Monogamy of Quantum
Entanglement
Fernando G.S.L. Brandão
ETH Zürich
Princeton EE Department, March 2013
Simulating quantum is hard
More than 25% of DoE
supercomputer power is devoted
to simulating quantum physics
Can we get a better handle on this
simulation problem?
Simulating quantum is hard,
secrets are hard to conceal
More than 25% of DoE
supercomputer power is devoted
to simulating quantum physics
All current cryptography is
based on unproven hardness
assumptions
Can we get a better handle on this
simulation problem?
Can we have better security
guarantees for our secrets?
Quantum Information 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 Computation:
Quantum Cryptography:
Use of engineered quantum
systems for performing
computation
Use of engineered quantum
systems for secret key
distribution
Exponential speed-ups over
classical computing
Unconditional security based
solely on the correctness of
quantum mechanics
E.g. Factoring (RSA)
Simulating quantum systems
The Two Holy Grails of QIS
Quantum Computation:
Quantum Cryptography:
Use of well
controlled quantum
State-of-the-art:
systems
for performing
+-5 qubits
computer
computations.
Use of engineered quantum
systems for secret key
distribution
Can prove (with high
Exponential
probability)speed-ups
that 15 = 3over
x5
classical computing
Unconditional security based
solely on the correctness of
quantum mechanics
The Two Holy Grails of QIS
Quantum Computation:
Quantum Cryptography:
Use of well
controlled quantum
State-of-the-art:
systems
for performing
+-5 qubits
computer
computations.
Use of well
controlled quantum
State-of-the-art:
systems
for secret
+-100
km key
distribution
Can prove (with high
Exponential
probability)speed-ups
that 15 = 3over
x5
classical computing
Still many technological
Unconditional
security based
challenges
solely on the correctness of
quantum mechanics
While waiting 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 communication, quantum algorithms, quantum fault
tolerance, quantum simulators, …)
While waiting 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 communication, quantum algorithms, quantum fault
tolerance, quantum simulators, …)
Answer 2: finding applications of QIS independent of building a
quantum computer
While waiting 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 communication, quantum algorithms, quantum fault
tolerance, quantum simulators, …)
Answer 2: finding applications 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:
y := (y1,..., yd )
T
y 2=
y y =1
Dynamics: L2 norm preserving linear maps,
unitary matrices: UUT = I
Observation: To any experiment with d outcomes we
associate d PSD matrices {Mk} such that ΣkMk=I
Born’s rule:
Pr(k) = y M k y
Quantum Mechanics II
probability theory based on PSD matrices
Mixed States: PSD matrix of unit trace:
r ³ 0, tr ( r ) =1
r = å pi yi yi
i
Dirac notation reminder:
y := (y ,..., y )
*
1
*
d
Dynamics: linear operations mapping density matrices to
density matrices (unitaries + adding ancilla + ignoring subsystems)
Born’s rule:
Pr(k) = tr ( r M k )
Quantum Entanglement
y
Pure States:
If
y
=
f
AB
Î
C
Ä
C
AB
d
Ä
j
A
B
l
, it’s separable
otherwise, it’s entangled.
Mixed States:
If
rAB Î D(C Ä C )
d
l
r = å pi yi yi Ä fi fi
i
otherwise, it’s entangled.
, it’s separable
A Physical Definition of
Entanglement
LOCC: Local quantum Operations and Classical Communication
Separable states can be created by LOCC:
r = å pi yi yi Ä fi fi
i
Entangled states cannot be created by LOCC:
non-classical correlations
Quantum Features
Superposition principle
interference
uncertainty principle
non-locality
Monogamy of Entanglement
Monogamy of Entanglement
Maximally Entangled State: f
AB
Correlations of A and B in f f
If ρABE is is s.t. ρAB = f f
AB
= ( 00 + 11 ) / 2
AB
cannot be shared:
, then ρABE = f f
AB
Ä rE
Only knowing that the quantum correlations 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 Distribution…
… as an application of entanglement monogamy
(Bennett, Brassard 84, Ekert 91)
- Using insecure channel Alice and Bob share entangled state ρAB
- Applying LOCC they distill f = ( 00 + 11 ) / 2
AB
Quantum Key Distribution…
… as an application of entanglement monogamy
(Bennett, Brassard 84, Ekert 91)
- Using insecure channel Alice and Bob share entangled state ρAB
- Applying LOCC they distill f = ( 00 + 11 ) / 2
AB
Let πABE be the state of Alice, Bob and the Eavesdropper.
