# Tag Info

22

One never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables, for which the commutator is non-zero (because they take negative values, they are not probability densities). So, to avoid negative probability ...

13

As Ernesto pointed out in his comment, I've answered your first question here (which was updated on the arXiv and published very recently. As for the question about the intuition behind negative probabilities, here is my warning if you don't already have tenure: don't go there. As Feynman pointed out (and Dirac much earlier) negative probabilities are a ...

7

It is not an complete answer, but (hopefully) it may help: $$\text{tr} \left( A B \right) = \pi \int W_A(r) W_B(r) dr = \pi \int P_A(r) Q_B(r) dr,$$ where $A$ and $B$ are self-adjoint operators, $r=(p,q)$ is a point of the phase space, $W(r)$ is a Winger function, $P(r)$ - Glauber-Sudarshan P-representation and $Q(r)$ - a Husimi Q representation. The ...

6

There are two works of Feynman about negative probabilities. It is hard to add something to that, if to look for introduction to the subject. R. P. Feynman, Negative probability in Quantum implications: Essays in honor of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987), Chap. 13, pp 235 – 248. R. P. Feynman, ...

5

The Wigner function is the fourier transform in one variable of the density matrix for the single particle $\rho(x,y)$. If you Fourier transform in y, you get the Wigner function $\rho(x,p)$. This is important to understand, because it explains why the Wigner function is at all interesting, and why it obeys simple dynamics. It also shows that it doesn't have ...

3

Citations are from here on Wikipedia. Is this really quantum mechanics? In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = |k|sinθ ≈ |k|θ in the small angle (paraxial) ...

3

Let us for simplicity work in 1D with $\hbar=1$. (The generalization to higher dimensions is straightforward.) Moreover, let us for simplicity take an operator $\hat{f}(\hat{X},\hat{P})$ without any ordering ambiguities, i.e., each monomial term in the symbol $f(x,p)$ depends only on either $x$ or $p$, but not on both. Then one possible motivation of ...

3

http://arxiv.org/abs/1202.3628 This is a very recent paper demystifying the negative probability density in the Wigner Function. I hope this helps.

2

Negative probabilities are only possible if they are invisible. They can only be associated with joint measurements. However, we must actually forbid joint measurements. This is only possible with the additional property of measurement disturbance aka Heisenberg's uncertainty principle as properly understood. If we measure a marginal value, the very act of ...

2

In the case when $B$ is a compact operator, a limiting process can be applied to get the operator's norm to any arbitrarily small precision. This process is a Hilbert space generalization of the linear algebra relation for a Hermitian matrix $A$. $||A||_{\infty} = \lim_{n\rightarrow \infty}|tr(A^n)|^{\frac{1}{n}}$. For the data given in the question the ...

2

Let me split the "equivalence" in two parts: Are there states with Wigner functions that are everywhere positive that show "quantum" behaviour? The answer to this question is "yes". One very famous example are Gaussian (bosonic) states - their Wigner function, by definition, is a Gaussian (which is obviously positive) for an intro see e.g. Adesso et ...

2

As Morgan pointed out, extended probabilities, which is the technical name, means joint probability distributions may have negative probabilities, but marginal probabilities never. But this is a stretch. How can it be a joint probability distribution if we can never measure complementary observables simultaneously? Extended probabilities also means we can ...

2

I'll answer my own question and hope this information is useful to someone. I'll take $\hbar = 1$ and will deal with systems of one degree of freedom (the generalisation should be obvious). The normalisation factors are very confusing, so I'll omit most of the reasons for them to be what they are. First, let's take a look at the good old continuous Wigner ...

1

You're effectively doing signal analysis. It's neither quantum mechanics nor a semi-classical approximation. Signal processing is (very) often not stochastic (random), unless thermal noise is an issue for accurate modeling of the apparatus. It's often easier to find a way to eliminate the effects of thermal noise than to calculate the effects of thermal ...

1

The Wigner function is used to describe joint probabilities between two sets of observables that do not commute. I could elaborate a bit more, but there is a lot of literature available where the authors explain this better than I can. A couple of useful articles from arXiv: Probabilistic aspects of Wigner function Negativity of the Wigner function as an ...

1

There is recent work that answers this question in special contexts. In particular, arXiv:1401.4174v1 establishes that for discrete systems of odd prime dimensions (i.e. we're talking about the discrete Wigner functions now, defined in arXiv:quant-ph/0401155v6 ), contextuality (a manifestly quantum property, see review by Mermin) is equivalent to negativity ...

1

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.) Is there a simple system (hopefully a simple harmonic ...

1

There is an obvious property of the Wigner function when one takes the limit $p\rightarrow\infty$. By the Riemann's lemma, the Wigner function is expected to go to zero in this limit but it does driven by the largest eigenvalue. Your question can be stated in the following way. The Wigner function for a generic operator $A$ can be defined as  ...

1

interesting... R. W. Spekkens, ''Negativity and contextuality are equivalent notions of nonclassicality'' from Matty Hoban answer then http://arxiv.org/pdf/0705.2742.pdf ...Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations.. ...allowing negative probabilities for the ...

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