Incompatibility of observables, channels and instruments in information theories

Abstract

Every theory of information, including classical and quantum, can be studied in the framework of operational probabilistic theories--where the notion of test generalizes that of quantum instrument, namely a collection of quantum operations summing to a channel, and simple rules are given for the composition of tests in parallel and in sequence. Here we study the notion of compatibility for tests of an operational probabilistic theory. Following the quantum literature, we first introduce the notion of strong compatibility, and then we illustrate its ultimate relaxation, that we deem weak compatibility. It is shown that the two notions coincide in the case of observation tests--which are the counterpart of quantum POVMs--while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a theory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.

11 pages, revtex4; bibliography is now displayed correctly

References in corpus (5)

• The uncertainty principle determines the non-locality of quantum mechanics
• The contextual fraction as a measure of contextuality
• Incompatibility of quantum channels
• Entanglement and superposition are equivalent concepts in any physical theory
• On the compatibility of quantum instruments

Cited by in corpus (12)

• Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication
• Measuring incompatibility and clustering quantum observables with a quantum switch
• Incompatibility as a resource for programmable quantum instruments
• Characterizing and quantifying the incompatibility of quantum instruments
• Contextual Measurement Model and Quantum Theory
• Measurement incompatibility is strictly stronger than disturbance
• Incompatibility of quantum instruments
• Joint-measurability and quantum communication with untrusted devices
• A Theory of Quantum Instruments
• Tsirelson's Inequality for the Precession Protocol is Maximally Violated by Quantum Theory
• Multi-object operational tasks for measurement incompatibility
• Minimal operational theories: classical theories with quantum features