Simulability of high-dimensional quantum measurements

TL;DR

This paper explores the conditions under which high-dimensional quantum measurements can be compressed into lower dimensions while preserving measurement statistics, linking simulability to measurement compatibility and providing practical construction methods.

Contribution

It introduces a new notion of simulability for quantum measurements, connecting measurement incompatibility with dimension reduction, and offers semi-definite programming techniques for constructing optimal simulation models.

Findings

Full quantum compression occurs if and only if measurements are jointly measurable.

Developed semi-definite programming methods for constructing simulation models.

Analytically constructed optimal models for noisy projective measurements.

Abstract

We investigate the compression of quantum information with respect to a given set $\mathcal{M}$ of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from $\mathcal{M}$ and an arbitrary quantum state $\rho$ are recovered exactly by first compressing $\rho$ into a lower dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set $\mathcal{M}$ is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative examples involving mutually unbiased basis, and develop a method based on semi-definite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.