· Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X...
Transcript of · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X...
![Page 1: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/1.jpg)
Quantum analogues of geometric inequalities
for Information Theory
Anna Vershynina
June 17, 2016Workshop on Mathematical Many-Body Theory and its Applications
Bilbao, Spain
based on a joint work with Robert Koenig and Stefan Huber
Technical University of Munich, Germany
![Page 2: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/2.jpg)
Outline of the talk
Introduction
Geometric inequalities in information theory
Classical vs quantum inequalities
Quantum inequalities
Application to the quantum Ornstein-Uhlenbeck semigroup
Optimizing entropy rates of the quantum attenuator semigroup
Entropy power inequality
Concavity of the entropy power of diffusion semigroup
Quantum isoperimetric inequality
![Page 3: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/3.jpg)
Geometric inequalities
Brunn-Minkowski inequality
A B A +B
A+B = a+ b|a ∈ A, b ∈ B
Rn = R2
vol(A)1/n vol(B)1/n+ vol(A+B)1/n≤
Isoperimetric inequality
A area(A) ≤ 14π length(∂A)2
![Page 4: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/4.jpg)
Geometry vs Information theory
Set A ⊂ Ω Random variable X on Ω
with prob. mass function px = Pr[X = x]
volume vol(A) entropy power 2H(X)
Shannon entropy H(X) = −∑px log px
![Page 5: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/5.jpg)
Geometry vs Information theory
Set A ⊂ Ω Random variable X on Ω
with prob. mass function px = Pr[X = x]
volume vol(A) entropy power 2H(X)
Shannon entropy H(X) = −∑px log px
Consider a set An = A× · · · ×A ∈ Ωn Consider n i.i.d. with PXn = PX · · · · · PXhas the following property:
∀ε > 0 ∃Mn,ε ⊂ Ωn s.t.
and Pr[Xn ∈Mn,ε] ≥ 1− ε
|Mn,ε| ∼(2H(X)
)nvol(An) = (vol(A))n
![Page 6: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/6.jpg)
Geometry vs Information theory
Set A ∈ Rn Random variable X on Rnwith prob. density function fX
volume vol(A) entropy power e2H(X)/n
entropy H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y has a density function
fX+Y (x) =∫fX(x− z)fY (z)dz
fX fY fX+Y
![Page 7: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/7.jpg)
Geometric inequalitiesBrunn-Minkowski inequality
vol(A)1/n + vol(B)1/n ≤ vol(A+B)1/n
Shannon‘s entropy power inequality [‘48]
e2H(X)/n + e2H(Y )/n ≤ e2H(X+Y )/n
![Page 8: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/8.jpg)
Geometric inequalitiesBrunn-Minkowski inequality
vol(A)1/n + vol(B)1/n ≤ vol(A+B)1/n
Shannon‘s entropy power inequality [‘48]
e2H(X)/n + e2H(Y )/n ≤ e2H(X+Y )/n
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
length(∂A) = limε→0area(A+Bε(0))−area(A)
ε“length“ limε→0
e2H(X+√εZ)/n−e2H(X)/d
ε
+ =A
Bε(0)
A+Bε(0)
fX f√εZ fX+√εZ
ε
![Page 9: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/9.jpg)
Geometric inequalitiesBrunn-Minkowski inequality
vol(A)1/n + vol(B)1/n ≤ vol(A+B)1/n
Shannon‘s entropy power inequality [‘48]
e2H(X)/n + e2H(Y )/n ≤ e2H(X+Y )/n
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
length(∂A) = limε→0area(A+Bε(0))−area(A)
ε“length“ limε→0
e2H(X+√εZ)/n−e2H(X)/d
ε
limε→0e2H(X+
√εZ)/n−e2H(X)/n
ε ≥ limε→0e2H(X)/n+e2H(
√εZ)/n−e2H(X)/n
ε = 2π e
![Page 10: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/10.jpg)
Geometric inequalitiesBrunn-Minkowski inequality
vol(A)1/n + vol(B)1/n ≤ vol(A+B)1/n
Shannon‘s entropy power inequality [‘48]
e2H(X)/n + e2H(Y )/n ≤ e2H(X+Y )/n
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
length(∂A) = limε→0area(A+Bε(0))−area(A)
ε“length“ limε→0
e2H(X+√εZ)/n−e2H(X)/d
ε
limε→0e2H(X+
√εZ)/n−e2H(X)/n
ε ≥ limε→0e2H(X)/n+e2H(
√εZ)/n−e2H(X)/n
ε = 2π e
ddε
∣∣ε=0
e2H(X+√εZ)/n = 2
ne2H(X)/n d
dε
∣∣ε=0
H(X +√εZ)
Fisher information
J(X) = 2 ddε
∣∣ε=0
H(X +√εZ)
![Page 11: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/11.jpg)
Geometric inequalitiesBrunn-Minkowski inequality
vol(A)1/n + vol(B)1/n ≤ vol(A+B)1/n
Shannon‘s entropy power inequality [‘48]
e2H(X)/n + e2H(Y )/n ≤ e2H(X+Y )/n
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
length(∂A) = limε→0area(A+Bε(0))−area(A)
ε“length“ limε→0
e2H(X+√εZ)/n−e2H(X)/d
ε
limε→0e2H(X+
√εZ)/n−e2H(X)/n
ε ≥ limε→0e2H(X)/n+e2H(
√εZ)/n−e2H(X)/n
ε = 2π e
ddε
∣∣ε=0
e2H(X+√εZ)/n = 2
ne2H(X)/n d
dε
∣∣ε=0
H(X +√εZ)
Fisher information
J(X) = 2 ddε
∣∣ε=0
H(X +√εZ)
Isoperimetric inequality for entropies
1nJ(X)e2H(X)/n ≥ 2π e
![Page 12: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/12.jpg)
Geometric analogue of Fisher information
x2
x1
A ε
ε
A+ ε ~x1
A+ ε ~x2
D(A1, A2) =“difference“ between A1 and A2
“Fisher Information“= d2
dε2
∣∣∣ε=0
∑2j=1D(A,A+ ε ~xj)
A
A
![Page 13: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/13.jpg)
Geometric analogue of Fisher information
x2
x1
A ε
ε
A+ ε ~x1
A+ ε ~x2
D(A1, A2) =“difference“ between A1 and A2
“Fisher Information“= d2
dε2
∣∣∣ε=0
∑2j=1D(A,A+ ε ~xj)
A
A
J(X) =∑nj=1
∂2
∂ε2
∣∣∣ε=0
D(f ||f ε ~xj )
distribution translated in ~xj direction by ε
de Bruijn‘s identity
= 2 ddε
∣∣ε=0
H(X +√εZ)
![Page 14: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/14.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
![Page 15: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/15.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
ρX + ρY ??
