Measurement incompatibility cannot be stochastically distilled

TL;DR

This paper proves that measurement incompatibility cannot be increased through stochastic filtering, impacting quantum measurement theory and steering experiments, with implications for nonphysical maps.

Contribution

It demonstrates that measurement incompatibility is non-increasing under stochastic filters and extends the analysis to EPR steering and nonphysical maps.

Findings

Incompatibility cannot be stochastically distilled or increased.

Maximum steerability can be explicitly calculated under local filters.

Results apply to positive but not completely positive maps.

Abstract

We show that the incompatibility of a set of measurements cannot be increased by subjecting them to a filter, namely, by combining them with a device that post-selects the incoming states on a fixed outcome of a stochastic transformation. This result holds for several measures of incompatibility, such as those based on robustness and convex weight. Expanding these ideas to Einstein-Podolsky-Rosen steering experiments, we are able to solve the problem of the maximum steerability obtained with respect to the most general local filters in a way that allows for an explicit calculation of the filter operation. Moreover, our results generalize to nonphysical maps, i.e., positive but not completely positive linear maps.