Open quantum systems are harder to track than open classical systems

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> 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 (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math>) is always possible with a bit (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math>), and gave a heuristic argument implying that a finite <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> should be sufficient for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>, although beyond <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> it would be necessary to have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>D</mml:mi></mml:math>. Our paper is concerned with rigorously investigating the relationship between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">m</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub></mml:math>, the smallest feasible <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math>. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">m</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>&gt;</mml:mo><mml:mi>D</mml:mi></mml:math>, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>-dimensional open quantum systems are provably harder to track than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> pure states. Moreover, we develop, and better justify, a new heuristic to guide our expectation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">m</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub></mml:math> as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>, taking into account the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math>. 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">m</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>, 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.