Accessible maps in a group of classical or quantum channels

Abstract

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 $G$ 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 $\mathcal{A}$ of accessible maps we show that this set is non-convex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set $\mathcal{A}$ covers a positive volume in the polytope of all convex combinations of the elements of the group.

arXiv comment25 pages, 6 figures

References in corpus (17)

• Quantifying Coherence
• Assessing non-Markovian dynamics
• Dividing Quantum Channels
• A Brief History of the GKLS Equation
• Birkhoff's polytope and unistochastic matrices, N=3 and N=4
• Generating random quantum channels
• Coherifying quantum channels
• Information flow versus divisibility for qubit evolution
• Divisibility of qubit channels and dynamical maps
• Convex Combinations of Pauli Semigroups: Geometry, Measure and an Application
• Memory kernel approach to generalized Pauli channels: Markovian, semi-Markov, and beyond
• From empirical data to continuous Markov processes: a systematic approach
• Pauli semigroups and unistochastic quantum channels
• Markovian semigroup from mixing non-invertible dynamical maps
• Quasi-inversion of quantum and classical channels in finite dimensions
• Log-Convex set of Lindblad semigroups acting on $N$-level system
• Necessary Criteria for Markovian Divisibility of Linear Maps