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Quantum Information and Computation

Some properties of partial fidelities

Alexey E. Rastegin

Published Nov 1, 2009• 5 citations
Publisher

Abstract

Basic properties of Uhlmann's partial fidelities are discussed. Statistical interpretation in terms of POVM measurements is established. Multiplicativity properties are considered. The relationship between partial fidelities and partitioned trace distances is derived. As it is shown, the partial fidelities cannot decrease under unistochastic quantum operations. Thus, the partial fidelities have good properties in the sense of their use as distinguishability measures.

DOI
10.26421/qic9.11-12-9

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Basic properties of Uhlmann's partial fidelities are discussed. Statistical interpretation in terms of POVM measurements is established. Multiplicativity properties are considered. The relationship between partial fidelities and partitioned trace distances is derived. As it is shown, the partial fidelities cannot decrease under unistochastic quantum operations. Thus, the partial fidelities have good properties in the sense of their use as distinguishability measures.
Abstract. 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 −1 | s(x)H = y −1 and x −1 | s(x) ⊥ H = 0.