We consider weak measurements (WMs) of a spin observable in quantum mechanics beyond the usual asymptotic regime. This is done by obtaining the exact time-dependent wave functions of the measuring apparatus for general nonideal measurements. Ideal strong measurements and the usual WM regime are obtained as the two extreme limiting cases of our exact treatment. We show that in the intermediate regime nonideal measurements lead to ``semiweak'' eccentric values that differ from the usual weak values. We further show that even in the WM regime the exact treatment leads to a meter behavior that can be markedly distinct from the one predicted by the usual WM formalism. We give several illustrations and discuss an application to the distinction of different realizations of the same density matrix.
We consider weak measurements (WMs) of a spin observable in quantum mechanics beyond the usual asymptotic regime. This is done by obtaining the exact time-dependent wave functions of the measuring apparatus for general nonideal measurements. Ideal strong measurements and the usual WM regime are obtained as the two extreme limiting cases of our exact treatment. We show that in the intermediate regime nonideal measurements lead to ``semiweak'' eccentric values that differ from the usual weak values. We further show that even in the WM regime the exact treatment leads to a meter behavior that can be markedly distinct from the one predicted by the usual WM formalism. We give several illustrations and discuss an application to the distinction of different realizations of the same density matrix.
We introduce weak measurements (WM) as a type of non-ideal measurement (NIM) coupling the system and the measuring device in a specific manner involving a weak interaction followed by post-selection. For the particular case of a WM measurement of spin, we solve the quantum dynamics for the coupled system-meter ensemble exactly for any type of non-ideal measurement. The standard WM regime is obtained as a limiting case; eccentric “semi-weak” values not only appear in other cases of NIM, but can also have a larger magnitude than the usual weak values. A couple of examples comparing the merits of the WM regime and of the exact treatment in situations of potential interest to quantum information applications are considered.
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.
We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without postselection yield the corresponding correlations of the Wigner function, offering direct quantum tomography through the moments of the canonical variables. Correlations in spin-21 sequential weak measurements coincide with those in strong measurements, they are constrained kinematically, and they are equivalent with single measurements. In sequential WMs with postselection, an anomaly occurs, different from the weak value anomaly of single WMs. In particular, the spread of polarization $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ as measured in double WMs of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ will diverge for certain orthogonal pre- and postselected states.
In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. Our results show that post-selection is extremely sensitive to the action of the meter on the system, highlighting the difference between the weak measurement regime where this back-action is negligible and the average meter position is a valid measurement result, and the intermediate regime where the meter statistics should not be misinterpreted as a measurement outcome. The transition between weak values and eigenvalues is facilitated by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter. The intermediate measurement regime thus reveals that the apparent contradictions between anomalous weak values and eigenvalues originate from the non-classical effects of back-action on the post-selected meter statistics.
Weak measurements have thus far been considered instrumental in the so-called direct measurement of the quantum wave function [4J. S. Lundeen, Nature (London) 474, 188 (2011).]. Here we show that a direct measurement of the wave function can be obtained by using measurements of arbitrary strength. In particular, in the case of strong measurements, i.e., those in which the coupling between the system and the measuring apparatus is maximum, we compared the precision and the accuracy of the two methods, by showing that strong measurements outperform weak measurements in both for arbitrary quantum states in most cases. We also give the exact expression of the difference between the original and reconstructed wave function obtained by the weak measurement approach; this will allow one to define the range of applicability of such a method.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.