Dynamics of an open N-state quantum system is often modeled with a Markovian master equation describing the evolution of the system density operator. By using generators of SU(N) group as a basis, the density operator can be transformed into a real-valued "coherence-vector." A generator of the dissipative evolution, so-called "Lindbladian," can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a nonhomogeneous system of N^{2}-1 real-valued linear ordinary differential equations. Now one can, e.g., implement standard high-performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace (norm) and Hermiticity of the density operator. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity O(N^{10}). The complexity can be reduced when the number of dissipative operators is independent of N, which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a specific scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with N=10^{3} states on a single node of a computer cluster.
While quantum master equations (QMEs) are the primary workhorse in quantum information science, quantum optics, spectroscopy, and quantum thermodynamics, bringing an arbitrary
N-level QME into Lindbladian form and verifying complete positivity of the associated quantum dynamical map remain open challenges for
N≥3. We explore and implement two independent methods to accomplish these tasks, which enable one to directly compute the Kossakowski matrix of an arbitrary Markovian QME from its Liouvillian. In the first method, due to Hall, Cresser, Li, and Andersson, the Kossakowski matrix elements are obtained by evaluating the action of the Liouvillian on the orthonormal SU(
N) basis matrices and then computing a sum of matrix-product traces. The second method, developed in this work, is based on the real
N-level coherence vector and relies on the Moore-Penrose pseudo-inverse of a rectangular matrix composed of the structure constants of SU
(N). We show that both methods give identical results, and apply them to establish the complete positivity of the partial secular Bloch-Redfield QME for the
Λ and V-systems driven by incoherent light. We find that the eigenvalues of the Kossakowski matrix of these seemingly different three-level systems are identical, implying close similarities of their dissipative dynamics. By facilitating the expression of multilevel Markovian QMEs in Lindblad form, our results enable testing the QMEs for complete positivity without solving them, as well as restoring complete positivity by keeping only non-negative eigenvalues of the Kossakowski matrix.
This work presents a structural reformulation of quantum dynamics based on a scalar coherence functional defined over a general state space. The framework distinguishes two mathematically natural classes of dynamics generated by such a functional:(i) gradient flows, which monotonically increase coherence and correspond to stabilization or selection processes, and(ii) antisymmetric flows, which conserve coherence and generate reversible dynamics. We show that standard unitary quantum mechanics arises as a special case when the state space is a complex Hilbert space, the coherence functional is chosen as the negative energy expectation, and the antisymmetric structure corresponds to the canonical complex structure. In this setting, the Schrödinger equation follows directly from the formalism, without postulating unitarity as an independent axiom. The framework is extended to open systems using the density-operator formalism and completely positive trace-preserving maps. Under standard assumptions (Markovianity, continuity, and semigroup structure), the reduced dynamics takes the Gorini–Kossakowski–Sudarshan–Lindblad form, with dissipative terms interpreted as coherence-stabilizing dynamics. A supporting lemma demonstrates exponential suppression of off-diagonal density matrix elements under dephasing Lindblad operators. Using the Wigner phase-space representation, we further show that conservative coherence dynamics reduces to classical Liouville transport in the semiclassical regime, with quantum corrections governed by the Moyal bracket expansion. This work does not propose a modification of quantum theory. Rather, it offers a structural reorganization of existing formalisms. All results remain fully compatible with standard quantum mechanics while providing a unified perspective on unitary evolution, decoherence, and the emergence of classical behavior. Related work. A companion technical note, Structural Decomposition of SRG Recursion into Flow, Phase, and Closure (Halldórsson & ChatGPT, 2025, Zenodo), develops a discrete dynamical-systems formulation of SRG-style recursion and connects it explicitly to Lemmas 2.5, 2.7, and 2.9 of the present work.
Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.
The concept of complete positivity fixes in considerable detail the mathematical structure of quantum Markovian master equations which describe the combined reversible-irreversible dynamics of open systems for the case of time-independent Hamilton operators. For many applications such as, for instance, problems of time-resolved spectroscopy, similar results would be of great value but involve time-varying external fields with a corresponding time-dependent Hamiltonian. For this reason, we propose a suitable extension of the Kossakowski generator to the time-dependent case by maintaining complete positivity and using arguments suggested by experimental facts. As an example, generalized Bloch-like equations can be obtained with three different relaxation times and a nonlinear dependence upon field strength.
Abstract We present a systematic formalism for the computation of the density matrix of an N-state quantum system in the presence of classical noise or a coupling to the environment. In this formalism, the density matrix of the system is given as an expansion in the generators of the SU(N) group with real coefficients. This leads to a system of master equations. The parameters in these equations may be approximately expressed in terms of the components of the Redfield tensor, when the Born and Markov approximations are valid. The general form of the solution of the system of master equations is established. All relaxation and dephasing rates are then very simply expressed as eigenvalues of a certain matrix. This gives the formulation its simplicity and makes it uniquely suitable for numerical computation. The spectral representation of the components of the Redfield tensor is derived in the case when the environment is a harmonic oscillator bath in thermal equilibrium. Beyond the Born approximation, the decoherence of the system is determined by the Lindblad formula for the Liouvillian superoperator. The Lindblad formulae of some models of multi-state quantum systems are also presented.