Some inequalities for quantum Tsallis entropy related to the strong subadditivity

In this paper we investigate the inequality $S_q(\rho_{123})+S_q(\rho_2)\leq S_q(\rho_{12})+S_q(\rho_{23}) \, (*)$ where $\rho_{123}$ is a state on a finite dimensional Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2\otimes \mathcal{H}_3,$ 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 $S_q(\rho_{123})+S_q(\rho_2)\leq S_q(\rho_{12})+S_q(\rho_{23})+f_q(\rho_{123})$, where $f_1(\rho_{123})=0$. 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.Comment: v2: the introductory part reorganized v3: the published versio

Conditional SIC-POVMs

In this paper, we examine a generalization of the symmetric informationally complete POVMs. SIC-POVMs are the optimal measurements for full quantum tomography, but if some parameters of the density matrix are known, then the optimal SIC POVM should be orthogonal to a subspace. This gives the concept of the conditional SIC-POVM. The existence is not known in general, but we give a result in the special case when the diagonal is known of the density matrix. © 1963-2012 IEEE

Complementarity and the algebraic structure of 4-level quantum systems

The history of complementary observables and mutual unbiased bases is reviewed. A characterization is given in terms of conditional entropy of subalgebras. The concept of complementarity is extended to non-commutative subalgebras. Complementary decompositions of a 4-level quantum system are described and a characterization of the Bell basis is obtained.Comment: 19 page

Means of positive matrices

Means of positive numbers are well-know but the theory of matrix means due to Kubo and Ando is less known. The lecture gives a short introduction to means, the emphasis is on matrices. It is shown that any two-variablemean of matrices can be extended to more variables. The n-variable-mean M ) is defined by a symmetrization procedure when the ntuple (A n (A 1 ; A 1 2 ; : : : ; A ; A 2 n ) is ordered, it is continuous and monotone in each variable. The geometric mean of matrices has a nice interpretation in terms of an information geometry and the ordering of the n-tuple is not necessary for the definition. It is conjectured that this strong condition might be weakened for some other means, too. Key Words: operator means, information geometry, logarithmic mean, geometric mean, positive matrices. AMS Classification Number: 47A64 (15A48, 47A63