Exponential Decay of Correlations Implies
Area Law:
Single-Shot Techniques
Fernando G.S.L. Brandão
ETH Zürich
Based on joint work with Michał Horodecki
Arxiv:1206.2947
Beyond i.i.d. in Info. Theory, Cambridge 08/01/2012
This talk:
• Go beyond the i.i.d. simplification used in the
previous talk.
• Contrast two views of single-shot info theory:
1. It’s the right info theory, i.i.d.ness is an abstraction
2. It only gives a reformulation of the problem
e.g. Hmax gives the compression rate or is it
defined as the compression rate?
This talk:
• Go beyond the i.i.d. simplification used in the
previous talk.
• Contrast two views of single-shot info theory:
1. It’s the right info theory, iid-ness is an abstraction
2. It only gives a reformulation of the problems
(e.g. Hmax gives the compression rate or is it
defined as the compression rate?)
This talk:
• Go beyond the i.i.d. simplification used in the
previous talk.
• Contrast two views of single-shot info theory:
1. It’s the right info theory, iid-ness is an abstraction
2. It only gives a trivial reformulation of the problems
(e.g. Hmax gives the compression rate or is it
defined as the compression rate?)
The result
X
Y
Thm 1 (B., Horodecki ‘12) If y
X = [1, r],
H
2-W(l )
max
1,...,n
has ξ-EDC, then for every
(X) £ l0 2O(x log(x )) + l
Cor(A : C) := max tr ((X ÄY )(r AC - r A Ä rC ))
X , Y £1
ξ-EDC (Expontial Decay of Correlations): For all regions A
and C separated by l sites,
Cor(A : C) £ 2-l/x
The proof sketch…
…was based on two ingredients:
1. State Merging Protocol (Horodecki, Oppenheim, Winter ‘05)
2. “Saturation of Mutual Information” (based on Hastings ‘07)
But we only have one copy of the state…
Single-shot info theory to rescue?
Single-Shot Entropies
1. Conditional min-entropy (Renner ‘05)
H min (A | B)r := max sup {l : 2 I A Ä s B ³ r AB }
-l
s
2. Smooth conditional min entropy
e
H min (A | B)r := max H min (A | B)r '
r '»e r
3. Smooth cond. max entropy (Tomamichel, Colbeck, Renner ‘09)
H
e
max
(A | C) := -H
e
max
(A | B), for y
4. Smooth max mutual info
I
e
max
(A : B) := H
e
max
e
min
(A)- H (A | B)
ABC
Single-Shot Entropies
1. Conditional min-entropy (Renner ‘05)
H min (A | B)r := max sup {l : 2 I A Ä s B ³ r AB }
-l
s
2. Smooth conditional min entropy
e
H min (A | B)r := max H min (A | B)r '
r '»e r
3. Smooth cond. max entropy (Tomamichel, Colbeck, Renner ‘09)
H
e
max
(A | C) := -H
e
max
(A | B), for y
4. Smooth max mutual info
I
e
max
(A : B) := H
e
max
e
min
(A)- H (A | B)
ABC
Single-Shot Entropies
1. Conditional min-entropy (Renner ‘05)
H min (A | B)r := max sup {l : 2 I A Ä s B ³ r AB }
-l
s
2. Smooth conditional min entropy
e
H min (A | B)r := max H min (A | B)r '
r '»e r
3. Smooth cond. max entropy (Tomamichel, Colbeck, Renner ‘09)
H
e
max
(A | C) := -H (A | B), for y
e
min
4. Smooth max mutual info
I
e
max
(A : B) := H
e
max
e
min
(A)- H (A | B)
ABC
Single-Shot Entropies
