arXiv:1504.01917 · Quantum Information Processing

Quantum inferring acausal structures and the Monty Hall problem

Dariusz Kurzyk, Adam Glos

Published Sep 8, 2016• 12 citations

Abstract

This paper presents a quantum version of the Monty Hall problem based upon the quantum inferring acausal structures, which can be identified with generalization of Bayesian networks. Considered structures are expressed in formalism of quantum information theory, where density operators are identified with quantum generalization of probability distributions. Conditional relations between quantum counterpart of random variables are described by quantum conditional operators. Presented quantum inferring structures are used to construct a model inspired by scenario of well-known Monty Hall game, where we show the differences between classical and quantum Bayesian reasoning.

DOI: 10.1007/s11128-016-1431-8

Related papers

Understanding the Monty Hall Problem Through a Quantum Measurement Analogy

Li, Mo

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.

arXiv:2506.00012May 17, 20250 citations

Positive operator valued measures and the quantum Monty Hall problem

Claudia Zander, M. Casas, Angel Ricardo Plastino +1 more

A quantum version of the Monty Hall problem, based upon the Positive Operator Valued Measures (POVM) formalism, is proposed. It is shown that basic normalization and symmetry arguments lead univocally to the associated POVM elements, and that the classical probabilities associated with the Monty Hall scenario are recovered for a natural choice of the measurement operators.

Anais da Academia Brasileira de CiênciasSep 1, 20067 citations

Quantum-Like Bayesian Networks for Modeling Decision Making

Catarina Moreira, Andreas Wichert

In this work, we explore an alternative quantum structure to perform quantum probabilistic inferences to accommodate the paradoxical findings of the Sure Thing Principle. We propose a Quantum-Like Bayesian Network, which consists in replacing classical probabilities by quantum probability amplitudes. However, since this approach suffers from the problem of exponential growth of quantum parameters, we also propose a similarity heuristic that automatically fits quantum parameters through vector similarities. This makes the proposed model general and predictive in contrast to the current state of the art models, which cannot be generalized for more complex decision scenarios and that only provide an explanatory nature for the observed paradoxes. In the end, the model that we propose consists in a nonparametric method for estimating inference effects from a statistical point of view. It is a statistical model that is simpler than the previous quantum dynamic and quantum-like models proposed in the literature. We tested the proposed network with several empirical data from the literature, mainly from the Prisoner's Dilemma game and the Two Stage Gambling game. The results obtained show that the proposed quantum Bayesian Network is a general method that can accommodate violations of the laws of classical probability theory and make accurate predictions regarding human decision-making in these scenarios.

Frontiers in PsychologyJan 26, 2016109 citations

The quantum monty hall problem

Giacomo Mauro D’Ariano, Richard David Gill, Michael Keyl +3 more

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.

arXiv:quant-ph/0202120Quantum Information and ComputationAug 1, 200239 citations

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

Michael Aksman

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].

Feb 21, 20260 citations

Quantum graphical models and belief propagation

Matthew Leifer, David Poulin

Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.

Jan 1, 200884 citations