Information-Theoretic Techniques in
Many-Body Physics
Day 1
Fernando G.S.L. Brandão
UCL
Based on joint work with A. Harrow and M. Horodecki
New Mathematical Directions for Quantum Info
Quantum Many-Body Systems
l
H = å H i Î (C
Quantum Hamiltonian
i=1
)
d Än
, H i £1
n
Cd
Hi
Interested in computing properties such as minimum
energy, correlations functions at zero and finite
temperature, dynamical properties, …
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
)
Classical vs Quantum Optimal
Assignments
Finding optimal assignment of CSPs is usually hard
(NP-hard)
Finding optimal assignment of quantum CSPs
(groundstates) seems even harder
(QMA-hard; See Daniel Nagaj’s talk)
Main difference: Optimal Assignment can be a
d Än
highly entangled state (unit vector in (C ) )
The Plan
Today: Product-State Approximations to Groundstates
- de Finetti theorem
- information theory approach
(entropies, chain rule, Pinsker’s inequality, info-complete meas., …)
Tomorrow: Groundstates in 1D
- area laws and matrix product states
- information theory approach
(decoupling, state merging, single-shot protocols, …)
Approximation Scale
We want to approximate the minimum energy
(i.e. minimum eigenvalue of H):
today
Small total error:
Small extensive error:
Mean-Field…
…consists in approximating the groundstate
by a product state y1 Ä… Ä yn
max å y1,… , yn H j y1,… , yn is a CSP
y ,… ,y
1
n
j
Mean-Field…
…consists in approximating the groundstate
by a product state y1 Ä… Ä yn
max å y1,… , yn H j y1,… , yn is a CSP
y ,… ,y
1
n
j
Mean-Field…
…consists in approximating the groundstate
by a product state y1 Ä… Ä yn
max å y1,… , yn H j y1,… , yn is a CSP
y ,… ,y
1
n
j
It’s a mapping from quantum Hamiltonians to CSPs
Successful heuristic in
Intuition:
Mean-Field good when
Quantum Chemistry (Hartree-Fock)
Condensed matter (e.g. BCS theory)
Many-particle interactions
Low entanglement in state
Hamiltonian on the Complete
Graph
Consider a Hamiltonian on the complete graph G of size n
Hij
The Hamiltonian is permutation symmetric:
with
Quantum de Finetti Theorems
(Stormer ’69, Hudson, Moody ’76)
Infinite Quantum de Finetti Theorem
(remember Graeme
Mitchison’s talk)
(Raggio, Werner ’89)
Connection of Infinite Quantum de Finetti with Mean-Field
(Caves, Fuchs, Sachs ’01)
Proof infinite de Finetti using info-complete measurements
(Koenig, Renner ’05)
Finite Quantum de Finneti Theorem
(Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a
permutation-symmetric state. Then
Product-States Approximation and
de Finetti Theorem
(Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a
permutation-symmetric state. Then
Product-States Approximation and
de Finetti Theorem
(Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a
permutation-symmetric state. Then
By de Finetti:
Product-States Approximation and
de Finetti Theorem
(Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a
permutation-symmetric state. Then
By de Finetti:
So
Product-states achieve error 2d2/n for mean-energy
The Role of Permutation
Symmetry
To apply quantum de Finetti we need a permutation-invariant
Hamiltonian.
Can we relax this assumption?
Can we show product states do a good job for models
not on the complete graph?
Product-State Approximation
without Symmetry
(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.
Ei
Deg
S(Xi)
: expectation over Xi
: degree of G
: entropy of
groundstate in Xi
X1
X2
size m
Product-State Approximation
without Symmetry
(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 states ψi in Xi s.t.
