A quantization procedure based on completely positive maps and Markov operators

Abstract

Abstract. We describe ω-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.

On the Structure of Quantum Markov Chains on Cayley Trees Associated with Open Quantum Random Walks

“…QMCs were introduced by L. Accardi in [1,2], and subsequently, they have been studied by many authors in the 1D case [3][4][5][6][7][8]. Important applications of QMCs have been investigated [7,[9][10][11][12].…”

“…QMCs were introduced by L. Accardi in [1,2], and subsequently, they have been studied by many authors in the 1D case [3][4][5][6][7][8]. Important applications of QMCs have been investigated [7,[9][10][11][12]. Namely, significant use cases of Markov chains in modeling collaborative interactions for detecting proteins within biological systems have been explored in prior studies, as evidenced by research, such as [13][14][15][16][17].…”