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arXiv:1602.05083 · WORLD SCIENTIFIC (EUROPE) eBooks

Quantum to Classical Transitions via Weak Measurements and Post-Selection

Eliahu Cohen, Yakir Aharonov

Published Nov 14, 2016• 10 citations
arXivHTMLPDF

Abstract

This work will incorporate a few related tools for addressing the conceptual difficulties arising from sewing together classical and quantum mechanics: deterministic operators, weak measurements and post-selection. Weak Measurement, based on a very weak von Neumann coupling, is a unique kind of quantum measurement with numerous theoretical and practical applications. In contrast to other measurement techniques, it allows to gather a small amount of information regarding the quantum system, with only a negligible probability of collapsing it. A single weak measurement yields an almost random outcome, but when performed repeatedly over a large ensemble, the averaged outcome becomes increasingly robust and accurate. Importantly, a long sequence of weak measurements can be thought of as a single projective measurement. I claim in this work that classical variables appearing in the macro-world, such as centre of mass, moment of inertia, pressure and average forces, result from a multitude of quantum weak measurements performed in the micro-world. Here again, the quantum outcomes are highly uncertain, but the law of large numbers obliges their convergence to the definite quantities we know from our everyday lives. By augmenting this description with a final boundary condition and employing the notion of "classical robustness under time-reversal" I will draw a quantitative borderline between the classical and quantum regimes. I will conclude by analyzing the role of macroscopic systems in amplifying and recording quantum outcomes.

DOI
10.1142/9781786341419_0012

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•Jan 1, 2006•6 citations
This work will incorporate a few related tools for addressing the conceptual difficulties arising from sewing together classical and quantum mechanics: deterministic operators, weak measurements and post-selection. Weak Measurement, based on a very weak von Neumann coupling, is a unique kind of quantum measurement with numerous theoretical and practical applications. In contrast to other measurement techniques, it allows to gather a small amount of information regarding the quantum system, with only a negligible probability of collapsing it. A single weak measurement yields an almost random outcome, but when performed repeatedly over a large ensemble, the averaged outcome becomes increasingly robust and accurate. Importantly, a long sequence of weak measurements can be thought of as a single projective measurement. I claim in this work that classical variables appearing in the macro-world, such as centre of mass, moment of inertia, pressure and average forces, result from a multitude of quantum weak measurements performed in the micro-world. Here again, the quantum outcomes are highly uncertain, but the law of large numbers obliges their convergence to the definite quantities we know from our everyday lives. By augmenting this description with a final boundary condition and employing the notion of "classical robustness under time-reversal" I will draw a quantitative borderline between the classical and quantum regimes. I will conclude by analyzing the role of macroscopic systems in amplifying and recording quantum outcomes.
The outcomes of a series of measurements, made on a quantum system, form a sequence of random events which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become "real", and the mean values of a quantity (a functional) is directly related to the frequencies with which the paths are travelled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one to distinguish between interfering alternatives, while leaving the interference between them intact.
We consider highly inaccurate measurements made on classical stochastic and quantum systems. In the quantum case such a \e{weak} measurement preserves coherence between the system's alternatives. We demonstrate that in both cases the information about the scenario realised in each individual trial is lost. However, ensemble parameters such as classical path probabilities, and quantum quasi-probabilities can be extracted from the obtained statistics. In both cases causality ensures that additional post-selection only redistributes individual outcomes between the system's final states. Quantum quasi-probabilities may change sign, which allows for anomalously large meter's (pointer's) reading for some final states. These, we show, result from mere \e{reshaping} of a broad distribution obtained earlier, and provide no \e{experimental evidence} of quantum variables taking, on rare occasions, exceptionally large values.
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
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 two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
Post-selection is the selection of a subset of an ensemble based on a measurement on that ensemble. In the last few years, post-selection has taken on new importance as a mechanism to induce the optical nonlinearities required for quantum logic gates. I present experiments and theory that explore post-selection in photonic systems. I then go on to describe an experimental implementation of Hardy's Paradox, which is one of the three great thought-experiments about entanglement and correlations in quantum mechanics (the others being the EPR and GHZ "paradoxes"). In the place of the electron--positron annihilation in the original proposal, I apply, for the first time, a post-selection-based absorptive photon switch that coworkers and I had previously developed. Weak measurement promises to be an important technique for characterizing post-selected quantum systems during their evolution. The technique minimizes disturbance to the system and therefore can be used in situ. In a third experiment, I attempt to resolve a recent debate: Which is more fundamental--complementarity or the uncertainty principle? Specifically, can interference be destroyed by a which-path measurement without transferring momentum to the particle? This question is difficult to investigate as any measurement of the momentum transfer causes collapse, severely disturbing the system. I circumvent this problem with the technique of weak measurement and show that which-path measurements do indeed induce a momentum transfer as required by the uncertainty principle. First, I experimentally demonstrate effective nonlinear absorption between two single photons. I show that this photon-exchange effect can be understood as a post-selection induced nonlinearity similar to the one that occurs in Hong-Ou-Mandel interference. In a fourth experiment, I use the latter theory and return to Hardy's Paradox to perform the first weak measurement of an entangled system. I post-select on the paradoxical measurement result while weakly measuring a set of observables. The results are surprising in that they no longer conflict; they now satisfy a classical-logic truth-table. I then proceed to develop theoretically a new method for performing multiparticle weak measurements that is unique because it does not require single-particle-level interactions. This theory provides new insight into the importance of the imaginary component of the weak measurement result.