Ei
Deg
S(Xi)
: expectation over Xi
: degree of G
: entropy of
groundstate in Xi
X1
X2
size m
Approximation in terms of degree
Implications to the quantum PCP problem (whether to compute
is QMA-hard ):
Shows that attempts to quantize Dinur’s proof of the PCP theorem
cannot work. Also gives a no-go for “quantum PCP + parallel
repetition of qCSP”
Approximation in terms of degree
Implications to the quantum PCP problem (whether to compute
is QMA-hard ):
Shows that attempts to quantize Dinur’s proof of the PCP theorem
cannot work. Also gives a no-go for “quantum PCP + parallel
repetition of qCSP”
Bound: ΦG < ½ - Ω(1/deg) implies product states work well on
highly expanding graphs (ΦG -> ½)
Obs: Restricted to 2-local models
(Aharonov, Lior ‘13) k-local, commuting models
Approximation in terms of degree
…shows mean field becomes exact in high dim
∞-D
1-D
2-D
3-D
See (Cirac, Kraus, Lewenstein) for
rotationally invariant systems
Approximation in terms of average
entanglement
Product-states do a good job if entanglement of
groundstate satisfies a subvolume law:
m < O(log(n))
X1
X2
X3
Approximation in terms of average
entanglement
If
states give error
, Pinsker’s inequality shows product
Approximation in terms of average
entanglement
If
states give error
, Pinsker’s inequality shows product
In constrast, if merely
shows product states give error
, the theorem
When does it fail?
E.g.
I - EPR EPR
Expander graph G(V, E)
with expansion ΦG
Intuition: Monogamy of
Entanglement
Quantum correlations are non-shareable
(see Aram Harrow’s and Thomas Vidick’s talks)
Cannot be highly entangled
with too many neighbors
S(Xi) quantifies how much
entangled Xi can be with the
rest
Intuition: Monogamy of
Entanglement
Quantum correlations are non-shareable
(see Aram Harrow’s and Thomas Vidick’s talks)
Cannot be highly entangled
with too many neighbors
S(Xi) quantifies how much
entangled Xi can be with the
rest
Proof uses information-theoretic techniques to make this
intuition precise
Inspired by classical information-theoretic ideas for bounding
convergence of Sum-Of-Squares hierarchy for CSPs
(Tan, Raghavendra ’10; Barak, Raghavendra, Steurer ‘10)
Mutual Information
1. Mutual Information
I( X :Y ) = D( pXY || pX Ä pY )
1. Pinsker’s inequality
1
I ( X :Y ) =
pXY - pX Ä pY
2ln 2
1. Conditional MI
1. Chain Rule
2
1
I(X :Y | Z) = I(X :YZ) - I(X : Z)
I(X :Y1… Yk ) = I(X :Y1 ) +… + I(X :Yk |Y1… Yk-1 )
5. Upper bound
4+5
Þ I( X :Yt |Y1… Yt-1 ) £ log(| X |) / k
for some t ≤ k
Quantum Mutual Information
1. Mutual Information
I( X :Y ) = D(r XY || r X Ä rY )
1. Pinsker’s inequality
1
I ( X :Y ) =
r XY - r X Ä rY
2ln 2
1. Conditional MI
1. Chain Rule
2
1
I(X :Y | Z) = I(X :YZ) - I(X : Z)
I(X :Y1… Yk ) = I(X :Y1 ) +… + I(X :Yk |Y1… Yk-1 )
5. Upper bound
4+5
Þ I( X :Yt |Y1… Yt-1 ) £ log(| X |) / k
for some t ≤ k
But…
…conditioning on quantum is problematic
For X, Y, Z random variables
No similar interpretation is known for I(X:Y|Z) with quantum Z
Conditioning Decouples
Idea that almost works. Suppose we have a distribution p(z1,…,zn)
1. Choose i, j1, …, jk at random from {1, …, n}.
Then there exists t<k such that
Define
j1
So
i
jk
j2
Conditioning Decouples
2. Conditioning on subsystems j1, …, jt causes, on average,
error <k/n and leaves a distribution q for which
, and so
By Pinsker:
Choosing k = εn
jt
j1
j2
Informationally Complete
Measurements
There exists a POVM M(ρ) = Σk tr(Mkρ) |k><k|
s.t. for all k and ρ1…k, σ1…k in D((Cd)k)
(18d)
-k /2
r1… k - s 1… k £ M ( r1… k ) - M (s 1… k )
Äk
1
(Lacien, Winter ‘12, Montanaro ‘12)
Äk
1
Proof Overview
1. Measure εn qudits with M and condition on outcomes.
Incur error ε.
2. Most pairs of other qudits would have mutual
information ≤ log(d) / ε deg(G) if measured.
3. Thus their state is within distance d3(log(d) / ε deg(G))1/2
of product.
4. Witness is a global product state. Total error is
ε + d6(log(d) / ε deg(G))1/2.
Choose ε to balance these terms.
