Monogamy(of(Quantum( Entanglement - …fernandobrandao.org/princeton-monogamy.pdf · Simulangquantumishard, secretsarehardtoconceal More(than(25%of DoE(supercomputerpower(is(devoted(to(simulang(quantum(physics((Can(we(getabeNer(handle(on(this

Monogamy of Quantum Entanglement

Fernando G.S.L. Brandão ETH Zürich

Princeton EE Department, March 2013

Simula2ng quantum is hard

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

Can we get a beNer handle on this

simula'on problem?

Simula2ng quantum is hard, secrets are hard to conceal

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

Can we get a beNer handle on this

simula'on problem?

All current cryptography is based on unproven hardness

assump2ons

Can we have be,er security guarantees for our secrets?

Quantum Informa2on Science…

… gives a path for solving both problems.

But it’s a long journey

QIS is at the crossover of

computer science, engineering and physics

Physics CS

Engineering

The Two Holy Grails of QIS

Quantum Computa2on:

Use of engineered quantum systems for performing

computa2on

Exponen2al speed-‐ups over classical compuGng

E.g. Factoring (RSA) SimulaGng quantum systems

Quantum Cryptography:

Use of engineered quantum systems for secret key

distribu2on

Uncondi2onal security based solely on the correctness of

quantum mechanics


Quantum Computa2on:

Use of well controlled quantum systems for performing

computa2ons.



Use of engineered quantum systems for secret key

distribu2on


quantum mechanics

State-‐of-‐the-‐art: +-‐5 qubits computer

Can prove (with high

probability) that 15 = 3 x 5


Quantum Computa2on:

Use of well controlled quantum systems for performing

computa2ons.



Use of well controlled quantum systems for secret key

distribu2on


quantum mechanics

State-‐of-‐the-‐art: +-‐5 qubits computer

Can prove (with high

probability) that 15 = 3 x 5

State-‐of-‐the-‐art: +-‐100 km

SGll many technological

challenges

While wai2ng 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 communicaGon, quantum algorithms, quantum fault tolerance, quantum simulators, …)




(quantum communicaGon, quantum algorithms, quantum fault tolerance, quantum simulators, …) Answer 2: finding applicaGons of QIS independent of building a quantum computer




(quantum communicaGon, quantum algorithms, quantum fault tolerance, quantum simulators, …) Answer 2: finding applicaGons of QIS independent of building a quantum computer

This talk

Quantum Mechanics I

States: norm-‐one vector in Cd:

Dynamics: L2 norm preserving linear maps, unitary matrices: UUT = I ObservaGon: To any experiment with d outcomes we associate d PSD matrices {Mk} such that ΣkMk=I Born’s rule:

ψ := (ψ1,...,ψd )T

Pr(k) = ψ Mk ψ

probability theory based on L2 norm instead of L1

ψ2= ψ ψ =1

Quantum Mechanics II

Mixed States: PSD matrix of unit trace: Dynamics: linear operaGons mapping density matrices to density matrices (unitaries + adding ancilla + ignoring subsystems)

Born’s rule:

Pr(k) = tr ρMk( )

probability theory based on PSD matrices

ρ ≥ 0, tr ρ( ) =1 ρ = pi ψi ψii∑

Dirac notaGon reminder: ψ := (ψ1*,...,ψd

* )

Quantum Entanglement

Pure States:

If , it’s separable otherwise, it’s entangled.

Mixed States:

If , it’s separable otherwise, it’s entangled.

ψAB= φ

A⊗ ϕ

B

ρ = pi ψi ψi ⊗ φi φii∑

ψAB∈Cd ⊗Cl

ρAB ∈ D(Cd ⊗Cl )

A Physical Defini2on of Entanglement

LOCC: Local quantum OperaGons and Classical CommunicaGon

Separable states can be created by LOCC: Entangled states cannot be created by LOCC:

non-‐classical correlaGons

ρ = pi ψi ψi ⊗ φi φii∑

Quantum Features SuperposiGon principle uncertainty principle

interference non-‐locality

Monogamy of Entanglement

Monogamy of Entanglement

Maximally Entangled State: CorrelaGons of A and B in cannot be shared:

If ρABE is is s.t. ρAB = , then ρABE = Only knowing that the quantum correlaGons of A and B are very strong one can infer A and B are not correlated with anything else. Purely quantum mechanical phenomenon!

