Fernando GSL Brand ão ETH Zürich Based on joint work with M Christandl and J Yard Journees Deferation de Reserche en Mathematiques de Paris CentreGT Informatique ID: 593770
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Slide1
The Complexity of Quantum Entanglement
Fernando
G.S.L.
Brand
ão
ETH Zürich
Based on joint work with M.
Christandl
and J. Yard
Journees
Deferation
de
Reserche
en
Mathematiques
de Paris Centre/GT
Informatique
Quantique
Paris, 09/05/2012Slide2
Problem 1: For M in H(Cd) (d x d matrix) compute
Very Easy!Problem 2: For M in H(Cd Cl), compute
Quadratic vs Biquadratic Optimization
This talk:
Best known algorithm
(and best hardness result) using ideas from
Quantum Information TheorySlide3
Problem 1: For M in H(Cd) (d x d matrix) compute
Very Easy!Problem 2: For M in H(Cd Cl), compute
Quadratic vs Biquadratic Optimization
This talk:
Best known algorithm
(and best hardness result) using ideas from
Quantum Information TheorySlide4
Problem 1: For M in H(Cd) (d x d matrix) compute
Very Easy!Problem 2: For M in H(Cd Cl), compute
Quadratic vs Biquadratic Optimization
This talk:
Best known algorithm
(and best hardness result) using ideas from
Quantum Information TheorySlide5
Outline
The Problem
Quantum States Quantum EntanglementThe Algorithm Parrilo-Lasserre Relaxation Monogamy of Entanglement
Quantum de Finetti TheoremApplications A new characterization of Quantum NP Small Set Expansion
Proof Ideas
Slide6
Quantum States
Pure States:
norm-one vector in Cd:Mixed States: positive semidefinite matrix of unit trace:
Dirac notation reminder:Slide7
Quantum M
easurements
To any experiment with d outcomes we associate d positive matrices {Mk} such that and calculate probabilities as E.g. For pure states,
Slide8
Quantum Entanglement
Pure States:
If , it’s separable otherwise, it’s entangled.
Mixed States: If it’s separable otherwise, it’s entangled.
Slide9
Quantum Entanglement
Pure States:
If , it’s separable otherwise, it’s entangled.
Mixed States: If , it’s separable otherwise, it’s entangled.
Slide10
A Physical Definition of Entanglement
LOCC: L
ocal quantum
O
perations and C
lassical Communication
Separable states
can be created by LOCC:Entangled states cannot be created by LOCC: non-classica
l correlationsSlide11
The Separability
Problem
Given is it separable or entangled? (Weak Membership: WSEP(ε, ||*||) Given
ρAB determine if it is separable, or ε-way from SEP
SEP
DSlide12
The P
roblem (for experimentalists)Slide13
The Problem (for experimentalists)Slide14
The Problem (for experimentalists)Slide15
Relevance
Quantum Cryptography Security only if state is entangled Quantum Communication Advantage over classical (e.g. teleportation, dense coding) only if state is entangledComputational Physics Entanglement responsible for difficulty of simulation of quantum systems Slide16
The problem of
deciding whether a state is entangledhas been considered since the early days of the field of quantum information theoryis regarded as a computationally difficult problemIn this talk I’ll discuss the fastest known algorithm forthis problem
Deciding EntanglementSlide17
The Separability
Problem (again)
Given is it separable or entangled? (Weak Membership: WSEP(ε, ||*||) Given
ρAB determine if it is separable, or ε-way from SEP
SEP
DSlide18
The Separability
Problem (again)
Given is it separable or entangled? (Weak Membership: WSEP(ε, ||*||) Given
ρAB determine if it is separable, or ε-way from SEP
SEP
D
Which norm should we use?Slide19
Norms on Quantum States
How to quantify the distance in Weak-Membership?