1. Conditional min-entropy (Renner ‘05)
H min (A | B)r := max sup {l : 2 I A Ä s B ³ r AB }
-l
s
2. Smooth conditional min entropy
e
H min (A | B)r := max H min (A | B)r '
r '»e r
3. Smooth cond. max entropy (Tomamichel, Colbeck, Renner ‘09)
H
e
max
(A | C) := -H (A | B), for y
e
min
4. Smooth max mutual info
I
e
max
(A : B) := H
e
max
e
min
(A)- H (A | B)
ABC
Single-Shot Counterparts 1:
Correlations vs Entropies
From i.i.d. merging: For y
ABC
H(C) > H(B) Þ Cor(A : C) ³ exp(-I(A : B))
Lemma: For y
ABC
and δ, ν > 0,
d
1. -H max
(A | C) ³ -2 log d + 5 Þ
d
Cor(A : C) ³ exp(-I max
(A : B) + 2 log d )
n
d
2. H max
(C) ³ 3H max (B) Þ
Cor(A : C) ³ exp(-3H
d
max
(B) - 6 log(d + n ))
-2
-2
Single-Shot Counterparts 1:
CorrelationsH(C)
vs-Entropies
H(AC) = -H(A | C) > 0
From i.i.d. merging: For y
ABC
H(C) > H(B) Þ Cor(A : C) ³ exp(-I(A : B))
Lemma: For y
ABC
and δ, ν > 0,
d
1. -H max
(A | C) ³ -2 log d + 5 Þ
d
Cor(A : C) ³ exp(-I max
(A : B) + 2 log d )
n
d
2. H max
(C) ³ 3H max (B) Þ
Cor(A : C) ³ exp(-3H
d
max
(B) - 6 log(d + n ))
-2
-2
Single-Shot Counterparts 1:
Correlations
vs Entropies
Independent
bounds as
From i.i.d. merging: For y
y ABC
I max (A : B) < c H max (B)
H(C) > H(B) Þ Cor(A : C) ³ exp(-I(A : B))
Lemma: For y
ABC
ABC
d
e
and δ, ν > 0,
d
1. -H max
(A | C) ³ -2 log d + 5 Þ
d
Cor(A : C) ³ exp(-I max
(A : B) + 2 log d )
n
d
2. H max
(C) ³ 3H max (B) Þ
Cor(A : C) ³ exp(-3H
d
max
(B) - 6 log(d + n ))
-2
-2
Single-Shot Merging
d
-H max
(A | C) ³ -2 log d + 5 Þ
d
Cor(A : C) ³ exp(-I max
(A : B) + 2 log d )
Follows from single-shot state merging:
(Dupuis, Berta, Wullschleger, Renner ‘10 )
A
B
y
ABC
C
“Environment”
EPR rate:
C. Com. Cost:
-H
d
max
(A | C) + 2 log d - 5 Û -H(A | C)
d
I max
(A : B) - 2 log d Û I(A : B)
Beyond Single-Shot Merging
H
n
max
(C) ³ 3H
d
max
(B) Þ
Cor(A : C) ³ exp(-3H
d
max
(B) - 6 log(d + n ))
-2
-2
Does not follows from single shot-merging. Might have
n
d
d
H max
(C) ³ 3H max
(B) but -H max
(A | C) << 0 !!
We operate in a regime where one cannot get EPR pairs.
Still one can get correlations by random measurements
It only happens in the single-shot regime, asymptotically
either one gets EPR pairs between A and C or decouples A
from C
Noisy Correlations by Random
Measurements
iid
H(A | C) = 0
Perfect correlations
EPR distillation
no correlations
(decoupling)
d
Single-shot H max (A | C) = 0
Perfect correlations
EPR distillation
d
H max
(A | B) = 0
Noisy correlations
no correlations
(decoupling)
Single-Shot Counterparts 2:
Saturation Mutual Info
Given a site s, for all l0, ε > 0 there is a region B2l := BL,l/2BC,lBR,l/2
of size 2l with 1 < l/l0 < 2O(1/ε) at a distance < l02O(1/ε) from s s.t.
I(BC,l:BL,l/2BR,l/2) < εl
A
BL
s
< l02O(1/ε)
BC
< l02O(1/ε)
BR
Single-Shot Counterparts 2:
Saturation Mutual Info
Given a site s, for all l0, ε > 0 there is a region B2l := BL,l/2BC,lBR,l/2
of size 2l with 1 < l/l0 < 2O(1/ε) at a distance < l02O(1/ε) from s s.t.
