The Expected Markov Property for Quantum Markov Fields

Luigi Accardi, Francesco Fidaleo

Luigi Accardi, Francesco Fidaleo

The Markov property for quantum lattice systems is investigated in terms of generalized conditional expectations. General properties of (particular cases of) quantum Markov fields, i.e. quantum Markov processes with multi-dimensional indices, are pointed out. In such a way, deep connections with the KMS boundary condition, as well as phenomena of phase transitions and symmetry breaking, naturally emerge.

Gabriel McCracken

Mou-Hsiung Chang

As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.

Gerd Niestegge

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.

Z. I. Bezhaeva, Валерий Иустинович Оселедец