Playing with Fidelities

The notion of partial fidelities as invented recently by A.Uhlmann for pairs of finite dimensional density matrices will be extended to the vNalgebraic context and is considered and thoroughly discussed in detail from a mathematical point of view. Especially, in the case of semifinite vN-algebras formulae and estimates for the partial fidelity between the functionals of a dense cone of inner derived normal positive linear forms are obtained. Also, some generalities on the notion of fidelity in quantum physics are collected in an appendix, and another system of mathematical axioms for fidelity over density operators, which is based on the concept of relative majorization and which is intimately related to complete positivity, is proposed. Basic notions and settings. In a vN -algebra M over a Hilbert space H call nonzero x ∈ M+ locally invertible if the linear operator y = x|s(x)H is invertible within s(x)Ms(x), where s(x) is the support of x. Let s(x)⊥ = 1− s(x), with the unit operator 1. The local inverse x−1 ∈ M+ of the locally invertible x then is defined as the unique element of M obeying x|s(x)H = y−1 and x|s(x)⊥H = 0. For simplicity, throughout the following we will tacitly agree in considering only vN -algebras acting on separable Hilbert spaces. In slightly modifying the settings from [50, 51] in the finite dimensional case, we let a set PAIRS(M) be defined as set of all pairs (a, b) of positive, locally invertible elements of M such that aba = a and bab = b hold. In addition, if q ∈ M\{0} is an orthoprojection, we let PAIRSq(M) ⊂ PAIRS(M) be defined as PAIRSq(M) = { (a, b) ∈ PAIRS(M) : s(a) ≈ s(b) ≈ q }