Since πAB = f f , πAB = f f Ä p E. Measuring A and B
AB
AB
in the computational basis give 1 bit of secret key.
Outline
Entanglement Monogamy
How general is the concept?
Can we make it quantitative?
Are there other applications/implications?
1. Detecting Entanglement and Sum-Of-Squares Hierarchy
2. Accuracy of Mean-Field Approximation
3. Other Applications
Quadratic vs Biquadratic Optimization
Problem 1: For M in H(Cd) (d x d matrix) compute
max x =1 x T Mx = max x =1 å M ij xi x*j
Very Easy!
i, j
Problem 2: For M in H(Cd Cl), compute
max x = y =1 (x Ä y)T M (x Ä y) = max x = y =1 å M ij;kl xi x *j yk yl*
ijkl
Next:
Best known algorithm (and best hardness result) using
ideas from Quantum Information Theory
Quadratic vs Biquadratic Optimization
Problem 1: For M in H(Cd) (d x d matrix) compute
max x =1 x T Mx = max x =1 å M ij xi x*j
Very Easy!
i, j
Problem 2: For M in H(Cd Cl), compute
max x = y =1 (x Ä y)T M (x Ä y) = max x = y =1 å M ij;kl xi x *j yk yl*
ijkl
Next:
Best known algorithm using monogamy of entanglement
The Separability Problem
•
Given
rAB Î D(C Ä C )
d
l
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
•
Given
rAB Î D(C Ä C )
d
l
is it entangled?
•
(Weak Membership: WSEP(ε, ||*||)) Given ρAB
determine if it is separable, or ε-away from SEP
Frobenius norm
X 2 = tr ( X X )
T
SEP
ε
D
1/2
The Separability Problem
•
Given
rAB Î D(C Ä C )
d
l
is it entangled?
•
(Weak Membership: WSEP(ε, ||*||)) Given ρAB
determine if it is separable, or ε-away from SEP
Frobenius norm
X 2 = tr ( X X )
T
SEP
ε
D
1/2
LOCC norm
r -s
LOCC
=
max tr(M (r - s ))
M ÎLOCC
The Separability Problem
•
Given
rAB Î D(C Ä C )
d
l
is it entangled?
•
(Weak Membership: WSEP(ε, ||*||)) Given ρAB
determine if it is separable, or ε-away from SEP
•
Dual Problem: Optimization over separable states
hSEP (M ) := max tr(Ms ) = max (x Ä y)T M(x Ä y)
s ÎSEP
x = y =1
The Separability Problem
•
Given
rAB Î D(C Ä C )
d
l
is it entangled?
•
(Weak Membership: WSEP(ε, ||*||)) Given ρAB
determine if it is separable, or ε-away from SEP
•
Dual Problem: Optimization over separable states
hSEP (M ) := max tr(Ms ) = max (x Ä y)T M(x Ä y)
s ÎSEP
•
x = y =1
Relevance: Entanglement is a resource in quantum
cryptography, quantum communication, etc…
Previous Work
When is ρAB entangled?
- Decide if ρAB is separable or ε-away from separable
Beautiful theory behind it (PPT, entanglement witnesses, etc)
Horribly expensive algorithms
dim of A
State-of-the-art: 2O(|A|log|B|) time complexity
Same for estimating hSEP (no better than exaustive search!)
