We establish a tight upper bound for the difference in von Neumann entropies between two quantum states,$\rho _{1}$and$\rho _{2}$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator$(\rho _{1} - \rho _{2})$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

This note is intended to address some inaccuracies in [1]. In Lemma 5.8, eq. (52) of this paper, we originally derived the following erroneous inequality:.

In this article, we generalize a proof technique by Alicki, Fannes and Winter and introduce a method to prove continuity bounds for entropic quantities derived from different quantum relative entropies. For the Umegaki relative entropy, we mostly recover known almost optimal bounds, whereas, for the Belavkin-Staszewski relative entropy, our bounds are new. Finally, we use these continuity bounds to derive a new entropic uncertainty relation. This is a short version of [1].

Based on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.

Quasi-factorization-type inequalities for the relative entropy have recently proven to be fundamental in modern proofs of modified logarithmic Sobolev inequalities for quantum spin systems. In this paper, we show some results of weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e. upper bounds for the BS-entropy between two bipartite states in terms of the sum of two conditional BS-entropies, up to some multiplicative and additive factors. A more detailed version of this paper is accessible at: https://arxiv.org/pdf/2101.10312.pdf

The property of superadditivity of the quantum relative entropy states that, in a bipartite system HAB= HA⊗ HB, for every density operator ρAB, one has D(ρAB||σA⊗ σB)≥ D(ρA||σA) + D(ρB||σB). In this paper, we provide an extension of this inequality for arbitrary density operators σAB. More specifically, we prove that α(σAB)· D(ρAB||σAB)≥D(ρA||σA)+ D(ρB||σB) holds for all bipartite states ρABand σAB, where α(σAB) = 1 + 2||σA-1/2⊗ σABσA-1/2⊗ σB-1/2- ||AB||∞.