Some operator monotone functions

We prove that the functions t -> (t^q-1)(t^p-1)^{-1} are operator monotone in the positive half-axis for 0 < p < q < 1, and we calculate the two associated canonical representation formulae. The result is used to find new monotone metrics (quantum Fisher information) on the state space of quantum systems.Comment: An author has quit and some material adde

Monotone trace functions of several variables

We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality \tau(x_1... x_n)\le\tau(y_1... y_n) for a trace \tau on a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices

Characterization of symmetric monotone metrics on the state space of quantum systems

The quantum Fisher information is a Riemannian metric, defined on the state space of a quantum system, which is symmetric and decreasing under stochastic mappings. Contrary to the classical case such a metric is not unique. We complete the characterization, initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed and tractable formula for the set of Morozova-Chentsov functions. In addition, we provide a continuously increasing bridge between the smallest and largest symmetric monotone metrics.Comment: Minor revision with new title and abstract as suggested by a refere

Operator monotone functions of several variables

We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship between operator convexity and operator monotonicity for functions of one variable is extended also to functions of several variables

Regular operator mappings and multivariate geometric means

We introduce the notion of regular operator mappings of several variables generalising the notion of spectral function. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives. As a main application of the theory we consider geometric means of k operator variables extending the geometric mean of k commuting operators and the geometric mean of two arbitrary positive definite matrices. We propose different types of updating conditions that seems natural in many applications and prove that each of these conditions, together with a few other natural axioms, uniquely defines the geometric mean for any number of operator variables. The means defined in this way are given by explicit formulas and are computationally tractable.Comment: Version to be publishe

Jensen's inequality for conditional expectations

We study conditional expectations generated by an abelian $C^*$-subalgebra in the centralizer of a positive functional. We formulate and prove Jensen's inequality for functions of several variables with respect to this type of conditional expectations, and we obtain as a corollary Jensen's inequality for expectation values

Convex multivariate operator means

The dominant method for defining multivariate operator means is to express them as fix-points under a contraction with respect to the Thompson metric. Although this method is powerful, it crucially depends on monotonicity. We are developing a technique to prove the existence of multivariate operator means that are not necessarily monotone. This gives rise to an entire new class of non-monotonic multivariate operator means.Comment: We discovered that an argument in the proof of the last part of the last theorem in the first version of the paper is plainly wrong. Based on examples we still have ground to believe that the theorem is correct, but this is now only a conjecture. Some mean inequalities are adde