I Can we find a quantum analogue of the Bertrand paradox?

[jk22]

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 ?

 

[DrClaude]

Suppose the presentator opens door 3

What does that mean quantum mechanically?

the intermediate state is then a mixture

Cos a|100>+sin a|010>

Where do the cos and sin come from?

 

[jk22]

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 ?

 

[DrClaude]

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?

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?

 

[naima]

The quantum Montu Hall game is described here
https://arxiv.org/abs/quant-ph/0202120

 

[Demystifier]

I like paradoxes of classical probability, and it's interesting to have their quantum analogues.

My favored paradox of classical probability is the Tuesday paradox
https://www.jesperjuul.net/ludologist/tuesday-changes-everything-a-mathematical-puzzle
It would be nice to have a quantum analogue of it.

But from a fundamental perspective, it would be much more interesting to have a quantum analogue of the Bertrand paradox
https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)