Frank Hansen

Metric adjusted skew information

We extend the concept of Wigner–Yanase–Dyson skew information to something we call “metric adjusted skew information” (of a state with respect to a conserved observable). This “skew information” is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova–Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the “λ-skew information,” parametrized by a λ ∈ (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.

Trace functions as Laplace transforms

We study trace functions on the form t→Trf(A+tB) where f is a real function defined on the positive half-line, and A and B are matrices such that A is positive definite and B is positive semidefinite. If f is non-negative and operator monotone decreasing, then such a trace function can be written as the Laplace transform of a positive measure. The question is related to the Bessis-Moussa-Villani conjecture.

Pressure-flow studies: An evaluation of within-testing reproducibility—validity of the measured parameters

The within‐examination variation in selected test parameters in repeated pressure‐flow studies was determined in a retrospective study of consecutive pressure‐flow examinations in 105 patients. It was further evaluated to see whether there was a systematic change in the measured parameters during retesting. To see if variation and reproducibility were influenced by the procedure of investigation, i.e., transurethral or suprapubic, patients were grouped according to the method employed. Finally, the effect of detrusor instability on the measurements was evaluated. Using the Abrams‐Griffiths nomogram, patients were classified as obstructed, equivocal, or unobstructed. The test‐retest variations in classification were evaluated.

Extensions of Lieb’s Concavity Theorem

AbstractThe operator function (A,B)→ Trf(A,B)(K*)K, defined in pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. We obtain, as a special case, a new proof of Lieb’s concavity theorem for the function (A,B)→ TrApK*BqK, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function

The fast track to Löwner’s theorem

(A,B)\to Tr\left[\frac{A}{A+\mu_1}K^*\frac{X1B}{B+\mu_2}K\right]

Pressure-flow studies: short-time repeatability.

in its natural domain D2(μ1,μ2), cf. Definition 3.

Jensen's Trace Inequality in Several Variables

The class

Inequalities for Quantum Skew Information

P_n(I)