L\"uders' and quantum Jeffrey's rules as entropic projections

TL;DR

This paper demonstrates that L"uders' rule and Jeffrey's rule in quantum mechanics can be derived as entropic projections, extending classical Bayesian updating principles to the quantum domain using relative entropy maximization.

Contribution

It introduces a quantum analogue of Jeffrey's rule and shows that quantum state updates via L"uders' rule are special cases of entropic projections, unifying measurement update rules under a common entropic framework.

Findings

L"uders' rule is a special case of quantum relative entropy maximization.

A quantum analogue of Jeffrey's rule is derived and discussed.

The results connect quantum measurement updates with Bayesian inference principles.

Abstract

We prove that the standard quantum mechanical description of a quantum state change due to measurement, given by Lueders' rules, is a special case of the constrained maximisation of a quantum relative entropy functional. This result is a quantum analogue of the derivation of the Bayes--Laplace rule as a special case of the constrained maximisation of relative entropy. The proof is provided for the Umegaki relative entropy of density operators over a Hilbert space as well as for the Araki relative entropy of normal states over a W*-algebra. We also introduce a quantum analogue of Jeffrey's rule, derive it in the same way as above, and discuss the meaning of these results for quantum bayesianism.