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This article introduces a concept of subharmonic projections for a quantum Markov semigroup, in view of characterizing the support projection of a stationary state in terms of the semigroup generator. These results, together with those of our previous article [J. Math. Phys. 42, 1296 (2001)], lead to a method for proving the existence of faithful stationary states. This is often crucial in the analysis of ergodic properties of quantum Markov semigroups. The method is illustrated by applications to physical models.

We provide two criteria on the existence of stationary states for quantum dynamical semigroups. The first one is based on the semigroup itself, while the second criterion is based on the generator which is in general unbounded and interpreted as a sesquilinear form. These results are illustrated by physical examples drawn from quantum optics.

This article introduces a concept of transience and recurrence for a Quantum Markov Semigroup and explores its main properties via the associated potential. We show that an irreducible semigroup is either recurrent or transient and characterize transient semigroups by means of the existence of non trivial superharmonic operators.

An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on B(h) as a numerical index measuring "how much far" they are from equilibrium. The entropy production is defined as derivative of the relative entropy of the one-step forward and backward evolution in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds.

Complete characterization of complete positivity preserving non-Markovian master equations is presented.

The paper is devoted to the study of nonlinear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born-Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator. . This reprint differs from the original in pagination and typographic detail. 1The interaction of the small system with the reservoir is simulated by L 1 = α 1 a, L 2 = α 2 a † , L 3 = α 3 N , L 4 = α 4 a 2 , L 5 = α 5 (a † ) 2 , and L 6 = α 6 N 2 , where β 1 , β 2 , β 3 are real numbers and α 1 , . . . , α 6 are complex numbers. Consider L k = 0 for all k ≥ 7. NONLINEAR STOCHASTIC SCHRÖDINGER EQUATIONS 3In Example 1, h represents the state space of a single mode of a quantized electromagnetic field and the vectors e n , with n ∈ Z + , provide the energy levels of the system. Because a destroys a photon, L 1 , L 4 model photon emissions. The operator a † describes the creation of a photon (see, e.g., [16,56]).Using (1), we can obtain a probabilistic representation of ρ t . Indeed, it is to be expected that ρ t = E|X t X t | (see, e.g., [6,31,51]). In Dirac notation, |x x|, with x ∈ h, stands for the linear operator defined by |x x|(z) = x, z x for any z ∈ h. Therefore, the probability that a measurement finds the system in the pure state x at time t ≥ 0 is E| x, X t | 2 assuming that E|X 0 X 0 | is the initial density operator and x is a vector of h of norm 1. On the other hand, the value of the observable A at time t T t (A) satisfies E X t , AX t = E X 0 , T t (A)X 0 (see, e.g., [3,34,35]). Recall that quantum observables are represented by self-adjoint operators in h.Second, X t is interpreted as the evolution of the state of a quantum system conditioned on continuous measurement (see, e.g., [4,8,19,30,51,59]). For instance, the following example describes the simultaneous monitoring of position and momentum of a quantum system whose evolution is governed by the Hamiltonian H (see, e.g., [32,57]).

In 2017 an emergency field effort was undertaken in an attempt to prevent the extinction of the world's most endangered marine mammal, the vaquita Phocoena sinus. The rescue effort involved 90 experts from 9 countries and cost US$ 5 million. Following a long decline due to entanglement in legal gillnet fisheries, the vaquita population had fallen from more than 200 to fewer than 30 individuals from 2008 to 2016, due to entanglement in an illegal gillnet fishery that supplies swim bladders of the endangered totoaba Totoaba macdonaldi to Chinese black markets. An emergency ban of gillnets and increased enforcement failed to slow the decline, triggering an emergency effort to catch vaquitas and place them under protection in captivity. Two animals were targeted and captured using light gill nets; a juvenile was released 4 h later because it appeared stressed, and an adult female died of capture myopathy. The program was suspended because of the risk of additional mortalities to the population. The lack of success in capturing vaquitas for temporary protection emphasizes the need to improve our understanding of the effects of chase, capture, handling and enclosure on cetaceans, and to consider intervention before populations reach critically low levels, when there is sufficient time to use phased, precautionary approaches. Furthermore, conservation approaches focused on single species must be integrated into broader efforts to conserve ecosystems and involve the human communities that depend on them.