Entropy power inequalities for qudits

TL;DR

This paper extends entropy power inequalities to finite-dimensional quantum systems (qudits), establishing new inequalities for various entropic functions and analyzing their implications for quantum information capacity bounds.

Contribution

It introduces a class of entropy power inequalities for qudits using a partial swap channel, generalizing previous continuous-variable quantum inequalities to finite dimensions.

Findings

Established EPI analogues for von Neumann, Rényi, and subentropy.

Proved a qudit analogue of the entropy photon number inequality.

Derived bounds on minimum output entropy and Holevo capacity for certain channels.

Abstract

Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: $$f(\sqrt{a}\,X + \sqrt{1-a}\,Y) \ge a f(X) + (1-a) f(Y) \quad \forall \, a \in [0,1].$$ Here, $X$ and $Y$ are continuous random variables and the function $f$ is either the differential entropy or the entropy power. K\"onig and Smith [arXiv:1205.3409] and De Palma, Mari, and Giovannetti [arXiv:1402.0404] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where $X$ and $Y$ are replaced by bosonic fields and the addition rule is the action of a beamsplitter with transmissivity $a$ on those fields. In this paper, we similarly establish a class of EPI analogues for $d$-level quantum systems (i.e. qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter $a \in [0,1]$ and acts like a finite-dimensional analogue of a beamsplitter with transmissivity $a$, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as $a$ is changed from zero to one. We obtain analogues of Shannon's EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but for a much larger class of functions as well. This class includes the R\'enyi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity.