Quantum Process Fidelity Bounds from Sets of Input States

TL;DR

This paper develops a semidefinite programming approach to bound quantum process fidelity using sets of input states, enabling efficient fidelity estimation without full tomography.

Contribution

It introduces a convex optimization framework for bounding process fidelity based on input state fidelities and characterizes bounds for symmetric POVM sets.

Findings

Bounds are convex functions of output state errors.

Fidelity bounds are linear for symmetric POVMs.

Method reduces the need for full process tomography.

Abstract

We investigate the problem of bounding the quantum process fidelity given bounds on the fidelities between target states and the action of a process on a set of pure input states. We formulate the problem as a semidefinite program and prove convexity of the minimum process fidelity as a function of the errors on the output states. We characterize the conditions required to uniquely determine a process in the case of no errors, and derive a lower bound on its fidelity in the limit of small errors for any set of input states satisfying these conditions. We then consider sets of input states whose one-dimensional projectors form a symmetric positive operator-valued measure (POVM). We prove that for such sets the minimum fidelity is bounded by a linear function of the average output state error. The minimal non-orthogonal symmetric POVM contains $d+1$ states, where $d$ is the Hilbert space dimension. Our bounds applied to these states provide an efficient method for estimating the process fidelity without the use of full process tomography.