Equation of motion for estimation fidelity of monitored oscillating qubits

Abstract

We study the convergence properties of state estimates of an oscillating qubit being monitored by a sequence of discrete, unsharp measurements. Our method derives a differential equation determining the evolution of the estimation fidelity from a single incremental step. When the oscillation frequency Ω is precisely known, the estimation fidelity converges exponentially fast to unity. For imprecise knowledge of Ω we derive the asypmtotic estimation fidelity.

Manipulation of Multi‐Level Quantum Systems via Unsharp Measurements and Feedback Operations

“…, and the effects should satisfy n Ê(k) n = Î, where Î is the identity operator. Generally speaking, after a sequence of unsharp measurements, the dynamical trajectory of post-measuredactual-system (DTPMAS) would deviate from the original one due to the back-action effect [35][36][37][38][39]. With such back-action effect, one can manipulate quantum states, including state initialization [53] and entanglement generation [54].…”

“…However, in certain cases, it is possible to design measurements that would not completely destroy quantum correlations, for instance, the unsharp measurement: a special positive operator‐valued measurement . Such measurements have been employed to track Rabi oscillations and investigate quantum process tomography in a driven two‐level system, as well as estimate full quantum state in multi‐level system . Despite these wide applications, there is also a bottleneck for quantum control by merely using measurements due to its unavoided back‐action.…”

“…On the other hand, unsharp measurement, a special positive operator-valued measure (POVM), has been widely used in quantum system control [29][30][31][32][33][34], including tracking Rabi oscillations [35][36][37][38], investigating quantum process tomography [39], estimating full quantum state [40], monitoring wave function [41]. For unsharp measurement, one needs to design general measurement operators, which has been proposed in linear optical qubit system [42].…”

Error correction of quantum system dynamics via measurement–feedback control

“…where the evolution time t k =kτ. Generally speaking, after a sequence of unsharp measurements, the evolution of actual system would deviate away from the original due to the backaction effect of measurement [37][38][39][40], as shown by the red dashed-dot line in figure 1(a). With such a back-action effect, one can manipulate quantum systems for state initialization [60] and entanglement generation [61].…”

“…Unsharp measurement (sometimes called weak measurement), a special positive operator-valued measure (POVM), has been widely used in quantum system control [33][34][35][36], including for tracking Rabi oscillations [37][38][39][40][41], investigating quantum process tomography [42], estimating full quantum state [43], and monitoring wave function [44]. For unsharp measurement operators, it must be proportional to unitary operators and its designation has been proposed in a linear optical qubit system [45].…”