5. General case follows by coarse graining sites
(can replace log(d) by Ei H(Xi))
Proof Overview
Let p(z1 ,
, zn ) = M
Än
(y
0
y0
)
…
previous argument
q: probability distribution obtained conditioning on zj1, …, zjt
Proof Overview
info complete measurement
(σ: state obtained by measuring M on j1, …, jt and conditioning
on the outcome). Choosing k = εn
Other Applications 1:
New Classical Algorithms for Q. Hamiltonians
Following same approach one obtains 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.
Other Applications 2:
New de Finetti Theorems
- Classical de Finetti without symmetry: For p(x1,…,xn)
with
- Q. version using info-complete measurement
- Q. version using locality constrained norms (see Aram’s talk)
- Version replacing uniform randomness by Santa-Vazirani source
(Ramanathan et al ‘13)
Thank you!
Information-Theoretic Techniques in
Many-Body Physics
Day 2
Fernando G.S.L. Brandão
UCL
Based on joint work with A. Harrow and M. Horodecki
New Mathematical Directions for Quantum Info
The Plan
Yesterday: Product-State Approximations to Groundstates
- de Finetti theorem
- information theory approach
(entropies, chain rule, Pinsker’s inequality, info-complete meas.)
Today: Groundstates in 1D
- matrix product states
- area law and exponential decay of correlations
- information theory approach
(decoupling, state merging, single-shot protocols)
(see Nilanjana Datta’s talk)
Quantum Many-Body Systems
l
H = å H i Î (C
Quantum Hamiltonian
i=1
)
d Än
, H i £1
n
Cd
Hi
Interested in computing properties such as minimum
energy, correlations functions, etc…
Approximation Scale
We want to approximate the minimum energy
(i.e. minimum eigenvalue of H):
today
Small total error:
Small extensive error:
Matrix Product States
(Fannes, Nachtergaele, Werner ‘92)
y
2
2
i1 =1
in =1
[1]
[n]
[l ]
=
...
tr
A
...A
i
,...,i
,
A
å å ( i1 in ) 1 n j Î Mat(D, D)
1,...,n
D : bond dimension
•
•
•
•
Only nD2 parameters.
Local expectation values computed in poly(D, n) time
Variational class of states for powerful DMRG
Generalization of product states (MPS with D=1)
Area Law in 1D
Let
C2
y
2 Än
Î
(C
) be a n-qubit quantum state
1,...,n
X
Y
Entanglement Entropy: E
( y ) := S(r
XY
X
)
Area Law: For all partitions of the chain (X, Y)
S(rX ) £ const
(Bekenstein ‘73, ….…, Eisert, Cramer, Plenio ’10)
MPS
X
For MPS,
Area Law
Y
MPS
X
Area Law
Y
For MPS,
If
is s.t.
then it has a MPS description of bound dim. D
(Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06)
MPS
Area Law
X
Y
For MPS,
If
is s.t.
then it has a MPS description of bound dim. D
(Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06)
(Approx. version) If
is s.t.
then it can be approximated by a MPS of bound dim. D up to
error ε
Def:
Exponential Decay of Correlations
Let
y
2 Än
Î
(C
) be a n-qubit quantum state
1,...,n
l
C
2
X
Y
Z
Correlation Function:
Cor(X : Z) := max tr ((M Ä N)(r XZ - r X Ä rZ ))
M , N £1
Exponential Decay of Correlations: There is ξ > 0 s.t. for all
cuts X, Y, Z with |Y| = l
Cor(X : Z) £ 2
-l/x
MPS
Let
y
EDC
≈
= å...åtr ( A ...A ) i ,..., i
2
2
[1]
i1
i1 =1
[n]
in
1
n
,
in =1
Define
(w.l.o.g.
and let λj be the second largest eingenvalue of
If λ is independent of n we say
)
and λ := max |λj|
is a gMPS
has (1/log(1/|λ|))-EDC
How good are MPS?
Negative results:
(Aharonov, Gottesman, Irani, Kempe ‘07)
1D Hamiltonians can be QMA-hard
(see Daniel Nagaj’s talk)
(Irani ’09; Gottesman, Hastings ‘09)
1D Hamiltonians with volume scaling of entanglement
(Irani, Gottesman ‘09)
1D Hamiltonians with translational-invariance still hard
… is there hope?
