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arXiv:2604.25953

A Monty-Hall test of non-contextual determinism with a single qutrit

Jorge Meza-Domínguez

quant-phmath-phSubmitted Apr 24, 2026• Announced Apr 28, 2026• 0 citations• Revised May 27, 2026 (v2)
v1 · Apr 24, 2026v2 · May 27, 2026v3 · Jun 13, 2026
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Abstract

We present a simple equality that distinguishes non-contextual deterministic hidden-variable theories (NCHV) from standard quantum mechanics using a single three-level system. The protocol is inspired by the Monty Hall puzzle: a coherent "discard" procedure followed by a projective measurement. In any NCHV theory that respects the Monty Hall condition (the discard never eliminates the true state), the probability of obtaining a chosen state after the discard is exactly 1/3. In contrast, quantum mechanics predicts 1/6, due to the preparation of coherent superpositions. Quantum mechanics thus violates the deterministic bound by a factor of two, providing a state-dependent non-contextuality inequality with a novel operational interpretation. Unlike standard Kochen-Specker proofs, which require 117 rays or 33 observables, our test uses only a single qutrit and two sequential measurements. An experimental implementation with photonic qutrits is proposed, including an analysis of experimental imperfections.

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New version, more rigourous

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•Sep 14, 2015•2 citations
We present a simple equality that distinguishes non-contextual deterministic hidden-variable theories (NCHV) from standard quantum mechanics using a single three-level system. The protocol is inspired by the Monty Hall puzzle: a coherent "discard" procedure followed by a projective measurement. In any NCHV theory that respects the Monty Hall condition (the discard never eliminates the true state), the probability of obtaining a chosen state after the discard is exactly 1/3. In contrast, quantum mechanics predicts 1/6, due to the preparation of coherent superpositions. Quantum mechanics thus violates the deterministic bound by a factor of two, providing a state-dependent non-contextuality inequality with a novel operational interpretation. Unlike standard Kochen-Specker proofs, which require 117 rays or 33 observables, our test uses only a single qutrit and two sequential measurements. An experimental implementation with photonic qutrits is proposed, including an analysis of experimental imperfections.
The Monty Hall problem is a classic probability puzzle known for its counterintuitive solution, revealing fundamental discrepancies between mathematical reasoning and human intuition. To bridge this gap, we introduce a novel explanatory framework inspired by quantum measurement theory. Specifically, we conceptualize the hosts' actions-opening doors to reveal non-prizes-as analogous to quantum measurements that cause asymmetric collapses of the probability distribution. This quantum-inspired interpretation not only clarifies why the intuitive misunderstanding arises but also provides generalized formulas consistent with standard Bayesian results. We further validate our analytical approach using Monte Carlo simulations across various problem settings, demonstrating precise agreement between theoretical predictions and empirical outcomes. Our quantum analogy thus offers a powerful pedagogical tool, enhancing intuitive understanding of conditional probability phenomena through the lens of probability redistribution and quantum-like measurement operations.
We show that a pair of two-qubit product measurements can be perfectly distinguished when accessed jointly, yet cannot be perfectly distinguished when their local components are distributed between distant parties restricted to local quantum operations and classical communication (LOCC). This establishes a measurement analogue of nonlocality without entanglement that, unlike its state-based counterpart, emerges in a remarkably simple setting. We further uncover a hierarchy of LOCC discrimination strategies: adaptive protocols, in which the probe used at one site may depend on classical information obtained from another, can strictly outperform restricted non-adaptive protocols, as demonstrated by a set of six bipartite qutrit product measurements. We also show that LOCC distinguishability of bipartite measurements can be intrinsically directional, depending on which party initiates the protocol. More generally, we construct pairs of nnn-qubit product measurements that are globally perfectly distinguishable but cannot be perfectly distinguished across any bipartition, even when all parties on either side of the bipartition are allowed arbitrary joint quantum operations. Finally, we show how quantum data hiding based on quantum-process nonlocality without entanglement can be realized using elementary optical components.
The linear combination of unitaries (LCU) is a fundamental quantum computing primitive used to embed non-unitary operators into larger systems via ancilla post-selection. In standard LCU implementations, only the zero-state ancilla outcome is retained, while all other measurement branches are discarded as unusable overhead. We study these discarded parts. Every computational-basis outcome of the ancilla projects the system onto a different linear combination of the target unitaries; collecting these outcome states in a matrix gives a factorization Φ=CXΦ=CXΦ=CX, where CCC is determined by the ancilla preparation and the rows of XXX are the states Ut∣ψ⟩U_t|ψ\rangleUt​∣ψ⟩, so that rank(Φ)≤Krank(Φ)\le Krank(Φ)≤K. Although this holds for any LCU circuit; to make the analysis easier, we use a simplified model circuit with Hadamard gates on the index register and a single rotation qubit in place of the coefficient preparation unitary, in which CCC is explicit. The structure suggests two complementary applications: (i) in an explicit amplitude-access model, classical low-rank matrix completion can reconstruct the full output (including the target) from a subset of its amplitude entries, while computational basis counts provide intensities and generally require additional phase information for full complex recovery; (ii) treating CCC as a secret key suggests a candidate quantum trapdoor function, whose security remains informal and depends on the difficulty of structured phase retrieval. The scheme thus turns the "junk" ancilla outcomes into a structured resource, possibly opening paths for further applications.
Quantum mechanics predicts the existence of correlations between composite systems that, although puzzling to our physical intuition, enable technologies not accessible in a classical world. Notwithstanding, there is still no efficient general method to theoretically quantify and experimentally detect entanglement of many qubits. Here we propose to detect entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. We witness entanglement without relying on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. To illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.
The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell's inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations.
The characterization of quantum coherence in the context of quantum information theory and its interplay with quantum correlations is currently subject of intense study. Several theoretical quantifiers of coherence have been proposed, and links to quantum correlation measures have been provided. We here propose an experimental strategy to witness the presence of entanglement in multipartite systems by evaluating coherence. We test our scheme by simulating coherence and entanglement detection in a three-qubit GHZ-diagonal state.