The one-time and two-time measurements are studied by employing the idea of Aharonov et al. that the strength of a measurement depends on the state of the measuring device in the context of von Neumann measurement theory. A measurement is classified as either a strong or a weak measurement depending on whether the interaction between the system and the measuring device is infinitely large or negligibly small. A single strong measurement yields one of the eigenvalues of the operator while a single weak measurement, in addition to the eigenvalues, has infinite other possible outcomes. Nevertheless, the expectation values of the strong and the weak one-time measurements are the same. However, the expectation values of the two-time measurements depend on the strength of the first measurement. For the strong case the result is consistent with the conventional result without having to introduce the state collapsing. For the weak case the result is the real part of the expectation value of the time-ordered two-time operator. The fact that this expectation value cannot be factorized indicates that the entanglement between the system and the measuring device introduced by a weak measurement cannot be neglected.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.
Weak measurement is a potentially powerful tool for studying post-selected quantum systems. The original approach to weak measurement requires a different von Neumann interaction between the quantum system and measurement pointer for weak values of every different observable. When studying weak values of properties of more than one particle, which is necessary for studying entanglement, these interactions can range from difficult to impossible to implement experimentally. For example, the measurement of properties on a pair of optical photons requires a strong Kerr-like nonlinearity at the two-photon level. In a recent work, we described a new method for extracting weak values of operators that can be expressed as products of two local, single-particle operators by studying the correlations in experimentally feasible local, single-particle measurements. In the present work, that theoretical approach is generalized so that weak values of products of N local, single-particle operators can be extracted from the correlations in N single-particle measurements.
We extend the idea of weak measurements to the general case, provide a complete treatment and obtain results for both the regime when the 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. We also find interesting quantities, the orthogonal weak values, which play the role of weak values for the case when the PPS are exactly 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.