A short note on the 0-fidelity

TL;DR

This paper establishes bounds on quantum process fidelity using the 0-fidelity measure, providing theoretical guarantees and evidence of tightness through semidefinite programming.

Contribution

It proves bounds on process fidelity based on 0-fidelity and demonstrates the tightness of these bounds via semidefinite programming.

Findings

Lower and upper bounds on process fidelity derived

Semidefinite program confirms bounds are tight

Provides theoretical validation for 0-fidelity approximation

Abstract

A recent article introduced a hierarchy of quantities called $k$-fidelities that approximate the quantum process fidelity with increasing accuracy. The lowest approximation in this hiearchy is the $0$-fidelity. The authors gave a protocol for estimating the $0$-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.