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arXiv:quant-ph/0202120 · Quantum Information and Computation

The quantum monty hall problem

Giacomo Mauro D’Ariano, Richard David Gill, Michael Keyl, Burkhard Kümmerer, Hans Maassen, Reinhard F. Werner

Published Aug 1, 2002• 39 citations
arXivHTMLPDF

Abstract

We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information.

DOI
10.26421/qic2.5-3

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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.
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.
Abstract In this work, the authors propose a quantum version of a generalized Monty Hall game, that is, one in which the parameters of the game are left free and not fixed on its regular values. The developed quantum scheme is then used to study the expected payoff of the player, using both a separable and an entangled initial‐state. In the two cases, the classical mixed‐strategy payoff is recovered under certain conditions. Lastly, the authors extend the quantum scheme to include multiple independent players, and use this extension to sketch two possible application of the game mechanics to quantum networks, specifically, two validated, multi‐party, key‐distribution quantum protocols.