Entanglement as a Witness of Quantum Coherence: A Bipartite Monty-Hall Protocol

Abstract

We present a bipartite protocol inspired by the Monty Hall puzzle that operationally distinguishes quantum coherence from classical ignorance. A principal qutrit is entangled with an ancillary qutrit via a controlled unitary, preparing $|\Psi ⟩=\frac{1}{\sqrt{3}}(|A,0⟩+|B,1⟩+|C,2⟩)$. A rank-1 projective discard then eliminates one basis state, leaving a coherent superposition of the two remaining states. Finally, the ancilla and qutrit are measured, yielding joint probabilities that encode the interplay between superposition and measurement back-action. We show that the conditional probability $P(B|\text{anc}=0)$ takes the value $1/4$ in both quantum mechanics and the classical ignorant-host model, making it unsuitable as a witness. The true quantum-classical separation emerges in conditional joint probabilities that correlate ancilla outcomes with specific discard operations. We define witnesses ${\mathcal{W}}_{i,j}=P(anc=i,qutrit=j\mid \text{discard }k)$ where $j$ differs from the ancilla-implied state. Quantum mechanics predicts $\mathcal{W}=1/4$, while any classical epistemic model with perfect initial correlations yields $\mathcal{W}=0$. We provide the explicit $9\times 9$ unitary matrix, a complete analysis of all measurement outcomes, and a detailed proof of the violation. The witness is fully immune to white noise and robust against moderate dephasing. The protocol requires only a single pair of entangled qutrits and sequential measurements -- no spatial separation, no multiple copies, and no complex sets of incompatible observables. This makes it suitable for advanced undergraduate laboratories and provides a pedagogically accessible test of the ontic-epistemic distinction in quantum foundations.

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