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THE QUANTUM MONTY HALL PROBLEM: SYMMETRY, ENTANGLEMENT, AND STATE MODIFICATION

Michael Aksman

Published Feb 21, 2026• 0 citations
Publisher

Abstract

The classical Monty Hall problem serves as a standard exercise in conditional probability. In this article, we examine its quantum mechanical extension to demonstrate how classical probability distributions are governed by symmetry and entanglement. By mapping the available choices to a three-level quantum system (qutrits) governed by SU(3) symmetry, the game is transformed into a formalized quantum system [1]. We establish that when the choices of the host and player are entangled, the act of revealing a door functions as a symmetry-breaking measurement that physically modifies the state space, illustrating that information acquisition in quantum systems is an active, geometric process with direct analogues in quantum field symmetries [2].

DOI
10.5281/zenodo.18724342

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The classical Monty Hall problem serves as a standard exercise in conditional probability. In this article, we examine its quantum mechanical extension to demonstrate how classical probability distributions are governed by symmetry and entanglement. By mapping the available choices to a three-level quantum system (qutrits) governed by SU(3) symmetry, the game is transformed into a formalized quantum system [1]. We establish that when the choices of the host and player are entangled, the act of revealing a door functions as a symmetry-breaking measurement that physically modifies the state space, illustrating that information acquisition in quantum systems is an active, geometric process with direct analogues in quantum field symmetries [2].
Abstract Quantum computing has emerged as a useful resource for winning random games. Monty Hall problem is one such game where the powers of quantum mechanics can be utilized for winning the prize. Our paper depicts this problem as a two-player game, where the two players, i.e. the host and the participant share an entangled quantum state. The entangled state is closely knit with the position of the prize containing door, and the decision of the player i.e. either switch or stay from his initial choice of door. The two players are part of one team as they share a quantum communication channel among them by sharing the entangled state as a quantum resource. This enables them to share share secret information about the position of the price as well as the strategy to opt for winning the game. The article shows a unique way of utilizing three different type of entangled states, 3-qubit symmetric W state, 3-qubit X state, and 5-qubit highly entangled Brown state. Further, the effectiveness of each quantum state as a source of communication in disguise is also compared. All the entangled states under study assure winning the price by the participant. However, each quantum state has different entanglement degree and offer varied merits as per the game setting.
A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than that obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qutrits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.
The emergence of quantum computing has made us utilize its powers in solving the Monty Hall problem. The problem is depicted as a two-player game, where the two players, i.e. the host and the participant share an entangled three-qubit quantum state between them. The quantum state of three qubits is closely knit with the fact that behind which door would one expect the prize of the game to be. The two players are part of one team as they share a quantum communication channel amongst them by sharing the entangled state as a quantum resource. The article shows a unique way of utilizing the effectiveness and entanglement features of the W state as a source of communication in disguise for solving the Monty Hall problem. It is shown that by sharing the W state, the players always win the prize. Moreover, since Noisy Intermediate-Scale Quantum technology is the need of the hour, the effects of decoherence are also observed on the winning prospects of the game.
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.
Abstract A quantum version of the Monty Hall problem is proposed inspired by an experimentally-feasible, quantum-optical set-up that resembles the classical game. The expected payoff of the player is studied by analyzing the classical expectation values of the obtained quantum probabilities. Results are examined by considering both entanglement and non-entanglement between player and host, and using two different approaches: random and strategy-based. We also discuss the influence of noise on the game outcome when the parties play through a noisy quantum channel. The experimental set-up can be used to quickly verify the counter-intuitive result of the Monty Hall problem, adding pedagogic value to the proposal.