Coarse-Graining of Observables

TL;DR

This paper introduces a formal framework for coarse-graining of probability measures and observables, exploring their coexistence, discretization, and sequential products, with applications to quantum observables like SIC and qubits.

Contribution

It extends the concept of coarse-graining to observables and instruments, clarifies their relationships, and connects coarse-graining with post-processing in finite cases.

Findings

Any two probability measures coexist.

Discretization of observables involves ndone stochastic kernels.

Coarse-graining corresponds to post-processing in finite observables.

Abstract

We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have \zeroone stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables.