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 K-min, the smallest feasible K. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with D > 2, K-min > 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 K-min 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, K-min similar to D-2, 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.
