Functions of positive type on locally compact abelian groups can be defined as positive functionals on group algebras, and play a remarkable role in probability theory and in classical statistical mechanics. By Bochner's celebrated theorem, indeed, they are Fourier-Stieltjes transforms of finite positive measures. Hence, a properly normalized nonzero function of positive type on (the group of translations on) phase space provides a realization of a classical state, so it may be called a function of classical positive type. A similar result holds in the quantum setting as well, where a generalized kind of functions of positive type on phase space — the so-called functions of quantum positive type — are related, via the Fourier-Plancherel transform, to the Wigner quasi-probability distributions. In this paper, we will argue that, as in the classical setting, the notion of function of quantum positive type is of a group-theoretical nature. Exploiting an interesting interplay between functions of classical and quantum positive type, we will then provide an interesting characterization of a class of semi-groups of operators that describe the evolution of certain open quantum systems which are of interest in quantum information science. Finally, a suitable extension of this framework to generalized phase spaces that are relevant for current applications will be proposed.
Functions of positive type on locally compact abelian groups can be defined as positive functionals on group algebras, and play a remarkable role in probability theory and in classical statistical mechanics. By Bochner's celebrated theorem, indeed, they are Fourier-Stieltjes transforms of finite positive measures. Hence, a properly normalized nonzero function of positive type on (the group of translations on) phase space provides a realization of a classical state, so it may be called a function of classical positive type. A similar result holds in the quantum setting as well, where a generalized kind of functions of positive type on phase space — the so-called functions of quantum positive type — are related, via the Fourier-Plancherel transform, to the Wigner quasi-probability distributions. In this paper, we will argue that, as in the classical setting, the notion of function of quantum positive type is of a group-theoretical nature. Exploiting an interesting interplay between functions of classical and quantum positive type, we will then provide an interesting characterization of a class of semi-groups of operators that describe the evolution of certain open quantum systems which are of interest in quantum information science. Finally, a suitable extension of this framework to generalized phase spaces that are relevant for current applications will be proposed.
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 .
An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions — classical and quantum — one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
Bochner's theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function.In the Wigner phase space picture, quantum Bochner's theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density.We extend this theorem to discrete phase space representations which possess enough symmetry.More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner's theorem.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl, SU(2), SU(1,1), and SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan P-function to the Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In SU(2) and SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale, t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.