Towards a Quantum Two Envelopes Problem

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Towards a Quantum Two Envelopes Problem

Postby gill1109 » Tue Mar 10, 2020 12:47 am

The famous Monty Hall problem (aka Three Doors problem) has a number of interesting quantum generalisations. So far there does not seem to exist an equally interesting quantum generalisation of the infamous Two Envelope Problem. I have just posted a new survey of the problem on arXiv. My paper is called "Anna Karenina and The Two Envelopes".
The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.

My challenge to those interested in the foundations of quantum mechanics: invent an exciting and challenging Quantum Two Envelopes Paradox (QTEP).

It should start with two observables P and Q say, defined on a Hilbert space H, and not commuting, and such that there exists an isomorphism on H under which the pair (P,Q) is mapped to the pair (Q,P). Alice is about to observe P and Bob is about to observe Q. However, if Alice goes ahead, then Bob's plans are disturbed ... Each wants to end up with as large a numerical value as possible. Now please invent a scenario (state and pair of observables) such that Alice and Bob both can reason that they should switch roles as to who does what when.
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