We study the problem of accessibility in a set of classical and quantum channels admitting a group structure. Group properties of the set of channels, and the structure of the closure of the analyzed group [Formula: see text] plays a pivotal role in this regard. The set of all convex combinations of the group elements contains a subset of channels that are accessible by a dynamical semigroup. We demonstrate that accessible channels are determined by probability vectors of weights of a convex combination of the group elements, which depend neither on the dimension of the space on which the channels act, nor on the specific representation of the group. Investigating geometric properties of the set [Formula: see text] of accessible maps we show that this set is nonconvex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set [Formula: see text] covers a positive volume in the polytope of all convex combinations of the elements of the group.
We study the problem of accessibility in a set of classical and quantum channels admitting a group structure. Group properties of the set of channels, and the structure of the closure of the analyzed group [Formula: see text] plays a pivotal role in this regard. The set of all convex combinations of the group elements contains a subset of channels that are accessible by a dynamical semigroup. We demonstrate that accessible channels are determined by probability vectors of weights of a convex combination of the group elements, which depend neither on the dimension of the space on which the channels act, nor on the specific representation of the group. Investigating geometric properties of the set [Formula: see text] of accessible maps we show that this set is nonconvex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set [Formula: see text] covers a positive volume in the polytope of all convex combinations of the elements of the group.
We analyze the set ANQ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an N-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ANQ is shown to be log-convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels, the super-decoherence commutes with the dynamics so that decohering a quantum accessible channel, we obtain a bistochastic matrix from the set ANC of classical maps accessible by a semigroup. Focusing on three-level systems, we investigate the geometry of the sets of quantum accessible maps, its classical counterpart, and the support of their spectra. We demonstrate that the set A3Q is not included in the set U3Q of quantum unistochastic channels, although an analogous relation holds for N = 2. The set of transition matrices obtained by super-decoherence of unistochastic channels of order N ≥ 3 is shown to be larger than the set of unistochastic matrices of this order and yields a motivation to introduce the larger sets of k-unistochastic matrices.
This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings is analyzed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assign preconditioners to matrices in numerical linear algebra. Thus, the known pre-conditioner maps are viewed as quantum channels in finite dimensions. In addition, the infinite-dimensional analogue of preconditioners is introduced and certain limit theorems are discussed; this is with an aim to analyze asymptotic methods in quantum channels analogous to problems in asymptotic linear algebra.
In recent times, there has been a growing scholarly focus on investigating the intricacies of quantum channel mixing. It has been commonly believed, based on intuition in the literature, that every generalized Pauli channel with dimensionality d could be represented as a convex combination of (d+1) generalized Pauli dephasing channels (see [Phys. Rev. A 103, 022605 (2021)] as a reference). To our surprise, our findings indicate the inaccuracy of this intuitive perspective. This has stimulated our interest in exploring the properties of convex combinations of generalized Pauli channels, beyond the restriction to just (d+1) generalized Pauli dephasing channels. We demonstrate that many previously established properties still hold within this broader context. For instance, any mixture of invertible generalized Pauli channels retains its invertibility. It's worth noting that this property doesn't hold when considering the Weyl channels setting. Additionally, we demonstrate that every Pauli channel (for the case of d=2) can be represented as a mixture of (d+1) Pauli dephasing channels, but this generalization doesn't apply to higher dimensions. This highlights a fundamental distinction between qubit and general qudit cases. In contrast to prior understanding, we show that non-invertibility of mixed channels is not a prerequisite for the resulting mapping to constitute a Markovian semigroup.
We investigate the emergence of non-Markovian dynamics in finite-dimensional open quantum systems governed by Weyl dynamical maps and their convex combinations. Using the Hermite normal form, we provide a complete classification of the subgroups of the discrete phase space Zd×Zd, establishing the algebraic framework underlying the Weyl maps. We characterize isotropic Weyl dynamical maps that generate Markovian semigroups and show that anisotropic Weyl maps with nonuniform weight distributions cannot possess the semigroup property. Furthermore, we analyze the role of convexity in the generation and suppression of memory effects. Remarkably, we prove that convex combinations of eternally non-Markovian Weyl dephasing maps can generate Markovian semigroups, demonstrating that non-Markovianity is not additive under mixing. Conversely, we establish a general condition under which convex mixtures of N distinct Weyl semigroups exhibit eternal non-Markovianity. In contrast to the qubit Pauli setting, we further identify the existence of irreducible eternally non-Markovian Weyl dephasing maps, namely, individual dynamical maps that display eternal memory effects without requiring any mixing mechanism. Finally, explicit qutrit examples illustrate the transition among Markovian, non-Markovian and eternally non-Markovian regimes. Our results uncover a fundamental connection among finite phase-space algebra, convex structures, and quantum memory effects, thereby extending the theory of non-Markovian dynamics beyond the Pauli framework.
Stochastic matrices and positive maps in matrix algebras have proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with ‘diamond’ sets of stochastic matrices and pseudo-positive maps are dealt with.
In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.