Efficient descriptions of open quantum systems can be obtained by performing an adiabatic elimination of the fast degrees of freedom and formulating effective operators for the slow degrees of freedom in reduced dimensions. Here, we perform the construction of effective operators in frequency space, and using the final value theorem or alternatively the Keldysh theorem, we provide a correction for the trace of the density matrix which takes into account the non trace-preserving character of the evolution. We illustrate our results with two different systems, ones where the eliminated fast subspace is constituted by a continuous set of states and ones with discrete states. Furthermore, we show that the two models converge for very large dissipation and at coherent population trapping points. Our results also provide an intuitive picture of the correction to the trace of the density matrix as a detailed balance equation.arXiv:1909.04211v3 [quant-ph]

We present a general strategy to derive entanglement criteria which consists in performing a mapping from qudits to qubits that preserves the separability of the parties and SU(2) rotational invariance. Consequently, it is possible to apply the well known positive partial transpose criterion to reveal the existence of quantum correlations between qudits. We discuss some examples of entangled states that are detected using the proposed strategy. Finally, we demonstrate, using our scheme, how some variance based entanglement witnesses can be generalized from qubits to higher dimensional spin systems.PACS numbers: 03.67.Mn,03.65.Ud A necessary and sufficient condition to assert the separability of a given general quantum state is an open problem. For bipartite quantum systems formed by two 2-level systems (qubits) or a 2-level system and a 3-level system (qutrit), the Peres-Horodecki criterion [1,2], that is the positivity of the partial transpose (PPT) of the quantum system density matrix, provides such a condition for separability. We still lack such a criterion to fully characterize separability in higher dimensional or in multipartite quantum systems. Although the PPT test constitutes a sufficient condition for detecting bipartite entanglement in the case of systems of arbitrary dimension, it requires not only local manipulation of each party, but also a full reconstruction of the state density matrix. Such requirements may be prohibitive from an experimental perspective. This is why entanglement witnesses [3], which are sufficient criteria for detecting bipartite or even multipartite entanglement based on less demanding measurements, have attracted so much interest lately [4].In this Letter, we present a general scheme to detect entanglement in systems of arbitrary (finite) dimension based on the mapping of qudits to qubits. The proposed mapping, which actually constitutes a general and operational formulation for dichotomization, preserves the separability of the subsystems, ensuring that it does not create entanglement that did not exist in the original system. Therefore, if the mapped qubits system is entangled, we can assert that the corresponding qudits are also entangled. In this way, the proposed mapping enables the application of entanglement criteria originally derived for qubit systems to qudit ones.We start by defining the mapping M U of a density operator ρ acting on the d-dimensional Hilbert space H (d) to a density operator σ = M U (ρ) acting on a 2-dimensional Hilbert space H