How quantum is a given state?
Posted by Arjendu on February 26, 2008
One of the puzzles I have been thinking about for a little bit (triggered by trying to explain to Referees that we were definitely seeing a quantum phenomenon) is trying to answer the question: Given a certain quantum state, can you quantify how quantum it is?
I have a sharp senior working on it (whenever he can not manufacture sufficient number of excuses relating to his two comps exercises in Math and Physics, his robotics projects, and general life excitement, sigh) right now. The starting point of our analysis is the idea that you can take the Wigner function corresponding to your quantum state, find out how much ‘negativity’ there is in that Wigner function and call the amount of negativity a measure of the quantum-ness of the state.
This seems fundamentally fair: Classical probability distributions in phase-space are positive-definite, and it is clear that Wigner function negativity comes from interference effects, so quantifying this should be a pretty good measure of quantum-ness. It is, except in a handful of cases, a numerical exercise, which is a bit of a pain, but that’s fine.
However, there is something counter-intuitive about what emerges from this calculation, and my instinct is that this should be a resolvable issue. Our project figuring this out is moving slowly however, and I’m not completely comfortable revealing all of our thinking so far on this particular project. So I’ll leave it as an exercise to the reader — take a look at that paper by Kenfack and Zycskowski (which has since been published in J. Phys. A) and see if you can find what’s weird about the one (semi-)analytic result they have on the harmonic oscillator.
And if you want to talk about it with us — and in particular about resolutions of the weirdness — that would be great; as I’ve said before, I’m always open to fresh collaborations/conversations.
What I like about this project is that it has all the hallmarks of a classic liberal-arts-college theorist project: Deep enough to be very provocative, but simple enough for an undergrad to make progress. Of course, I could just be completely out of the loop on some critical literature. We shall see.