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Physical Review A

Weak values in continuous weak measurements of qubits

Lupei Qin, Pengfei Liang, Xin-Qi Li

Published Jul 23, 2015• 8 citations
Publisher

Abstract

For continuous weak measurements of qubits, we obtain exact expressions for weak values (WVs) from the postselection restricted average of measurement outputs, by using both the quantum-trajectory equation (QTE) and the quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the ``very'' weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert, and Vaidman (AAV), in the sense that our result incorporates nonperturbative correction which could be important when the AAV WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful results for homodyne measurement in a circuit-QED system, which allows for measuring the real and imaginary parts of the AAV WV by simply tuning the phase of the local oscillator. This advantage can be exploited as an efficient state-tomography technique.

DOI
10.1103/physreva.92.012119

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•Jul 1, 2021•0 citations
For continuous weak measurements of qubits, we obtain exact expressions for weak values (WVs) from the postselection restricted average of measurement outputs, by using both the quantum-trajectory equation (QTE) and the quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the ``very'' weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert, and Vaidman (AAV), in the sense that our result incorporates nonperturbative correction which could be important when the AAV WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful results for homodyne measurement in a circuit-QED system, which allows for measuring the real and imaginary parts of the AAV WV by simply tuning the phase of the local oscillator. This advantage can be exploited as an efficient state-tomography technique.
For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory- equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV's WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV's WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.
Weak values as introduced by Aharonov, Albert, and Vaidman (AAV) are ensemble-average values for the results of weak measurements. They are interesting when the ensemble is preselected on a particular initial state and postselected on a particular final measurement result. It is shown that weak values arise naturally in quantum optics, as weak measurements occur whenever an open system is monitored (as by a photodetector). The quantum-trajectory theory is used to derive a generalization of AAV's formula to include (a) mixed initial conditions, (b) nonunitary evolution, (c) a generalized (nonprojective) final measurement, and (d) a non-back-action-evading weak measurement. This theory is applied to the recent cavity-QED experiment demonstrating wave particle duality [G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, Phys. Rev. Lett. 85, 3149 (2000)]. It is shown that the ``fractional-order'' correlation function measured in that experiment can be recast as a weak value in a form as simple as that introduced by AAV.
Compared with the quantum trajectory equation (QTE), the quantum Bayesian approach has the advantage of being more efficient to infer a quantum state under monitoring, based on the integrated output of measurements. For weak measurement of qubits in circuit quantum electrodynamics (cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest. Elegant work towards this task was carried out by Korotkov in 'bad-cavity' and weak-response limits (Korotkov 2011 Quantum Bayesian approach to circuit QED measurement (arXiv: 1111.4016 )). In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of Korotkov to more general system parameters. The obtained Bayesian rule is in full agreement with Korotkovʼs result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.
The weak-value (WV) measurement proposed by Aharonov, Albert, and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this paper, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain a nonperturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV prediction, we find that the SNR asymptotically gets worse when the AAV WV Aw{A}_{w}Aw​ becomes large, i.e., in the case $g|{A}_{w}{|}^{2}\ensuremath{\gg}1$, where ggg is the measurement strength. (ii) With the increase of ggg (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach---actually the former can slightly exceed---the SNR of the standard measurement. However, with a further increase of ggg, the WV technique will become less efficient than the standard measurement, despite the fact that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet the difference will become more prominent with the increase of ggg. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The nonperturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.
The contribution of the detector dynamics to the weak measurement is analyzed. According to the usual theory [Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)] the outcome of a weak measurement with preselection and postselection can be expressed as the real part of a complex number: the weak value. By accounting for the Hamiltonian evolution of the detector, here we find that there is a contribution proportional to the imaginary part of the weak value to the outcome of the weak measurement. This is due to the coherence of the probe being essential for the concept of complex weak value to be meaningful. As a particular example, we consider the measurement of a spin component and find that the contribution of the imaginary part of the weak value is sizable.
Weak measurements introduced by Aharonov, Albert and Vaidman (AAV) can provide informations about the system with minimal back action. Weak values of product observables (commuting) or higher moments of an observable are informationally important in the sense that they are useful to resolve some paradoxes,realize strange quantum effects, construct density matrices, etc. In this work, we show that it is possible to access the higher moment weak values of an observable using weak values of that observable with pairwise orthogonal post-selections. Although the higher moment weak values of an observable are inaccessible with Gaussian pointer states, our method allows any pointer state. We have calculated product weak values in a bipartite system for any given pure and mixed pre selected states. Such product weak values can be obtained using only the measurements of local weak values (which are defined as single system weak values in a multi-partite system). As an application, we use higher moment weak values and product weak values to reconstruct unknown quantum states of single and bipartite systems, respectively. Robustness of our method which occurs due to inappropriate choices of quantum observables and noisy post-selections is also discussed here. Our method can easily be generalized to the multi-partite systems.