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arXiv:1010.4875

Environmental influences on Quantum Monty Hall problem

Salman Khan, Muhammad Asif Ramzan, Majid Khan

Published Oct 23, 2010• 0 citations
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Abstract

We reformulate the quantum Monty Hall problem in the presence of decoherence. The decoherence destroys the fairness of the game. A new Nash equilibrium for a particular strategy profile in the presence of decoherence emerges. It is shown that in the presence of decoherence under the action of amplitude damping channel, Bob's winning probability is always higher than three-forth, irrespective of Alice's strategy, if he does not switch to the other door and always wins for a fully decohered case of the channel. Depolarizing channel damps up Bob's winning probability and gets better off if he sticks to his current selection. Phase damping channel leaves the winning probability unaffected. Unlike the classical and the quantum forms of the game, Bob's dominant strategy in the presence of decoherence is not switching.

DOI
10.48550/arxiv.1010.4875

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We reformulate the quantum Monty Hall problem in the presence of decoherence. The decoherence destroys the fairness of the game. A new Nash equilibrium for a particular strategy profile in the presence of decoherence emerges. It is shown that in the presence of decoherence under the action of amplitude damping channel, Bob's winning probability is always higher than three-forth, irrespective of Alice's strategy, if he does not switch to the other door and always wins for a fully decohered case of the channel. Depolarizing channel damps up Bob's winning probability and gets better off if he sticks to his current selection. Phase damping channel leaves the winning probability unaffected. Unlike the classical and the quantum forms of the game, Bob's dominant strategy in the presence of decoherence is not switching.
We study the effect of decoherence on quantum Monty Hall problem under the influence of amplitude damping, depolarizing, and dephasing channels. It is shown that under the effect of decoherence, there is a Nash equilibrium of the game in case of depolarizing channel for Alice's quantum strategy. Whereas in case of dephasing noise, the game is not influenced by the quantum channel. For amplitude damping channel, Bob's payoffs are found symmetrical about a decoherence of 50% and the maximum occurs at this value of decoherence for his classical strategy. However, it is worth-mentioning that in case of depolarizing channel, Bob's classical strategy remains always dominant against any choice of Alice's strategy.
We are considering a quantum version of the penny flip game, whose implementation is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. We focus our attention on the phase damping, amplitude damping and amplitude raising channels. Our results show that the Pauli strategy is no longer a Nash equilibrium under decoherence. We attempt to optimize the players’ control pulses in the aforementioned setup to allow them to achieve higher probability of winning the game compared with the Pauli strategy.
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.
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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.