The suboptimality ratio of projective measurements restricted to low-rank subspaces

TL;DR

This paper analyzes how restricting projective measurements to low-rank subspaces affects quantum state discrimination, showing the suboptimality ratio is polylogarithmic in the subspace dimension, independent of the full space.

Contribution

It provides a theoretical bound on the suboptimality ratio for low-rank projective measurements, using probabilistic methods and trace inequalities, advancing understanding of measurement limitations in quantum algorithms.

Findings

Suboptimality ratio is independent of full space dimension.

Ratio is at most polylogarithmic in low-rank subspace dimension.

Probabilistic approach and trace inequalities are key techniques.

Abstract

Limitations in measurement instruments can hinder the implementation of some quantum algorithms. Understanding the suboptimality of such measurements with restrictions may then lead to more efficient measurement policies. In this paper, we theoretically examine the suboptimality arising from a Procrustes problem for minimizing the average distance between two fixed quantum states when one of the states has been measured by a Projective Measurement (PM). Specifically, we compare optima when we can only use PMs that are aligned with a low-rank subspace where the quantum states are supported, and when we can measure with the full set of PMs. For this problem, we show that the suboptimality ratio is independent of the dimension of the full space, and is at most polylogarithmic in the dimension of the low-rank subspace. In the proof of this result, we use a probabilistic approach and the main techniques include trace inequalities related to projective measurements, and operator norm bounds for equipartitions of Parseval frames, which are of independent interest.