![Page 16: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/16.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
ρXλY = Tr2
(Uλ(ρX ⊗ ρY )U†λ
)beamsplitter with transmissivity λ ∈ [0, 1]
![Page 17: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/17.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
ρXλY = Tr2
(Uλ(ρX ⊗ ρY )U†λ
)beamsplitter with transmissivity λ ∈ [0, 1]
Quantum entropy power inequality
λeS(X)/n + (1− λ)eS(Y )/n ≤ eS(XλY )/n
[Koenig, Smith 14] λ = 1/2
[de Palma et al 15]
Brunn-Minkowski inequality
vol(A)1/n + vol(B)1/n
Shannon entropy power inequality
e2H(X)/d + e2H(Y )/d ≤ e2H(X+Y )/d
≤ vol(A+B)1/n
![Page 18: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/18.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
ρXλY = Tr2
(Uλ(ρX ⊗ ρY )U†λ
)beamsplitter with transmissivity λ ∈ [0, 1]
Quantum entropy power inequality
λeS(X)/n + (1− λ)eS(Y )/n ≤ eS(XλY )/n
[Koenig, Smith 14] λ = 1/2
[de Palma et al 15]
Brunn-Minkowski inequality
vol(A)1/n + vol(B)1/n
Shannon entropy power inequality
e2H(X)/d + e2H(Y )/d ≤ e2H(X+Y )/d
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
Isoperimetric inequality for entropies
1nJ(X)e2H(X)/n ≥ 2π e
≤ vol(A+B)1/n
???
![Page 19: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/19.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
Brunn-Minkowski inequality
vol(A)1/n + vol(B)1/n
Shannon entropy power inequality
e2H(X)/d + e2H(Y )/d ≤ e2H(X+Y )/d
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
Isoperimetric inequality for entropies
1nJ(X)e2H(X)/n ≥ 2π e
≤ vol(A+B)1/n
f ?t ρ =∫f(ξ)W (
√tξ)ρW †(
√tξ)dξ
classical-quantum
![Page 20: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/20.jpg)
Geometry Classical Quantum
Set A ∈ Rn Random variable X on Rn
with prob. density function fX
volume vol(A) entropy power e2H(X)/n
H(X) = −∫Rn fX(x) log fX(x)dx
addition A+B convolution: X + Y
fX+Y (x) =∫fX(x− z)fY (z)dz
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
S(ρ) = −Tr(ρ log ρ)
N(ρ) = expS(ρ)/n
Brunn-Minkowski inequality
vol(A)1/n + vol(B)1/n
Shannon entropy power inequality
e2H(X)/d + e2H(Y )/d ≤ e2H(X+Y )/d
Isoperimetric inequality
area(A) ≤ 14π length(∂A)2
Isoperimetric inequality for entropies
1nJ(X)e2H(X)/n ≥ 2π e
≤ vol(A+B)1/n
f ?t ρ =∫f(ξ)W (
√tξ)ρW †(
√tξ)dξ
classical-quantum
Classical-quantum entropy power inequality
teH(f)/n + eS(ρ)/n ≤ eS(f?tρ)/n t ≥ 0
Isoperimetric inequality for entropies
1nJ(ρ)eS(ρ)/n ≥ 4π e
![Page 21: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/21.jpg)
Classical vs Quantum information theory
Entropy
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
H(X) = H(f) = −∫f(x) log f(x)dx S(ρ) = −Tr(ρ log ρ)
Entropy Power N(f) = exp2H(f)/n N(ρ) = expS(ρ)/n
Relative entropy D(f ||g) =∫f(x) log f(x)
g(x)dx D(ρ||σ) = Tr(ρ log ρ− ρ log σ)
Classical QuantumX - Rn-valued r.v. with a prob. density function fX .
Shannon von Neumann
![Page 22: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/22.jpg)
Classical vs Quantum information theory
Entropy
ρ - n-mode state.[Qj , Pk] = −[Pk, Qj ] = iδj,kI[Qj , Qk] = [Pj , Pk] = 0 1 ≤ j, k,≤ n
H(X) = H(f) = −∫f(x) log f(x)dx S(ρ) = −Tr(ρ log ρ)
Entropy Power N(f) = exp2H(f)/n N(ρ) = expS(ρ)/n
Relative entropy D(f ||g) =∫f(x) log f(x)
g(x)dx D(ρ||σ) = Tr(ρ log ρ− ρ log σ)
Fisher information matrix For ~θ ∈ Rn, define
f (θ)(x) = f(x− θ)
J(f (θ))∣∣θ=θ0
=(
∂2
∂θi∂θj
∣∣θ=θ0
D(f (θ0)||f (θ)
))2ni,j=1
Fisher information
J(f) = Tr(J(f (θ))
∣∣θ=θ0
)
Classical QuantumX - Rn-valued r.v. with a prob. density function fX .
Shannon von Neumann
![Page 23: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/23.jpg)
Quantum Fisher information
W (~θ) = ei√2π ~θ·(σ ~R),
ρ(θ) = W (θ)ρW †(θ)
~R = (Q1, P1, . . . , Qn, Pn), σ =
(0 1−1 0
)⊕n.
For ~θ ∈ R2n the Weyl displacement operators are defined as
where
Translated state is
![Page 24: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/24.jpg)
Quantum Fisher information
W (~θ) = ei√2π ~θ·(σ ~R),
ρ(θ) = W (θ)ρW †(θ)
~R = (Q1, P1, . . . , Qn, Pn), σ =
(0 1−1 0
)⊕n.
For ~θ ∈ R2n the Weyl displacement operators are defined as
where
Translated state is
Weyl operators translate position and momentum operators
W (~θ)QjW (~θ) = Qj + θI W (~θ)PjW (~θ) = Pj + θI
![Page 25: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/25.jpg)
Quantum Fisher information
W (~θ) = ei√2π ~θ·(σ ~R),
ρ(θ) = W (θ)ρW †(θ)
J(ρ(θ))∣∣θ=θ0
=(
∂2
∂θi∂θj
∣∣θ=θ0
D(ρ(θ0)||ρ(θ)
))2ni,j=1
J(ρ) = Tr(J(ρ(θ))
∣∣θ=θ0
)
~R = (Q1, P1, . . . , Qn, Pn), σ =
(0 1−1 0
)⊕n.
