For a Markovian (in the strongest sense) open quantum system it is possible, by continuously monitoring the environment, to perfectly track the system; that is, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension D, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension K of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit (D=2) is always possible with a bit (K=2), and gave a heuristic argument implying that a finite K should be sufficient for any D, although beyond D=2 it would be necessary to have K>D. Our paper is concerned with rigorously investigating the relationship between D and Kmin, the smallest feasible K. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with D>2, Kmin>D, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond D=2, D-dimensional open quantum systems are provably harder to track than D-dimensional open classical systems. We stress that this result allows complete freedom in choice of monitoring scheme, including adaptive monitoring which is, in general, necessary to implement a physically realizable ensemble (as it is known) of just K pure states. Moreover, we develop, and better justify, a new heuristic to guide our expectation of Kmin as a function of D, taking into account the number L of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries (that we recently introduced elsewhere, [New J. Phys. https://doi.org/10.1088/1367-2630/ab14b2]) makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles in D=3. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, Kmin∼D2, implying a quadratic gap between the classical and quantum tracking problems. Explicit adaptive monitoring schemes that realize the discovered finite ensembles are obtained numerically, thus facilitating future experimental investigations.
Hamiltonian extensions in quantum metrology