1D gapped models
n
Cd
Hi
Given
Let
Then Hn is gapped if
(remember Toby Cubbit’s talk)
1D gapped models
Area Law
(Hastings ’07
Arad, Kitaev,
Landau, Vazirani ’12)
Groundstate
Gapped model
(Hastings ’05)
EDC
gMPS
(Landau, Vidick,
Vazirani ‘12)
Do we need the gap?
EDC
gMPS
Area Law
E.g. there are gapless Hamiltonians with a mobility gap/dynamical
localization, which imply EDC in the groundstate
(Hastings ’10; Hamza, Sims, Stolz ‘11)
Do we need the gap?
EDC
gMPS
(B., Horodecki ’12)
Area Law
Do we need the gap?
EDC
gMPS
(B., Horodecki ’12)
Area Law
Area Law vs. Decay of Correlations
Exponential Decay of Correlations suggests Area Law
Area Law vs. Decay of Correlations
Exponential Decay of Correlations suggests Area Law:
y
l = O(ξ)
X
ξ-EDC implies
Z
Y
rXZ » 2
-l/x
r X Ä rZ
XYZ
Area Law vs. Decay of Correlations
Exponential Decay of Correlations suggests Area Law:
y
l = O(ξ)
X
ξ-EDC implies
y
XYZ
Z
Y
rXZ » 2
-l/x
» 2-l/x (U Y1Y2 ®Y ÄI XZ ) p
X is only entangled with Y!
XYZ
r X Ä rZ which implies
XY1
u
Y2 Z
(by Uhlmann’s theorem)
Area Law vs. Decay of Correlations
Exponential Decay of Correlations suggests Area Law:
y
l = O(ξ)
X
ξ-EDC implies
y
XYZ
Z
Y
rXZ » 2
XYZ
-l/x
» 2-l/x (U Y1Y2 ®Y ÄI XZ ) p
r X Ä rZ which implies
XY1
u
Y2 Z
(by Uhlmann’s theorem)
X is only entangled with Y! Alas, the argument is wrong…
Uhlmann’s thm require 1-norm:
r AC - r A Ä rC 1 = 2 max tr ( M ( r AC - r A Ä rC ))
0<M<I
Area Law vs. Decay of Correlations
Exponential Decay of Correlations suggests Area Law:
y
l = O(ξ)
X
ξ-EDC implies
y
XYZ
Z
Y
rXZ » 2
XYZ
-l/x
» 2-l/x (U Y1Y2 ®Y ÄI XZ ) p
r X Ä rZ which implies
XY1
u
Y2 Z
(by Uhlmann’s theorem)
X is only entangled with Y! Alas, the argument is wrong…
Uhlmann’s thm require 1-norm:
M ¹ X ÄY
r AC - r A Ä rC 1 = 2 max tr ( M ( r AC - r A Ä rC ))
0<M<I
Data Hiding States
Well distinguishable globally, but poorly distinguishable locally
(DiVincenzo, Leung, Terhal ’02)
Ex. 1 Antisymmetric Werner state ωAB = (I – F)/(d2-d)
w AB -w A Ä wB 1 » 1/ 2
Cor(A : B) £1/ d,
Ex. 2 Random state
y
Cor(X :Y ) £ 2-W(l),
X
XYZ
with |X|=|Z| and |Y|=l
S(X) » (n - l) / 2
Y
Z
What data hiding implies?
1. Intuitive explanation is flawed
What data hiding implies?
1. Intuitive explanation is flawed
2. No-Go for area law from exponential decaying correlations?
What data hiding implies?
1. Intuitive explanation is flawed
2. No-Go for area law from exponential decaying correlations?
What data hiding implies?
1. Intuitive explanation is flawed
2. No-Go for area law from exponential decaying correlations?
3. We fixed a partition; EDC gives us more…
What data hiding implies?
1. Intuitive explanation is flawed
2. No-Go for area law from exponential decaying correlations?
3. We fixed a partition; EDC gives us more…
4. It’s an interesting quantum information problem:
How strong is data hiding in quantum states?
