Gaussian states play a major role in the field of quantum information with continuous variables. For any practical implementation of such states it is therefore important to provide a customer with a condition to test whether a given state is a Gaussian one.

The well-known criterion for a pure state to be a Gaussian one is the positivity of the corresponding Wigner function at any point of the phase space. However, this simple and fundamental criterion of Gaussianity is not easy to use in practice. Recently, M. Genoni and co-workers [1] have suggested a more practical criteria to detect quantum non-Gaussianity, which is valid for a general quantum state and based on the value of the Wigner function at the origin of the phase space.

Within this project we use the entire Wigner function corresponding to a density matrix \(\hat{\rho}\). We introduce a new condition based on the mean value $$f^{(\mathrm{max})}(c) \equiv \max_{\hat{\rho}} {\rm Tr}\left[(\mathrm{e}^{-c\hat{x}^2} + \mathrm{e}^{-c\hat{p}^2})\hat{\rho}\right].$$ to distinguish a non-Gaussian state from a Gaussian one. Here \(c\) is a positive parameter.

Fig. 1: Maximum of the mean value \(f(c)\) as a function of the parameter \(c\) in log-log-scaled axes. The four lines correspond to different set of states we are maximizing over: classical states (green), Gaussian states (black), all physical states (blue) and an analytic approximation to the optimal state (red).

Theorem (Necessary condition for quantum non-Gaussianity)

Let \(\hat{\rho}\) be the density matrix of a given physical state. If there exists a value \(c>0\) , for which $$f_{\mathrm{G}}^{(\mathrm{max})}(c) < {\rm Tr}\left[(\mathrm{e}^{-c\hat{x}^2} + \mathrm{e}^{-c\hat{p}^2})\hat{\rho}\right] \leq f^{(\mathrm{max})}(c), $$ then the \(\hat{\rho}\) describes a non-Gaussian state.


L. Happ, M.A. Efremov, W.P. Schleich


H. Nha (Korea Institute for Advanced Study, School of Computational Sciences, and Department of Physics, Texas A&M University at Qatar)


Alexander von Humboldt Stiftung
Texas A&M University Institute for Advanced Study (TIAS)


[1] M. G. Genoni, M.L. Palma, T. Tufarelli, S. Olivares, M.S. Kim, and M.G.A. Paris, Detecting quantum non-Gaussianity via the Wigner function, Phys. Rev. A 87, 062104 (2013)
[2] L. Happ, M.A. Efremov, H.Nha, and W.P. Schleich, in preparation (2016)