Lasserre ’01)

We’ll derive the SDP hierachy by a quantum argument

Entanglement Monogamy

Classical correlaons are shareable:

Given separable state σ AB = ∑ pj ψ j ψ j ⊗ ϕ j ϕ j j Consider the symmetric extension

⊗k B1 σ = p ψ ψ ⊗ ϕ ϕ B2 AB1,...,Bk ∑ j j j j j B3 j B4 A …

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

Classical correlaons are shareable:

⊗k

σ AB ,...,B = pj ψ j ψ j ⊗ ϕ j ϕ j 1 k ∑ j B1 B2 B3 B Def. ρAB is k-extendible if there is ρAB1…Bk 4 A … s.t for all j in [k], tr\ Bj (ρAB1…Bk) = ρAB Bk

Separable states are k-extendible for every k Entanglement Monogamy

Quantum correlaons are non-shareable:

ρAB separable iﬀ ρAB k-extendible for all k

Follows from: Quantum de Fine Theorem (Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)

Monogamy of entanglement: Very useful concept in general, applicaon e.g. in quantum key distribuon

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].

1 I. INTRODUCTION1 2 00 2 AB 2 1 R| | 0000⇥ ⇥AB = ⇥ 0000 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? probability 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 Entanglement Monogamy

Quantave version: For any k-extendible ρAB, 2 $ B ' min ρ −σ ≤ O& ) σ ∈SEP 1 & ) % k ( - Follows from: Finite quantum de Fine Theorem (Christandl, König, Mitchson, Renner ‘05) Entanglement Monogamy

Quantave version: For any k-extendible ρAB, 2 $ B ' min ρ −σ ≤ O& ) σ ∈SEP 1 & ) % k ( - Follows from: Finite quantum de Fine Theorem (Christandl, König, Mitchson, Renner ‘05) Close to opmal: % B ( there is a k-ext state ρ s.t. min ρ −σ ≥ Ω' * AB σ ∈SEP 1 & k )

For other norms (||*||2, ||*||LOCC, …) no beer bound known. Exponenally Improved de Fine type bound

(B., Christandl, Yard ‘10) For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC 1 $ log A '2 min ρ −σ ≤ O& ) σ ∈SEP % k (

Bound proporonal to the (square root) of the number of qubits: exponenal improvement over previous bound How long does it take to check if a k- extension exists?

• Search for a symmetric extension is a semideﬁnite program (Doherty, Parrilo, Spedalieri ‘04)

∃ π ≥ 0 : π = ρ ∀ j ∈ [k] AB1,...,Bk ABj AB • Can be solved in poly(n) me in the number of variables n

• n = |A|2|B|2k • Our bound implies k = O(ε-2log|A|)

• Time Complexity: poly(|A||B|2k) = exp(O(ε-2log|A|log|B|)) Does it work for 1-norm? % B ( • There are k-extendible states s.t. min ρ −σ 1 ≥ Ω' * σ ∈SEP & k ) • For such states the SDP hierarchy only gives good soluons for k = O(|B|), which requires exponenal me 1 $ log A '2 • But we know also: min O ρ −σ LOCC ≤ & ) σ ∈SEP % k (

• So, hard instances are always “data hiding” states, i.e. min ρ −σ >> min ρ −σ σ ∈SEP 1 σ ∈SEP LOCC Does it work for 1-norm? % B ( • There are k-extendible states s.t. min ρ −σ 1 ≥ Ω' * σ ∈SEP & k ) • For such states the SDP hierarchy only gives good soluons for k = O(|B|), which requires exponenal me 1 $ log A '2 • But we know also: min O ρ −σ LOCC ≤ & ) σ ∈SEP % k (

• So, hard instances are always “data hiding” states, i.e. min ρ −σ >> min ρ −σ σ ∈SEP 1 σ ∈SEP LOCC Algorithm for Best Separable State

-2 2 The idea Opmize over k=O(log|A|ε (||X||2) ) extension of ρAB by SDP

maxtr(π AB X) : π AB ,...,B ≥ 0, π AB = π AB ∀ j ∈ [k] π 1 1 k j 1

This is precisely the Parrilo-Lasserre hierarchy for the problem! (wrien in a somewhat diﬀerent form)

