State transformations after quantum fuzzy measurements

TL;DR

This paper confronts how state transformations should be updated after fuzzy filter measurements by introducing a fuzzyfication procedure as a dynamical map in the Heisenberg picture, yielding a dual completely positive Schrödinger map $\hat{\rho} \mapsto \hat{\rho}_{post}(B;\hat{\rho})=\mathcal{E}^{*}(B,\hat{\rho})$ that respects covariance. It develops explicit discrete and continuous formulations for the state transformer, including Kraus-like operators and Gaussian kernels, and compares them with the standard operational quantum physics (OQP) update, highlighting when the two diverge in moments and entropy. The work provides concrete examples (discrete Gaussian and continuous Gaussian kernels) to illustrate how fuzzy measurements shift higher moments and entropy, while leaving some first-moment results invariant. The findings suggest that the fuzzy approach can yield testable predictions that differ from the projection-postulate-driven OQP, potentially informing both the interpretation of quantum measurements and the design of experiments probing measurement-induced state updates.

Abstract

Using a standard fuzzification procedure and the dynamical map in Heisenberg picture, a new expression for the state transformation after a fuzzy filter measurement, subject to covariance conditions, was obtained and some calculations were done to distinguish its properties from the those of the usual solution.