An effective toy model in $M_n(\mathbb{C})$ for selective measurements in quantum mechanics

TL;DR

This paper introduces a simple toy model within matrix algebra to simulate the process of selective quantum measurements, capturing the transition from decohered to pure states through a two-step dynamical process.

Contribution

It proposes a novel two-step dynamical framework combining Lindblad evolution and a non-linear toy model to describe measurement-induced state transitions in quantum mechanics.

Findings

The model reproduces the emergence of pure states as asymptotic fixed points.

Probabilities of outcomes correspond to volumes of attractor regions in the state space.

The approach provides a conceptual bridge between decoherence and state collapse.

Abstract

The non-selective and selective measurements of a self-adjoint observable $A$ in quantum mechanics are interpreted as `jumps' of the state of the measured system into a decohered or pure state, respectively, characterized by the spectral projections of $A$. However, one may try to describe the measurement results as asymptotic states of a dynamical process, where the non-unitarity of time evolution arises as an effective description of the interaction of the measured system with the measuring device. The dynamics we present is a two-step dynamics: the first step is the non-selective measurement or decoherence, which is known to be described by the linear, deterministic Lindblad equation. The second step is a process from the resulted decohered state to a pure state, which is described by an effective non-linear `randomly chosen' toy model dynamics: the pure states arise as asymptotic fixed points, and their emergent probabilities are the relative volumes of their attractor regions.