Representation of superoperators in double phase space

Abstract

Abstract. Operators in quantum mechanics -either observables, density or evolution operators, unitary or not -can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner-Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi-Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a "double" Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived.

Translations and reflections on the torus: Identities for discrete Wigner functions and transforms

“…However, here we want to stress the complete phase space symmetry between these two representations. This symmetry has provided a framework for the representation of superoperators in double phase space [19], that we review in the discrete case in Section 6…”

“…In a recent publication [19], we studied the Wigner-Weyl representation of superoperators in the continuum case. In close analogy to the ordinary reflections and translations in "single" phase space it was possible to define reflection and translation superoperators in a phase space with doubled dimensions, maintaining in that space all the structure of the usual Heisenberg-Weyl group.…”

“…In Section 2 we recall the classical affine group of translations and reflections and its quantization in terms of Heisenberg-Weyl operators in continuum quantum mechanics. The main difference with the previous [19] treatment is the labelling of translations by half-chords [20]. This has the advantage of providing very compact and symmetrical group properties, simplifying many formulae and leading in a natural way to the transition to the discrete case, which we treat in detail in Section 3 .…”

“…Section 4 is devoted to projections of centers and chords and the peculiarities arising from the redundant operator bases. In Section 5 we rewrite the pure state identities of [19] identifying the differences that arise because of the discreteness. They involve quadratic and quartic relationships between Wigner and chord distributions.…”

“…They involve quadratic and quartic relationships between Wigner and chord distributions. In Section 6 we show, following [19], that reflection and translation superoperators can be defined and used as lagrangian coordinates in double phase space, thus allowing the representation of general superoperators either as double center or double chord discrete arrays, in exact correspondence to the representation of operators in "single" phase space.…”