# Open quantum systems are harder to track than open classical systems

## TL;DR

This paper rigorously proves that tracking open quantum systems with Hilbert space dimension greater than two requires more classical memory than the system's dimension, confirming that quantum systems are inherently harder to monitor than classical ones.

## Contribution

The paper confirms that for quantum systems with dimension greater than two, the minimal classical memory needed exceeds the system's dimension, and introduces a heuristic relating this to the number of Lindblad operators and symmetries.

## Key findings

- For D>2, K_min > D, proving quantum tracking is harder.
- Supports heuristic that K_min scales as D^2 without symmetries.
- Numerical search confirms the heuristic for D=3.

## Abstract

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.