By Cauchy Schwartz: tr(X(ρ −σ )) ≤ ρ −σ X 2 2 By de Fine Bound: −1 m ax tr(π X) ≥ max tr(σ X) ≥ max tr(π X)− X O(k log | A |) π ∈ k−Ext σ ∈SEP π ∈k−Ext 2 Applicaon 1: Quantum NP

QMA

Quantum Computer

A language L is in QMA if for every x in L: QMA: - YES instance: Merlin can convince Arthur with probability > 2/3 - NO instance: Merlin cannot convince Arthur with probability > 1/3 QMA

- Quantum analogue of NP (or MA) - Local Hamiltonian Problem, N-representability, …

Is QMA a robust complexity class? (Aharonov, Regev ‘03) superveriﬁers don’t help

(Marrio, Watrous ‘05) Exponenal ampliﬁcaon with ﬁxed proof size

(Beigi, Shor, Watrous ‘09) logarithmic size interacon doesn’t help New Characterizaon QMA

Corollary QMA doesn’t change allowing k = O(1) diﬀerent proofs if the veriﬁer can only apply LOCC measurements in the k proofs New Characterizaon QMA

Corollary QMA doesn’t change allowing k = O(1) diﬀerent proofs if the veriﬁer can only apply LOCC measurements in the k proofs

Def QMAm(k): analogue of QMA with k proofs and proof size m New Characterizaon QMA

Corollary QMA doesn’t change allowing k = O(1) diﬀerent proofs if the veriﬁer can only apply LOCC measurements in the k proofs

Def QMAm(k): analogue of QMA with k proofs and proof size m

Def LOCCQMAm(k): analogue of QMA with k proofs, proof size m and LOCC veriﬁcaon procedure along the k proofs. QMA(k)

Def QMAm(k): A language L is in QMAm(k) if there is a quantum poly-me circuit that for every instance x implements the measurement {Ax, I - Ax} such that

• Completeness: If x in L, there exists k proofs, each of m qubits, s.t.

tr(Ax ( ψ1 ψ1 ⊗ ψ2 ψ2 ⊗... ⊗ ψk ψk )) ≥ 2 / 3 • Soundness: If x not in L, for any k states,

tr(Ax ( ψ1 ψ1 ⊗ ψ2 ψ2 ⊗... ⊗ ψk ψk )) ≤1/ 3

Def 2 LOCCQMAm(k): Likewise, but {Ax, I - Ax} must be LOCC New Characterizaon QMA

Corollary QMA = LOCCQMA(k), k = O(1)

LOCCQMAm(2) contained in QMAO(m2)

Contrast: QMAm(2) not in QMAO(m2-δ) for any δ>0 unless Quantum ETH* fails (Harrow and Montanaro ’10) -- based on Aaronson et al ‘08

1/2 And: SAT has a LOCCQMAO(log(n))(n ) protocol (Chen and Drucker ’10)

* Quantum ETH: SAT cannot be solved in 2o(n) quantum me

New Characterizaon QMA

Corollary QMA = LOCCQMA(k), k = O(1)

LOCCQMAm(2) contained in QMAO(m2)

Contrast: QMAm(2) not in QMAO(m2-δ) for any δ>0 unless Quantum ETH* fails

Follows from QMAn1/2(2) protocol for SAT with n clauses (Harrow and Montanaro ’10 – built on Aaronson et al ’08)

1/2 And: SAT has a LOCCQMAO(log(n))(n ) protocol (Chen and Drucker ’10) * Quantum ETH: SAT cannot be solved in 2o(n) quantum me

New Characterizaon QMA

Corollary QMA = LOCCQMA(k), k = O(1)

LOCCQMAm(2) contained in QMAO(m2)

Idea to simulate LOCCQMAm(2) in QMA:

-2 • Arthur asks for proof ρ on AB1B2…Bk with k = mε • He symmetrizes the B systems and applies the original

veriﬁcaon prodedure to AB1 Correcteness m de Fine bound implies: min ρAB −σ ≤ = ε σ ∈SEP 1 LOCC k

Applicaon 2: Small Set Expansion

Small Set Expansion Problem: Given a graph determine whether all sets of sublinear size expand almost perfectly.

