Entropy power inequalities for qudits (Q2814213)

Strangely enough for someone -- as this reviewer -- involved in the search for possible \textit{classical analogs of quantum systems}, the main offspring of decades of inquiries about the foundations of quantum mechanics appears to be the burgeoning domain of quantum computing and information where instead the principal interest seems to lie in finding \textit{quantum analogs of classical ideas}: a stark reminder of how completely the perspective has tilted. Entropy is no exception to this rule.NEWLINENEWLINEAs the authors claim at the very beginning of their paper, the ``inequalities between entropic quantities play a fundamental role in information theory and have been employed effectively in finding bounds on optimal rates of various information-processing tasks. Shannon's entropy power inequality (\textit{EPI}) is one such inequality and it has proved to be of relevance in studying problems not only in information theory, but also in probability theory and mathematical physics. It has been used, for example, in finding upper bounds on the capacities of certain noisy channels (e.g., the Gaussian broadcast channel) and in proving convergence in relative entropy for the central limit theorem.'' It is understandable then their interest in broadening the scope of the already available results in the field of quantum information. In particular it is remarkable that -- exactly as in the classical field, where these results have been readily established only for absolutely continuous random variables, while proving more elusive for discrete random variables -- the best known quantum analogs (the \textit{EPI} proper, and the so called \textit{entropy photon number inequality}: \textit{EPnI}) have been previously proved only in the continuous variable setting: consequently in the present paper the authors address the open problem of giving the inequalities for general quantum \(d\)-level systems a.k.a.\ \textit{qudits}. It is clearly stated about these quantum results, however, that neither the previous ones, nor the author's present results are true \textit{generalizations} of the classical Shannon's \textit{EPI}'s because the ``quantum inequalities do not reduce to the classical [ones] for commuting states'': they are instead properly classified just as \textit{analogs}.NEWLINENEWLINEA remarkable point is that, even in the classical case, ``the form of the \textit{EPI} ... motivates the definition of an operation ... on the space of random variables, given by the following scaled addition rule: NEWLINE\[NEWLINE X\boxplus_aY:=\sqrt{a}\,X+\sqrt{1-a}\,Y\qquad\quad\forall a\in[0,1] NEWLINE\]NEWLINE ... With this notation, [\textit{EPI}] can be written as NEWLINE\[NEWLINE H(X\boxplus_aY)\geq aH(X)+(1-a)H(Y)\qquad\quad\forall a\in[0,1]\text{''} NEWLINE\]NEWLINE where \(H\) is the Shannon entropy. As a consequence a pre-requisite to the author's inquiry is the formulation of ``an addition rule for \(d\)-level systems (\textit{qudits}) in the form of a quantum channel \(\mathcal{E}_a\), which we call the \textit{partial swap channel} that acts on the two input quantum states.'' The new quantum addition rule, again denoted by \(\boxplus_a\), combines a pair of \(d\)-dimensional quantum states \(\rho\) and \(\sigma\) as follows NEWLINE\[NEWLINE \rho\boxplus_a\sigma:=a\rho+(1-a)\sigma-\sqrt{a(1-a)}i[\rho,\sigma] NEWLINE\]NEWLINE where \([\rho,\sigma]\) is the usual commutator. It is proved also that \(\rho\boxplus_a\sigma=\mathcal{E}_a(\rho\otimes\sigma)\) for some quantum channel \(\mathcal{E}_a\), implying that \(\rho\boxplus_a\sigma\) is a valid state of a \textit{qudit}: similar to its analogs in the continuous-variable classical and quantum settings, it results in an interpolation between the two states which it combines, as the parameter \(a\) is changed from \(1\) to \(0\). The main results (in Corollary \(7\)) consist then in a number of inequalities, as for instance NEWLINE\[NEWLINE H(\rho\boxplus_a\sigma)\geq aH(\rho)+(1-a)H(\sigma) NEWLINE\]NEWLINE where now \(H\) is the von Neumann entropy of finite-dimensional quantum systems.NEWLINENEWLINEAll this seems to make the stated results contingent on the choice of this addition rule, and in some sense also explains why they are not simple \textit{generalizations}, but rather quantum \textit{analogs}. Needless to say, however, the interest of the new inequalities lies in their possible role for further inquiries (as for the classical inequalities), and to this end it is important to remark that in the Section \textit{VII} an application of the new quantum \textit{EPI} is already produced in the form of ``lower bounds on the minimum output entropy and upper bounds on the Holevo capacity for a class of single-input channels that are formed from the channel \(\mathcal{E}_a\) by fixing the second input state.''NEWLINENEWLINEIt is finally remarkable from a mathematical point of view that the (long and elaborated) proof is not achieved by adopting methods analogous to those used in proving the classical \textit{EPI}'s (as was the case for the previous inequalities for continuous-variable quantum systems), but rather relies on completely different tools, namely, spectral majorization and concavity of functions: ideas properly introduced in a section of preliminaries.