The main aim of this work is to provide a criterion for conservativity of minimal solutions of time-dependent master equations with unbounded coefficients (Quantum Dynamical Evolution System, QDES). In Section 2 we prove a criterion for conservativity of a QDES without using resolvents and in Section 3 we give sufficient conditions for conservativity of the class of QDES whose completely positive part is a time-dependent bounded sesquilinear form on a Hilbert space K t densely and continuously embedded in H and satisfies an extension property.
The main aim of this work is to provide a criterion for conservativity of minimal solutions of time-dependent master equations with unbounded coefficients (Quantum Dynamical Evolution System, QDES). In Section 2 we prove a criterion for conservativity of a QDES without using resolvents and in Section 3 we give sufficient conditions for conservativity of the class of QDES whose completely positive part is a time-dependent bounded sesquilinear form on a Hilbert space K t densely and continuously embedded in H and satisfies an extension property.
We prove new a priori estimates for the resolvent of a class of minimal Quantum Dynamical Semigroup (QDS).These estimates simplify the proof of the unital property of QDS and suggest a continuity condition for time-dependent in- finitesimal generators to ensure existence and conservativity of the Markov master evolution equation. \S 1. INTRODUCTION.The theory of the Markov mas ter equation has been intensively studied dur- ing the recent years [1][2][3][4][5][6][7][8][9][10].Important applications of this theory were contributed to qu.antum chemistry [3] and quantum optics [4-6].Numerical Monte-Carlo and Runge-Kutta algorithms for these equations are discussed in [7-8].One of efficient analytical tools in the theory of master equation is the interaction representation technique [6-7], where the Markov m..aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov mas ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra B(H) of bounded operators or states from the algebra T(H) of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup Pt(⋅) acting in B(H) is called a Quantum Dynamical Semigroup (QDS) if it is completely positive, normal, conservative, and ultraweak continuous [2,11] .Normal and ultraweak continuity properties mean respectively that l.u.b. Pt(xn)=Pt(1.u.b.xn) and Tr{ρ(Pt(B)−B)}arrow0 as tarrowO for any ρ∈T(H) and B∈B(H) .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator I in the algebra B(H) of observables: Pt(I)≡I∀t≥0 ; in the Schr\"odinger representation, it means the conservation of the trace of an initial state ρ∈T(H) during the evolution: TrTt(β)≡bρt≥0 , where Tt : T(H)arrowT(H) and Pt : B(H)arrowB(H) are dual semigroups: Tr{τt(ρ)B}=Tr{ρPt(B)} .The semigroup Tt=Pt1 is called the predual of the semigroup Pt .The dagger "\dagger '' is used to denote predual operators and maps.The map. P(⋅) with the predual P↑(⋅) is referred to as a completely positive (CP) if ∑i,jTr{P(Bi∗Bj)σjσi}∗≥0 or equivalently ∑i,jTr{B∗iBjPt(σjσi)∗}≥0 (1.1)
It is shown that if the decoherence matrix corresponding to a qubit master equation has a block-diagonal real part, then the evolution is determined by a one-dimensional oscillator equation. Further, when the full decoherence matrix is block-diagonal, then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator Hamiltonian or Lagrangian. When the solution of the oscillator equation is not known, an explicit sufficient condition for complete positivity can still be obtained, based on a Hamiltonian/Lagrangian inequality. A rotational form-invariance property is used to characterize the evolution via a single first-order nonlinear differential equation, enabling some further exact results to be obtained. A class of master equations is identified for which complete positivity reduces to the simpler condition of positivity.
We derive sufficient criteria that guarantee convergence of open quantum system dynamics to a unique long-time state independent of the initial conditions. We refer to this behavior as complete relaxation. Focusing on systems described by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equations on finite-dimensional Hilbert spaces, our approach uses only elementary mathematical arguments. For autonomous systems, we recover three previously derived relaxation criteria, which can be expressed as algebraic conditions on the Lindblad operators and the Hamiltonian entering the GKSL equation. For periodically driven systems, we obtain a new criterion involving a similar algebraic condition and a local continuity requirement on the time-dependent GKSL generator. This result implies an earlier criterion for complete relaxation formulated in Ref. [J. Phys. A 52, 43LT01 (2019)]. We also discuss implications of our approach for periodically modulated classical master equations, as well as extensions to driven systems without a periodicity constraint.
In this thesis we study the problem of continuous measurement on an open quantum system.The starting point of our analysis is the time-evolution of an open quantum system where the evolution of the environment is not explicitly described, i.e. traced out.More explicitly we restrict to time-evolutions having the structure of a semigroup or evolution system.That is, for time-points 0 ≤ r ≤ s ≤ t there exists a two-parameter family of completely-positive maps E * (t, s), such that E * (t, s)E * (s, r) = E * (t, r) and the systems state evolved according to ρ(tPhysically this means that in our Ansatz the interaction between system and environment is already fixed and can only be weak.The main goal of this thesis is now to describe, mathematically rigorous, all continuous measurements compatible with the given evolution.A measurement is compatible with the given time-evolution, if the evolution does not change when measurement results are ignored.In other words the measurement is indirect and does not further disturb the systems evolution.Our results can be interpreted as a rigorous construction of continuous matrix product states, and are related to the Hudson-Parthasarathy quantum stochastic calculus.The analysis in this thesis separates into two parts.In the first we study the class of quantum systems having an evolution system structure as described above.Precisely we study the theory of minimal solutions to a Cauchy equation with a generator of Gorini-Kossakowski-Sudarshan-Lindblad type.We do this for the case of time-dependent and unbounded generators.This analysis culminates in the description of a counterexample due to Holevo of a semigroup which is not the minimal solution to such a standard Lindblad equation.In the second part we construct a description of all measurements compatible with a given evolution and thus achieve our main goal.The analysis starts with the discrete-time version of the problem, i.e.where the time-points r, s, t in the above description of the evolution are chosen from a finite set.This discrete-time problem is completely solved by Stinespring dilation theory.We then construct our solution of the continuous-time case as a refinement limit over arbitrary discretizations of the evolution.Hence, our construction gives a continuous-time analogue of the Stinespring dilation for semigroups and evolution systems.It can be interpreted as a complete quantum description of the information the system radiates into the environment, i.e. it yields a quantum state of the environment.We complement the description of the state with a description of the observables for continuous measurements and the calculation expectation values.
Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.