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|)) time algorithm with ε=Ω(1)
and any ν+μ>0 for unless ETH fails
ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time
(Impagliazzo&Paruti ’99)
Quasipolynomial-time Algorithm
(B., Christandl, Yard ’11) There is an algorithm with run time
exp(O(ε-2log|A|log|B|)) for WSEP(||*||, ε)
Quasipolynomial-time Algorithm
(B., Christandl, Yard ’11) There is an algorithm with run time
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|)
Quasipolynomial-time Algorithm
(B., Christandl, Yard ’11) For M in 1-LOCC, can compute hSEP(M)
within additive error ε in time exp(O(ε-2log|A|log|B|))
M in 1-LOCC: M =
åA
k
k
Ä Bk , Ak , Bk ³ 0, Bk £ I, å Ak £ I
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 = å Ak Ä Bk , Ak , Bk ³ 0, M £ I
k
Algorithm: SoS Hierarchy
h (M ) = max
åM x x y y
SEP
*
*
ij;kl i j k l
x = y =1
ijkl
Polynomial optimization over hypersphere
Sum-Of-Squares (Parrilo/Lasserre) hierarchy:
gives sequence of SDPs that approximate hSEP(M)
- Round k takes time dim(M)O(k)
- Converge to hSEP(M) when k -> ∞
SoS is the strongest SDP hierarchy known for
polynomial optimization (connections 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
åM x x y y
SEP
*
*
ij;kl i j k l
x = y =1
ijkl
Polynomial optimization 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
åM x x y y
SEP
*
*
ij;kl i j k l
x = y =1
ijkl
Polynomial optimization 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 optimization (connections with SoS proof system,
real algebraic geometric, Hilbert’s 17th problem, …)
We’ll derive SoS hierarchy exploring ent. monogamy
Classical Correlations are Shareable
Given separable state s AB = å p j y j y j Ä j j j j
j
Consider the symmetric extension
s AB1,...,Bk = å p j y j y j Ä j j j j
Äk
A
j
Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all
j in [k], tr\ Bj (ρAB1…Bk) = ρAB
B1
B2
B3
B4
…
Bk
Monogamy Defines Entanglement
(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)
ρAB separable iff ρAB is k-extendible for all k
SoS as optimization over
k-extensions
(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is
equivalent to optimization over k-extendible states
(plus PPT (positive partial transpose) test):
maxtr(M p ) : $ s AB1...Bk ³ 0, tr(s ) =1, s ABj = p "j
How close to separable are
k-extendible states?
(Koenig, Renner ‘04) De Finetti Bound
r AB - s AB
If ρAB is k-extendible smin
ÎSEP
2 ö
æ
B
÷
£ Wç
ç k ÷
è
ø
Improved de Finetti Bound (B., Christandl, Yard ’11)
If ρAB is k-extendible:
min r AB - s AB
s ÎSEP
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
How close to separable are
k-extendible states?
(Koenig, Renner ‘04) De Finetti Bound
r AB - s AB
If ρAB is k-extendible smin
ÎSEP
2 ö
æ
B
÷
£ Wç
ç k ÷
è
ø
Improved de Finetti Bound (B., Christandl, Yard ’11)
If ρAB is k-extendible:
min r AB - s AB
s ÎSEP
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
How close to separable are
k-extendible states?
Improved de Finetti Bound
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
min r AB - s AB
If ρAB is k-extendible:
s ÎSEP
k=2ln(2)ε-2log|A| rounds of SoS
solves WSEP(ε) with a SDP of size
SEP
|A||B|k = exp(O(ε-2log|A| log|B|))
ε
D
Proving…
Improved de Finetti Bound
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
If ρAB is k-extendible:
min r AB - s AB
s ÎSEP
Proof is information-theoretic
Mutual Information:
I(A:B)ρ := H(A) + H(B) – H(AB)
H(A)ρ := -tr(ρ log(ρ))
Proving…
Improved de Finetti Bound
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
If ρAB is k-extendible:
min r AB - s AB
s ÎSEP
Proof is information-theoretic
Mutual Information:
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<k: I(A:Bl|B1…Bl-1) < log|A|/k
Proving…
Improved de Finetti Bound
æ 4 ln 2 log A ö
£ç
÷
k
è
ø
1/2
If ρAB is k-extendible:
min r AB - s AB
s ÎSEP
Proof is information-theoretic
Mutual Information:
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<k: I(A:Bl|B1…Bl-1) < log|A|/k
What does it imply?
Quantum Information?
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 Information?
Nature isn't classical, dammit, and if
you want to make a simulation of information 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
• 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)
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)
Proving the Bound
Thm (B., Christandl, Yard ’11)
1
I(A : B | C) ³
min r AB - s AB
2 ln 2 s ÎSEP
2
Obs: I(A:B|E)≥0 is strong subadditivity 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
For ρAB k-extendible: 2 log A ³
min r AB - s AB
2 ln 2 s ÎSEP
2
Conditional Mutual Information
Bound
1
I(A : B | C) ³
min r AB - s AB
2 ln 2 s ÎSEP
• Coding Theory
Strong subadditivity of von Neumann
entropy as state redistribution rate
(Devetak, Yard ‘06)
• Large Deviation Theory
Hypothesis testing for entanglement
(B., Plenio ‘08)
(
I(A : B | C)r ³ER¥ (r A:BE ) - ER¥ (r A:E )³ D1-LOCC (r A:B )³
1
min r A:B - s
s
2 ln 2 ÎSEP
2
1-LOCC
2
)
hSEP equivalent to
1.