Introduced in (Raghavendra, Steurer ’09), where it was conjectured to be a hard problem. It’s closely related to Khot’s Unique Games Conjecture Applicaon 2: Small Set Expansion

Small Set Expansion Problem: Given a graph determine whether all sets of sublinear size expand almost perfectly.

Introduced in (Raghavendra, Steurer ’09), where it was conjectured to be a hard problem. It’s closely related to Khot’s Unique Games Conjecture

• (Barak, B., Harrow, Kelner, Steurer, and Zhou ‘12) connecon of the Small Set Expansion Problem to the Best-Separable-State Problem for a LOCC operator (via the 2->4 norm of a projector)

• Can show that the SDP hierarchy gives a subexponenal-me algorithm for the small set expansion problem, matching the performance of the algorithm of (Arora, Barak and Steurer ‘10) 2

I. INTRODUCTION

d B A = C ,d N, A := d, B = C| | ⇧ ⌃ | | ⇧

A B AB AB := A B = C| | C| | = C| | ⇥ ⇧ ⇥ ⇧

AB | AB ⌃

= ⇤ ⇤ | AB | A ⇥ | B

0, tr =1 Sys ⌅ Sys

0, tr =1 ⌅

= p i i AB i A ⇥ B ⌥i

= p i (i )T AB i A ⇥ B ⌥i

i A

i B

1 AB =Proof Techniques( 0 A 0 B + 1 A 1 B) | 2 | ⇥ | | ⇥ | k-extendible log A min AB ⇥AB const. | | AB separable || || ⇤ k • Coding Theory 1000 Strong subaddivity of von Neumann 0000 entropy as state redistribuon rate AB = AB = ⇥(Devetak, Yard ‘06) | ⌥ | 0000 ⇧ 0000⌃ • Large Deviaon Theory ⇧ ⌃ ⇤ ⌅ Hypothesis tesng of separable states (B., Plenio ‘08) 1000 • Entanglement Measure Theory 0000 AB = ⇥ Squashed Entanglement 0000 (Christandl, Winter ’04) ⇧ 0000⌃ ⇧ ⌃ ⇤ ⌅ I(A:B|E)

Condional Mutual Informaon: Measures the correlaons of A and B relave to E in ρABE

I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ

Always posive: I(A:B|E)ρ ≥ 0 (strong-subaddivity of entropy)

(Lieb, Ruskai ‘73) When does it vanish?

I(A:B|E)ρ = 0 iﬀ ρABE is a “Quantum Markov Chain State”

(Hayden, Jozsa, Petz, Winter ‘04)

A B E E.g. ρABE = ∑ pk ρk ⊗σ k ⊗ k k k Approximate version??? … New Inequality for I(A:B|E)

Thm: (B., Christandl, Yard ’10) 2 I(A : B | E) ≥ Ω min ρAB −σ AB (σ ∈SEP )

• Either LOCC or 2-norm

• Obs: The statement fails badly for 1-norm!

• The monogamy bound follows from this inequality and the chain rule (via an entanglement measure called squashed entanglement) Summary

• Tesng separability is rather easy

• Family of Parrilo-Lasserre SDP relaxaons converge in log(n) rounds; proof by a quantum argument – new approach to proving fast convergence of SDP hierarchies.

• New Pinsker type lower bound for I(A:B|E)

• QMA is robust Open Problems

• Is there a polynomial algorithm in 2-norm?

• Can we close the LOCC norm vs. trace norm gap in the results? (hardness vs. algorithm, LOCCQMA(k) vs QMA(k))

• Are there more applicaons of the bound on the convergence of the SDP relaxaon? Can we prove a quasipolynomial me algorithm for Small set Expansion? And for unique games or other UG-hard prioblems?

• Can we put new problems in QMA using QMA = LOCCQMA(k)?