φAB= 00 + 11( ) / 2

φ φAB

φ φAB

φ φAB⊗ ρE

Quantum Key Distribu2on… … as an applica2on of entanglement monogamy

-‐  Using insecure channel Alice and Bob share entangled state ρAB -‐  Applying LOCC they disGll

φAB= 00 + 11( ) / 2

(BenneN, Brassard 84, Ekert 91)

Quantum Key Distribu2on… … as an applica2on of entanglement monogamy

-‐  Using insecure channel Alice and Bob share entangled state ρAB -‐  Applying LOCC they disGll

Let πABE be the state of Alice, Bob and the Eavesdropper. Since πAB = , πAB = . Measuring A and B in the computaGonal basis give 1 bit of secret key.

φAB= 00 + 11( ) / 2

φ φAB

φ φAB⊗ π E

(BenneN, Brassard 84, Ekert 91)

Outline Entanglement Monogamy

How general is the concept? Can we make it quanGtaGve? Are there other applicaGons/implicaGons?

1. Detec2ng Entanglement and Sum-‐Of-‐Squares Hierarchy 2. Accuracy of Mean-‐Field Approxima2on 3. Other Applica2ons

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

Very Easy!

Problem 2: For M in H(Cd Cl), compute

Quadra2c vs Biquadra2c Op2miza2on

Next: Best known algorithm (and best hardness result) using

ideas from Quantum InformaGon Theory

max x =1 xTMx =max x =1 Mijxix j

*

i, j∑

max x = y =1(x⊗ y)T M (x⊗ y) =max x = y =1 Mij;kl xix j*

ijkl∑ ykyl

*

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

Very Easy!

Problem 2: For M in H(Cd Cl), compute

Quadra2c vs Biquadra2c Op2miza2on

Next: Best known algorithm using monogamy of entanglement

max x =1 xTMx =max x =1 Mijxix j

*

i, j∑

max x = y =1(x⊗ y)T M (x⊗ y) =max x = y =1 Mij;kl xix j*

ijkl∑ ykyl

*

The Separability Problem

•  Given

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


SEP D ε


•  Given



SEP D ε X

2= tr XTX( )

1/2Frobenius norm


•  Given



SEP D ε X

2= tr XTX( )

1/2

ρ −σLOCC

=

maxM∈LOCC

tr(M (ρ −σ ))

Frobenius norm

LOCC norm


•  Given

is it entangled? •  (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-‐away from SEP •  Dual Problem: OpGmizaGon over separable states


hSEP (M ) := maxσ∈SEPtr(Mσ ) = max

x = y =1(x⊗ y)T M (x⊗ y)


•  Given

is it entangled? •  (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-‐away from SEP •  Dual Problem: OpGmizaGon over separable states

•  Relevance: Entanglement is a resource in quantum cryptography, quantum communicaGon, etc…


hSEP (M ) := maxσ∈SEPtr(Mσ ) = max

x = y =1(x⊗ y)T M (x⊗ y)

Previous Work

When is ρAB entangled?

-‐ Decide if ρAB is separable or ε-‐away from separable BeauGful theory behind it (PPT, entanglement witnesses, etc) Horribly expensive algorithms

State-‐of-‐the-‐art: 2O(|A|log|B|) Gme complexity Same for esGmaGng hSEP (no beNer than exausGve search!)

dim of A

Hardness Results

When is ρAB entangled?

