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arXiv:1903.11369 · Journal of Physics A Mathematical and Theoretical

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

Paolo Aniello

Published Jun 13, 2019• 6 citations
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Abstract

Abstract 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.

DOI
10.1088/1751-8121/ab29be

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Abstract 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.
Abstract A notion of stochastic product of quantum states — a binary operation on the set of density operators preserving the convex structure — is discussed. We describe, in particular, a class of group-covariant, associative stochastic products: the twirled products. Each binary operation in this class can be constructed by means of a square integrable projective representation of a locally compact group, a probability measure on this group and a fiducial density operator in the Hilbert space of the representation. By suitably extending this operation from the convex set of density operators to the full Banach space of trace class operators, one obtains a Banach algebra, which is commutative in the case where the relevant group is abelian.
A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.
Abstract A quantum stochastic product is defined as a binary operation on the convex set of quantum states that preserves the convex structure. We discuss a class of group-covariant, associative stochastic products, the twirled products, having remarkable connections with quantum measurement theory and with the theory of open quantum systems. By extending this binary operation from the density operators to the full Banach space of trace class operators, one obtains a Banach algebra. In the case where the covariance group is the group of phase-space translations, one has a quantum convolution algebra. The expression of the quantum convolution in terms of Wigner distributions and of the associated characteristic functions is analyzed.
Abstract In classical statistical mechanics, physical states (probability measures) are embedded in the Banach algebra of complex Borel measures on phase space, where the algebra product is realized by convolution. Convolution is state-preserving; namely, the convolution of two classical states is a state too. This is a special case of the convolution algebra of all complex measures on a locally compact group. A natural problem is whether an analogous structure may emerge in the quantum setting. By resorting to a group-theoretical construction, a quantum counterpart of the convolution of probability measures — the twirled product, or quantum convolution — can be introduced, yielding a group-covariant, associative binary operation on the states of a quantum system, that preserves the convex structure of this set. The analogy with the classical setting becomes striking in the case where the symmetry group is abelian. We focus, in particular, on the quantum convolution product stemming from the group of phase-space translations.
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.
A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral