Informationally Overcomplete POVMs for Quantum State Estimation and Binary Detection

TL;DR

This paper demonstrates that overcomplete POVMs in quantum mechanics provide robustness in state estimation and binary detection, with a tradeoff between the number of system copies and measurement elements, supported by simulation results.

Contribution

It applies classical frame theory to quantum operator spaces, showing robustness and a tradeoff in quantum state estimation and detection using overcomplete POVMs.

Findings

Robustness in quantum state estimation with overcomplete POVMs.

Existence of a tradeoff between copies and measurement elements.

Simulation evidence of improved binary detection performance.

Abstract

It is well-known in classical frame theory that overcomplete representations of a given vector space provide robustness to additive noise on the frame coefficients of an unknown vector. We describe how the same robustness can be shown to exist in the context of quantum state estimation. A key element of the discussion is the application of classical frame theory to operator-valued vector spaces, or operator spaces, which arise naturally in quantum mechanics. Specifically, in the problem we describe the frame vectors are represented by the elements of an informationally complete or overcomplete (IC or IOC) POVM, the frame coefficients are represented by the outcome probabilities of a quantum measurement made on an unknown state, and the error on the frame coefficients arises from finite sample size estimations of the probabilities. We show that with this formulation of the problem, there is a tradeoff in estimation performance between the number of copies of the unknown system and the number of POVM elements. Lastly, we present evidence through simulation that the same tradeoff is present in the context of quantum binary state detection -- the probability of error can be reduced either by increasing the number of copies of the unknown system or by increasing the number of POVM elements.