Operators versus functions: from quantum dynamical semigroups to tomographic semigroups

Quantum mechanics can be formulated in terms of phase-space functions, according to Wigner's approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup 'in disguise', namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The 'disguised counterparts' of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups. We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.Comment: 12 page

Star products: a group-theoretical point of view

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 Groenewold-Moyal star product. In the second example, the link with wavelet analysis is clarified.Comment: 42 pages; conclusions added; a few references adde

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.Comment: 13 page

Symmetry witnesses

A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner's theorem, the set of pure states, the rank-one projections, is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the set of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e., reasoning in terms of quantum states, the sets of uniform density operators of corresponding fixed rank are symmetry witnesses too.Comment: 15 page

Playing with functions of positive type, classical and quantum

A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the Fourier transform of a probability measure. Therefore, considering the group of translations on phase space, a suitably normalized phase-space function of positive type can be regarded as a realization of a classical state. Thus, it may be called a function of classical positive type. Replacing the ordinary convolution on phase space with the twisted convolution, one obtains a noncommutative algebra of functions whose positive functionals we may call functions of quantum positive type. In fact, by a quantum version of Bochner's theorem, a continuous function of quantum positive type is, up to normalization, the (symplectic) Fourier transform of a Wigner quasi-probability distribution; hence, it can be regarded as a phase-space realization of a quantum state. Playing with functions of positive type, classical and quantum, one is led in a natural way to consider a class of semigroups of operators, the classical-quantum semigroups. The physical meaning of these mathematical objects is unveiled via quantization, so obtaining a class of quantum dynamical semigroups that, borrowing terminology from quantum information science, may be called classical-noise semigroups.Comment: 19 page

A class of stochastic products on the convex set of quantum states

We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a so-called twirled stochastic algebra. This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.Comment: 42 page

Robustness of geometric phase under parametric noise

We study the robustness of geometric phase in the presence of parametric noise. For that purpose we consider a simple case study, namely a semiclassical particle which moves adiabatically along a closed loop in a static magnetic field acquiring the Dirac phase. Parametric noise comes from the interaction with a classical environment which adds a Brownian component to the path followed by the particle. After defining a gauge invariant Dirac phase, we discuss the first and second moments of the distribution of the Dirac phase angle coming from the noisy trajectory.Comment: 1 figure, 9 pages, comments welcom

Trace class operators and states in p-adic quantum mechanics

Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators and, in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well. The analogies, but also the several (highly non-trivial) differences, with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.Comment: 70 pages; minor changes, typos correcte

Bipartite quantum systems: on the realignment criterion and beyond

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.Comment: 19 pages, 4 figures, improved versio