Some properties of an extended Wigner function

N. C. Dias

The set of non-negative Wigner functions is a convex set which, in the finite dimensional case, is compact and equal to the convex hull of its extreme points. Thus, in finite dimensions, these particular (extreme) Wigner functions can be used to generate the entire set of non-negative Wigner functions. We show that the extreme Wigner functions can be identified by the properties of their null sets, and discuss the problem of constructing these states explicitly. As a by-product, we also elaborate on the properties of the interior and the boundary of the set of non-negative Wigner functions. Finally, if time permits, we will discuss the difficulties of extending some of these results to the infinite dimensional case.

Francisco Soto, P. Claverie

Nicolas J. Cerf, Anaelle Hertz, Zacharie Van Herstraeten

Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.

Stanisław Mrówczyński

The Wigner function, which provides a phase-space description of quantum systems, has various applications in quantum mechanics, quantum kinetic theory, quantum optics, radiation transport and others. The concept of the Wigner function has been extended to quantum fields, scalar and electromagnetic. Then, one deals with the Wigner functional which gives a distribution of field and its conjugate momentum. We introduce here the Wigner functional of fermionic fields of the values in a Grassmann algebra. Properties of the functional are discussed and its equation of motion, which is of the Liouville form, is derived.

Sibel Başkal, Young S Kim, Marilyn E. Noz