In the present paper we give proof that the information-transmission capacity of the approximate position measurement with the oscillator energy constraint, which underlies noisy Gaussian homodyning in quantum optics, is attained on Gaussian encoding. The proof is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. We hope that this method should work also for other models lying out of the scope of the "threshold condition" ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable.In the present paper we give proof that the information-transmission capacity of the approximate position measurement with the oscillator energy constraint, which underlies noisy Gaussian homodyning in quantum optics, is attained on Gaussian encoding. The proof is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. We hope that this method should work also for other models lying out of the scope of the "threshold condition" ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable.

arXiv:2204.10626Apr 22, 20222 citations

Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation

A. S. Holevo, M. E. Shirokov

We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state $ρ$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ valid for any state $σ$ , where $C_ρ$ is a constant depending on the rank of $ρ$ . The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $ρ$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ valid for any state $σ$ , where $C_ρ$ is a constant depending on the Schmidt rank of $ρ$ . In both cases the optimal constant $C_ρ$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}ρ$ , in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$ . The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state $ρ$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the rank of $ρ$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $ρ$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the Schmidt rank of $ρ$. In both cases the optimal constant $C_ρ$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}ρ$, in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

quant-pharXiv:2609.12667Sep 14, 20260 citations

Log-majorization and matrix norm inequalities with application to quantum information

Fumio Hiai

arXiv:2402.16067Acta Scientiarum MathematicarumJun 14, 20245 citations

ρ

with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality

\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,

valid for any state

σ

, where

C_ρ

is a constant depending on the rank of

ρ

. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state

ρ

with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality

\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,

valid for any state

σ

, where

C_ρ

is a constant depending on the Schmidt rank of

ρ

. In both cases the optimal constant

C_ρ

is equal to the optimal constant

K_{d}

in the log-Sobolev inequality for the complete graph with

d

vertices: in the first case

d=\mathrm{rank}ρ

, in the second one

d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B

. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.