We extend the idea of weak measurements to the general case, provide a complete treatment, and obtain results for both the regime when the preselected and postselected states (PPS) are almost orthogonal and the regime when they are exactly orthogonal. We surprisingly find that for a fixed interaction strength, there may exist a maximum signal amplification and a corresponding optimum overlap of PPS to achieve it. For weak measurements in the orthogonal regime, we find interesting quantities that play the same role that weak values play in the nonorthogonal regime.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
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 becomes large, i.e., in the case $g|{A}_{w}{|}^{2}\ensuremath{\gg}1$, where
g is the measurement strength. (ii) With the increase of
g (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
g, 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
g. (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 notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al. (Phys. Rev A 41,11 (1990)), hereafter referred to as AAV. We have performed an experiment similar to one proposed by Knight et al. (1990) and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity. We suggest that preselection and postselection measurement techniques could be useful for study of physical phenomena that might be otherwise unobservable.
We study quantum measurements with preselection and postselection, and derive the precise expressions of the measurement results without any restriction on the coupling strength. For a qubit system, we derive the maximum measurement results by choosing appropriate initial and finial states. Significant amplification effect is obtained when the interaction between the system and the measuring device is very weak; and typical ideal quantum measurement results are obtained when the interaction is strong. The improvement of the signal-to-noise ratio (SNR) and the enhancement of the measure sensitivity (MS) by weak measurements are studied. Without considering the probability decrease due to postselection, the SNR and the MS can both be significantly improved by weak measurements; surprisingly, neither SNR nor MS can be effectively improved when the probability decrease is considered.
The weak measurement proposed by Aharonov and his colleagues extracts information about a physical quantity of the system by the postselection as shifts of the argument of the probe wave function. The more the postselected state is orthogonal to the preselected state, the larger the shift determined by the weak value becomes. Recently, the signal amplification by the weak measurement has been extensively studied. In the present work, we explicitly obtain the optimal probe wave function and the amplification factor for a given weak value, which is calculated from the experimental setup. It is shown that the amplification factor has no upper bound, in contrast to the Gaussian probe wave function, and that the amplified signal is sharp.