Exponential Decaying Correlations
Imply Area Law
X
X
Thm 1 (B., Horodecki ‘12) If y
X and m,
2-W( m )
max
S
c
1,...,n
has ξ-EDC then for every
(X) £ 2O(x log(x )) + m
Exponential Decaying Correlations
Imply Area Law
X
X
Thm 1 (B., Horodecki ‘12) If y
X and m,
2-W( m )
max
S
Obs1: Implies
c
1,...,n
has ξ-EDC then for every
(X) £ 2O(x log(x )) + m
S(X) £ 2O(x log(x ))
Obs2: Only valid in 1D…
Obs3: Reproduces bound of Hastings for GS 1D gapped Ham.,
using EDC in such states
Exponential Decaying Correlations
Imply Area Law
X
X
Thm 1 (B., Horodecki ‘12) If y
X and m,
2-W( m )
max
S
Obs1: Implies
c
1,...,n
has ξ-EDC then for every
(X) £ 2O(x log(x )) + m
S(X) £ 2O(x log(x ))
Obs2: Only valid in 1D…
Obs3: Reproduces bound of Hastings for GS 1D gapped Ham.,
using EDC in such states
Exponential Decaying Correlations
Imply Area Law
X
X
Thm 1 (B., Horodecki ‘12) If y
X and m,
2-W( m )
max
S
c
1,...,n
has ξ-EDC then for every
(X) £ 2O(x log(x )) + m
Obs4: Implies stronger form of EDC: For l > exp(O(ξlogξ))
and split ABC with |B|=l
r AC - rA Ä rC 1 £ 2
-l/x
EDC
gMPS
X
(Cor. Thm 1) If y
gMPS
X
c
1,...,n
has ξ-EDC then for every ε>0 there is
ye with poly(n, 1/ε) bound dim. s.t.
y ye ³1- e
Random States Have EDC?
l
X
y
XYZ
Z
: Drawn from Haar measure
cor(X : Z) £ 2
S(X) » S(Z) » (n - l) / 2
w.h.p, if size(X) ≈ size(Z):
and
Y
-W(l)
Small correlations in a fixed partition do not imply area law.
Random States Have EDC?
l
X
y
XYZ
Y
: Drawn from Haar measure
cor(X : Z) £ 2
S(X) » S(Z) » (n - l) / 2
w.h.p, if size(X) ≈ size(Z):
and
Z
-W(l)
Small correlations in a fixed partition do not imply area law.
But we can move the partition freely...
Random States Have Big Correl.
l
X
y
Y
Let size(XY) < size(Z). W.h.p.
X is decoupled from Y.
XYZ
: Drawn from Haar measure
Z
r XY - t X Ä t Y 1 £ 2
I
, t X :=
|X|
-W(n)
Random States Have Big Correl.
l
X
y
Y
Let size(XY) < size(Z). W.h.p.
X is decoupled from Y.
Extensive entropy, but
also large correlations:
XYZ
: Drawn from Haar measure
Z
r XY - t X Ä t Y 1 £ 2
I
, t X :=
|X|
-W(n)
Random States Have Big Correl.
l
X
y
Y
XYZ
: Drawn from Haar measure
Z
Let size(XY) < size(Z). W.h.p.
r XY - t X Ä t Y 1 £ 2
I
, t X :=
|X|
-W(n)
X is decoupled from Y.
Extensive entropy, but
also large correlations:
F
XZ1
UZ®Z1Z2 y
XYZ
»F
:Maximally entangled state between XZ1.
XZ1
ÄF
YZ2
(Uhlmann’s theorem)
Random States Have Big Correl.
l
X
y
Y
XYZ
: Drawn from Haar measure
Z
Let size(XY) < size(Z). W.h.p.
r XY - t X Ä t Y 1 £ 2
I
, t X :=
|X|
-W(n)
X is decoupled from Y.
Extensive entropy, but
also large correlations:
F
XZ1
UZ®Z1Z2 y
XYZ
»F
XZ1
ÄF
YZ2
(Uhlmann’s theorem)
:Maximally entangled state between XZ1.
Cor(X:Z) ≥ Cor(X:Z1) = Ω(1) >> 2-Ω(n) : long-range correlations!
Random States Have Big Correl.
l
y
XYZ
: Drawn from Haar measure
ThisXreasoningY hints at the idea of the general proof:
Z
I
r
t
Ä
t
£
2
LetWe’ll
size(XY)
< size(Z).
