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

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.

Tristochastic operations and products of quantum states

“…The simplest example of a tristochastic tensor inducing an associative product reads A ijk = r (i −j −k)| modN for any value of indices i, j, k, where ⃗ r is a probability vector and the subtraction of indices is taken modulo N. A more involved example can be interpreted as a 'classical analogue' of the twirled product proposed by Aniello [13,14],…”

“…It is worth mentioning, that definition 3.1 can be regarded as a special case of example 7 proposed by Aniello in [14], where the quantum channel is used without resorting to the Choi-Jamiołkowski isomorphism.…”

“…The question of whether a quantum product is commutative or associative, depends on the choice of the dynamical matrix D and is analogous to the classical problem. It is worth mentioning that the twirled product proposed and extensively studied by Aniello [13][14][15], which for density matrices ρ, σ and ϑ and the Lie group SU(N) in fundamental representation reads…”

“…However, we provide here a construction of quantum product of arbitrary density matrices of the same size, which results in a legitimate density matrix. A construction of convolution of quantum states, called twirled product, was recently proposed by Aniello [13][14][15], while other techniques were used to generate a composition of bi-partite density matrices [16] and convolution of quantum superoperators [17]. However, we believe that the extension of those products, with associativity possibly replaced by other properties, might result in new useful applications, especially in the areas where the associativity of convolution is not crucial.…”

“…In this work, we focus on the, generally non-associative, product of density matrices introduced in [14], in analogy to the classical map (1.1). Such a product can be treated as a quantum channel, Ω N ⊗ Ω N → Ω N , where Ω N denotes the set of density matrices of size N.…”