Some applications of the Hermit-Hadamard inequality for log-convex functions in quantum divergence

One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.

On Composite Quantum Hypothesis Testing

We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $$\rho ^{\otimes n}$$ ρ ⊗ n against convex combinations of quantum states $$\sigma ^{\otimes n}$$ σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

Numerical finite-key analysis of quantum key distribution

Quantum key distribution (QKD) allows for secure communications safe against attacks by quantum computers. QKD protocols are performed by sending a sizeable, but finite, number of quantum signals between the distant parties involved. Many QKD experiments, however, predict their achievable key rates using asymptotic formulas, which assume the transmission of an infinite number of signals, partly because QKD proofs with finite transmissions (and finite-key lengths) can be difficult. Here we develop a robust numerical approach for calculating the key rates for QKD protocols in the finite-key regime in terms of two semi-definite programs (SDPs). The first uses the relation between conditional smooth min-entropy and quantum relative entropy through the quantum asymptotic equipartition property, and the second uses the relation between the smooth min-entropy and quantum fidelity. The numerical programs are formulated under the assumption of collective attacks from the eavesdropper and can be promoted to withstand coherent attacks using the postselection technique. We then solve these SDPs using convex optimization solvers and obtain numerical calculations of finite-key rates for several protocols difficult to analyze analytically, such as BB84 with unequal detector efficiencies, B92, and twin-field QKD. Our numerical approach democratizes the composable security proofs for QKD protocols where the derived keys can be used as an input to another cryptosystem.

Quantum Relative Entropy of Tagging and Thermodynamics

Thermodynamics establishes a relation between the work that can be obtained in a transformation of a physical system and its relative entropy with respect to the equilibrium state. It also describes how the bits of an informational reservoir can be traded for work using Heat Engines. Therefore, an indirect relation between the relative entropy and the informational bits is implied. From a different perspective, we define procedures to store information about the state of a physical system into a sequence of tagging qubits. Our labeling operations provide reversible ways of trading the relative entropy gained from the observation of a physical system for adequately initialized qubits, which are used to hold that information. After taking into account all the qubits involved, we reproduce the relations mentioned above between relative entropies of physical systems and the bits of information reservoirs. Some of them hold only under a restricted class of coding bases. The reason for it is that quantum states do not necessarily commute. However, we prove that it is always possible to find a basis (equivalent to the total angular momentum one) for which Thermodynamics and our labeling system yield the same relation.

Tensorization of the strong data processing inequality for quantum chi-square divergences

It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E. For a fixed channel E and a state σ, the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ, which is characterized by the strong data processing inequality (SDPI). Among various input states ρ, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2) for arbitrary quantum-classical channels.

Some trace inequalities for exponential and logarithmic functions

Consider a function [Formula: see text] of pairs of positive matrices with values in the positive matrices such that whenever [Formula: see text] and [Formula: see text] commute [Formula: see text] Our first main result gives conditions on [Formula: see text] such that [Formula: see text] for all [Formula: see text] such that [Formula: see text]. (Note that [Formula: see text] is absent from the right side of the inequality.) We give several examples of functions [Formula: see text] to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables [Formula: see text] instead of just [Formula: see text] alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy [Formula: see text], and two others, the Donald relative entropy [Formula: see text], and the Belavkin–Stasewski relative entropy [Formula: see text]. They are known to satisfy [Formula: see text]. We prove that the Donald relative entropy provides the sharp upper bound, independent of [Formula: see text] on [Formula: see text] in a number of cases in which [Formula: see text] is homogeneous of degree [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text]. We also investigate the Legendre transforms in [Formula: see text] of [Formula: see text] and [Formula: see text], and show how our results for these lead to new refinements of the Golden–Thompson inequality.