Operation fidelity explored by numerical range of Kraus operators

TL;DR

This paper investigates the statistical properties of operation fidelity in quantum channels using the joint numerical range of Kraus operators, providing analytic expressions and applications for quantum device characterization.

Contribution

It introduces a novel approach using the joint numerical range of Kraus operators to analyze operation fidelity, including deriving explicit distributions for specific channels.

Findings

Derived analytic expressions for fidelity distributions in certain channels

Identified extreme values and probability distributions of operation fidelity

Showed how fidelity distributions can distinguish quantum operations

Abstract

Present-day quantum devices require precise implementation of desired quantum channels. To characterize the quality of implementation one uses the average operation fidelity $F$, defined as the fidelity between an initial pure state and its image with respect to the analyzed operation, averaged over an ensemble of pure states. We analyze statistical properties of the operation fidelity for low-dimensional channels and study its extreme values and probability distribution, both of which can be used for statistical channel discrimination. These results are obtained with help of the joint numerical range of the set of Kraus operators representing a channel. Analytic expressions for the density $P\left(F\right)$ are derived in some particular cases including unitary and mixed unitary channels as well as quantum maps represented by commuting Kraus operators. Measured distributions of operation fidelity can be used to distinguish between two quantum operations.