Time Reflected Markov Processes

Abstract

A classical stochastic process which is Markovian for its past filtration is also Markovian for its future filtration. We show with a counterexample based on quantum liftings of a finite state classical Markov chain that this property cannot hold in the category of expected Markov processes. Using a duality theory for von Neumann algebras with weights, developed by Petz on the basis of previous results by Groh and Kümmerer, we show that a quantum version of this symmetry can be established in the category of weak Markov processes in the sense of Bhat and Parthasarathy. Here time reversal is implemented by an anti-unitary operator and a weak Markov process is time reversal invariant if and only if the associated semigroup coincides with its Petz dual. This construction allows one to extend to the quantum case, both for backward and forward processes, the Misra–Prigogine–Courbage internal time operator and to show that the two operators are intertwined by the time reversal anti-automorphism.

Generators of Detailed Balance Quantum Markov Semigroups

“…When this happens we can find particular GKSL representations of L as in (1), that we call privileged, with H commuting with ρ and the L k , L * k 's eigenvalues of the modular automorphism (Definition 20) i.e. ρL k ρ −1 = λ k L k .…”

“…The case s = 1/2, sometimes called symmetric, however, is also interesting (see Goldstein and Lindsay [9]). Indeed, as wrote Accardi and Mohari [1] (p.409), "it is worth characterizing the class of Markov semigroup such that T t = T t " in full generality also for the dual semigroup with respect to the "symmetric" scalar product a, b = tr(ρ 1/2 a * ρ 1/2 b) (Petz's duality). Note that the "symmetric" dual semigroup is always a QMS.…”

On two quantum versions of the detailed balance condition

“…satisfying L = L , investigated by Accardi and Mohari [2], Goldstein and Lindsay [18], Cipriani [8], Park [23] and the references therein.…”

“…The norm continuous semigroup T = ( T t ) t≥0 generated by L is called the dual semigroup of T and satisfies the corresponding equation tr(ρxT t (y)) = tr(ρ T t (x)y) for all t ≥ 0. Since L is conditionally completely positive by condition (2), the quantum detailed balance condition implies that the dual semigroup T of T is still a QMS. As a consequence, all the maps T t commute with the modular group (σ t ) t∈R associated with ρ (see Prop.…”

Детальный Баланс, Обращение Времени И Генераторы Квантовых Марковских Полугрупп

“…Двойственная полугруппа может быть также определена для сим-метричного скалярного произведения на B(h) (a, b) → tr(ρ 1/2 a * ρ 1/2 b) (см. [17]- [19]). Двойственная полугруппа всегда является полугруппой вполне положительных отоб-ражений.…”