· Geometry Classical Quantum Set A 2Rn Random variable X on R n with prob. density function f X...

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Quantum analogues of geometric inequalities for Information Theory Anna Vershynina June 17, 2016 Workshop 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

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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

∣∣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 +

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

∣∣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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

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 +

References

S. Huber, R. Koenig, A. Vershynina. Geometric inequalities from phase-space translations. In preparation

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