• Are there more applicaon of the inequality for I(A:B|E)? Thank you! Proof Outline Relave Entropy of Entanglement

The proof is largely based on the properes of the following entanglement measure:

Def Relave Entropy of Entanglement (Vedral, Plenio ‘99) E (ρ ⊗n ) E∞ ( ) : lim R AB R ρAB = ER (ρAB ) := min S(ρ ||σ ) n→∞ n σ ∈SEP

S(ρ ||σ ) := tr(ρ(logρ − logσ )) Entanglement Hypothesis Tesng

Given (many copies) of ρAB, what’s the opmal probability of disnguishing it from a separable state? Entanglement Hypothesis Tesng

Given (many copies) of ρAB, what’s the opmal probability of disnguishing it from a separable state?

Def Rate Funcon: D(ρAB) is maximum number r s.t. there exists {Mn, I-Mn} , 0 < Mn < I,

−nr ⊗n min tr(M nσ ) ≤ 2 , tr(MρAB ) ≥ Ω(1) σ ∈SEP

DLOCC(ρAB) : deﬁned analogously, but now {M, I-M} must be LOCC Entanglement Hypothesis Tesng

Given (many copies) of ρAB, what’s the opmal probability of disnguishing it from a separable state?

Def Rate Funcon: D(ρAB) is maximum number r s.t there exists {Mn, I-Mn} , 0 < Mn < I,

−nr ⊗n min tr(M nσ ) ≤ 2 , tr(MρAB ) ≥ Ω(1) σ ∈SEP

DLOCC(ρAB) : deﬁned analogously, but now {M, I-M} must be LOCC

∞ (B., Plenio ‘08) D(ρAB) = ER (ρAB)

Obs: Equivalent to reversibility of entanglement under non-entangling operaons (B., Plenio ‘08) Proof in 1 Line

(i) (ii) (iii) ∞ ∞ 2 I(A : B | E)ρ ≥ ER (ρA:BE )− ER (ρA:E ) ≥ DLOCC (ρA:B ) ≥ Ω min ρA:B −σ ABE (σ ∈SEP LOCC ) Proof in 1 Line

(i) (ii) (iii) ∞ ∞ 2 I(A : B | E)ρ ≥ ER (ρA:BE )− ER (ρA:E ) ≥ DLOCC (ρA:B ) ≥ Ω min ρA:B −σ ABE (σ ∈SEP LOCC )

Relave entropy of Entanglement plays a triple role:

(i) Quantum Shannon Theory: State redistribuon Protocol (Devetak and Yard ‘07) (ii) Large Deviaon Theory: Entanglement Hypothesis Tesng (B. and Plenio ‘08) (iii) Entanglement Theory: Faithfulness bounds First Inequality

(i) I(A : B | E) E∞ ( ) E∞ ( ) ρABE ≥ R ρA:BE − R ρA:E

Non-lockability: ER (ρA:BE ) ≤ ER (ρA:E )+ 2log B (Horodecki3 and Oppenheim ‘04) State Redistribuon: How much does it cost to redistribute a quantum system? ½ I(A:B|E) ⊗n ⊗n A B E A E BF F ψ A:BE:F → ψ A:E:BF

Proof (i): ⊗n Apply non-lockability to and use ρA:BE state redistribuon to trace out B at a rate of ½ I(A:B|E) qubits per copy Second Inequality (ii) ∞ ∞ ER (ρA:BE )− ER (ρA:E ) ≥ DLOCC (ρA:B )

Equivalent to: D(ρA:BE ) ≥ D(ρA:E )+ DLOCC (ρA:B )

Monogamy relaon for entanglement hypothesis tesng

Proof (ii)

Use opmal measurements for ρAE and ρAB achieving D(ρAE) and DLOCC(ρAB), resp., to construct a measurement for ρA:BE achieving D(ρA:BE) Third Inequality (iii) 2 DLOCC (ρA:B ) ≥ Ω min ρA:B −σ (σ ∈SEP LOCC )

Pinsker type inequality for entanglement hypothesis tesng

Proof (iii) minimax theorem + marngale like property of the set of separable states Thank you!