QUANTUM f-DIVERGENCES AND ERROR CORRECTION

Abstract

Quantum f -divergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f -divergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f -divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f -divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f -divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

“…So, by A −1 and A 0 we respectively mean the generalized inverse and the support projection of A. A quantum counterpart of Csiszár's f -divergence is introduced as follows [22]. For an operator A ∈ L + (H), let Λ A and Υ A denote the left and the right multiplications by A, respectively, defined as…”

“…Basic properties of the quantity (3.4) are discussed in the paper [22]. Using the function f α (z) = z α − z /(α − 1), we actually obtain the quantity (2.18).…”

“…Note that the inequality (3.6) has generally been established in [22] under weaker conditions on the maps. From the monotonicity (3.6) we can derive simple upper bounds on the quantum f -divergence in terms of classical one.…”

Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

“…We mention that this example shows that the statement Proposition 2.12 in [4] asserting that the f -divergences are continuous on L + (H ) in their second variables is false.…”

Some quantum $f$-divergence inequalities for convex functions of self-adjoint operators

“…In the important case of finite dimensional space H and the generalized inverse P −1 , numerous properties of the quantum f -divergence, mostly in the case when f is operator convex, have been obtained in the recent papers [15], [16], [20], [21] and the references therein.…”