Talk:Husimi Q representation

To technical

I literally have no idea what this article is about.Circuitboardsushi (talk) 04:01, 6 January 2012 (UTC)

Probability Interpretation

Strictly speaking, isn't this statement wrong?: "Thus Q does not represent the probability of mutually exclusive states, as needed in the third axiom of probability theory." I'm pretty sure that the the Husimi function is exactly the probability of the result of a POVM composed of phase-space wavepackets (i.e. the coherent states, indexed by alpha). These are certainly mutually exclusive outcomes, and can (if the proper effect operators are chosen) correspond to the post-measurement conditional states of the system measured. A better sentence might be "the state rho is very different from the state sum_alpha [ Q(alpha) | alpha >< alpha | ], and so Q cannot be interpreted as a probability distribution for the actual quantum state of the system". Jess (talk) 04:33, 1 February 2014 (UTC)

Yes and no. The statement is quite clumsy ("points" would be preferable to "states"). Husimi does, of course, produce correct expectation values, when used properly, and that is why it is there! The statement echoes something not quite explicit: when re-expressed in phase-space variables, x, and p, the Husimi gives the mis-impression (often repeated in print) that it solves the Wigner distribution "problem of negative probabilities", which it does, and "so" it produces bona fide probabilities, which it does not. Even though it is positive semi-definite, it still needs suitable operator dressings of the observables to produce suitable expectation values, and so it is not a "classical probability distribution" serving as the probability measure of arbitrary phase-space functions, as it is often misunderstood to do. Throwing an observable f(α) into the integral w.r.t. α with a measure Q(α) does not produce the correct quantum expectation value of the observable f(α). The reason is that the uncertainty principle does not allow nearby phase-space points to be statistically independent contingencies--the 3rd theorem. Indeed, the statement might say something about contiguous variables α not specifying mutually exclusive options, and so a distribution in the space of α s failing Kolmogorov's 3rd axiom. You might try your improvements, sufficiently judiciously to reflect that point... anything to let the reader know that the weirdness of QM is still operative in full force and manifests itself inexorably, unforgivingly, despite appearances of a distribution in classical c-number variables α. Cuzkatzimhut (talk) 11:25, 1 February 2014 (UTC)