Probability Distribution for the Backward-Evolving Quantum State Concerning a Monitored Qubit

The evolution of two-level quantum systems (qubits) in the presence of continuous-time weak measurements can be described by stochastic differential equations, which is dominated by the Hamiltonian and decoherence channels. In more concrete terms, the conventional density matrix evolves according to the stochastic master equation in the Schrödinger picture. Particularly, for a monitored qubit, the deterministic submanifolds concerning the dynamical evolution of the density matrix has been previously obtained. As a complement, We have recently developed deterministic submanifolds for the backward-evolving quantum state for a monitored qubit. In this paper, in order to better understand the behavior of a monitored qubit, by using the Bloch sphere representation and the properties of Pauli matrices, we provide a closed-form expression for probability distribution over the deterministic submanifolds of the backward evolving dynamics in different scenarios. We further analyze the backward-evolving trajectories of the quantum state by taking into account the analytic solutions and the corresponding submanifolds under heterodyne/homodyne detection associated with non-demolition or fluorescence measurements.