Quasiprobabilities and Nonclassicality

Abstract

Bell’s theorem indicates all any physical theories of local hidden variables contradict quantum mechanics. No physical expectations made by hidden variables can a reproduce all of the quantum phenomena. Nonlocality is the typical example of such remarkable quantum phenomena. Bell’s theorem enables us to understand that we can discriminate quantum stuffs from the classical. By “classical” we mean that a given phenomenon can be simulated by some hidden variable theory that assumes the locality principle and the objectivism (or realism). As it is clear that the hidden variable theory is complementary to the quantum theory in making deeper and wider our understanding, it is a natural question to ask if one can further formulate any hidden variable theory so as to judge and determine the classical parts of given physical phenomena. On the other hand, most of the quantum effects have been studied by means of sole of quantum theory. For instance, how much entangled a given state is, and how a quantum state is teleported to a remote place. In this work, we will propose a newly defined mathematical method. For the purpose, we will assume macroscopic reality, a typical criterion of classical physics, such that macroscopic properties are not altered by past nor future observations. We consider two descriptions by a hidden variable theory and a quantum theory. We will define a Wigner-like function by a hidden variable and simply it is probability function. By a quantum theory, we will define one in a similar way and show that such a function has negative component, not allowed for any probabilities. In other words, such a negativity of Wigner function is a signature of quantumness. It is notable that this is a similar behavior on quantum optics.