#011: Determine the relationship between quantum and classical moment problems
Category: Probability theory - Tags: Quantum mechanics, Statistics, Measure theory
Determine the correct equivalent formulations between classical and quantum moment problems.
Mathematical problem. In probability theory (and statistics) the moments of a distribution over a stochastic variable \(X\) are the expectation values \(m_i = E[X^i]\) of all the powers of \(X\). The moment problem is determining whether given all the moments of the distribution one can reconstruct the probability distribution. That is, given a sequence \(\{m_i\}_{i=0}^{\infty}\), determine whether a positive Borel measure \(\mu\) exists such that: \(m_i = \int_{-\infty}^{+\infty} x^i d\mu\). Note that not all sequences \(\{m_i\}\) allow for a solution. In general, the moment problem may have multiple solutions (i.e. two different probability distributions can have the same moments). The multivariate case does not seem to have a clear solution.
In quantum mechanics, there is an anologue problem, which aims to reconstruct a mixed state from expectation value of observables. For example, one starts from \(m_ij = E[X^iP^j]\) and tries to find a suitable \(\rho\) such that \(m_{ij} = Tr\[(X^iP^j + P^jX^i)/2 \rho]\).
Alternatively, one can formulate the problem in terms of quasi-probability distributions on phase space. Note that there are multiple representations, since there is no single product of observables. Therefore, depending on what permutations are chosen, one gets different quasi-probability distributions. It would be nice to have a summary of these details. The Wigner function should correspond to a choice where the product is defined with all permutations.
One can now ask whether there are moment values that can solve both problems. This essentially is asking whether there are quantum expectations that give a semi-positive Wigner functions. The general question seem to not have an answer. For pure states, it seems that only gaussian states have semi-positive Wigner functions.
Another interesting aspect is that pure quantum states always solve the moment problem uniquely, as their topology is induced by the semi-norms defined by the expectations. Given that pure states have zero entropy, a possible equivalent in classical mechanics would be to ask whether the moment problem is uniquely fixed once a set value of entropy is given.
Physical significance. Given that the solutions to the moment problem for both classical and quantum mechanics exist only for gaussian states, it should be clear that complete hidden variable theory has to be ruled out. However, it would be interesting to understand exactly how and why. As for the classical limit, we would like to show that, once the entropy tends to infinity, distributions become classical. This would mean that the moment problem would be approximately solvable in both cases.