Inﬁnite Dimensional Analysis, Quantum Probability and Related Topics Vol. 17, No. 3 (2014) 1450020 (12 pages) c  World Scientiﬁc Publishing Company DOI: 10.1142/S0219025714500209 Support projection of state and a quantum L´ evy–Austin–Ornstein theorem Skander Hachicha Universit´ e TUNIS-EL MANAR, Ecole Nationale des Ing´ enieurs de Tunis, Tunisia skander.hachicha@ipeit.rnu.tn Received 17 December 2013 Accepted 9 April 2014 Published 29 July 2014 Communicated by F. Fagnola We prove two characterizations of the support projection of a state evolving under the action of a quantum Markov semigroup and a quantum analogue of the L´ evy–Austin– Ornstein theorem. We discuss applications to open quantum systems. Keywords : Quantum Markov semigroup; support of a state; L´ evy–Austin–Ornstein theorem. AMS Subject Classiﬁcation: Primary: 81S22; Secondary: 46N50, 60J28, 82C10 1. Introduction A classical theorem due to P. L´ evy (Ref. 18 Theorem II.8.1, p. 362 of Ref. 18) says that transition probabilities p ij (t)= P{X t = j | X 0 = i} of a classical time-con- tinuous Markov chain (X t ) t≥0 with values in a countable set I , for all pair i, j ∈ I , are either strictly positive for all t> 0 or vanish for all t> 0. D.G. Austin proved it independently; another proof, more analytic, was later given by Ornstein (see Chung 6 ). This result clearly means that, for every initial state i, the support of the discrete density of random variables X t , namely the set of states j for which p ij (t) > 0, is constant in time for t> 0. The same phenomenon may happen for certain diﬀusion processes, for instance those that are nondegenerate, but not for all diﬀusion processes. Indeed, the two- dimensional diﬀusion process (t, B t ) t≥0 where (B t ) t≥0 is a standard one-dimensional Brownian motion has disjoint supports for any pair of times s = t. We postpone a discussion to Sec. 6. In quantum probability, quantum information and open quantum systems a probability density corresponds to a density matrix, namely a positive, trace-one 1450020-1