For ~θ ∈ R2n the Weyl displacement operators are defined as
where
Translated state is
Weyl operators translate position and momentum operators
W (~θ)QjW (~θ) = Qj + θI
Fisher information matrix
Fisher information
W (~θ)PjW (~θ) = Pj + θI
![Page 26: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/26.jpg)
Inequalities
Classical
(fX , fY )→ fX+Y (z) =∫fX(z − x)fY (x)dx
Quantum
(f, ρ)→ f ?t ρ =∫f(ξ)W (
√tξ)ρW †(
√tξ)dξt = 1 [Werner ‘84]
![Page 27: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/27.jpg)
Inequalities
Classical
(fX , fY )→ fX+Y (z) =∫fX(z − x)fY (x)dx
the Fisher information ineq.
Quantum
(f, ρ)→ f ?t ρ =∫f(ξ)W (
√tξ)ρW †(
√tξ)dξ
for λ ∈ [0, 1]
J(√
λX +√
1− λY)≤ λJ(X) + (1− λ)J(Y )
Stam inequality
J (X + Y )−1 ≥ J(X)−1 + J(Y )−1
the Fisher information ineq.
for ω =√tωc + ωq
ω2J(f ?t ρ) ≤ ω2cJ(f) + ω2
qJ(ρ)
Stam inequality
J(f ?t ρ)−1 ≥ tJ(f)−1 + J(ρ)−1
t = 1 [Werner ‘84]
![Page 28: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/28.jpg)
Inequalities
Classical
(fX , fY )→ fX+Y (z) =∫fX(z − x)fY (x)dx
the Fisher information ineq.
Quantum
(f, ρ)→ f ?t ρ =∫f(ξ)W (
√tξ)ρW †(
√tξ)dξ
Define Liouvillean L(ρ) = −π∑2nj=1[Rj , [Rj , ρ]].
for λ ∈ [0, 1]
J(√
λX +√
1− λY)≤ λJ(X) + (1− λ)J(Y )
Stam inequality
J (X + Y )−1 ≥ J(X)−1 + J(Y )−1
the Fisher information ineq.
for ω =√tωc + ωq
ω2J(f ?t ρ) ≤ ω2cJ(f) + ω2
qJ(ρ)
Stam inequality
J(f ?t ρ)−1 ≥ tJ(f)−1 + J(ρ)−1
ρ(t) = etL(ρ) : ddtρ(t) = L(ρ(t))
etL(ρ) = 1(2π)n
∫e−‖
~ξ‖2/2W (√t~ξ) ρW †(
√t~ξ)d~ξ
For Z - Gaussian r.v. : fZ(~ξ) = (2π)−ne−|ξ|2/2
fZ ?t ρ = etL(ρ)
Z - Gaussian r.v. : fZ(~ξ) = (2π)−n/2e−|ξ|2/2
t = 1 [Werner ‘84]
Consider
Note that
![Page 29: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/29.jpg)
Inequalities
de Bruijn identity
J(X +
√tZ)
= 2 ∂∂tH
(X +
√tZ)
J(ρ) = 2 ddtS
(etL(ρ)
)∣∣t=0
Classical Quantum
Entropy power inequality
N(X + Y ) ≥ N(X) +N(Y ) N(f ?t ρ) ≥ tN(f) +N(ρ)
Concavity of the entropy power
d2
dε2
∣∣∣∣ε=0
N(X +√εZ) ≤ 0 d2
dt2
∣∣∣∣t=0
N(etL(ρ)) ≤ 0
Fisher information isoperimetric inequality
ddε
∣∣∣∣ε=0
[1n J(X +
√εZ)
]−1 ≥ 1 ddt
∣∣∣∣t=0
[12nJ(etL(ρ))
]−1≥ 1
Isoperimetric inequality
1nJ(X)N(X) ≥ 2πe 1
nJ(ρ)N(ρ) ≥ 4πe
etL(ρ) = fZ ?t ρ
[Koenig, Smith ‘14]
![Page 30: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/30.jpg)
Inequalities
de Bruijn identity
J(X +
√tZ)
= 2 ∂∂tH
(X +
√tZ)
J(ρ) = 2 ddtS
(etL(ρ)
)∣∣t=0
Classical Quantum
Entropy power inequality
N(X + Y ) ≥ N(X) +N(Y ) N(f ?t ρ) ≥ tN(f) +N(ρ)
Concavity of the entropy power
d2
dε2
∣∣∣∣ε=0
N(X +√εZ) ≤ 0 d2
dt2
∣∣∣∣t=0
N(etL(ρ)) ≤ 0
Fisher information isoperimetric inequality
ddε
∣∣∣∣ε=0
[1n J(X +
√εZ)
]−1 ≥ 1 ddt
∣∣∣∣t=0
[12nJ(etL(ρ))
]−1≥ 1
Isoperimetric inequality
1nJ(X)N(X) ≥ 2πe 1
nJ(ρ)N(ρ) ≥ 4πe
Tight for Gaussian states:ωn a 1-mode G. thermal st.with a mean-photon number n
J(ωn)N(ωn) = 4π(n+1n
)nlog(n+1n
)n+1→4πe, as n→∞
etL(ρ) = fZ ?t ρ
[Koenig, Smith ‘14]
![Page 31: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/31.jpg)
Quantum Inequalities
Rescaling
fX ?t ρ = f√tX ?1 ρ
If fX is a prob. density function of a r.v. X, then
here the r.v.√tX is given by f√tX(ζ) = f(ζ/
√t)/(√t)2n
AdditionfX ?1 (fY ?1 ρ) = fX+Y ?1 ρ
where (fX , fY )→ fX+Y (z) =∫fX(z − x)fY (x)dx
Translation Let ωc, ωq > 0, and t ≥ 0. Then for all θ ∈ R2n
(f ?t ρ)(ωθ) = f (ωcθ) ?t ρ(ωqθ) ω =
√tωc + ωq
![Page 32: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/32.jpg)
Quantum Inequalities
Data Processing Inequality
D (f ?t ρ‖ g ?t σ) ≤ D (f‖ g) +D (ρ‖σ)
Rescaling
fX ?t ρ = f√tX ?1 ρ
If fX is a prob. density function of a r.v. X, then
