No abstract

We study the twirling semigroups of (super)operators, namely, certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form [Formula: see text] where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0∩H. We give a new selfcontained proof of the following result: the induced representation [Formula: see text] is square-integrable if and only if the orbit G[x0] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of [Formula: see text].

Using the notions of frame transform and of square integrable projective representation of a locally compact group G, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group G × G. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase space formulation' of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.2. The image, through the isometry DT , of the product of operators in B 2 (H) is a 'star product of functions' in Ran DT .3. The standard expectation value formula of quantum mechanics -namely,where andρ are, respectively, a bounded selfadjoint operator and a density operator (a positive trace class operator of unit trace) in H -admits, in this framework, a suitable expression in terms of C-valued functions 'on phase space'.The adjoint QT of the isometry DT , like the Weyl map, has a simple integral expression and can be regarded as a 'quantization map'. The paper is organized as follows. In Sect. 2, we discuss the notion of 'frame transform' and its main consequences. In Sect. 3, we briefly review the definition of the Wigner distribution and its relation with projective representations. Next, in Sect. 4, we recall the basic properties of square integrable projective representations, tools that are fundamental for the definition of the (generalized) Wigner transform and of its reverse arrow, the (generalized) Weyl map, see Sect. 5; we will also argue that the generalized Wigner transform is not, in general, a frame transform. Our analysis will culminate in the introduction of the class of frame transforms mentioned before -Sect. 6 -and in the discussion of the main consequences from the point of view of quantum mechanics, see Sect. 7. In Sect. 8, we consider a remarkable example that allows to show the link of our results with the formalism of s-parametrized quasi-distributions developed by Cahill and Glauber [18]. Eventually, in Sect. 9, a few conclusions are drawn.

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product explicit formulae can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Grönewold-Moyal star product. In the second example, the link with wavelet analysis is clarified.

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

Inspired by the 'computable cross norm' or 'realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.

We introduce the notion of square integrable group representation modulo a relatively central subgroup and, establishing a link with square integrable projective representations, we prove a generalization of a classical theorem of Duflo and Moore. As an example, we apply the results obtained to the Weyl-Heisenberg group.