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Register a new userOn the metric character of the quantum Jensen-Shannon divergence
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In a recent paper, the generalization of the Jensen Shannon divergence (JSD) in the context of quantum theory has been studied (Phys. Rev. A 72, 052310 (2005)). This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of simulations for mixed states, that its square root verifies the triangle inequality.
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The notion of distance in Hilbert space is relevant in many scenarios. In particular, distances between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.
This is the 10th and final chapter of my book on Quantum Information, based on the course I have been teaching at Caltech since 1997. An earlier version of this chapter (originally Chapter 5) has been available on the course website since 1998, but this version is substantially revised and expanded. Topics covered include classical Shannon theory, quantum compression, quantifying entanglement, accessible information, and using the decoupling principle to derive achievable rates for quantum protocols. This is a draft, pre-publication copy of Chapter 10, which I will continue to update. See the URL on the title page for further updates and drafts of other chapters, and please send me an email if you notice errors.
We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterized by $alphain [0,2]$), which in turn (for $alpha=1$) yields a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti emph{et al.} a decade ago (emph{Metric character of the quantum Jensen-Shannon divergence}, Phy. Rev. A, textbf{79}, (2008).) A somewhat more intricate argument also establishes metric properties of Jensen-Renyi divergences (for $alpha in (0,1)$), and outlines a technique that may be of independent interest.
Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical IID sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e. simulations in which the sender retains what would escape into the environment in an ordinary simulation), on non-tensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.
Fawzi and Fawzi recently defined the sharp Renyi divergence, $D_alpha^#$, for $alpha in (1, infty)$, as an additional quantum Renyi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit ${alpha} to 1$ of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Renyi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences, and for which we prove various properties including the data-processing inequality.
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