here the r.v.√tX is given by f√tX(ζ) = f(ζ/
√t)/(√t)2n
AdditionfX ?1 (fY ?1 ρ) = fX+Y ?1 ρ
where (fX , fY )→ fX+Y (z) =∫fX(z − x)fY (x)dx
Translation Let ωc, ωq > 0, and t ≥ 0. Then for all θ ∈ R2n
(f ?t ρ)(ωθ) = f (ωcθ) ?t ρ(ωqθ) ω =
√tωc + ωq
![Page 33: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/33.jpg)
Stam Inequality
TheoremLet ωc, ωq ∈ R, and t ≥ 0. Then ω2J(f ?t ρ) ≤ ω2
cJ(f) + ω2qJ(ρ)
In particular,J(f ?t ρ)−1 − J(ρ)−1 − tJ(f)−1 ≥ 0
for ω =√tωc + ωq
![Page 34: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/34.jpg)
Stam Inequality
TheoremLet ωc, ωq ∈ R, and t ≥ 0. Then ω2J(f ?t ρ) ≤ ω2
cJ(f) + ω2qJ(ρ)
In particular,J(f ?t ρ)−1 − J(ρ)−1 − tJ(f)−1 ≥ 0
Proof
By Data processing inequality
Tr(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣~θ= ~θ0
)≤ Tr
(J(f (ωc
~θ); ~θ)∣∣∣~θ= ~θ0
+ J(ρ(ωq~θ); ~θ)
∣∣∣∣~θ= ~θ0
)Since, (f ?t ρ)
(ω~θ)= f (ωc
~θ) ?t ρ(ωq~θ),
for ω =√tωc + ωq
Tr
(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣∣~θ=~0
)= Tr
(J(f ?t ρ(ω
~θ); ~θ)
∣∣∣∣~θ=~0
)= ω2J(f ?t ρ)
![Page 35: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/35.jpg)
Stam Inequality
TheoremLet ωc, ωq ∈ R, and t ≥ 0. Then ω2J(f ?t ρ) ≤ ω2
cJ(f) + ω2qJ(ρ)
In particular,J(f ?t ρ)−1 − J(ρ)−1 − tJ(f)−1 ≥ 0
Proof
By Data processing inequality
Tr(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣~θ= ~θ0
)≤ Tr
(J(f (ωc
~θ); ~θ)∣∣∣~θ= ~θ0
+ J(ρ(ωq~θ); ~θ)
∣∣∣∣~θ= ~θ0
)Since, (f ?t ρ)
(ω~θ)= f (ωc
~θ) ?t ρ(ωq~θ),
for ω =√tωc + ωq
Tr
(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣∣~θ=~0
)= Tr
(J(f ?t ρ(ω
~θ); ~θ)
∣∣∣∣~θ=~0
)= ω2J(f ?t ρ)
Since, J(f (ωc~θ); ~θ) = ω2
cJ(f (~θ); ~θ), and J(ρ(ωq
~θ); ~θ) = ω2qJ(ρ(
~θ); ~θ)
Tr
(J(f (ωc
~θ); ~θ)
∣∣∣∣~θ=~0
)+ Tr
(J(ρ(ωq
~θ); ~θ)
∣∣∣∣~θ=~0
)= ω2
cJ(f) + ω2qJ(ρ)
![Page 36: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/36.jpg)
Stam Inequality
TheoremLet ωc, ωq ∈ R, and t ≥ 0. Then ω2J(f ?t ρ) ≤ ω2
cJ(f) + ω2qJ(ρ)
In particular,J(f ?t ρ)−1 − J(ρ)−1 − tJ(f)−1 ≥ 0
Proof
By Data processing inequality
Tr(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣~θ= ~θ0
)≤ Tr
(J(f (ωc
~θ); ~θ)∣∣∣~θ= ~θ0
+ J(ρ(ωq~θ); ~θ)
∣∣∣∣~θ= ~θ0
)Since, (f ?t ρ)
(ω~θ)= f (ωc
~θ) ?t ρ(ωq~θ),
for ω =√tωc + ωq
Tr
(J(f (ωc
~θ) ?t ρ(ωq~θ)); ~θ)
∣∣∣∣~θ=~0
)= Tr
(J(f ?t ρ(ω
~θ); ~θ)
∣∣∣∣~θ=~0
)= ω2J(f ?t ρ)
Since, J(f (ωc~θ); ~θ) = ω2
cJ(f (~θ); ~θ), and J(ρ(ωq
~θ); ~θ) = ω2qJ(ρ(
~θ); ~θ)
Tr
(J(f (ωc
~θ); ~θ)
∣∣∣∣~θ=~0
)+ Tr
(J(ρ(ωq
~θ); ~θ)
∣∣∣∣~θ=~0
)= ω2
cJ(f) + ω2qJ(ρ)
Taking ωc =√tJ(f)−1
J(ρ)−1+tJ(f)−1 and ωq = J(ρ)−1
J(ρ)−1+tJ(f)−1 , we obtain J(f ?t ρ)−1 − J(ρ)−1 − tJ(f)−1 ≥ 0
![Page 37: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/37.jpg)
Quantum Fisher information isoperimetric inequality
ddt
∣∣∣∣t=0
[12nJ(etL(ρ))
]−1≥ 1
![Page 38: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/38.jpg)
Quantum Fisher information isoperimetric inequality
ddt
∣∣∣∣t=0
[12nJ(fZ ?t ρ)
]−1≥ 1
Recall that for a Gaussian r.v. Z we have etL(ρ) = fZ ?t ρ
![Page 39: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/39.jpg)
Quantum Fisher information isoperimetric inequality
Proof:
ddt
∣∣∣∣t=0
[12nJ(fZ ?t ρ)
]−1≥ 1
Take f = fZ in q. Stam inequality:
1t
J(fZ ?t ρ)−1 − J(ρ)−1
≥ J(fZ)−1 = (2n)−1
Take a limit t→∞.
Recall that for a Gaussian r.v. Z we have etL(ρ) = fZ ?t ρ
![Page 40: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/40.jpg)
Quantum Fisher information isoperimetric inequality
Proof:
ddt
∣∣∣∣t=0
[12nJ(fZ ?t ρ)
]−1≥ 1
Take f = fZ in q. Stam inequality:
1t
J(fZ ?t ρ)−1 − J(ρ)−1
≥ J(fZ)−1 = (2n)−1
Take a limit t→∞.