-‐ 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|)) Gme algorithm with ε=Ω(1) and any ν+μ>0 for unless ETH fails

ETH (ExponenGal Time Hypothesis): SAT cannot be solved in 2o(n) Gme (Impagliazzo&ParuG ’99)

Quasipolynomial-‐2me Algorithm

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


(B., Christandl, Yard ’11) There is an algorithm with run Gme exp(O(ε-‐2log|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|)


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

k∑ ,M ≤ I

M = Ak ⊗ Bk, Ak,Bk ≥ 0k∑ ,Bk ≤ I, Ak ≤ I

k∑

Algorithm: SoS Hierarchy hSEP (M ) =max x = y =1 Mij;kl xix j

*

ijkl∑ ykyl

*

Polynomial opGmizaGon over hypersphere

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

-‐ Round k takes Gme dim(M)O(k) -‐  Converge to hSEP(M) when k -‐> ∞

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

We’ll derive SoS hierarchy by a quantum argument


*

ijkl∑ ykyl

*



-‐ Round k SDP has size dim(M)O(k) -‐  Converge to hSEP(M) when k -‐> ∞


*

ijkl∑ ykyl

*



-‐ Round k SDP has size dim(M)O(k) -‐  Converge to hSEP(M) when k -‐> ∞

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

We’ll derive SoS hierarchy exploring ent. monogamy

Classical Correla2ons are Shareable

Given separable state Consider the symmetric extension

σ AB1,...,Bk= pj ψ j ψ j ⊗

j∑ ϕ j ϕ j

⊗k

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

A

B1 B2 B3 B4

Bk …

σ AB = pj ψ j ψ j ⊗j∑ ϕ j ϕ j

Monogamy Defines Entanglement (Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89) ρAB separable iff ρAB is k-‐extendible for all k

2-extendible

separable states= extendible

3-extendible

137-extendible

2

Measure Esq ED KD EC EF ER E�R EN

normalisation y y y y y y y yfaithfulness y n ? y y y y nLOCC monotonicity y y y y y y y yasymptotic continuity y ? ? ? y y y nconvexity y ? ? ? y y y nstrong superadditivity y y y ? n n ? ?subadditivity y ? ? y y y y ymonogamy y ? ? n n n n ?

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

I. INTRODUCTION

⇥ R|AB|2�1

⌃⇧⌅⌥�

11|AB|

⇤min(|A|,|B|i,j |ii⌃⇧jj|

|AB|

|00⌃

O

�log |A|�2

⇥

⌅�

Squashed entanglement is the quantum analogue of the intrinsic information, which is definedas

I(X;Y ⇤Z) := infPZ̄|Z

I(X;Y |Z̄),

for a triple of random variables X,Y, Z [16]. The minimisation extends over all conditional prob-ability distributions mapping Z to Z̄. It has been shown that the minimisation can be restrictedto random variables Z̄ with size |Z̄| = |Z|[17]. This implies that the minimum is achieved and in

all quantum states

3

⇥AB =

�

⇧⇧⇤

12 0 0 1

20 0 0 00 0 0 012 0 0 1

2

⇥

⌃⌃⌅

� > 0

⇤

⇧⇥

⇥

⌅

⇥�AB =

�

⇧⇧⇤

12 0 0 00 0 1

2 00 1

2 0 00 0 0 1

2

⇥

⌃⌃⌅

⇥AB =

�

⇧⇧⇤

13 0 0 1

60 1

6 0 00 0 1

6 016 0 0 1

3

⇥

⌃⌃⌅

⇥AB =

�

⇧⇧⇤

23 0 0 1

30 0 0 00 0 0 013 0 0 1

3

⇥

⌃⌃⌅

⇥AB =

�

⇧⇧⇤

38 0 0 2

80 1

8 0 00 0 1

8 028 0 0 3

8

⇥

⌃⌃⌅

⇥AB1B2···Bk =⌥

i

pi⇥iA � ⇥iB1

� ⇥iB2� · · ·� ⇥iBk

|�⌃AB = |0⌃A � |0⌃B

search for a 2-extension, 3-extension......How close to separable is if a k-extension is found? How long does it take to check if a k-extension exists?