W.h.p. leads
, by
t X :=
XY to large
X
Y 1
show
large entropy
correlations
-W(n)
a random
X ischoosing
decoupled
from Y. measurement that decouples A and B
Extensive entropy, but
also large correlations:
F
XZ1
UZ®Z1Z2 y
XYZ
»F
XZ1
ÄF
YZ2
(Uhlmann’s theorem)
:Maximally entangled state between XZ1.
Cor(X:Z) ≥ Cor(X:Z1) = Ω(1) >> 2-Ω(n) : long-range correlations!
|X|
Entanglement Distillation by
Decoupling
We apply the state merging protocol to show large entropy
implies large correlations
A
B
y
ABE
E
State merging protocol: Given y ABC Alice can distill
-S(A|B) = S(B) – S(AB) EPR pairs with Bob by making
a random measurement with N≈ 2I(A:E) elements, with
I(A:E) := S(A) + S(E) – S(AE), and communicating the
outcome to Bob. (Horodecki, Oppenheim, Winter ‘05)
Entanglement Distillation by
Decoupling
We apply the state merging protocol to show large entropy
implies large correlations
Än
Disclaimer: only works for y
y
ABC
y ABE
A
ABC
B
E
Let’s cheat for a while and pretend it works for a
single copy, and later deal with this issue
State merging protocol: Given y ABC Alice can distill
-S(A|B) = S(B) – S(AB) EPR pairs with Bob by making
a random measurement with N≈ 2I(A:E) elements, with
I(A:E) := S(A) + S(E) – S(AE), and communicating the
outcome to Bob. (Horodecki, Oppenheim, Winter ‘05)
Optimal Decoupling
State merging protocol works by applying a random
measurement {Pk} to A in order to decouple it from E:
y
ABE
j
ABE
log( # of Pk’s )
# EPR pairs:
A
µ ( Pk Ä idBE ) y
ABE
j AE - t A Ä j E 1 » 0
» I(A : E)
log X » S(B) - S(AB)
B
y
ABE
E
Distillation Bound
l
E
XYZ
Z
Y
X
A
y
B
(
S(Z) > S(Y ) Þ Cor(X : Z) ³ O 2
-I ( X:Y )
)
Distillation Bound
l
E
B
(
S(Z) > S(Y ) Þ Cor(X : Z) ³ O 2
S(Z) – S(XZ) > 0
(EPR pair distillation
by random measurement)
XYZ
Z
Y
X
A
y
-I ( X:Y )
)
Prob. of getting one of the
2I(X:Y) outcomes in random
measurement
Proof Strategy
We apply previous result to prove
EDC -> Area Law in 3 steps:
1. Get area law from EDC under assumption there is a
region with “subvolume” law
2. Get region with “subvolume” law from assumption
there is a region of “small mutual information”
3. Show there is always a region of “small mutual info”
1. Area Law from Subvolume Law
l
X
Y
y
Z
(
S(Z) > S(Y ) Þ Cor(X : Z) ³ O 2
XYZ
-I ( X:Y )
)
1. Area Law from Subvolume Law
l
X
Y
y
Z
(
S(Z) £ S(Y ) Ü Cor(X : Z) < O 2
XYZ
-I ( X:Y )
)
1. Area Law from Subvolume Law
l
X
Y
y
Z
(
S(Z) £ S(Y ) Ü Cor(X : Z) < O 2
Suppose S(Y) < l/(4ξ)
XYZ
-I ( X:Y )
(“subvolume law” assumption)
)
1. Area Law from Subvolume Law
l
X
Y
y
Z
(
S(Z) £ S(Y ) Ü Cor(X : Z) < O 2
XYZ
-I ( X:Y )
)
Suppose S(Y) < l/(4ξ)
(“subvolume law” assumption)
Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2-l/ξ < 2-I(X:Y)
1. Area Law from Subvolume Law
l
X
Y
y
Z
(
S(Z) £ S(Y ) Ü Cor(X : Z) < O 2
XYZ
-I ( X:Y )
)
Suppose S(Y) < l/(4ξ)
(“subvolume law” assumption)
Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2-l/ξ < 2-I(X:Y)
Thus: S(Z) < S(Y)
2. Subvolume Law from Small
Mutual Info
YL
YC
YR
l/2
l
l/2
2. Subvolume Law from Small
Mutual Info
R
YL
YC
YR
R
l/2
l
l/2
R := all except YLYCYR : y
YLYCYR R
Suppose I(YC: YLYR) < l/(4ξ) (small mutual information assump.)
