This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew-informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew-information is the Hessian of a quasi-entropy. The skewinformation and some inequalities are extended to a von Neumann algebra setting. 2000Mathematics Subject Classification. Primary 15A63, 94A17; Secondary 47N50, 46L60.

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, 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 Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. 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 Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.

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.

The original geometric mean of two positive definite operators A and B is given by A#B = A1/2(A-1/2BA-1/2)1/2A1/2. In this article we provide a new proof to construct from the two-variable geometric mean to the multivariable mean via symmetrization introduced by Lawson and Lim [5]. Finally we provide an algorithm to find three-variable geometric mean via symmetrization, which plays an important role to construct higher-order geometric means.

In this paper we present a new weighted, multivariable operator mean of positive definite operators over an arbitrary Hilbert space which provides us the first generally applicable extension of the classical Kubo–Ando theory of 2-variable operator means. The construction is a weighted extension of the Bini–Meini–Poloni symmetrization process originally given for the matrix geometric mean. Here to be able to consider such an iterative procedure, we need a weighted version of every Kubo–Ando mean in two variables. Therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional Hilbert space that provides weighted counterparts to every (not-necessarily symmetric) Kubo–Ando mean and also agrees with the most well known weighted operator means.

Matrices are important in many areas of both pure and applied mathematics. In particular, they play essential roles in quantum probability and quantum information.

A linear mapping is essentially a matrix if the vector space is finite-dimensional. In this book the vector space is typically a finite-dimensional complex Hilbert space.

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.

Some recent papers formulated sufficient conditions for the decomposition of matrix variances [6,10].A statement was that if we have one or two observables, then the decomposition is possible.In this paper we consider an arbitrary finite set of observables and we present a necessary and sufficient condition for the decomposition of the matrix variances.

Let $$(a, b) \subset {\mathbb R}$$ be an interval. A function $$f: (a, b) \rightarrow {\mathbb R}$$ is said to be monotone for $$n \times n$$ matrices if $$f(A) \le f(B)$$ whenever $$A$$ and $$B$$ are self-adjoint $$n \times n$$ matrices, $$A \le B$$ and their eigenvalues are in $$(a, b)$$ . If a function is monotone for every matrix size, then it is called matrix monotone or operator monotone.

A citation from von Neumann: “The object of this note is the study of certain properties of complex matrices of $$n$$ th order: $$A=(a_{ij})_{i,j=1}^n$$ , $$n$$ being a finite positive integer: $$n=1,2,\dots $$ . Together with them we shall use complex vectors of $$n$$ th order (in $$n$$ dimensions): $$x=(x_i)_{i=1}^n$$ .” This classical subject in matrix theory is exposed in Sects. 6.2 and 6.3 after discussions on vectors in Sect. 6.1. This chapter also contains several matrix norm inequalities as well as majorization results for matrices, which were mostly developed more recently.

The study of numerical means has been a popular subject for centuries, and the inequalities

Let $$A\in \mathbb {M}_n({\mathbb C})$$ and $$p(x):=\sum _i c_i x^i$$ be a polynomial. It is quite obvious that by $$p(A)$$ we mean the matrix $$\sum _i c_i A^i$$ . So the functional calculus is trivial for polynomials.

Most of the statements and definitions in this chapter are formulated in the Hilbert space setting.

We show that the variance is its own concave roof. For rank-2 density matrices and operators with zero diagonal elements in the eigenbasis of the density matrix, we prove analytically that the quantum Fisher information is four times the convex roof of the variance. Strong numerical evidence suggests that this statement is true even for operators with nonzero diagonal elements or density matrices with a rank larger than $2.$ We also find that within the different types of generalized quantum Fisher information considered in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)], after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case.