Abstract The notion of convolution of two probability vectors, corresponding to a coincidence experiment can be extended to a family of binary operations determined by (tri)stochastic tensors, to describe Markov chains of a higher order. The problem of associativity, commutativity, and the existence of neutral elements and inverses for such operations acting on classical states is analyzed. For a more general setup of multi-stochastic tensors, we present the characterization of their probability eigenvectors. Similar results are obtained for the quantum case: we analyze tristochastic channels, which induce binary operations defined in the space of quantum states. Studying coherifications of tristochastic tensors we propose a quantum analogue of the convolution of probability vectors defined for two arbitrary density matrices of the same size. Possible applications of this notion to construct schemes of error mitigation or building blocks in quantum convolutional neural networks are discussed.
Abstract The notion of convolution of two probability vectors, corresponding to a coincidence experiment can be extended to a family of binary operations determined by (tri)stochastic tensors, to describe Markov chains of a higher order. The problem of associativity, commutativity, and the existence of neutral elements and inverses for such operations acting on classical states is analyzed. For a more general setup of multi-stochastic tensors, we present the characterization of their probability eigenvectors. Similar results are obtained for the quantum case: we analyze tristochastic channels, which induce binary operations defined in the space of quantum states. Studying coherifications of tristochastic tensors we propose a quantum analogue of the convolution of probability vectors defined for two arbitrary density matrices of the same size. Possible applications of this notion to construct schemes of error mitigation or building blocks in quantum convolutional neural networks are discussed.
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 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 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.
Many alternative approaches to construct quantum channels with large entangling capacity were proposed in the past decade, resulting in multiple isolated gates. In this work, we put forward a novel one, inspired by convolution, which provides greater freedom of nonlocal parameters. Although quantum counterparts of convolution have been shown not to exist for pure states, several attempts with various degrees of rigorousness have been proposed for mixed states. In this work, we follow the approach based on coherifications of multi-stochastic operations and demonstrate a surprising connection to gates with high entangling power. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. Furthermore, we establish new, continuous classes of bipartite 2-unitary matrices of dimension d 2 for d = 7 and d = 9 , with 2 and 4 free nonlocal parameters beyond simple phasing of matrix elements, corresponding to perfect tensors of rank 4 or 4-partite absolutely maximally entangled states.
We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors γ(β) depending on a parameter β, we introduce a notion of convergence as β→∞ for the corresponding semigroups of γ(β)-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature β. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi α-entropies as β→∞.