Recall that for a Gaussian r.v. Z we have etL(ρ) = fZ ?t ρ
Optimality: for n = 1, let ωn be a Gaussian thermal state with a mean-photon number n
Then S(ωn) = g(n) = (n + 1) log(n + 1)− n logn
Under L: etL(ωn) = ωnt with nt = n + 2πt
Therefore, J(ωn) = 2 ddt
∣∣∣∣t=0
S(etL(ωn)) = 4π log(n+1n
)J(etL(ωn)) = 4π log
(1 + 1
n+2πt
)ddt
∣∣∣∣t=0
[12nJ(etL(ρ))
]−1= 1
n(n+1) log−2(1 + 1
n
)→ 1 as n→∞
![Page 41: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/41.jpg)
Concavity of the quantum entropy power
d2
dt2
∣∣∣∣t=0
N(etL(ρ)) ≤ 0
![Page 42: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/42.jpg)
Concavity of the quantum entropy power
d2
dt2
∣∣∣∣t=0
N(fZ ?t ρ) ≤ 0
Recall that for a Gaussian r.v. Z we have etL(ρ) = fZ ?t ρ
![Page 43: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/43.jpg)
Concavity of the quantum entropy power
d2
dt2
∣∣∣∣t=0
N(etL(ρ)) ≤ 0
Proofde Bruijn identity: J(ρ) = 2 d
dtS(etL(ρ)
)∣∣t=0
Therefore:
d2
dt2
∣∣∣∣t=0
N(etL(ρ)) = N(ρ)[
12nJ(ρ)
]2+ 1
2nddtJ(etLρ)
∣∣∣∣t=0
ddtN(etL(ρ)) = 1
nN(etL(ρ)) ddtS(etL(ρ))
d2
dt2N(etL(ρ)) = 1n2N(etL(ρ))
(ddtS(etL(ρ))
)2+ 1
nN(etL(ρ)) d2
dt2S(etL(ρ))
From quantum Fisher information isoperimetric inequality:
12nJ(ρ)2 + d
dtJ(etLρ)
∣∣∣∣t=0
≤ 0
![Page 44: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/44.jpg)
Entropy power inequality
N(f ?t ρ) ≥ tN(f) +N(ρ)
![Page 45: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/45.jpg)
Entropy power inequality
N(etL(ρ)) ≥ N(ρ) + t 2πe
![Page 46: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/46.jpg)
Entropy power inequality
N(f ?t ρ) ≥ tN(f) +N(ρ)
Proof: Consider δ := expS(ρ)/n+t expH(f)/nexpS(f?tρ)/n
EPI in equivalent to δ ≤ 1
![Page 47: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/47.jpg)
Entropy power inequality
N(f ?t ρ) ≥ tN(f) +N(ρ)
Proof:
Classical heat diffusion semigroup on R2n Cc = 12
∑2nj=1
∂2
∂~ξ2j
Define
µ→ EA(µ) := exp(S(eµL(ρ))/n
)ν → EB(ν) := exp
(H(eνCc(f))/n
)ξ → EC(ξ) := exp
(S(eξL(f ?t ρ)))/n
)Consider
µ(s) = EA(µ(s)) , µ(0) = 0
ν(s) = EB(ν(s)) , ν(0) = 0
ξ(s) := µ(s) + tν(s)
Define
δ(s) := EA(µ(s))+tEB(ν(s))EC(ξ(s))
EPI is equivalent to δ(0) ≤ 1
Consider δ := expS(ρ)/n+t expH(f)/nexpS(f?tρ)/n
EPI in equivalent to δ ≤ 1
![Page 48: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/48.jpg)
Entropy power inequality
N(f ?t ρ) ≥ tN(f) +N(ρ)
Proof:
Classical heat diffusion semigroup on R2n Cc = 12
∑2nj=1
∂2
∂~ξ2j
exp(H(etCc(f))/d
)= (2πe)t+O(1)
exp(S(etL(ρ))/d
)= (2πe)t+O(1)
Define
µ→ EA(µ) := exp(S(eµL(ρ))/n
)ν → EB(ν) := exp
(H(eνCc(f))/n
)ξ → EC(ξ) := exp
(S(eξL(f ?t ρ)))/n
)Consider
µ(s) = EA(µ(s)) , µ(0) = 0
ν(s) = EB(ν(s)) , ν(0) = 0
ξ(s) := µ(s) + tν(s)
Define
δ(s) := EA(µ(s))+tEB(ν(s))EC(ξ(s))
EPI is equivalent to δ(0) ≤ 1
Consider δ := expS(ρ)/n+t expH(f)/nexpS(f?tρ)/n
EPI in equivalent to δ ≤ 1
Step 1 lims→∞ δ(s) = 1
(in the limit t→∞)
![Page 49: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/49.jpg)
Entropy power inequality
N(f ?t ρ) ≥ tN(f) +N(ρ)
Proof:
Classical heat diffusion semigroup on R2n Cc = 12
∑2nj=1
∂2
∂~ξ2j
Define
µ→ EA(µ) := exp(S(eµL(ρ))/n
)ν → EB(ν) := exp
(H(eνCc(f))/n
)ξ → EC(ξ) := exp
(S(eξL(f ?t ρ)))/n
)Consider
µ(s) = EA(µ(s)) , µ(0) = 0
ν(s) = EB(ν(s)) , ν(0) = 0
ξ(s) := µ(s) + tν(s)
Define
δ(s) := EA(µ(s))+tEB(ν(s))EC(ξ(s))
EPI is equivalent to δ(0) ≤ 1
Consider δ := expS(ρ)/n+t expH(f)/nexpS(f?tρ)/n
EPI in equivalent to δ ≤ 1
Step 1 lims→∞ δ(s) = 1 Step 2 δ(s) ≥ 0 for all s ≥ 0
eξL(f ?t ρ) = eνCc(f) ?t eµL(ρ)
Note that
![Page 50: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/50.jpg)
Isoperimetric Inequality for Entropies1nJ(ρ)N(ρ) ≥ 4πe
![Page 51: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/51.jpg)
Isoperimetric Inequality for Entropies1nJ(ρ)N(ρ) ≥ 4πe
Proof:from de Bruijn identity:
ddt
∣∣∣∣t=0
N(ρ(t)) = 12nJ(ρ)N(ρ)
denoting ρ(t) = etL(ρ), t ≥ 0, the entropy power inequality yields:
1t [N(ρ(t))−N(ρ)] ≥ 2πe
take a limit t→∞.
