Nonclassical character of statistical mixtures of the single-photon and vacuum optical states

Nonclassical states of the electromagnetic field have for many years provided an excellent playground for testing fundamental concepts of quantum mechanics [1]. Recently, nonclassical light has also become an important asset to the rapidly developing applied fields of quantum optics, such as quantum communication and quantum information technology. Its applications, to name only a few, include quantum cryptography [2], interferometric measurements [3] and linear-optics quantum computation [4]. Upon this background, it is becoming more important to have simple criteria for identifying a particular quantum state as nonclassical. This question has attracted strong interest from the first days of quantum optics and a number of solutions, both for the single-mode and multi-mode cases, have been proposed [6–10]. Although there exists a commonly accepted formal definition of a nonclassical state, no necessary and sufficient criterion has been proposed that would allow verification of an optical state’s nonclassical character in a simple experiment. For single-mode optical fields, nonclassical states are commonly defined as those that cannot be represented as a statistical mixture of coherent states. This can be reformulated in terms of the Glauber-Sudarshan P -function [5]: if the latter is positive definite, i.e., if it can be interpreted as a probability density, then the state possesses a classical analog. The practical application of the above definition requires complete information about the quantum state in question so that the P function can be reconstructed. In experimental practice, however, only partial information about a quantum state is normally available and it is often degraded by detection noise and losses. The goal of a nonclassicality criterion is to enable a conclusion about the character of a quantum state based only on this partial information. Known signatures of nonclassicality, frequently associated with a particular method of quantum-state characterization, include sub-Poissonian photon statistics [6], squeezing [7], photon number oscillations [8], and negative values of the Wigner function [9,11]. All of these nonclassicality conditions are sufficient but not necessary and leave wide classes of quantum states outside their scope. For example, the Fock states are subPoissonian, but exhibit quadrature noises above the shot noise level; phase-squeezed states are nonclassical, but possess super-Poissonian photon statistics and positive Wigner functions. This situation was recently improved by Vogel, who showed that a quantum state is nonclassical if the absolute magnitude of the state’s characteristic function exceeds that of the vacuum state at any point in inverse phase space [12]. This modification of the traditional squeezing criterion covers a very wide set of quantum optical states. As noticed by Diósi, however, there are quantum states that are nonclassical but do not satisfy the Vogel criterion, so the latter is still not a necessary condition for nonclassicality [13]. The Vogel criterion is now very relevant because of recent progress in quantum homodyne tomography of highly nonclassical optical states. In a recent experiment, Lvovsky and co-workers prepared the singlephoton Fock state |1〉 by conditional measurements on a photon pair produced via parametric downconversion [11]. The phase-averaged Wigner function reconstructed in the measurement showed a strong dip around the phase-space origin, reaching classically-impossible negative values. Although negativity of the Wigner function is very