Euclidean Norm (Hilber-Schmidt): ||X||2 = tr(XTX)1/2
Trace Norm ||X||1 = tr((XTX)1/2)
Obs: ||X||1 ≥||X||2≥d
-1/2||X||1Slide20
The LOCC Norm
Operational interpretation trace norm:
||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))optimal bias of distinguishing the two states by quantum measurements For ρ
AB, σAB define ||ρ – σ||
LOCC = 2 max 0<M<I
tr(M(ρ
– σ)) : {M, I - M} in LOCC
LOCC: L
ocal quantum
O
perations and
C
lassical
C
ommunicationSlide21
The LOCC Norm
Operational interpretation trace norm:
||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))optimal bias of distinguishing the two states by quantum measurements For ρ
AB, σAB define the LOCC norm ||ρ – σ||LOCC = 2 max 0<M<I
tr(M(ρ – σ)) : {M, I - M} in LOCCOptimal bias of distinguishing two states by LOCC measurementsE.g. (one-way LOCC)
Slide22
Optimization O
ver Separable States
(Best Separable State BSS(ε)) Given estimate
to additive error ε Slide23
Previous Work
When is
ρAB entangled? - Decide if ρAB is separable or ε-away from separableBeautiful theory behind it (PPT, entanglement witnesses, etc)
Horribly expensive algorithmsState-of-the-art: 2O(|A|log|B|log (1/ε)) time complexity
for either ||*||2 or ||*||1 norms
(Doherty, Parrilo, Spedalieri ‘04)Slide24
Hardness Results
When is
ρAB entangled? - 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))
time
algorithm for membership in any convex set within
ε
=
Ω
(1) trace distance
to SEP and any
ν+μ
>0,
unless ETH fails
ETH
(Exponential Time Hypothesis): SAT cannot be solved in 2
o(n)
time
(
Impagliazzo&Paruti
’
99
)Slide25
Hardness Results
When is
ρAB entangled? - 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
))
time
algorithm for membership in any convex set within
ε
=
Ω
(1) trace distance
to SEP and any
ν+μ
>0,
unless ETH fails
ETH
(Exponential Time Hypothesis): SAT cannot be solved in 2
o(n)
time
(
Impagliazzo&Paruti
’
99
)Slide26
Hardness Results
When is
ρAB entangled? - 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
))
time
algorithm for membership in any convex set within
ε
=
Ω
(1) trace distance
to SEP and any
ν+μ
>0,
unless ETH fails
ETH
(Exponential Time Hypothesis): SAT cannot be solved in 2
o(n)
time
(
Impagliazzo&Paruti
’
99
)Slide27
Hardness Results
When is
ρAB entangled? - 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
|))
time
algorithm for membership in any convex set within
ε
=
Ω
(1)
trace distance
to SEP, and any
ν+μ
>0,
unless ETH fails
ETH
(Exponential Time Hypothesis): SAT cannot be solved in 2
o(n)
time
(
Impagliazzo&Paruti
’
99
)Slide28
Algorithms for BSS
Estimate with additive error
ε
State-of-the-art: 2O((|A|+|B|)log (1/ε)) time complexity
Exhaustive search over ε-nets on A and B! Slide29
Hardness Results for BSS
(
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|log1-μ|
B|||M||∞)) time algorithm for any
ν+μ>0 and constant ε,
unless ETH fails
Estimate with additive error
ε
Slide30
Main Result 1: Weak Membership
(B.,
Christandl, Yard ‘10) There is a exp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*|
LOCC) Slide31
Main Result 1: Weak Membership
(B.,
Christandl, Yard ‘10) There is a exp(O(ε-2log|A|log|B|)) time 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 f
or ε = 1/
polylog(|A||B|), unless ETH failsSlide32
Main Result 2: Best Separable State
(BCY ‘10)
There is a exp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε)For M in LOCC, there is a
exp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε)Slide33
Main Result 2: Best Separable State
(BCY ‘10)
There is a exp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε)For M in LOCC, there is a
exp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε)
Contrast with:(Harrow, Montanaro ‘10)
No exp(O(log
1-ν|A|log1-μ|B
|||M||∞)) time algorithm for any
ν+μ>0 and constant ε,
unless ETH fails, even for separable M: . Remember: Part 2 works for Slide34
Main Result 2: Best Separable State
(BCY ‘10)
There is a exp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε)For M in LOCC, there is a
exp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε)
Contrast with:(Harrow, Montanaro ‘10)
No exp(O(log
1-ν|A|log1-μ|B
|||M||∞)) time algorithm for any
ν+μ>0 and constant ε,
unless ETH fails, even for separable M: . Remember: Part 2 works for
Quantum Info Remark:
The difficulty to show optimality of the algorithm is the existence of separable measurements that are
not
LOCC, a well studied phenomena in quantum information (e.g.
