  For a Markovian open quantum system it is possible, by continuously
monitoring the environment, 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
$K_{\rm min}$, the smallest feasible $K$. We confirm the long-standing
conjecture of Karasik and Wiseman that, for generic systems with $D>2$, $K_{\rm
min}>D$, 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. Moreover,
we develop, and better justify, a new heuristic to guide our expectation of
$K_{\rm min}$ 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 makes it tractable to conduct a numerical
search, using the method of polynomial homotopy continuation, to find finite
physically realizable ensembles (as they are known) 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, $K_{\rm
min} \sim D^2$, implying a quadratic gap between the classical and quantum
tracking problems.
