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

• Published in: IET control theory & applications Vol. 17; no. 1; pp. 92 - 97
• Main Authors: Miao, Zibo, Zhang, Zigui, Li, Kai, Pan, Yu, Sarlette, Alain
• Format: Journal Article
• Language: English
• Published: Stevenage John Wiley & Sons, Inc 01.01.2023; Institution of Engineering and Technology
• Subjects: Amplitudes; Differential equations; Efficiency; Hilbert space; Manifolds (mathematics); Parameter estimation; Physics; Quantum theory; Quaternions; Qubits (quantum computing); Representations; Time measurement
• ISSN: 1751-8644; 1751-8652
• DOI: 10.1049/cth2.12363

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