Quantum Information

Monty Hall

Simulation

The Quantum Monty Hall Problem

The Monty Hall problem is a well-known problem in statistics, which time and again leads to controversies because of its counter-intuitive solution. Here we consider a quantum version, which illustrates nicely some differences between classical and quantum information. As in the classical case a simulation helps to understand the solution.

You will find on this site:

An explanation of the quantum version of the problem, in parallel with a description of the classical version. Here we assume familiarity with basic quantum mechanics.
A Simulation of the quantum version, in the form of a Java applet with instructions and some hints on what experiments you might try.
A paper explaining the theory of the problem.
A reminder of the solution of the classical problem, with links and references.
A description of an even more quantum version, in which the prior information of the show master is encoded in an entangled state.

The game

	Classical	Quantum
	The basic setting of the problem is a game show. There is a prize hidden behind one of three doors, which the player (called P here) can get, if he opens the correct door. His opponent is the host of the show (called Q for quiz master), who basically tries to confuse the player.	In the quantum version the three doors become a quantum system described in a three dimensional Hilbert space, called the game space. Opening a door is the same as choosing a direction in that space, and making a measurement of the associated one-dimensional projection. If the result is positve, the player gets the prize.
1	Before the game begins, the producers of the show put the prize behind some randomly chosen door. Q knows where the prize is.	The producers choose a vector in the game space, inform the host Q about their choice, and prepare the system in the corresponding pure state.
2	The player P is now asked to choose a door, which is, however, not opened at this stage.	P chooses a direction in the game space.
3	Q opens a door other than the one chosen by P. Since he knows where the prize is, he can and does avoid opening the door with the prize.	Q chooses a direction in the game space, which is orthogonal to the direction chosen by P in the previous round, and also orthogonal to the prize vector (known only to him). He makes a measurement along this direction, but is certain not to hit the prize.
4	This leaves two closed doors. P now chooses one, opens it, and if he is lucky, he gets the prize.	This leaves a two dimensional Hilbert space of possible directions for the prize. P now chooses one, and makes a yes/no-measurement of the corresponding projection. If the result is yes, he gets the prize.
What should P do?
	Stick to his choice from stage 2, or switch to the third door?	Stick to his choice from stage 2, or pick another direction, perhaps the orthogonal complement of his previous choice?

Questions and comments

Last modified: 07 Mar 2001

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