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Annalen der Physik

Self‐Fulfilling Prophesy: Manipulation of Multi‐Level Quantum Systems via Unsharp Measurements and Feedback Operations (Ann. Phys. 11/2019)

Du Ran, Wu‐Jiang Shan, Zhi‐Cheng Shi, Zhen‐Biao Yang, Jie Song, Yan Xia

Published Nov 1, 2019• 0 citations
Publisher

Abstract

Self-fulfilling prophesy consists of a sequence of measurement-feedback operations, as represented here. In article number 1900063, Du Ran, Zhi-Cheng Shi, Yan Xia, and co-workers formulate the general unitary feedback operation with respect to the eigenstates of the density matrix of target state and its post-measurement state. Such a design ensures the superposition state generation and quantum system dynamical elicitation for arbitrary dimension quantum systems. This work may provide new insight on the application of measurement-feedback control.

DOI
10.1002/andp.201970039

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Self-fulfilling prophesy consists of a sequence of measurement-feedback operations, as represented here. In article number 1900063, Du Ran, Zhi-Cheng Shi, Yan Xia, and co-workers formulate the general unitary feedback operation with respect to the eigenstates of the density matrix of target state and its post-measurement state. Such a design ensures the superposition state generation and quantum system dynamical elicitation for arbitrary dimension quantum systems. This work may provide new insight on the application of measurement-feedback control.
Abstract In this paper, we generalize the application of a recently proposed quantum control technique called self-fulfilling prophesy (Uys et al 2018 Phys. Rev. A 97 060102) to achieve high-fidelity target quantum dynamics which are disturbed by unwanted external couplings, based on sequential unsharp measurements and tailored unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well-designed based on the eigenstates of the density matrices of the expected (without external couplings) dynamical states and their corresponding post-measurement states. By means of quantum state estimation, the mechanism of the measurement–feedback scheme is explained generally. For illustrative examples, we show that the dynamical trajectory errors caused by both constant couplings and time-varying couplings are substantially reduced in typical two-level and multi-level systems, i.e. the high-fidelity quantum dynamics are achieved. Furthermore, we discuss the influence of external coupling strength and measurement strength on system dynamics. Crucially, the scheme is universal and can be applied to any dimension systems. Thus, it may find applications in quantum information processing.
Measurement combined with feedback that aims to restore a presumed premeasurement quantum state will yield this state after a few measurement-feedback cycles even if the actual state of the system initially had no resemblance to the presumed state. Here we introduce this mechanism of self-fulfilling prophecy and show that it can be used to prepare finite-dimensional quantum systems in target states or force them into target dynamics. Using two-level systems as an example, we demonstrate that self-fulfilling prophecy protects the system against noise and tolerates imprecision of feedback up to the level of the measurement strength. By means of unsharp measurements the system can be driven deterministically into arbitrary, smooth quantum trajectories.
In this paper, we propose a scheme to eliminate the influence of noises on system dynamics, by means of a sequential unsharp measurements and unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well designed based on the eigenstates of the density matrices of the exact (noiseless) dynamical states and its corresponding post-measurement states. For illustrative examples, we show that the dynamical trajectory errors caused by both static and non-static noises are successfully eliminated in typical two-level and multi-level systems, i.e., the high-fidelity quantum dynamics can be maintained. Furthermore, we discuss the influence of noise strength and measurement strength on the degree of precise quantum control. Crucially, the measurement-feedback scheme is quite universal in that it can be applied to precise quantum control for any dimension systems. Thus, it naturally finds extensive applications in quantum information processing.
It is notoriously known that in contrast to the measure of classical macro systems, the quantum measurement will in general cause quantum state collapse. In this note, we will present some further results on stabilizing control of quantum systems with measurement being involved. To be precise, we will study the effects of the open-loop control and the feedback control in preparing an arbitrarily desired eigenstate of an effective Hamiltonian, respectively. For the open-loop control case, it is shown that no matter how to select the measurement channel, control channel and the control law, one cannot prepare all the eigenstates of the effective Hamiltonian with arbitrary high fidelity for all sufficient large time. While for the feedback case, we not only demonstrate how to select the measurement channel properly to achieve the control target, but also show if not, what cannot be done by the measurement-based feedback control.
Abstract The recently proposed quantum control method called self‐fulfilling prophesy is investigated in multi‐level cases, based on a sequence of measurement‐feedback operations. The feedback operation is elaborately designed with respect to the eigenstates of the density matrix of the target state and its post‐measurement state. This design procedure is suitable for the generation of any predefined coherent superpositions of multiple quantum states and elicitation of desired quantum dynamics. For the sake of clearness, arbitrarily prescribed superposition states in three‐ and four‐level systems are prepared, and quantum dynamics achieved as desired. The simulation results indicate that the scheme tolerates modest imprecisions of feedback operation and is robust against sudden perturbations. Thus, the scheme enables new insight on quantum manipulations in a variety of multi‐level systems to be achieved.
A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is presented.