8,511 research outputs found

### 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

### 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

### 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

### 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

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