Abstract. In this paper we investigate the inequality S q (ρ 123 ) + S q (ρ 2 ) S q (ρ 12 ) + S q (ρ 23 )( * ) where ρ 123 is a state on a finite dimensional Hilbert space H 1 ⊗ H 2 ⊗ H 3 , and S q is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (*), which is an inequality of relative quasi-entropies. We derive an inequality of the form S q (ρ 123 ) + S q (ρ 2 ) S q (ρ 12 ) + S q (ρ 23 ) + f q (ρ 123 ) , where f 1 (ρ 123 ) = 0 . Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that (*) does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.Mathematics subject classification (2010): 46N50, 94A15, 46L30, 47L90.
We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every monotone Bregman divergence is jointly convex.
The optimal measurement configuration, i.e., the optimal input quantum state and measurement in the form of a POVM with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Paulimatrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/N , where N is the number of used measurements.
ABSTRACT. In this paper we determine those bijective maps of the set of all positive definite n × n complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.
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