Bennett et al ‘98
). Here we have a new computational-complexity motivation for further studying the problem!Slide35
The Algorithm
We consider the a
Parrilo-Lasserre hierarchy of SDP relaxations to the problem introduced in (Doherty, Parrilo and Spedalieri ’01)We prove it converges to a good approximate solution in a O(log|B
|) number of rounds. Previously convergence only in Ω(|B|) rounds was known.Slide36
Optimization O
ver Separable States (again)
(Best Separable State BSS(ε)) Given estimate to additive error
ε This is a polynomial optimization problem. One can calculate a sequence of SDP approximations to it following the approach of (Parrilo ‘00, Lasserre ’
01) We’ll derive the SDP hierachy by a quantum argument Slide37
Entanglement Monogamy
Classical correlations are shareable
: Given separable state Consider the symmetric extension
Def. ρAB is k-extendible if there is ρAB1…Bk s.t
for all j in [k], tr\ Bj
(ρAB1
…Bk) = ρAB
A
B
1
B
2
B
3
B
4
B
k
…Slide38
Entanglement Monogamy
Classical correlations are shareable
:Def. ρ
AB is k-extendible if there is ρAB1…Bk s.t for all j in [k], tr
\ Bj (ρAB1
…Bk
) = ρAB
Separable states are k-extendible for every k
A
B
1
B
2
B
3
B
4
B
k
…Slide39
Entanglement Monogamy
Quantum correlations are non-shareable:
ρAB separable iff ρAB k-extendible for all k Follows from: Quantum de
Finetti Theorem (Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’
89)Monogamy of entanglement: Very useful concept in general, application e.g. in quantum key distributionSlide40Slide41
Entanglement Monogamy
Quantitative version
: For any k-extendible ρAB,
Follows from: Finite quantum de Finetti
Theorem (Christandl, König
, Mitchson
, Renner ‘05) Slide42
Entanglement Monogamy
Quantitative version
: For any k-extendible ρAB,
Follows from: Finite quantum de Finetti
Theorem (Christandl, König
, Mitchson, Renner ‘05)
Close to optimal: there is a k-
ext state ρAB s.t.
For
other norms (||*||2, ||*||LOCC
, …) no better bound known. Slide43
Exponentially Improved de
Finetti
type bound(B., Christandl, Yard ‘10) For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC
Bound proportional to the (square root) of the
number of qubits: exponential improvement over previous boundSlide44
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)Can be solved in poly(n) time in the number of variables n
n = |A|2|B|2kOur bound implies k = O(ε-2log|A|)
Time Complexity: poly(|A||B|2k) = exp(O(
ε-2log|A
|log|B|))Slide45
Does it work for 1-norm?
There are
k-extendible states s.t.For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential timeBut we know also:So, hard instances
are always “data hiding” states, i.e.Slide46
Does it work for 1-norm?
There are
k-extendible states s.t.For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential timeBut we know also:So, hard instances
are always “data hiding” states, i.e.Slide47
Does it work for 1-norm?
There are
k-extendible states s.t.For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential timeBut we know also:So, hard instances
are always “data hiding” states, i.e.Slide48
Does it work for 1-norm?
There are
k-extendible states s.t.For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential timeBut we know also:So, hard instances
are always “data hiding” states, i.e.Slide49
Algorithm for Best Separable State
The idea
Optimize over k=O(log|A|ε-2 (||X||2)2) extension of ρ
AB by SDPThis is precisely the Parrilo-Lasserre hierarchy for the problem! (written in a somewhat different form)By Cauchy Schwartz: By de F
inetti Bound: Slide50
Application 1: Quantum NPSlide51
QMASlide52
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(Marriott, Watrous ‘05) Exponential amplification with fixed proof size(Beigi
, Shor, Watrous ‘09) logarithmic size interaction doesn’t helpSlide53
New Characterization QMA
Corollary
QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurementsin the k proofsSlide54
New Characterization QMA
Corollary
QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurementsin the k proofs
Def QMAm(k): analogue of QMA with k proofs and proof size
mSlide55
New Characterization QMA
Corollary
QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurementsin the k proofs
Def QMAm(k): analogue of QMA with k proofs and proof size
mDef
LOCCQMAm(k)
: analogue of QMA with k proofs, proof size m
and LOCC verification procedure along the k proofs.Slide56
QMA(k)
Def
QMAm(k): A language L is in QMAm(k) if there is a quantum poly-time circuit that for every instance x implements the measurement {Ax, I - Ax} such thatCompleteness: If x in L, there exists k proofs, each of m qubits, s.t.
