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