Injective norm of 3-index tensors
2.
Minimum output entropy quantum channel,
Optimal acceptance probability in QMA(2), ….
4. 2->q norms of projectors: P
P
2®4
2®q
:= max x
=1
2
Px
q
= hSEP (M ), M = å( P k k P ) Ä ( P k k P)
k
(||*||2->q for q > 2 are hypercontractive norms: useful in
Markov Chain Monte Carlo (rapidly mixing condition), 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.
A
4
2®4
£x£ A
4
2®4
+e A
2
2®2
A
2
2®¥
2. nO(ε) -level SoS solves ε-Small Set Expansion Problem
(closely related to unique games).
Alternative algorithm to (Arora, Barak, Steurer ’10)
3. Open Problem:
Improvement in bound (additive -> multiplicative error) would
solve SSE in exp(O(log2(n))) time.
Can formulate it as a quantum information-theoretic 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.
A
4
2®4
£x£ A
4
2®4
+e A
2
2®2
A
2
2®¥
2. nO(ε) -level SoS solves ε-Small Set Expansion Problem
(closely related to unique games).
Alternative algorithm to (Arora, Barak, Steurer ’10)
3. Open Problem:
Improvement in bound (additive -> multiplicative error) would
solve SSE in exp(O(log2(n))) time.
Can formulate it as a quantum information-theoretic 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.
A
4
2®4
£x£ A
4
2®4
+e A
2
2®2
A
2
2®¥
2. nO(ε) -level SoS solves ε-Small Set Expansion Problem
(closely related to unique games).
Alternative algorithm to (Arora, Barak, Steurer ’10)
3. Open Problem:
Improvement in bound (additive -> multiplicative error) would
solve SSE in exp(O(log2(n))) time.
Can formulate it as a quantum information-theoretic problem
Outline
Entanglement Monogamy
How general is the concept?
Can we make it quantitative?
Are there other applications/implications?
1. Detecting Entanglement and Sum-Of-Squares Hierarchy
2. Accuracy of Mean Field Approximation
3. Other applications
Constraint Satisfaction Problems vs
Local Hamiltonians
k-arity CSP:
Variables {x1, …, xn}, alphabet Σ
Constraints: c j
Assignment: s
: S ® {0,1}
k
:[n] ® S
Unsat := min å c j (s (x j1 ),..., s (x jk ))
s
j
Constraint Satisfaction Problems vs
Local Hamiltonians
qudit
H1
k-arity CSP:
k-local Hamiltonian H:
Variables {x1, …, xn}, alphabet Σ
n qudits in (C d )
Constraints: c j
Assignment: s
: S ® {0,1}
k
:[n] ® S
Unsat := min å c j (s (x j1 ),..., s (x jk ))
s
j
Än
(
Constraints: H j Î Her (C
æ
ö
qUnsat := E0 çç å H j ÷÷
è j
ø
E0 : min eigenvalue
)
d Äk
)
C. vs Q. Optimal Assignments
Finding optimal assignment of
CSPs can be hard
C. vs Q. Optimal Assignments
Finding optimal assignment of
CSPs can be hard
Finding optimal assignment of
quantum CSPs can be even harder
(BCS, Laughlin states for FQHE,…)
Main difference: Optimal Assignment can be a
d Än
highly entangled state (unit vector in (C ) )
Optimal Assignments:
Entangled States
Non-entangled state:
(a
)
(
0 + b1 1 Ä ... Ä an 0 + bn 1
1
e.g. - Ä ... Ä Entangled states:
e.g.
(- ¯ - ¯ - ) /
åc
i1 ,...,in
i1 ,...,in
i1 ,...,in
2
To describe a general entangled state of n spins
requires exp(O(n)) bits
)
How Entangled?
Given bipartite entangled state y
the reduced state on A is mixed:
AB
Î Cn Ä Cm
r A ³ 0, tr(r A ) =1
The more mixed ρA, the more entangled ψAB:
Quantitatively: E(ψAB) := S(ρA) = -tr(ρA log ρA)
Is there a relation between the amount of entanglement in
the ground-state and the computational complexity of the
model?
