## 1 Quantum Information and The Wigner Function  
When we extend the classical idea of informatics, or even the idea of physically-based informatics to the quantum world, we uncover two basic threads: (1) quantum computing, and (2) quantum information as a description of quantum communications. The idea of using quantum mechanics to provide more computing power than classical computers seems to have begun around 1981-2 [1], but [PERSON] put the ideas on a more firm basis a few years later [2]. [PERSON] reformulated the [PERSON] argument [3] with an assertion that every finite realizable physical system could be simulated in a perfect manner by a universal model computing machine of finite size, and even by a quantum mechanical version of this universal machine. The fact that the quantum computer can be far more efficient than a classical version relies upon the concept of entanglement, or quantum superposition [4]. Information in such a quantum computer is naturally quantum information, but we tend to use this term more for the transmission of information in a manner in which it cannot be intercepted. Naturally, the ideas of transmitting bits using quantum-based encryption grew along with the ideas of quantum computing, so the two together become quantum information processing [5; 6].  
Important to each of these applications of quantum mechanics is the use of entanglement. When two bodies interact in a quantum manner they become entangled, which implies that they are now described by a single wave function and are no longer two distinct bodies. Even after they have ceased to interact and have moved some distance away, they remain entangled until some decoherence process acts upon them [7]. However, in experiments, one must face the fact that entanglement is difficult to measure. There is no physical variable whose eigenvalue yields the entanglement. Hence, people have worked for years to devise measures of entanglement [8]. Most of these, however, do not provide a clear visualization of the entanglement. But, it has been demonstrated that the Wigner function does provide a clear visualization of entanglement [7]. And, experimentalists have devised methods of reconstructing the Wigner function from output data of the experiments, particularly in optics (for a review on recent advances see [9]). One such approach involves the study of entanglement in the Jaynes-Cummings model [10]. This model studies the interaction between a cavity mode, treated as a harmonic oscillator, and a two-level atom. When the field in the cavity is in a coherent state, the atom is prepared in an excited state. It is found that the atomic and field states become rapidly entangled, and then subsequently disentangle at one-half the so-called revival time [11]. Thus, the maximum entanglement occurs when the atom Rabi oscillations are at maximum amplitude. It was subsequently demonstrated that the Wigner function can be written as a shifted expectation value of the optical parity operator [12]. Other optical experiments have shown that it is possible to measure directly a significant fraction of the Wigner function, when a spontaneous parametric down conversion of a photon produces a pair of entangled photons [13]. In some implementations of superconducting qubits, the entanglement can be used to follow the evolution of the Wigner function of the reduced density matrix of the qubits [14]. Indeed, it is now clear that the Wigner function can be used to exhibit directly the non-classical behavior through its non-Gaussian and negative excursions [15].