We establish the general form of the generator of a completely positive dynamical semigroup of an N-level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2-level system.

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We analyze two recently proposed measures of non-Markovianity: one based on the concept of divisibility of the dynamical map and the other one based on distinguishability of quantum states. We provide a toy model to show that these two measures need not agree. In addition, we discuss possible generalizations and intricate relations between these measures.

We analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing evolution of such a system may be described either by non-local master equation with memory kernel or equivalently by equation which is local in time. These two descriptions are complementary: if one is simple the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point 't0'. This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular, nevertheless, the corresponding dynamics is perfectly regular. Remarkably, singularities of generator may lead to interesting physical phenomena like revival of coherence or sudden death and revival of entanglement. [2]. It turns out that the popular Markovian approximation which does not take into account memory effects is not sufficient for modern applications and todays technology calls for truly nonMarkovian approach. Non-Markovian dynamics was recently studied in [3][4][5][6][7][8][9][10][11][12][13][14][15]. Interestingly, several measures of non-Markovianity were proposed during last year [16][17][18][19].The standard approach to the dynamics of open system uses the Nakajima-Zwanzig projection operator technique [20] which shows that under fairly general conditions, the master equation for the reduced density matrix ρ(t) takes the form of the following non-local equationin which quantum memory effects are taken into account through the introduction of the memory kernel K(t): this simply means that the rate of change of the state ρ(t) at time t depends on its history (starting at t = t 0 ). Usually, one takes t 0 = 0, however, in this letter we shall keep 't 0 ' arbitrary. An alternative and technically much simpler scheme is the time-convolutionless (TCL) projection operator technique [1,21,22] in which one obtains a first-order differential equation for the reduced density matrix. The advantage of the TCL approach consists in the fact that it yields an equation of motion for the relevant degrees of freedom which is local in time and which is therefore often much easier to deal with than the Nakajima-Zwanzig non-local master equation (1).An essential step to derive TCL from (1) relies on the existence of certain operator inverse [22]. However, this inverse needs not exist and then the method does not work [1,22]. Moreover, even if it exists the corresponding local in time TCL generator is usually defined by the perturbation series (see e.g. detailed discussion in [1]) in powers of the coupling strength characterizing the system. However in general the perturbative approach leads to significant problems. For example the dynamical map needs not be completely positive if one takes only finite number of terms from the perturbative expansion.In the present paper we take a different path. We show that any solution of the non-local equation (1) always satisfies equation which does not involve the integral memory k...

We study a model of an N-level atom coupled linearly to an infinite free boson bath whose time correlation functions are Gaussian. We prove that, in the limit when the decay time of the correlations of the bath goes to zero, the reduced dynamics of the atom is given by a completely positive dynamical semigroup. By varying the interaction parameters, any such semigroup can be obtained in the limit. We also discuss the formally analogous situation of an N-level atom whose Hamiltonian contains an external fluctuating Gaussian stationary contribution.

Bloch-vector spaces for N -level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU(N ). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.