![Page 52: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/52.jpg)
Isoperimetric Inequality for Entropies1nJ(ρ)N(ρ) ≥ 4πe
Optimality: let ρ = ωn be a Gaussian thermal state with a mean photon number n
Then S(ωn) = g(n) = (n + 1) log(n + 1)− n logn
![Page 53: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/53.jpg)
Isoperimetric Inequality for Entropies1nJ(ρ)N(ρ) ≥ 4πe
Optimality: let ρ = ωn be a Gaussian thermal state with a mean photon number n
Then S(ωn) = g(n) = (n + 1) log(n + 1)− n logn
Under L the state evolves as etL(ωn) = ωnt where nt = n + 2πt
In particular, by de Bruijn identity
J(ωn) = 2 ddt
∣∣∣∣t=0
S(etL(ωn)) = 4π log(n+1n
)
![Page 54: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/54.jpg)
Isoperimetric Inequality for Entropies1nJ(ρ)N(ρ) ≥ 4πe
Optimality: let ρ = ωn be a Gaussian thermal state with a mean photon number n
Then S(ωn) = g(n) = (n + 1) log(n + 1)− n logn
Under L the state evolves as etL(ωn) = ωnt where nt = n + 2πt
In particular, by de Bruijn identity
J(ωn) = 2 ddt
∣∣∣∣t=0
S(etL(ωn)) = 4π log(n+1n
)N(ωn) = exp(S(ωn)/1) = (n+1)n+1
nn
Also
J(ωn)N(ωn) = 4π(n+1n
)nlog(n+1n
)n+1 →n→∞
4πe
Therefore
![Page 55: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/55.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Quantum Attenuator:L−(ρ) = aρa† − 1
2a†a, ρ
Quantum Amplifier L+(ρ) = a†ρa− 12aa
†, ρ
n = 1
Let ρ = ωn be a Gaussian thermal state with a mean photon number n. Then ρ(t) = ωn(t)
Under L−n−(t) = e−tn
Under L+
n+(t) = etn + et − 1
![Page 56: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/56.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Quantum Attenuator:L−(ρ) = aρa† − 1
2a†a, ρ
Quantum Amplifier L+(ρ) = a†ρa− 12aa
†, ρ
n = 1
Let ρ = ωn be a Gaussian thermal state with a mean photon number n. Then ρ(t) = ωn(t)
Under L−n−(t) = e−tn
Under L+
n+(t) = etn + et − 1
One-parameter group of CPTP maps eLµ,λt≥0, generated by
Lµ,λ = µ2L− + λ2L+ for µ > λ > 0
Then ωn(t) is
nt = e−(µ2−λ2)tn + (1− e−(µ2−λ2)t)n∞ n∞ = tr(σµ,λn) = λ2
µ2−λ2
![Page 57: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/57.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Quantum Attenuator:L−(ρ) = aρa† − 1
2a†a, ρ
Quantum Amplifier L+(ρ) = a†ρa− 12aa
†, ρ
n = 1
Let ρ = ωn be a Gaussian thermal state with a mean photon number n. Then ρ(t) = ωn(t)
Under L−n−(t) = e−tn
Under L+
n+(t) = etn + et − 1
One-parameter group of CPTP maps eLµ,λt≥0, generated by
Lµ,λ = µ2L− + λ2L+ for µ > λ > 0
Then ωn(t) is
nt = e−(µ2−λ2)tn + (1− e−(µ2−λ2)t)n∞ n∞ = tr(σµ,λn) = λ2
µ2−λ2
Conjecture
D(etLµ,λ(ρ)‖σµ,λ) ≤ e−(µ2−λ2)tD(ρ‖σµ,λ) for all t ≥ 0
Fixed point of Lµ,λ is σµ,λ = (1− ν)∑∞n=0 ν
n|n〉〈n|, with ν = λ2/µ2
![Page 58: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/58.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Quantum Attenuator:L−(ρ) = aρa† − 1
2a†a, ρ
Quantum Amplifier L+(ρ) = a†ρa− 12aa
†, ρ
n = 1
Let ρ = ωn be a Gaussian thermal state with a mean photon number n. Then ρ(t) = ωn(t)
Under L−n−(t) = e−tn
Under L+
n+(t) = etn + et − 1
One-parameter group of CPTP maps eLµ,λt≥0, generated by
Lµ,λ = µ2L− + λ2L+ for µ > λ > 0
Then ωn(t) is
nt = e−(µ2−λ2)tn + (1− e−(µ2−λ2)t)n∞ n∞ = tr(σµ,λn) = λ2
µ2−λ2
Conjecture
D(etLµ,λ(ρ)‖σµ,λ) ≤ e−(µ2−λ2)tD(ρ‖σµ,λ) for all t ≥ 0
To this end we will focus on −ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) for ζ = µ2 − λ2
Fixed point of Lµ,λ is σµ,λ = (1− ν)∑∞n=0 ν
n|n〉〈n|, with ν = λ2/µ2
![Page 59: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/59.jpg)
Quantum Ornstein-Uhlenbeck semigroupLemma
−ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
![Page 60: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/60.jpg)
Quantum Ornstein-Uhlenbeck semigroup
D(ρ‖σµ,λ) = −S(ρ)− (log ν)n− log(1− ν)
We have
nt = e−(µ2−λ2)tn + (1− e−(µ2−λ2)t)n∞ n∞ = tr(σµ,λn) = λ2
µ2−λ2
And soddt
∣∣∣t=0
nt = −(µ2 − λ2) (n− n∞)
Therefore,
− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ)− (log ν)(µ2 − λ2)n + λ2 log ν
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
n = Tr(ρn)
Proof
![Page 61: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/61.jpg)
Quantum Ornstein-Uhlenbeck semigroup
D(ρ‖σµ,λ) = −S(ρ)− (log ν)n− log(1− ν)
We have
nt = e−(µ2−λ2)tn + (1− e−(µ2−λ2)t)n∞ n∞ = tr(σµ,λn) = λ2
µ2−λ2
And soddt
∣∣∣t=0
nt = −(µ2 − λ2) (n− n∞)
Therefore,
− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ)− (log ν)(µ2 − λ2)n + λ2 log ν
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
For ζ = µ2 − λ2 we obtain the desired equality
n = Tr(ρn)
Proof
![Page 62: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/62.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Theorem−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≥ −ζn log(1 + 1/n) + δ for ζ = µ2 − λ2, n = Tr(ρn)
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
