Contravariant Densities, Complete Distances and Relative Fidelities for Quantum Channels

TL;DR

This paper introduces contravariant trace-densities for quantum states, establishing new formulas for distances and fidelities between quantum channels with operational interpretations, advancing quantum information theory.

Contribution

It develops a novel framework of contravariant densities for quantum channels, providing explicit formulas for CB-norm and Helinger-type distances, and extends their operational meaning.

Findings

Explicit CB-norm distance formulas in terms of contravariant densities

Introduction of Helinger-type distance related to channel purification

Proved equivalence of Helinger distance and CB distance for certain channels

Abstract

Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace-densities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for the generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states.