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Entropy power inequalities for qudits

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(a X+1−a Y)≥af(X)+(1−a)f(Y)∀ a∈[0,1]. Here, X and Y are continuous random variables and the function f is either the differential entropy or the entropy power. Konig and Smith [IEEE Trans. Inf. Theory 60(3), 1536–1548 (2014)] and De Palma, Mari, and Giovannetti [Nat. Photonics 8(12), 958–964 (2014)] 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 beam splitter 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 ∈ [0, 1] and acts like a finite-dimensional analogue of a beam splitter with transmissiv...