Soundness: If x not in L, for any k states, Def
2 LOCCQMAm(k): Likewise, but {Ax, I - Ax
} must be LOCCSlide57
New Characterization 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
And:
SAT has a LOCCQMAO(log(n))(n1/2
) protocol* Quantum ETH:
SAT cannot be solved in 2o(n)
quantum time(Harrow and Montanaro
’
10) -- based on Aaronson et al ‘08
(Chen and
Drucker
’
10)Slide58
New Characterization 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 QMAn
1/2(2) protocol for SAT with n clauses
And: SAT has a LOCCQMAO(log(n))(n
1/2) protocol
* Quantum ETH: SAT cannot be solved in 2o(n)
quantum
time
(
Harrow and
Montanaro
’
10 – built on Aaronson et al
’
08)
(Chen and
Drucker
’
10)Slide59
New Characterization QMA
Corollary
QMA = LOCCQMA(k), k = O(1) LOCCQMAm(2) contained in QMAO(m2)
Idea to simulate LOCCQMAm
(2) in QMA:Arthur asks for proof ρ on
AB1B
2…Bk with
k = mε-2
He symmetrizes the B
systems and applies the original verification prodedure to AB1
Correcteness
de Finetti bound implies: Slide60
Application 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 ConjectureSlide61
Application 2: Small Set Expansion
(Barak, B., Harrow,
Kelner, Steurer, and Zhou ‘12) connection 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 subexponential-time algorithm for the small set expansion problem, matching the performance of the algorithm of (
Arora, Barak and Steurer ‘10)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 ConjectureSlide62
Proof Techniques
Coding Theory
Strong subadditivity of von Neumann entropy as state redistribution rate (Devetak, Yard ‘06)Large Deviation Theory
Hypothesis testing of separable states (B., Plenio ‘08)Entanglement Measure Theory Squashed Entanglement (Christandl
, Winter ’04)Slide63
I(A:B|E)
Conditional Mutual Information:
Measures the correlations of A and B relative to E in ρABE I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ
Always positive: I(A:B|E)ρ ≥ 0 (strong-subadditivity of entropy)When does it vanish? I(A:B|E)ρ = 0
iff ρABE is a “Quantum Markov Chain State”
E.g.
Approximate version??? …
(Hayden,
Jozsa
, Petz
, Winter ‘04)
(Lieb, Ruskai ‘73)Slide64
New Inequality for I(A:B|E)
Thm
: (B., Christandl, Yard ’10)
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)Slide65
Summary
Testing
separability is rather easyFamily of Parrilo-Lasserre SDP relaxations 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 robustSlide66
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 applications
of the bound on the convergence of the SDP relaxation? Can we prove a quasipolynomial time 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 application
of the inequality for
I(A:B|E)? Slide67
Thank you!Slide68
Proof OutlineSlide69
Relative Entropy of Entanglement
The proof is largely based on the properties of the following
entanglement measure:Def Relative Entropy of Entanglement (Vedral, Plenio ‘99)Slide70
Entanglement Hypothesis Testing
Given (many copies) of
ρAB, what’s the optimal probability of distinguishing it from a separable state?Slide71
Entanglement Hypothesis Testing
Given (many copies) of
ρAB, what’s the optimal probability of distinguishing it from a separable state?
Def Rate Function: D(ρAB) is maximum number r s.t. there exists {Mn
, I-Mn} , 0 < Mn < I,
D
LOCC
(ρ
AB) : defined analogously, but now {M, I-M} must be
LOCCSlide72
Entanglement Hypothesis Testing
Given (many copies) of
ρAB, what’s the optimal probability of distinguishing it from a separable state?
Def Rate Function: D(ρAB) is maximum number r s.t there exists {Mn
, I-Mn} , 0 < Mn < I,
D
LOCC
(ρ
AB) : 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 operations (B.,
Plenio
‘08)Slide73
Proof in 1 LineSlide74
Proof in 1 Line
(i) Quantum Shannon Theory: State redistribution Protocol
(ii) Large Deviation Theory: Entanglement Hypothesis Testing (iii) Entanglement Theory: Faithfulness bounds Relative entropy of Entanglement plays a triple role:
(Devetak and Yard ‘07)
(B. and
Plenio ‘08)Slide75
First Inequality
Non-
lockability:
(Horodecki3 and Oppenheim ‘04)State Redistribution: How much does it cost to redistribute a quantum system?
AB
E
F
A
E
B
F
½ I(A:B|E)
Proof (
i
):
Apply
non-
lockability
to and use
state redistribution
to trace out B at a rate of ½ I(A:B|E)
qubits
per copy Slide76
Second Inequality
Equivalent to:
Monogamy relation for entanglement hypothesis testing
Proof (ii)
Use optimal measurements for ρAE
and ρAB
achieving D(ρAE
) and DLOCC(ρ
AB), resp., to construct a measurement
for ρA:BE achieving D(
ρA:BE)Slide77
Third Inequality
Pinsker
type inequality for entanglement hypothesis testing Proof (iii) minimax theorem + martingale like property
of the set of separable statesSlide78
Thank you!