Quasilinear evolution versus von Neumann selective measurement

TL;DR

This paper proposes a new quantum measurement approach replacing the von Neumann projection with quasilinear evolution, maintaining core quantum principles while avoiding instantaneous collapse, and explores its implications through numerical and analytical methods.

Contribution

It introduces a nonlinear generalization of the von Neumann equation for quantum measurement, preserving ensemble equivalence and no-signalling, and compares it with standard measurement.

Findings

Numerical solutions show agreement with von Neumann measurement in fundamental properties.

Identifies narrow parameter regions where outcomes differ, suggesting potential experimental tests.

Discusses the Stern-Gerlach experiment within the new measurement framework.

Abstract

In this article, we introduce a new form of quantum selective measurement in which the von Neumann projection postulate is replaced by quasilinear evolution, governed by a nonlinear generalization of the von Neumann equation. We demonstrate that this equation preserves the equivalence of quantum ensembles and, consequently, satisfies the no-signalling principle, ensuring consistency with both quantum mechanics and Einstein causality. Our approach eliminates the need for instantaneous, discontinuous state collapse and provides a unified description of the postmeasurement quantum state reduction as a form of quantum state evolution. Notably, it does not require invoking concepts such as the quantum state assigned to a classical apparatus. At the same time, the stochastic character of selective measurement and the Born rule remain unchanged. We present several numerical solutions of the evolution equation for quasilinear selective measurement in two-level quantum systems and compare them with the standard von Neumann projection. The results demonstrate agreement between the two measurement schemes in their fundamental properties. Furthermore, we investigate phenomena associated with the structural instability of the evolution equation and identify very narrow parameter regions in which the outcomes deviate from those predicted by the von Neumann projection. These regions may offer opportunities to test the proposed approach experimentally. Finally, using specific analytical solutions, we discuss the Stern-Gerlach experiment within the framework of quasilinear measurement.