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

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 $\hat{I}\pm =\pm 1$, initialized in the pure distribution $p(\hat{I}\pm )=\frac{1}{2}(1+\hat{I}\pm )$. The measurement interaction is modeled by a quasi-bistochastic process $S(b{b}^{\prime }{\hat{I}}^2\mid a{a}^{\prime }\hat{I}\pm )$ - 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 $p(a{a}^{\prime })p(\hat{I}\pm )$, it produces a collapsed state $p(b{b}^{\prime }\mid {\hat{I}}^2)$, yielding the measurement outcome ${\hat{I}}^2$ with the correct quantum-mechanical probability $p({\hat{I}}^2)$. 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.

12 pages