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Laser Physics

Weak measurements as an instance of non-ideal measurements

A. K. Pan, A. Matzkin

Published Sep 3, 2012• 3 citations
Publisher

Abstract

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.

DOI
10.1134/s1054660x12100167

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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.
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.
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.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
'Weak measurements'—involving a weak unitary interaction between a quantum system and a meter followed by a projective measurement—are investigated when the system has a non-Hermitian Hamiltonian. We show in particular how the standard definition of the 'weak value' of an observable must be modified. These studies are undertaken in the context of bound-state scattering theory, a non-Hermitian formalism for which the involved Hilbert spaces are unambiguously defined and the metric operators can be explicitly computed. Numerical examples are given for a model system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
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.