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Register a new userThe $alpha to 1$ Limit of the Sharp Quantum Renyi Divergence
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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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We prove decomposition rules for quantum Renyi mutual information, generalising the relation $I(A:B) = H(A) - H(A|B)$ to inequalities between Renyi mutual information and Renyi entropy of different orders. The proof uses Beigis generalisation of Reisz-Thorin interpolation to operator norms, and a variation of the argument employed by Dupuis which was used to show chain rules for conditional Renyi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Renyi mutual information, generalising the original relation by Hall.
We derive an inequality for the linear entropy, that gives sharp bounds for all finite dimensional systems. The derivation is based on generalised Bloch decompositions and provides a strict improvement for the possible distribution of purities for all finite dimensional quantum states. It thus extends the widely used concept of entropy inequalities from the asymptotic to the finite regime, and should also find applications in entanglement detection and efficient experimental characterisations of quantum states.
We evaluate the variance of coefficients of the characteristic polynomial of the quantum evolution operator for chaotic 4-regular quantum graphs (networks) via periodic orbits without taking the semiclassical limit. The variance of the n-th coefficient is precisely determined by the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) with n bonds that fall in the following classes: those with no self-intersections, and those where all the self-intersections consist of two sections of the pseudo orbit crossing at a single vertex (2-encounters of length zero).
This paper studies forward and reverse projections for the R{e}nyi divergence of order $alpha in (0, infty)$ on $alpha$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {em et al.} in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoes proved a Pythagorean inequality for R{e}nyi divergences on $alpha$-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of forward projection is proved for probability measures on a general alphabet. For $alpha in (1, infty)$, the proof relies on a new Apollonius theorem for the Hellinger divergence, and for $alpha in (0,1)$, the proof relies on the Banach-Alaoglu theorem from functional analysis. Further projection results are then obtained in the finite alphabet setting. These include a projection theorem on a specific $alpha$-convex set, which is termed an {em $alpha$-linear family}, generalizing a result by Csiszar for $alpha eq 1$. The solution to this problem yields a parametric family of probability measures which turns out to be an extension of the exponential family, and it is termed an {em $alpha$-exponential family}. An orthogonality relationship between the $alpha$-exponential and $alpha$-linear families is established, and it is used to turn the reverse projection on an $alpha$-exponential family into a forward projection on a $alpha$-linear family. This paper also proves a convergence result of an iterative procedure used to calculate the forward projection on an intersection of a finite number of $alpha$-linear families.
We study the entropy production of the sandwiched Renyi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Renyi divergence of any order ${alpha} in (0, infty)$. This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810-1869] for the quantum relative entropy (i.e., ${alpha} = 1$). Moreover, we show that the sandwiched Renyi divergence of any order ${alpha} in (0, infty)$ decays exponentially fast under the time-evolution of such a Lindblad equation.
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