The geometry of Wigner's function

Nandor L. Balazs

J. Bertrand, P. Bertrand

Fernando Parisio

By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function ( W ). A non-integral representation is presented in terms of a quadratic form W ∝ V † F V , where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it.

Sibel Başkal, Young S Kim, Marilyn E. Noz

Juan Jesús Jesús Díaz Guevara, Isabel Sainz, A. B. Klimov

Abstract We have developed a general algebraic method for constructing a nonredundant Wigner function for N-ququart systems using the structure of the underlying algebraic ring and the associated non-euclidian geometry of the 4 N × 4 N pseudo phase-space. Using the stabilizer formalism we show that the Wigner function satisfies the tomographic condition and possesses a generalized covariance under some specific Clifford symplectic transformations allowing to represent a wide class of ququart stabilizer states in form of delta-functions. We establish a relation between the Wigner function of N-ququart states represented in 4 N × 4 N grid with its 2N-qubit counterpart in a F 2 2 N × F 2 2 N discrete phase space and discuss the corresponding drawbacks.

Georges Chevalier