## 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.

## Matt Leifer said

You have to be careful here. The Wigner function of an EPR state is positive, since it is a limit of Gaussian states, but I think we ought to be unhappy with a definition of “quantumness” in which it comes out as classical. For that matter, it is possible to do much of quantum information science, e.g. teleportation, quantum error correction, etc., with only Gaussian states. Despite its usefulness, I think that taking Wigner function negativity too seriously as an indicator of nonclassicality is a red herring.

Another difficulty is that there are several different types of pseudo-probability distribution (Wigner, Husimi, etc.) and so why should any one of them be privileged in the definition of “quantumness”? You might want to say that a state is “quantum” if there is no pseudo-probability representation in which it is positive, but unfortunately, for any given state, you can always find a representation in which it is positive. In fact, you can even find representations in which ALL states are positive, the trick being that some of the functions representing measurement outcomes have to become negative instead.

A better question (I think) is to ask when a SET of states and measurement outcomes has to be negative in all possible representations. After all, a quantum experiment usually involves many possible states and measurements and it is the experiment as a whole that should be taken as an indicator of a quantum effect, rather than a mathematical object that just represents part of it.

This notion of “quantumness” turns out to be equivalent to a refined notion of contextuality proposed by Rob Spekkens. Here are some relevant references:

http://arxiv.org/abs/quant-ph/0406166

http://arxiv.org/abs/0710.5549

http://arxiv.org/abs/0711.2658

http://arxiv.org/abs/quant-ph/0604155

There are a few people in my group thinking about this and similar issues, so feel free to get in touch.

## arjendu said

Thanks, Matt.

One point I should’ve made in my original post — it was so ‘obvious’ to me that I didn’t bother specifying, and of course, the other perspective was so obvious to you as a quantum information theorist (I assume, given your references) that you didn’t think of it: I am talking about uni-partite systems.

With that constraint, the only positive-definite Wigner function is one for a Gaussian (‘Hudson’s theorem’). And while you can do all sorts of cool quantum things with Gaussians in bi-or-multipartite systems, in a uni-partite system, there’s nothing a quantum Gaussian can do that a classical Gaussian can’t.

The question about privileging the Wigner function still holds. All I can say is that Wigner functions are extremely useful in that they allow us to retain some modicum of classical intuition AND the ability to recover the original quantum density matrix (unlike the Husimi function, which throws away information), so the quantum-classical correspondence/ quantum-chaos community finds it hard NOT to use it …

Linking this to experimentally measurable stuff is critical, of course, and actually part of how I have been thinking about it. I’d be happy to discuss in further with you if you like, but don’t feel comfortable posting those raw-er ideas here.

Thanks again for the comment.

## Bob and Adam’s excellent quantum adventure « Confused at a higher level said

[…] earlier that my students Bob and Adam had done some interesting work in figuring out how to resolve a problem in using the negativity of a Wigner function to measure how quantum a given state was. We shipped the paper to Phys. Rev. today, as well as […]