Rethinking Collapse: Coupling Quantum States to Classical Bits with quasi-probabilities

TL;DR

This work proposes a minimal, probabilistic quantum-measurement framework in which a single qubit, represented by a positive classical distribution $p(aa')$, couples to a classical meter bit via a quasi-bistochastic map. The core insight is that nonclassical measurement features arise from the dynamics (the quasi-probabilistic interaction) rather than from the quantum state space, which remains positive; this is formalized through a SIC-frame / generalized Bloch vector representation that recasts quantum states and channels as affine transformations on classical-like distributions. The model reproduces standard measurement statistics and post-measurement collapse while highlighting the inevitable negativity in the measurement operator, a feature tied to the geometry of 3D measurement directions. It further clarifies how classicality can emerge as an interface between quantum and classical probabilistic descriptions, suggests conditions under which positivity can be preserved in repeated measurements, and discusses connections to foundational ideas such as Quantum Darwinism and potential applications in quantum simulation and control.

Abstract

We propose a formulation of quantum measurement within a modified framework of frames, in which a quantum system - a single qubit - is directly coupled to a classical measurement bit. The qubit is represented as a positive probability distribution over two classical bits, a and a', denoted by p(aa'). The measurement apparatus is described by a classical bit $α= \pm 1$, initialized in the pure distribution $p(α) = \frac{1}{2}(1 + α)$. The measurement interaction is modeled by a quasi-bistochastic process $ S(bb'β\mid aa'α)$ - a bistochastic map that may include negative transition probabilities, while acting on an entirely positive state space. When this process acts on the joint initial state $p(aa')p(α)$, it produces a collapsed state $p(bb'\midβ)$, yielding the measurement outcome $β$ with the correct quantum-mechanical probability $p(β)$. This approach bypasses the von Neumann chain of infinite couplings by treating the measurement register classically, while capturing the nonclassical nature of measurement through the quasi-bistochastic structure of the interaction.