Truncated Wigner approximation as non-positive Kraus map

Abstract

We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator , which does not describe a physical state. The rate of appearance of negative eigenvalues of can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.