A key lesson of the decoherence program is that information flowing out from
an open system is stored in the quantum state of the surroundings.
Simultaneously, quantum measurement theory shows that the evolution of any open
system when its environment is measured is nonlinear and leads to pure states
conditioned on the measurement record. Here we report the discovery of a
fundamental relation between measurement and entanglement which is
characteristic of this scenario. It takes the form of a scaling law between the
amount of entanglement in the conditional state of the system and the
probabilities of the experimental outcomes obtained from measuring the state of
the environment. Using the scaling, we construct the distribution of
entanglement over the ensemble of experimental outcomes for standard models
with one open channel and provide rigorous results on finite-time
disentanglement in systems coupled to non-Markovian baths. The scaling allows
the direct experimental detection and quantification of entanglement in
conditional states of a large class of open systems by quantum tomography of
the bath.