2. Subvolume Law from Small
Mutual Info
R
YL
YC
YR
R
l/2
l
l/2
R := all except YLYCYR : y
YLYCYR R
Suppose I(YC: YLYR) < l/(4ξ) (small mutual information assump.)
ξ-EDC implies Cor(YC : R) < exp(-l/(2ξ)) < exp(-I(YC:YLYR))
2. Subvolume Law from Small
Mutual Info
R
YL
YC
YR
R
l/2
l
l/2
R := all except YLYCYR : y
YLYCYR R
Suppose I(YC: YLYR) < l/(4ξ) (small mutual information assump.)
ξ-EDC implies Cor(YC : R) < exp(-l/(2ξ)) < exp(-I(YC:YLYR))
From distillation bound H(YLYCYR) = H(R) < H(YLYR)
2. Subvolume Law from Small
Mutual Info
R
YL
YC
YR
R
l/2
l
l/2
R := all except YLYCYR : y
YLYCYR R
Suppose I(YC: YLYR) < l/(4ξ) (small mutual information assump.)
ξ-EDC implies Cor(YC : R) < exp(-l/(2ξ)) < exp(-I(YC:YLYR))
From distillation bound H(YLYCYR) = H(R) < H(YLYR)
Finally H(YC) ≤ H(YC) + H(YLYR) – H(YLYCYR) = I(YC:YLYR) ≤ l/(4ξ)
Getting Area Law
Z
To prove area law for Z it suffices to find a not-so-far
and not-so-large region YLYCYR with small mutual
information
We show it with
not-so-far = not-so-large = exp(O(ξ))
Getting Area Law
YL
YC
YR
Z’
Z
Since I(YC: YLYR) < l/(4ξ)
with
l = exp(O(ξ))
by part 2, H(YC) < l/(4ξ)
by part 1, H(Z’) < l/(4ξ)
by subadditivity and Araki-Lieb: H(X) < exp(O(ξ))
3. Getting Small Mutual Info.
Lemma (Saturation Mutual Info.) Given a site s, for all ε > 0
there is a region Y2l := YL,l/2YC,lYR,l/2 of size 2l with 1 < l < 2O(1/ε) at
a distance < 2O(1/ε) from s s.t.
I(YC,l:YL,l/2YR,l/2) < εl
X
YL
s
< 2O(1/ε)
YC
YR
< 2O(1/ε)
Proof: Easy adaptation of result used by Hastings in his area law
proof for gapped Hamiltonians
(based on successive applications of subadditivity)
Making it Work
So far we have cheated, since merging only works for many
copies of the state. To make the argument rigorous, we use
single-shot information theory (see Nilanjana Datta’s talk)
Single-Shot State Merging
State Merging
(Dupuis, Berta, Wullschleger, Renner ‘10)
+ New bound on correlations
by random measurements
Saturation max- Mutual Info.
Saturation
Mutual Info.
Proof much more involved; based on
- Quantum substate theorem,
- Quantum equipartition property,
- Min- and Max-Entropies Calculus
- EDC Assumption
Overview
• Condensed Matter (CM) community always knew EDC implies
area law
Overview
• Condensed Matter (CM) community always knew EDC implies
area law
• Quantum information (QI) community gave a counterexample
(hiding states)
Overview
• Condensed Matter (CM) community always knew EDC implies
area law
• Quantum information (QI) community gave a counterexample
(hiding states)
• QI community sorted out the trouble they gave themselves
(this talk)
Overview
• Condensed Matter (CM) community always knew EDC implies
area law
• Quantum information (QI) community gave a counterexample
(hiding states)
• QI community sorted out the trouble they gave themselves
(this talk)
• CM community didn’t notice either of these minor perturbations
”EDC implies Area Law”
stays true!
Conclusions and Open problems
• EDC implies Area Law and MPS parametrization in 1D.
• Proof uses state merging protocol and single-shot
information theory: Tools from QIT useful to address
problem in quantum many-body physics.
1. Can we improve the dependency of entropy with
correlation length?
2. Can we prove area law for 2D systems? HARD!
3. Can we decide if EDC alone is enough for 2D area law?
4. See arxiv:1206.2947 for more open questions