3

n = |A|2|B|O⇣

log |A|�2

⌘

= eO(��2 log |A| log |B|)

poly(n) = eO(��2 log |A| log |B|)

⇥AB = |�⌃⇧�|AB =

�

⇧⇧⇤

1 0 0 00 0 0 00 0 0 00 0 0 0

⇥

⌃⌃⌅

⇥AB =

�

⇧⇧⇤

1 0 0 00 0 0 00 0 0 00 0 0 0

⇥

⌃⌃⌅

⇥AB = |�⌃⇧�|AB =

�

⇧⇧⇤

12 0 0 1

20 0 0 00 0 0 012 0 0 1

2

⇥

⌃⌃⌅

⇥AB =

�

⇧⇧⇤

12 0 0 1

20 0 0 00 0 0 012 0 0 1

2

⇥

⌃⌃⌅

� > 0

⇥

⌅�

�

⇤

⇥�AB =

�

⇧⇧⇤

12 0 0 00 0 1

2 00 1

2 0 00 0 0 1

2

⇥

⌃⌃⌅

SoS as op2miza2on over k-‐extensions

(Doherty, Parrilo, Spedalieri ‘01) k-‐level SoS SDP for hSEP(M) is equivalent to opGmizaGon over k-‐extendible states (plus PPT (posiGve parGal transpose) test):

max tr(Mπ ) : ∃ σ AB1...Bk≥ 0, tr(σ ) =1, σ ABj

= π ∀j

How close to separable are k-‐extendible states?

(Koenig, Renner ‘04) De Fine~ Bound If ρAB is k-‐extendible

Improved de Fine~ Bound (B., Christandl, Yard ’11)

If ρAB is k-‐extendible:

minσ∈SEP

ρAB −σ AB ≤4 ln2 log A

k$

%&

'

()

1/2

minσ∈SEP

ρAB −σ AB ≤ΩB 2

k

%

&

''

(

)

**


(Koenig, Renner ‘04) De Fine~ Bound If ρAB is k-‐extendible

Improved de Fine~ Bound (B., Christandl, Yard ’11)


minσ∈SEP


k$

%&

'

()

1/2

minσ∈SEP

ρAB −σ AB ≤ΩB 2

k

%

&

''

(

)

**


Improved de Fine~ Bound


minσ∈SEP


k$

%&

'

()

1/2

k=2ln(2)ε-‐2log|A| rounds of SoS solves WSEP(ε) with a SDP of size

|A||B|k = exp(O(ε-‐2log|A| log|B|)) SEP D ε

Proving…

minσ∈SEP


k$

%&

'

()

1/2

Proof is informaGon-‐theoreGc

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



Proving…

minσ∈SEP


k$

%&

'

()

1/2



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) For some l<k: I(A:Bl|B1…Bl-‐1) < log|A|/k

(chain rule)



Proving…

minσ∈SEP


k$

%&

'

()

1/2



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) For some l<k: I(A:Bl|B1…Bl-‐1) < log|A|/k

(chain rule)

What does it imply?



Quantum Informa2on?

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 Informa2on?

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.�

Bad news�•  Only definition I(A:B|C)=H(AC)

+H(BC)-H(ABC)-H(C) �

•  Can’t condition on quantum information.�

�

•  I(A:B|C)ρ ≈ 0 doesn’t imply ρ is approximately separable�in 1-norm (Ibinson, Linden, Winter ’08)�

Good news�•  I(A:B|C) still defined�

•  Chain rule, etc. still hold�

•  I(A:B|C)ρ=0 implies ρ is separable�

(Hayden, Jozsa, Petz, Winter‘03)�

informa2on theory

Obs: I(A:B|E)≥0 is strong subaddiGvity 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)

For ρAB k-‐extendible:

Proving the Bound

Thm (B., Christandl, Yard ’11)

