From the paper – Quantum version of the Monty Hall problem
A.P. Flitney
, D. Abbott
Centre for Biomedical Engineering (CBME) and Department of Electrical and Electronic Engineering,
Adelaide University, SA 5005, Australia
(February 1, 2008)

  • Quantum Strategies and Entanglement:
    • If the initial state involves no entanglement, quantum strategies offer no advantage over classical mixed strategies.
    • With entanglement, if one player can use quantum strategies while the other cannot, the quantum player has an advantage.
  • Classical Monty Hall Problem:
    • In the classical problem, the player has a higher chance (2/3) of winning if they switch doors after the host reveals an empty door.
  • Quantum Monty Hall Problem:
    • In the quantum version, Alice (the banker) and Bob (the player) can choose quantum states represented by qutrits.
    • If both players have access to quantum strategies, there is no Nash equilibrium in pure strategies, but a Nash equilibrium exists in quantum mixed strategies.
  • Initial State and Operators:
    • The game starts in an initial state ∣ψi⟩|ψ_i\rangle, which evolves through a series of operators representing the strategies of Alice and Bob.
    • The operators involved include Alice’s choice operator A^Â, Bob’s initial choice operator B^B̂, the box-opening operator O^Ô, and Bob’s switching or not-switching operator (S^Ŝ and N^N̂ respectively).
  • Results with and without Entanglement:
    • Without entanglement, the quantum game results in the same payoffs as the classical game.
    • With maximal entanglement, every quantum strategy has a counterstrategy, leading to a situation where the average payoffs mirror those of the classical game.
  • Conclusions:
    • Quantum strategies do not provide an advantage in the Monty Hall problem if the initial state is maximally entangled.
    • The classical winning strategy (switching doors) remains optimal in the quantum game when both players can use quantum strategies.