We show that the time evolution of entanglement under incoherent environment coupling can be faithfully recovered by monitoring the system according to a suitable measurement scheme.PACS numbers: 03.67.Mn,03.65.Yz,42.50.Lc Quantum information processing requires the ability to produce entangled states and coherently perform operations on them. Under realistic laboratory conditions, however, entanglement is degraded through uncontrolled coupling to the environment. It is of crucial practical importance to quantify this degradation process [1][2][3], though also extremely difficult in general, due to the intricate mathematical notions upon which our understanding of entanglement relies [4][5][6]. Up to now, no general observable is known which would complement such essentially formal concepts with a specific experimental measurement setup.In the present Letter, we come up with a dynamical characterization of entanglement, through the continuous observation of a quantum system which evolves under incoherent coupling to an environment. We show that, at least for small, yet experimentally relevant systems, there is an optimal measurement strategy to monitor the entanglement of the time evolved, mixed system state. Mixed state entanglement is then given as the average entanglement of the pure states generated by single realisations of the optimal measurement-induced, stochastic time evolution.Consider a bipartite quantum system composed of subsystems A and B, interacting with its environment. Due to this coupling, an initially pure state |Ψ 0 of the composite system will evolve into a mixed state ρ(t), in a way governed by the master equatioṅwhere the Hamiltonian H generates the unitary system dynamics. The superoperators L k describe the effects of the environment on the system, and, for a Markovian bath, have the standard form [7]where the operators J k depend on the specific physical situation under study.To extract the time evolution of entanglement under this incoherent dynamics, one solution is to evaluate a given entanglement measure M (ρ) for the solution ρ(t), at all times t. One starts from one of the known pure state measures M (Ψ) [5,6,8], together with a pure state decomposition of ρ,where the p i are the positive, normalized weights of each pure state |Ψ i . The most naive generalization for a mixed state would then be to consider the averagewhich, however, is not suitable, since the decomposition (3) is not unique: M would thus give rise to different values of entanglement for different valid decompositions of ρ [9], inconsistently with the general requirements for a bona fide entanglement measure [5,6]. The proper definition of M (ρ) therefore is the infimum of all possible averages M [10], but holds two main drawbacks: (i) it turns into a hard numerical problem for higher dimensional or multipartite systems, and, (ii) even for bipartite qubits, where analytical solutions for some measures M (ρ) are known [8], there is no obvious interpretation of this optimal decomposition, in physical terms. Our a...

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