Mean-Field…
…consists in approximating groundstate
by a product state y1 Ä… Ä yn
max y1,… , yn H y1,… , yn
y1,… ,yn
Successful heuristic in
Folklore:
Mean-Field good when
is a CSP
Quantum Chemistry (e.g. Hartree-Fock)
Condensed matter (e.g. BCS theory)
Many-particle interactions
Low entanglement in state
Mean-Field Good When
(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction
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 interaction
graph G(V, E) and |E| local terms.
Let {Xi} be a partition of the sites with each Xi having m sites.
X1
X2
m < O(log(n))
X3
Mean-Field Good When
(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 products states ψi in Xi s.t.
1
y1 ,...,ym H y1 ,...,ym
|E|
deg(G) : degree of G
Ei
: expectation over Xi
S(Xi) : entropy of
groundstate in Xi
æ 6 1
S( X i ) ö
£ e0 (H ) + Wç d
Ei
÷
m ø
è deg(G)
X1
1/8
X2
m < O(log(n))
X3
Approximation in terms of degree
…shows mean field becomes exact in high dim
1-D
2-D
3-D
∞-D
Approximation in terms of degree
1
y1 ,...,ym H y1 ,...,ym
|E|
æ 6 1
S( X i ) ö
£ e0 (H ) + Wç d
Ei
÷
m ø
è deg(G)
1/8
Relevant to the quantum PCP problem:
(Aharonov, Arad, Landau, Vazirani ’08; Freedman, Hastings ‘12; Bravyi,
DiVincenzo, Loss, Terhal ’08, Aaronson ‘08)
Is approximating e0(H) to constant accuracy
quantum NP-hard?
It shows that quantum generalizations of PCP theorem
and parallel repetition cannot both be true
(assuming “quantum NP”= QMA ≠ NP)
Approximation in terms of average
entanglement
1
y1 ,...,ym H y1 ,...,ym
|E|
æ 6 1
S( X i ) ö
£ e0 (H ) + Wç d
Ei
÷
m ø
è deg(G)
1/8
Mean-field works well if entanglement
of groundstate satisfies a
subvolume law:
Connection of amount of
entanglement in groundstate
and computational
complexity of the model
X1
S( X i )
Ei
= o(1)
m
m < O(log(n))
X2
X3
Approximation in terms of average
entanglement
1
y1 ,...,ym H y1 ,...,ym
|E|
æ 6 1
S( X i ) ö
£ e0 (H ) + Wç d
Ei
÷
m ø
è deg(G)
1/8
Systems with low entanglement are expected to be simpler
So far only precise in 1D:
Area law for entanglement -> Succinct classical description
(MPS)
Here:
Good: arbitrary lattice, only subvolume law
Bad: Only mean energy approximated well
New Algorithms for Quantum
Hamiltonians
Following same approach we also obtain 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.
Proof Idea: Monogamy of
Entanglement
Cannot be highly entangled
with too many neighbors
Xi
Entropy S(Xi) quantifies how
entangled it can be
Proof uses information-theoretic techniques (chain rule of
conditional mutual information, informationally complete
measurements, etc) to make this intuition precise
Inspired by classical information-theoretic 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 quantitative?
Are there other applications/implications?
1. Detecting Entanglement and Sum-Of-Squares Hierarchy
2. Accuracy of Mean Field Approximation
3. Other applications
Other Applications
(B., Horodecki ‘12)
Proving entanglement area law from exponential decay
of correlations for 1D states
(Almheiri, Marolf, Polchinski, Sully ’12)
Black hole firewall controversy
(Vidick, Vazirani ’12; Bartett, Colbeck, Kent ’12; …)
Device independent key distribution
(Vidick and Ito ‘12)
Entangled multiple proof systems: NEXP in MIP*
Conclusions
Entanglement Monogamy is a distinguishing feature of
quantum correlations
(quantum engineering)
It’s at the heart of quantum cryptography
(convex optimization)
Can be used to detect entanglement efficiently and understand
Sum-Of-Squares hierarchy better
(computational physics)
Can be used to bound efficiency of mean-field approximation
(quantum many-body physics)
Can be used to prove area law from exponential decay of
correlations
Thank you!