Detecting Quantum States with a Positive Wigner Function beyond Mixtures of Gaussian States

We propose a criterion giving a sufficient condition for quantum states of a harmonic oscillator not to be expressible as a convex mixture of Gaussian states. This nontrivial property is inherent to, e.g., a single-photon state and the criterion thus allows one to reveal a signature of the state even in quantum states with a positive Wigner function. The criterion relies on directly measurable photon number probabilities and enables detection of this manifestation of a single-photon state in quantum states produced by solid-state single-photon sources in a weak coupling regime.

Figure 1

Photon number distribution and Wigner function density plot of the optimal pure Gaussian state maximizing the probability of finding Fock state $|1⟩$.

Figure 2

Comparison between criterion based on negativity of the Wigner function (white area below dash-dotted curve) and the proposed criterion (area below solid curve) for the single photon generated by the emitter in a weak coupling regime. See text for details.