Sufficient condition for a quantum state to be genuinely quantum non-Gaussian

Abstract

We show that the expectation value of the operator $\hat{{ \mathcal O }}\equiv \exp (-c{\hat{x}}^{2})+\exp (-c{\hat{p}}^{2})$ defined by the position and momentum operators $\hat{x}$ and $\hat{p}$ with a positive parameter c can serve as a tool to identify quantum non-Gaussian states, that is states that cannot be represented as a mixture of Gaussian states. Our condition can be readily tested employing a highly efficient homodyne detection which unlike quantum-state tomography requires the measurements of only two orthogonal quadratures. We demonstrate that our method is even able to detect quantum non-Gaussian states with positive–definite Wigner functions. This situation cannot be addressed in terms of the negativity of the phase-space distribution. Moreover, we demonstrate that our condition can characterize quantum non-Gaussianity for the class of superposition states consisting of a vacuum and integer multiples of four photons under more than 50 % signal attenuation.

Keywords

• Gaussian state
• Wigner functions
• quantum information
• continuous variables
• non-Gaussian states
• quadrature measurements