  Finding optimal and noise robust probe states is a key problem in quantum
metrology. In this paper we propose Markov dynamics as a possible mechanism for
generating such states, and show how the Heisenberg scaling emerges for systems
with multiple `dynamical phases' (stationary states), and noiseless channels.
We model noisy channels by coupling the Markov output to `environment'
ancillas, and consider the scenario where the environment is monitored to
increase the quantum Fisher information of the output. In this setup we find
that the survival of the Heisenberg limit depends on whether the environment
receives `which phase' information about the memory system.

Quantum Physics