I(BC,l:BL,l/2BR,l/2) < εl
A
BL
s
< l02O(1/ε)
BC
BR
< l02O(1/ε)
Single-shot Let Ψ have ξ-EDC. Given a site s, for all l0, ε, δ > 0
there is a region B2l := BL,l/2BC,lBR,l/2 of size 2l with 1 < l/l0 <
exp(log(1/ε)/ε) at a distance < l0exp(log(1/ε)/ε) from s s.t.
Iδmax(BC,l:BL,l/2BR,l/2) < 8εl/δ + δ
Saturation von Neumann Mutual
Information
Given a site s, for all l0, ε > 0 there is a region B2l := BL,l/2BC,lBR,l/2
of size 2l with 1 < l/l0 < 2O(1/ε) at a distance < l02O(1/ε) from s s.t.
I(BC,l:BL,l/2BR,l/2) < εl
I(BC : BL BR ) £ el Û H(BC )+ H(BL BR ) - H(BC BL BR ) £ el
Proof by contradiction: If it doesn’t hold, for all l:
H(BC,l BL,l BR,l ) £ H(BC,l )+ H(BL,l BR,l )- el
H(2l)
H (l)
H (l)
Saturation von Neumann Mutual
Information
Given a site s, for all l0, ε > 0 there is a region B2l := BL,l/2BC,lBR,l/2
of size 2l with 1 < l/l0 < 2O(1/ε) at a distance < l02O(1/ε) from s s.t.
I(BC,l:BL,l/2BR,l/2) < εl
I(BC : BL BR ) £ el Û H(BC )+ H(BL BR ) - H(BC BL BR ) £ el
Proof by contradiction: If it doesn’t hold, for all l:
H(BC,l BL,l BR,l ) £ H(BC,l )+ H(BL,l BR,l )- el
Interating:
H(2l)
H (l)
H (l)
H(2l) < 2H(l) – εl < 4 H(l/2) – 2εl < 8 H(l/4) – 3εl < …
… which implies H(l) < l H(2) - εl log(l)
Saturation max Mutual Information
Single-shot Let Ψ have ξ-EDC. Given a site s, for all l0, ε, δ > 0
there is a region B2l := BL,l/2BC,lBR,l/2 of size 2l with 1 < l/l0 <
exp(log(1/ε)/ε) at a distance < l0exp(log(1/ε)/ε) from s s.t.
Iδmax(BC,l:BL,l/2BR,l/2) < 8εl/δ + δ
We do not know how to prove it without exponential decay
of correlations.
Proof breaks down completely:
1. Imax not linear combination of entropies
1. Maybe use bound: Imax(A:B) < Hmax(A)+Hmax(B)-Hmin(AB)
But arbitrary gaps between Hmax and Hmin
Saturation max Mutual Information
The idea: Use EDC to relate single-shot and i.i.d.
1. From the q. substate theorem (Jain, Radhakrishnan, Sen ’07)
1 d /2
d
I max (BC : BL BR ) £ H max (BC ) + H(BL BR ) - H(BC BL BR ) + d
(
d
)
Suffices to prove: H max (BC,l )+ H(BL,l BR,l ) £ H(BC,l BL,l BR,l )+ el
d /2
Saturation max Mutual Information
The idea: Use EDC to relate single-shot and i.i.d.
1. From the q. substate theorem (Jain, Radhakrishnan, Sen ’07)
1 d /2
d
I max (BC : BL BR ) £ H max (BC ) + H(BL BR ) - H(BC BL BR ) + d
(
d
)
Suffices to prove: H max (BC,l )+ H(BL,l BR,l ) £ H(BC,l BL,l BR,l )+ el
d /2
d /2
H max (l)
H (l)
H(2l)
Saturation max Mutual Information
The idea: Use EDC to relate single-shot and i.i.d.
1. From the q. substate theorem (Jain, Radhakrishnan, Sen ’07)
1 d /2
d
I max (BC : BL BR ) £ H max (BC ) + H(BL BR ) - H(BC BL BR ) + d
(
d
)
Suffices to prove: H max (BC,l )+ H(BL,l BR,l ) £ H(BC,l BL,l BR,l )+ el
d /2
d /2
H(2l)
£
H
2. Suppose not: For all l,
max (l)+ H(l)- el
d /2
We must relate H max
(l) to H (l). In general not possible
But we have EDC…
Relating max to vNeumann entropy
By subadditivity:
e
d
H max
(BC,l ) £ å H max
(Yk ) + f (e )
k
But by EDC, rZk,1,...,Zk,m - rZk,1 Ä... Ä rZk,m
(
d
d
Thus H max (Yk ) » H max rZk,1 Ä... Ä r Zk,m
1
is small.
