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 $a^{\prime }$, denoted by $p\left(a{a}^{\prime }\right)$. The measurement apparatus is described by a classical bit $\alpha =\pm 1$, initialized in the pure distribution $p\left(\alpha \right)=\frac{1}{2}(1+\alpha )$. The measurement interaction is modeled by a quasi-bistochastic process $S(b{b}^{\prime }\beta \mid a{a}^{\prime }\alpha )$—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 $p\left(a{a}^{\prime }\right)p\left(\alpha \right)$, it produces a collapsed state $p(b{b}^{\prime }\mid \beta )$, yielding the measurement outcome $\beta$ with the correct quantum-mechanical probability $p\left(\beta \right)$. We also show that the same framework extends consistently to two qubits, including local measurements on entangled states and a joint Bell-basis measurement. 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.

Close

PrevNext

View figure in article

• Quantum correlations, foundations & formalism
• Quantum measurements