A celebrated phase-space description of nonclassicality in single-mode quantum oscillators is based on the presence of negative regions of the Wigner function.
Since the Wigner function is a normalized but not positive-definite quasi-probability distribution, its negativity has no classical counterpart.

The Wigner function $W(x,p)$ associated with a quantum state $ \rho $ is defined as

\[W(x,p) = \frac{1}{\pi\hbar} \int_{-\infty}^{+\infty} \langle x+y|\rho|x-y\rangle \,e^{-\,\frac{2ipy}{\hbar}} \,dy .\]

It satisfies the normalization condition

\[\iint dx\,dp\, W(x,p) = 1 ,\]

but in general \(W(x,p) \ngeq 0 .\)

The Wigner negativity of a quantum state $ \rho $ is defined as the total phase-space volume of the negative part of the Wigner function. It is quantified by

\[\nu[\rho] = \left( \iint dx\,dp\, |W(x,p)| \right) - 1 .\]

This quantity vanishes if and only if the Wigner function is everywhere nonnegative.

  • If $W(x,p)\ge0$ for all $(x,p)$, the state admits a classical phase-space description and \(\nu[\rho]=0 .\)

  • If $W(x,p)<0$ in some region of phase space, genuine quantum interference effects appear and \(\nu[\rho]>0 .\)

Thus, Wigner negativity is a direct witness of nonclassicality.

Example: Single-photon Fock state $|1\rangle$

For the single-photon Fock state, \(\rho = |1\rangle\langle 1| ,\) the Wigner function is

\[W_1(x,p) = \frac{2}{\pi} \left(2x^2+2p^2-1\right) e^{-2(x^2+p^2)} .\]

This function takes negative values in the region

\[x^2+p^2 < \frac12 .\]

As a result, the Wigner negativity is strictly positive,

\[\nu[|1\rangle] > 0 ,\]

signaling the intrinsic nonclassical character of the single-photon state.

Relation to non-Gaussianity

  • All Gaussian states have nonnegative Wigner functions and therefore zero Wigner negativity.
  • A nonzero value of $ \nu[\rho] $ always implies non-Gaussianity.
  • However, the converse is not guaranteed: some non-Gaussian states may still have a positive Wigner function.

Hence, Wigner negativity is a stronger but more restrictive indicator than entropy-based measures of non-Gaussianity.

Interpretation

Wigner negativity captures phase-space quantum interference effects that cannot be reproduced by any classical probability distribution.
It plays a central role in understanding the resources required for quantum information processing and quantum computational advantage in continuous-variable systems.