![Page 63: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/63.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Theorem−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≥ −ζn log(1 + 1/n) + δ
From Isoperimetric inequality for entropies
−S(ρ) ≤ AJ(ρ)− (2 + log(4πA))
for A > 0
for ζ = µ2 − λ2, n = Tr(ρn)
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
Proof
![Page 64: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/64.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Theorem−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≥ −ζn log(1 + 1/n) + δ
From Isoperimetric inequality for entropies
−S(ρ) ≤ AJ(ρ)− (2 + log(4πA))
for A > 0
−S(ρ) ≤ log
14πeJ(ρ)
using log x ≤ x− 1
log(
14πeJ(ρ)
)= log
(AJ(ρ)4πeA
)= log
(1
4πeA
)+ log
(AJ(ρ)
)≤ AJ(ρ)− 2− log(4πA)
for ζ = µ2 − λ2, n = Tr(ρn)
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
Proof
![Page 65: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/65.jpg)
Quantum Ornstein-Uhlenbeck semigroup
Theorem−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≥ −ζn log(1 + 1/n) + δ
From Isoperimetric inequality for entropies
−S(ρ) ≤ AJ(ρ)− (2 + log(4πA))
for A > 0
S(ρ) ≥ −2πAJ−(ρ)− 2πAJ+(ρ) + 2 + log(4πA)
Gaussian optimality: J−(ρ) ≥ J−(ωn) = −2n log(1 + 1/n)
for ζ = µ2 − λ2, n = Tr(ρn)
With the choice of A = λ2
4π(µ2−λ2) we obtain the desired bound
D(ρ‖σµ,λ) = −S(ρ)− (log ν)n− log(1− ν)
Lemma−ζD(ρ‖σµ,λ)− d
dt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
J±(ρ) := 2 ddtS(etL±(ρ)) J(ρ) = 2π(J−(ρ) + J+(ρ))
Proof
![Page 66: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/66.jpg)
Fast convergence of qOU semigroup µ2 = 2, λ2 = 1
Exampleddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≤ −D(ρ‖σ√2,1) for all states ρ s.t. tr(ρn) . 0.67
or S(ρ) & 2.4In particular,D(etL
√2,1(ρ)‖σ√2,1) ≤ e−tD(ρ‖σ√2,1)
![Page 67: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/67.jpg)
Fast convergence of qOU semigroup µ2 = 2, λ2 = 1
Example
−ζD(ρ‖σ√2,1)− ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≥ −n log(1 + 1/n) + 2− 2 log(2)
The RHS is monotonically decreasing, and for n . 0.67, is non-negative.
ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≤ −D(ρ‖σ√2,1) for all states ρ s.t. tr(ρn) . 0.67
or S(ρ) & 2.4In particular,D(etL
√2,1(ρ)‖σ√2,1) ≤ e−tD(ρ‖σ√2,1)
Proof From the last Theorem we have
![Page 68: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/68.jpg)
Fast convergence of qOU semigroup µ2 = 2, λ2 = 1
Example
−ζD(ρ‖σ√2,1)− ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≥ −n log(1 + 1/n) + 2− 2 log(2)
The RHS is monotonically decreasing, and for n . 0.67, is non-negative.
We had that
−ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≤ −D(ρ‖σ√2,1) for all states ρ s.t. tr(ρn) . 0.67
or S(ρ) & 2.4In particular,D(etL
√2,1(ρ)‖σ√2,1) ≤ e−tD(ρ‖σ√2,1)
Proof From the last Theorem we have
![Page 69: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/69.jpg)
Fast convergence of qOU semigroup µ2 = 2, λ2 = 1
Example
−ζD(ρ‖σ√2,1)− ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≥ −n log(1 + 1/n) + 2− 2 log(2)
The RHS is monotonically decreasing, and for n . 0.67, is non-negative.
We had that
−ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
From [Buscemi et al.16] we have J+(ρ) ≥ 2
ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≤ −D(ρ‖σ√2,1) for all states ρ s.t. tr(ρn) . 0.67
or S(ρ) & 2.4In particular,D(etL
√2,1(ρ)‖σ√2,1) ≤ e−tD(ρ‖σ√2,1)
Proof From the last Theorem we have
![Page 70: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/70.jpg)
Fast convergence of qOU semigroup µ2 = 2, λ2 = 1
Example
−ζD(ρ‖σ√2,1)− ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≥ −n log(1 + 1/n) + 2− 2 log(2)
The RHS is monotonically decreasing, and for n . 0.67, is non-negative.
We had that
−ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) = µ2
2 J−(ρ) + λ2
2 J+(ρ) + ζS(ρ) + λ2 log ν + ζ log(1− ν)
From [Buscemi et al.16] we have J+(ρ) ≥ 2
From [De Palma et al. 16] we have
−ζD(ρ‖σµ,λ)− ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≥ F (S0) + λ2 + λ2 log ν + ζ log(1− ν)
Also, for S0 & 2.06, RHS is non-negative
ddt
∣∣∣t=0
D(etL√
2,1(ρ)‖σ√2,1) ≤ −D(ρ‖σ√2,1) for all states ρ s.t. tr(ρn) . 0.67
or S(ρ) & 2.4In particular,D(etL
√2,1(ρ)‖σ√2,1) ≤ e−tD(ρ‖σ√2,1)
Proof From the last Theorem we have
Function F (S) is monotonically increasing for S & 0.5 for µ2 = 2, λ2 = 1
![Page 71: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/71.jpg)
Optimality of the convergence rate
ConjectureD(etLµ,λ(ρ)‖σµ,λ) ≤ e−(µ2−λ2)tD(ρ‖σµ,λ) for all t ≥ 0
![Page 72: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/72.jpg)
Optimality of the convergence rate
ConjectureD(etLµ,λ(ρ)‖σµ,λ) ≤ e−(µ2−λ2)tD(ρ‖σµ,λ) for all t ≥ 0
Tightness:for any Gaussian state
ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) ≤ −(µ2 − λ2)D(ρ||σµ,λ)
Moreover, for any ζ > µ2 − λ2 there exists a Gaussian state ρ s.t.