I(A :B |C) ≥ 12 ln2

minσ∈SEP

ρAB −σ AB2

2 log A ≥k

2 ln2minσ∈SEP

ρAB −σ AB2

Condi2onal Mutual Informa2on Bound

•  Coding Theory Strong subaddiGvity of von Neumann entropy as state redistribuGon rate (Devetak, Yard ‘06) •  Large DeviaGon Theory Hypothesis tesGng for entanglement (B., Plenio ‘08)

( )

I(A :B |C) ≥ 12 ln2

minσ∈SEP

ρAB −σ AB2

I(A :B |C)ρ ≥ER∞(ρA:BE )−ER

∞(ρA:E )≥D1−LOCC (ρA:B )≥1

2 ln2minσ∈SEP

ρA:B −σ 1−LOCC

2

hSEP equivalent to

1.  InjecGve norm of 3-‐index tensors

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

4.  2-‐>q norms of projectors:

(||*||2-‐>q for q > 2 are hypercontracGve norms: useful in Markov Chain Monte Carlo (rapidly mixing condiGon), etc)

P2→q:=max x 2=1

Pxq

P2→4

= hSEP (M ), M = P k k P( )⊗ P k k P( )k∑

Quantum Bound on SoS Implies

1. For n x n matrix A can compute with O(log(n)ε-‐3) rounds of SoS a number x s.t.

2. nO(ε) -‐level SoS solves ε-‐Small Set Expansion Problem (closely related to unique games). AlternaGve algorithm to (Arora, Barak, Steurer ’10) 3. Open Problem: Improvement in bound (addiGve -‐> mulGplicaGve error) would solve SSE in exp(O(log2(n))) Gme.

Can formulate it as a quantum informaGon-‐theoreGc problem

A2→4

4≤ x ≤ A

2→4

4+ε A

2→2

2 A2→∞

2

(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)





A2→4

4≤ x ≤ A

2→4

4+ε A

2→2

2 A2→∞

2






A2→4

4≤ x ≤ A

2→4

4+ε A

2→2

2 A2→∞

2




1. Detec2ng Entanglement and Sum-‐Of-‐Squares Hierarchy 2. Accuracy of Mean Field Approxima2on 3. Other applica2ons

Constraint Sa2sfac2on Problems vs Local Hamiltonians

k-‐arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

cj :Σk → {0,1}

minσ

cj (σ (x j1 ),...,σ (x jk ))j∑

σ :[n]→Σ

Constraint Sa2sfac2on Problems vs Local Hamiltonians

k-‐arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

cj :Σk → {0,1}

minσ

cj (σ (x j1 ),...,σ (x jk ))j∑

σ :[n]→Σ

k-‐local Hamiltonian H:

n qudits in

Constraints:

qUnsat := E0 : min eigenvalue

Cd( )⊗n

H j ∈ Her Cd( )⊗k( )

E0 H jj∑"

#$$

%

&''

H1

qudit

C. vs Q. Op2mal Assignments Finding opGmal assignment of CSPs can be hard

C. vs Q. Op2mal Assignments Finding opGmal assignment of CSPs can be hard

Finding opGmal assignment of quantum CSPs can be even harder

(BCS, Laughlin states for FQHE,…)

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

⊗n

Op2mal Assignments: Entangled States

↑ ↓ − ↓ ↑( ) / 2

Non-‐entangled state:

e.g. Entangled states:

e.g.

↑ ⊗ ...⊗ ↑

a1 0 +b1 1( )⊗ ...⊗ an 0 +bn 1( )

ci1,...,in i1,...,ini1,...,in

∑

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

How Entangled?

ψAB∈Cn ⊗Cm

ρA ≥ 0, tr(ρA) =1

Given biparGte entangled state the reduced state on A is mixed: The more mixed ρA, the more entangled ψAB: QuanGtaGvely: E(ψAB) := S(ρA) = -‐tr(ρA log ρA) Is there a relaGon between the amount of entanglement in the ground-‐state and the computaGonal complexity of the model?

