Incompatibility of quantum instruments

TL;DR

This paper investigates the concept of incompatibility among quantum instruments, exploring its implications for POVMs, channels, and non-disturbance, and establishing a new equivalence with postprocessing of complementary instruments.

Contribution

It introduces a comprehensive framework for understanding quantum instrument incompatibility, including new relations to non-disturbance and postprocessing equivalence.

Findings

Instrument incompatibility affects POVMs and channels.

Compatibility is equivalent to postprocessing of complementary instruments.

Examples illustrate various classes of incompatible instruments.

Abstract

Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.