)
By q. equipartition property (Tomamichel, Colbeck, Renner ’08)
(
)
(
d
H max
rZk,1 Ä... Ä rZk,m » H rZk,1 Ä... Ä rZk,m
)
Using Single-Shot Lemmas to show
the main result
1. Area law from subvolume law
2. Getting subvolume law
Area Law from Subvolume Law
(cheating)
l
y
A
B
ABC
C
H(C) > H(B) Þ Cor(A : C) ³ exp(-I(A : B))
Area Law from Subvolume Law
(cheating)
l
y
A
B
ABC
C
H(C) £ H(B) Ü Cor(A : C) < exp(-I(A : B))
Area Law from Subvolume Law
(cheating)
l
y
A
B
ABC
C
H(C) £ H(B) Ü Cor(A : C) < exp(-I(A : B))
Suppose H(B) < l/(4ξ). Since I(A:B) < 2H(B) < l/(2ξ), if
state y ABC has ξ-EDC then Cor(A:C) < 2-l/ξ < 2-I(A:B)
Thus: H(C) < H(B) : Area Law for C!
Area Law from Subvolume Law
(cheating)
l
y
A
B
ABC
C
H(C) £ H(B) Ü Cor(A : C) < exp(-I(A : B))
Suppose H(B) < l/(4ξ). Since I(A:B) < 2H(B) < l/(2ξ), if
state y ABC has ξ-EDC then Cor(A:C) < 2-l/ξ < 2-I(A:B)
Thus: H(C) < H(B) : Area Law for C!
It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)
Area Law from Subvolume Law
Could consider single-shot
(cheating)
l
merging. But remember
y ABC
I max (A : B) < 2H max (B)
d
A
B
e
C
H(C) £ H(B) Ü Cor(A : C) < exp(-I(A : B))
Suppose H(B) < l/(4ξ). Since I(A:B) < 2H(B) < l/(2ξ), if
state y ABC has ξ-EDC then Cor(A:C) < 2-l/ξ < 2-I(A:B)
Thus: H(C) < H(B) : Area Law for C!
It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)
Area Law from Subvolume Law
l
A
H
n
max
y
C
B
(C) ³ 3H
ABC
d
max
(B) Þ Cor(A : C) ³ exp(-3H
d
max
(B))
Area Law from Subvolume Law
l
A
H
n
max
y
C
B
(C) £ 3H
ABC
d
max
(B) Ü Cor(A : C) £ exp(-3H
d
max
(B))
Area Law from Subvolume Law
l
A
H
n
max
y
C
B
(C) £ 3H
ABC
d
max
(B) Ü Cor(A : C) £ exp(-3H
Suppose Hmax(B) < l/(4ξ). If state y
then Cor(A:C) < 2-l/ξ < 2-3H (B)
ABC
has ξ-EDC
max
Thus: Hmax(C) < Hmax(B) : Area Law for C!
d
max
(B))
Area Law from Subvolume Law
l
A
H
n
max
y
C
B
(C) £ 3H
ABC
d
max
(B) Ü Cor(A : C) £ exp(-3H
Suppose Hmax(B) < l/(4ξ). If state y
then Cor(A:C) < 2-l/ξ < 2-3H (B)
ABC
d
max
(B))
has ξ-EDC
max
Thus: Hmax(C) < Hmax(B) : Area Law for C!
