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arXiv:1111.2478

A note on the measures of process fidelity for non-unitary quantum operations

Joydip Ghosh

Published Nov 10, 2011• 7 citations
arXivHTMLPDF

Abstract

Various fidelity measures can be defined between two quantum processes especially when at least one of them is non-unitary. In this paper we consider two such measures of state averaged process fidelity, put forward an efficient procedure to compute them and investigate their mathematical properties as well as their relationship.
DOI
10.48550/arxiv.1111.2478

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Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories and quantum computers, quantum communications, and quantum phase-space simulations.
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A recent article introduced a hierarchy of quantities called kkk-fidelities that approximate the quantum process fidelity with increasing accuracy. The lowest approximation in this hiearchy is the 000-fidelity. The authors gave a protocol for estimating the 000-fidelity and showed numerical evidence that it approximates the process fidelity. In this note, we prove lower and upper bounds on the process fidelity as linear functions of the 0-fidelity. By solving a semidefinite program, we provide evidence that the lower bound is tight.