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18 - Weak Values and Quantum Nonlocality
- from Part III - Nonlocality: Illusion or Reality?
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- By Yakir Aharonov, Tel Aviv University and Chapman University, Eliahu Cohen, Tel Aviv University
- Edited by Mary Bell, Shan Gao, Chinese Academy of Sciences, Beijing
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- Book:
- Quantum Nonlocality and Reality
- Published online:
- 05 September 2016
- Print publication:
- 19 September 2016, pp 305-314
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Summary
Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.”
Introduction
It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles.
In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes.
The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].
3 - Protective measurement, postselection and the Heisenberg representation
- from Part I - Fundamentals and applications
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- By Yakir Aharonov, Tel Aviv University and Chapman University, Eliahu Cohen, Tel Aviv University
- Edited by Shan Gao
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- Book:
- Protective Measurement and Quantum Reality
- Published online:
- 05 January 2015
- Print publication:
- 22 January 2015, pp 28-38
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Summary
Classical ergodicity retains its meaning in the quantum realm when the employed measurement is protective. This unique measuring technique is re-examined in the case of post-selection, giving rise to novel insights studied in the Heisenberg rep-resentation. Quantum statistical mechanics is then briefly described in terms of two-state density operators.
Introduction
In classical statistical mechanics, the ergodic hypothesis allows us to measure position probabilities in two equivalent ways: we can either measure the appropriate particle density in the region of interest or track a single particle over a long time and calculate the proportion of time it spent there. As will be shown below, certain quantum systems also obey the ergodic hypothesis when protectively measured. Yet, since Schrödinger's wave function seems static in this case [1, 2, 3], and Bohmian trajectories were proven inappropriate for calculating time averages of the particle's position [4, 5], we will perform our analysis in the Heisenberg representation.
Indeed, quantum theory has developed along two parallel routes, namely the Schrödinger and Heisenberg representations, later shown to be equivalent. The Schrödinger representation, due to its mathematical simplicity, has become more common. Yet, the Heisenberg representation offers some important insights which emerge in a more natural way, especially when employing modular variables [6]. For example, in the context of the two-slit experiment it sheds a new light on the question of momentum exchange [7, 8, 9]. Recently studied within the Heisenberg representation are also the double Mach–Zehnder interferometer [10] and the N-slit problem [11]. As can be concluded from [11], the Heisenberg representation prevails in emphasizing the non-locality in quantum mechanics, thus providing us with insights about this aspect of quantum mechanics as well.
Equipped with the backward evolving state-vector within the framework of two-state-vector formalism (TSVF) [12], the Heisenberg representation becomes even more powerful since the time evolution of the operators includes now information from the two boundary conditions.