ddt
∣∣∣t=0
D(etLµ,λ(ρ)‖σµ,λ) > −ζD(ρ||σµ,λ)
![Page 73: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/73.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
![Page 74: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/74.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
Let r.v. X has density hX(x) = |g(x)|2 and r.v. Y has density hY (x) = e−π|x|2
. Then
H(X|Y )
∫hX log hXdx−
∫hX log hY dx
![Page 75: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/75.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
Let r.v. X has density hX(x) = |g(x)|2 and r.v. Y has density hY (x) = e−π|x|2
. Then
H(X|Y )
∫hX log hXdx−
∫hX log hY dx
≤ 14π4
∫| 5 h
1/2X |2dx− n−
∫hX log hY dx
![Page 76: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/76.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
Let r.v. X has density hX(x) = |g(x)|2 and r.v. Y has density hY (x) = e−π|x|2
. Then
H(X|Y ) ≤ 14π4
∫| 5 h
1/2X |2dx− n−
∫hX log hY dx
14πJ(X) +πE|X|2
![Page 77: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/77.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
Let r.v. X has density hX(x) = |g(x)|2 and r.v. Y has density hY (x) = e−π|x|2
. Then
H(X|Y ) ≤ 14πJ(X)− n+ πE|X|2
![Page 78: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/78.jpg)
Log-Sobolev inequality
∫|f |2 log |f |2e−π|x|2dx ≤ 1
π
∫| 5 f |2e−π|x|2dx for
∫|f |2e−π|x|2dx = 1
Let g(x) = f(x)e−π|x|2/2. Then
Gross‘ logarithmic Sobolev inequality is[Carlen ‘91]
∫|g|2 log |g|2dx ≤ 1
π
∫| 5 g|2dx− n with
∫|g|2dx = 1
Let r.v. X has density hX(x) = |g(x)|2 and r.v. Y has density hY (x) = e−π|x|2
. Then
H(X|Y ) ≤ 14πJ(X)− n+ πE|X|2
D(ρ‖σµ,λ) = −S(ρ)− (log ν)n− log(1− ν) ≤ AJ(ρ)− 2− log(1− ν)− log(4πA) + n log 1ν
Quantum case:
ν = λ2/µ2 < 1
A > 0
![Page 79: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/79.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)
![Page 80: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/80.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)Step 1 (Correspondence to classical problem)
Let p = (p0, p1, . . . , ) be a prob. distribution
For ρ =∑n pn|n〉〈n| we have
ρ(t) = etL−(ρ) =∑n pn(t)|n〉〈n|
with pn(t) = −npn(t) + (n+ 1)pn+1(t)
![Page 81: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/81.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)Step 1 (Correspondence to classical problem)
Let p = (p0, p1, . . . , ) be a prob. distribution
For ρ =∑n pn|n〉〈n| we have
ρ(t) = etL−(ρ) =∑n pn(t)|n〉〈n|
with pn(t) = −npn(t) + (n+ 1)pn+1(t)Denote (C−(p))n = −npn + (n+ 1)pn+1
Then p(t) = etC−(p) - pure-death processTheorem
infρ:Tr(nρ)≤nddt
∣∣∣t=0
S(etL−(ρ)) = infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p))
![Page 82: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/82.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)Step 1 (Correspondence to classical problem)
Let p = (p0, p1, . . . , ) be a prob. distribution
Denote (C−(p))n = −npn + (n+ 1)pn+1
Then p(t) = etC−(p) - pure-death processTheorem
infρ:Tr(nρ)≤nddt
∣∣∣t=0
S(etL−(ρ)) = infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p))
H(p) := −∑∞k=0 pk log pk
![Page 83: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/83.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)Step 1 (Correspondence to classical problem)
Let p = (p0, p1, . . . , ) be a prob. distribution
Denote (C−(p))n = −npn + (n+ 1)pn+1
Then p(t) = etC−(p) - pure-death processTheorem
infρ:Tr(nρ)≤nddt
∣∣∣t=0
S(etL−(ρ)) = infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p))
Step 2 (Classical problem)
infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p)) = −n log(1 + 1n )
![Page 84: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/84.jpg)
Gaussian optimality for energy-constrained entropy rates
Quantum Attenuator: L−(ρ) = aρa† − 12a†a, ρ
Theoreminfρ:Tr(nρ)≤n
ddt
∣∣t=0
S(etL−(ρ)) = −n log(1 + 1
n
)Step 1 (Correspondence to classical problem)
Let p = (p0, p1, . . . , ) be a prob. distribution
Denote (C−(p))n = −npn + (n+ 1)pn+1
Then p(t) = etC−(p) - pure-death processTheorem
infρ:Tr(nρ)≤nddt
∣∣∣t=0
S(etL−(ρ)) = infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p))
Step 2 (Classical problem)
infp:Ep[N ]≤nddt
∣∣∣t=0
H(etC−(p)) = −n log(1 + 1n )
Step 3 (Gaussian optimality) Let ωn be a Gaussian thermal state with a mean-photon number nThen
ddt
∣∣∣t=0
S(etL−(ωn)) = −n log(1 + 1
n
)
![Page 85: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/85.jpg)
Thank you!
![Page 86: · Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X volume vol(A) entropy power e2H(X)=n H(X) = R Rn f X(x)log f X(x)dx addition A +](https://reader034.fdocuments.in/reader034/viewer/2022052023/60383851c8a7f62560736fe0/html5/thumbnails/86.jpg)
References
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R. Koenig and G. Smith. The entropy power inequality for quantum systems. Infor- mation Theory, IEEETransactions on, 60(3):1536-1548, March 2014.
G. De Palma, D. Trevisan, and V. Giovannetti. Gaussian states minimize the output entropy of the one-modequantum attenuator. 2016. arXiv:1605.00441
F. Buscemi, S. Das, and M. M. Wilde. Approximate reversibility in the context of entropy gain, information gain, andcomplete positivity. 2016. arXiv:1601.01207
G. De Palma, A. Mari, S. Lloyd, and V. Giovannetti. Multimode quantum entropy power inequality. Phys. Rev. A,91:032320, Mar 2015
R. Werner. Quantum harmonic analysis on phase space. Journal of Mathematical Physics, 25(5):1404-1411, 1984