It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with max entropy < l/(4ξ)
Getting Subvolume Law (cheating)
“It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)”
A
BL
s
< l02O(1/ε)
BC
< l02O(1/ε)
BR
R := all except BLBCBR : y
BL BC BR R
Hastings lemma with ε = 1/(4ξ), gives I(BC:BLBR) < l/(4ξ)
Then from state merging protocol and ξ-EDC:
Cor(BC : R) £ 2-l/2x £ 2-I (BC :BL BR ) Þ S(R) £ S(BL BR )
Getting Subvolume Law (cheating)
“It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)”
A
BL
s
< l02O(1/ε)
BC
< l02O(1/ε)
BR
R := all except BLBCBR : y
BL BC BR R
Hastings lemma with ε = 1/(4ξ), gives I(BC:BLBR) < l/(4ξ)
Then from state merging protocol and ξ-EDC:
Cor(BC : R) £ 2-l/2x £ 2-I (BC :BL BR ) Þ S(R) £ S(BL BR )
Finally: S(BC) ≤ S(BC) + S(BLBR) – S(R) = I(BC:BLBR) ≤ l/(4ξ)
Getting Subvolume Law
“It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)”
A
BL
s
BC
BR
R := all except BLBCBR : y
Single-shot Hastings lemma with ε = 1/(8ξ) gives
< l02O(1/ε)
< l02O(1/ε)
BL BC BR R
Iδmax(BC:BLBR) < l/(2ξ)
Then from single-shot merging and ξ-EDC:
Cor(BC : R) £ 2
-l/2x
£2
d
-I max
(BC :BL BR )
d
Þ H max (BC | R) £ 0
Getting Subvolume Law
“It suffices to prove that nearby the boundary of C there is a
region of size < l02O(ξ) with entropy < l/(4ξ)”
A
BL
s
BC
BR
R := all except BLBCBR : y
Single-shot Hastings lemma with ε = 1/(8ξ) gives
< l02O(1/ε)
< l02O(1/ε)
BL BC BR R
Iδmax(BC:BLBR) < l/(2ξ)
Then from single-shot merging and ξ-EDC:
Cor(BC : R) £ 2
-l/2x
£2
d
-I max
(BC :BL BR )
d
Þ H max (BC | R) £ 0
d
Cor(A :Getting
C) £ exp(-ISubvolume
max (A : B))
Law
d
Þ
H
(A
|
C)
³
0
max
“It suffices to prove that nearby the boundary of C there is a
y ABC
for of size
region
< l02O(ξ) with entropy < l/(4ξ)”
A
BL
s
BC
BR
R := all except BLBCBR : y
Single-shot Hastings lemma with ε = 1/(8ξ) gives
< l02O(1/ε)
< l02O(1/ε)
BL BC BR R
Iδmax(BC:BLBR) < l/(2ξ)
Then from single-shot merging and ξ-EDC:
Cor(BC : R) £ 2
-l/2x
£2
d
-I max
(BC :BL BR )
d
Þ H max (BC | R) £ 0
d
Cor(BC : Getting
R) £ exp(-I max
(BC : BL BR ))
Subvolume
d
max to
C
suffices
Þ
H
“It
Law
(B prove
| R) ³that
0 nearby the boundary of C there is a
for ofysize < l02O(ξ) with entropy < l/(4ξ)”
region
BL BC BR R
A
BL
s
BC
BR
R := all except BLBCBR : y
Single-shot Hastings lemma with ε = 1/(8ξ) gives
< l02O(1/ε)
< l02O(1/ε)
BL BC BR R
Iδmax(BC:BLBR) < l/(2ξ)
Then from single-shot merging and ξ-EDC:
Cor(BC : R) £ 2
-l/2x
£2
d
-I max
(BC :BL BR )
d
Þ H max (BC | R) ³ 0
Getting Subvolume Law
Cor(BC : R) £ 2
H
d
max
-l/2x
d
max
d
max
£2
d
-I max
(BC :BL BR )
d
Þ H max (BC | R) ³ 0
d
max
d
min
(BC ) £ H (BC )+ H (BC | R)
= H (BC )- H (BC | BL BR )
e
= Imax
(BC : BL BR ) £ l / (2x )
By duality of min and max entropies
(Tomamichel, Colbeck, Renner ‘09)
Conclusions
• EDC implies Area Law in 1D.
• Single-shot info theory useful in physics, here condensedmatter physics/quantum many-body theory
• Single-shot is more than to replace von Neumann by either
min or max entropy: new regimes
• Single-shot theory is less structured, but gives a unified way
of addressing different types of structures (iid, permutation
invariance, ergodicity, exponential decay of correlations)
Thanks!