Quantum R\'{e}nyi Entropy Functionals for Bosonic Gaussian Systems

TL;DR

This paper introduces a quantum Rényi entropy power inequality for bosonic Gaussian systems, extending classical concepts to quantum regimes and providing tools for quantum channel capacity estimation and entanglement detection.

Contribution

It develops a quantum Rényi entropy power inequality for bosonic Gaussian systems using quantum convolution, a novel approach in quantum information theory.

Findings

Derived a quantum Rényi-$p$ entropy power inequality for bosonic Gaussian systems.

Connected the inequality to quantum channel capacity bounds.

Proposed applications in entanglement witnessing and quantum communication.

Abstract

In this study, the quantum R\'{e}nyi entropy power inequality of order $p>1$ and power $\kappa$ is introduced as a quantum analog of the classical R\'{e}nyi-$p$ entropy power inequality. To derive this inequality, we first exploit the Wehrl-$p$ entropy power inequality on bosonic Gaussian systems via the mixing operation of quantum convolution, which is a generalized beam-splitter operation. This observation directly provides a quantum R\'{e}nyi-$p$ entropy power inequality over a quasi-probability distribution for $D$-mode bosonic Gaussian regimes. The proposed inequality is expected to be useful for the nontrivial computing of quantum channel capacities, particularly universal upper bounds on bosonic Gaussian quantum channels, and a Gaussian entanglement witness in the case of Gaussian amplifier via squeezing operations.