Non-homogeneous quantum Markov states and quantum Markov fields

Abstract

The program relative to the investigation of quantum Markov states for general one-dimensional spin models is carried on, following the strategy developed in the last years. In such a way, the emerging structure is fully clarified. This analysis is a starting point for the solution of the basic (still open) problem concerning the construction of a theory of quantum Markov fields, i.e. quantum Markov processes with multi-dimensional indices.

Markov Property – Recent Developments on the Quantum Markov Property

“…In the one-dimensional case where the order plays a crucial role, the structure arising from a quantum Markov state is fully understood. Following previous results of [5,6], a splitting of a Markov state into a classical part, and a purely quantum part was obtained in [3]. This result allowed us to provide a reconstruction theorem for quantum Markov states on chains.…”

“…Recently, taking into account results contained in [5,6], the emerging structure has been fully understood, see [3]. Here we report the main results relative to quantum Markov states on chains.…”

“…Namely, the Markov property for a locally faithful state ϕ is characterized by the existence of a very explicit nearest neighbour Hamiltonian (13) canonically associated to ϕ. Such a potential generates a oneparameter group of automorphisms of the quasi-local algebra A, admitting ϕ as a KMS-state, see [3], Theorem 5.3.…”

“…Theorem 5 (Theorem 3.2 of [3]) Let ϕ be a Markov state on the quasi-local algebra A w.r.t. the sequence {E j } j − ≤j<j + of transition expectations.…”

“…Recent advances in the structure theory of Markov states on chains ( [3,6]) have suggested a natural multi-dimensional generalization of the notion of Markov state, see [4]. Such a notion has the advantage of being entirely expressible in terms of Umegaki conditional expectations with additional localization properties.…”