We introduce the Lie algebra of super-operators associated with a quantum filter, specifically emerging from the Stratonovich calculus. In classical filtering, the analogue algebra leads to a geometric theory of nonlinear filtering which leads to well-known results by Brockett and by Mitter characterizing potential models where the curse-of-dimensionality may be avoided, and finite dimensional filters obtained. We discuss the quantum analogue to these results. In particular, we see that, in the case where all outputs are subjected to homodyne measurement, the Lie algebra of super-operators is isomorphic to a Lie algebra of system operators from which one may approach the question of the existence of finite-dimensional filters.

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与量子滤波器相关的估计李代数

我们介绍了与量子滤波器相关的超级算子的李代数，特别是从 Stratonovich 演算中出现的。在经典滤波中，模拟代数导致非线性滤波的几何理论，这导致了 Brockett 和 Mitter 的著名结果，其中描述了可以避免维数灾难的潜在模型，并获得了有限维滤波器。我们讨论了这些结果的量子模拟。特别是，我们看到，在所有输出都经过零差测量的情况下，超级算子的李代数与系统算子的李代数同构，从中可以解决有限维滤波器的存在性问题.