Mean-‐Field… …consists in approxima2ng groundstate by a product state is a CSP

ψ1 ⊗…⊗ ψn

maxψ1,…,ψn

ψ1,…,ψn H ψ1,…,ψn

Successful heurisGc in Quantum Chemistry (e.g. Hartree-‐Fock) Condensed maNer (e.g. BCS theory)

Folklore: Mean-‐Field good when Many-‐parGcle interacGons Low entanglement in state

Mean-‐Field Good When (B., Harrow ‘12) Let H be a 2-‐local Hamiltonian on qudits with interacGon

graph G(V, E) and |E| local terms.

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

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

X1

X3 X2

m < O(log(n))

Mean-‐Field Good When

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

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

1| E |

ψ1,...,ψm H ψ1,...,ψm ≤ e0 (H )+Ω d 6 1deg(G)

EiS(Xi )m

#

$%

&

'(

1/8

deg(G) : degree of G Ei : expectaGon over Xi S(Xi) : entropy of groundstate in Xi

X1

X3 X2

m < O(log(n))

Mean-‐Field Good When

Approxima2on in terms of degree

1-D �

2-D �

3-D �

∞-D �

…shows mean field becomes exact in high dim

1| E |


EiS(Xi )m

#

$%

&

'(

1/8

Approxima2on in terms of degree

Relevant to the quantum PCP problem:

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

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

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

1| E |


EiS(Xi )m

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Approxima2on in terms of average entanglement

EiS(Xi )m

= o(1)Mean-‐field works well if entanglement of groundstate saGsfies a subvolume law:

ConnecGon of amount of entanglement in groundstate and computaGonal complexity of the model

X1

X3 X2

m < O(log(n))

Approxima2on in terms of average entanglement

Systems with low entanglement are expected to be simpler

So far only precise in 1D:

Area law for entanglement -‐> Succinct classical descripGon (MPS) Here: Good: arbitrary la~ce, only subvolume law

Bad: Only mean energy approximated well

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New Algorithms for Quantum Hamiltonians

Following same approach we also obtain polynomial Gme algorithms for approximaGng 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) quanGfies how entangled it can be

Proof uses informaGon-‐theoreGc techniques (chain rule of condiGonal mutual informaGon, informaGonally complete measurements, etc) to make this intuiGon precise

Inspired by classical informaGon-‐theoreGc ideas for bounding convergence of SoS hierarchy for CSPs (Tan, Raghavendra ’10; Barak, Raghavendra, Steurer ‘10)

Xi



1. Detec2ng Entanglement and Sum-‐Of-‐Squares Hierarchy 2. Accuracy of Mean Field Approxima2on 3. Other applica2ons

Other Applica2ons

(B., Horodecki ‘12) Proving entanglement area law from exponenGal decay of correlaGons for 1D states (Almheiri, Marolf, Polchinski, Sully ’12) Black hole firewall controversy (Vidick, Vazirani ’12; BarteN, Colbeck, Kent ’12; …) Device independent key distribuGon (Vidick and Ito ‘12) Entangled mulGple proof systems: NEXP in MIP*

Conclusions Entanglement Monogamy is a dis2nguishing feature of quantum correla2ons

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

(convex opGmizaGon) Can be used to detect entanglement efficiently and understand Sum-‐Of-‐Squares hierarchy beNer

(computaGonal physics) Can be used to bound efficiency of mean-‐field approximaGon

(quantum many-‐body physics) Can be used to prove area law from exponenGal decay of correlaGons QIT useful to bound efficiency of mean-‐field theory -‐ Cornering quantum PCP -‐ Poly-‐Gme algorithms for planar and dense Hamiltonians

Thank you!

Documents

Monogamy(of(Quantum( Entanglement - …fernandobrandao.org/princeton-monogamy.pdf · Simulangquantumishard, secretsarehardtoconceal More(than(25%of DoE(supercomputerpower(is(devoted(to(simulang(quantum(physics((Can(we(getabeNer(handle(on(this