Can we find a quantum analogue of the Bertrand paradox?

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Discussion Overview

The discussion revolves around the exploration of a quantum analogue to the Bertrand paradox, focusing on the implications of quantum mechanics in a game-theoretic context, specifically relating to a variant of the Monty Hall problem. Participants examine the quantum states involved, the probabilities of winning, and the interpretations of these probabilities within the framework of quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes a quantum version of the Monty Hall problem, suggesting an initial state of |100>+|010>+|001> divided by sqrt3 and discusses the resulting probabilities after a door is opened.
  • Another participant questions the meaning of the intermediate state being a mixture and asks for clarification on the origin of the cosine and sine coefficients in the quantum state representation.
  • A participant suggests that the opening of a door provides knowledge about the possible states, leading to a mixture of the first two superposed states, but expresses uncertainty about the measurement process involved.
  • There is a repeated inquiry about the manipulation of the quantum state and how the coefficients are determined for the presentator's choice.
  • One participant shares a reference to a paper that describes the quantum Monty Hall game, indicating interest in the formal treatment of the problem.
  • Another participant expresses a general interest in paradoxes of classical probability and the potential for quantum analogues, specifically mentioning the Bertrand paradox as a topic of interest.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement, particularly regarding the interpretation of quantum states and the implications of the probabilities calculated. There is no consensus on the measurement process or the manipulation of quantum states.

Contextual Notes

Participants express uncertainty about the assumptions underlying the quantum states and the specific measurements being performed. The discussion highlights the complexity of translating classical probability paradoxes into the quantum realm without resolving the mathematical or conceptual steps involved.

jk22
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I considered a quantum version of the problem

There is one winning position so the initial state is |100>+|010>+|001> divided by sqrt3

Suppose the presentator opens door 3 the intermediate state is then a mixture

Cos a|100>+sin a|010>

We suppose finally the player chooses door 2 hence the end state were |010>

Going through those steps we can compute the probabilities pi*pf=(cos a+sin a)^2/3*sin^2 a

We find the extremas to be .06 up to .48

How to interprete those probabilities ? Does it mean that the game can be won only 48% of the time and hence it would be a lucrative game for the presentator ?
 
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jk22 said:
Suppose the presentator opens door 3
What does that mean quantum mechanically?

jk22 said:
the intermediate state is then a mixture

Cos a|100>+sin a|010>
Where do the cos and sin come from?
 
It probably means that now we have the knowledge its 1 in door 1 or 2 hence the first two superposed states.

The cos and sin are unknown coefficient that will be found afterwards for the presentator to choose where to put the 1 ?
 
jk22 said:
It probably means that now we have the knowledge its 1 in door 1 or 2 hence the first two superposed states.
I don't understand how you can do that. What measurement are you doing?

jk22 said:
The cos and sin are unknown coefficient that will be found afterwards for the presentator to choose where to put the 1 ?
Again, how do you do that? What kind of manipulation are you making on the quantum state?
 
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