Deterministic submanifolds for the backward‐evolving quantum state concerning a monitored qubit

The evolution of a quantum system subject to continuous weak measurement is described by stochastic differential equation (SDE). It has been shown that the conventional density matrix ρ t $\rho _t$ characterized such SDE, under experimentally relevant settings, remains within deterministically evolving manifold low dimension, whose associated equations can be computed explicitly. In this paper, as complement, it demonstrated effect E $E_t$ , at time express its during interval [ T ] $[t,T]$ also confined manifold. particular, using Bloch sphere representation together with properties Pauli matrices (as quaternions), deterministic submanifolds for concerning monitored qubit are given stays on curve in homodyne amplitude detection scenario, and moving surface heterodyne detection. This characterization should interest parameter estimation towards obtaining computationally efficient filters real-time control.