Classically, probability distributions are nonnegative real measures over the sp
ID: 2287998 • Letter: C
Question
Classically, probability distributions are nonnegative real measures over the space of all possible outcomes which add up to 1. What they mean is open to debate between Bayesians, frequentists and ensemble interpretations. A degenerate distribution is the least random distribution with a probability of 1 for a given fixed event, and 0 for everything else.
What is the analog of a classical probability distribution in quantum mechanics? Is it a wave function augmented with the Born interpretation for probabilities, or is it the density matrix? Does a pure density matrix correspond to a degenerate distribution?
Explanation / Answer
I'll reproduce my comment from above here.
The point is perhaps that quantum theory states are not analogs of probability distributions -- at least not exactly. I suggest arxiv.org/abs/quant-ph/0101012v4 as an interesting attempt to squeeze the two as close together as they can be (it's 34 pages, but Lucien Hardy is relatively easy reading). I'm not as happy as I'd like with this response, which is why it's a comment, not an Answer.
I decided I was being lazy, and looked at Lucien's paper for what I might take to be its relevance to your Question. Lucien takes quantum pure states to be analogous enough to degenerate probabilities to use the idea. Where that becomes interesting is the way he can then characterize the difference between classical probabilistic states and quantum states, given this starting point. The distinguishing feature is his fifth axiom,
"Axiom 5 Continuity. There exists a continuous reversible transformation on a system between any two pure states of that system."
IMO, this is definitely a curious way to construct things. It has a distinct failing, that it's limited to finite-dimensional Hilbert spaces and probability distributions over a finite set of outcomes, and AFAIK no-one has extended Lucien's analysis to infinite dimensional Hilbert spaces and probability spaces, which somewhat diminishes its interest unless you in any case work only with finite dimensional Hilbert spaces (as you might if you work in quantum information).
The point I'd make about this is that this is an interesting partial analogy, although I do not know that any more directly related-to-experiment use has been made of it. It may well be worth thinking in terms of this partial analogy some more, but my personal assessment has been that this is not something worth hanging my hat on exclusively. On the other hand, it's only if one immerses oneself in a way of thinking in a committed way that new results come from anywhere, and it's because people have made different choices than I make that we get different results.
In any case, I think Lucien's paper is relevant for you. It has, to me, the same feel as the way you have asked your Question