This chapter is based on a talk given at the Science and Ultimate Reality meeting in March, 2002, in honour of John Archibald Wheeler. In it, I discuss some questions related to what can and cannot be said about the history of a quantum mechanical system. Relying heavily on the weak- measurement formalism of Aharonov and coworkers, I argue that there is much to be learned about a system based both on its preparation and on subsequent postselection. This is illustrated with examples from a number of past, present, and future experiments from our lab, ranging from tests of quantum "paradoxes" to studies of nonlocality to non-deterministic implementations of logic operations on quantum information. The connection between weak measurements and generalized probability theory is discussed, along with some of the counterintuitive features of these "probabilities." Conclusions are for the most part left to the reader.
This chapter is based on a talk given at the Science and Ultimate Reality meeting in March, 2002, in honour of John Archibald Wheeler. In it, I discuss some questions related to what can and cannot be said about the history of a quantum mechanical system. Relying heavily on the weak- measurement formalism of Aharonov and coworkers, I argue that there is much to be learned about a system based both on its preparation and on subsequent postselection. This is illustrated with examples from a number of past, present, and future experiments from our lab, ranging from tests of quantum "paradoxes" to studies of nonlocality to non-deterministic implementations of logic operations on quantum information. The connection between weak measurements and generalized probability theory is discussed, along with some of the counterintuitive features of these "probabilities." Conclusions are for the most part left to the reader.
Introduction A volume in honor of a visionary thinker such as John Archibald Wheeler is a rare license to exercise in the kind of speculation and exploration for which Wheeler is famous, but which most of the rest of us usually feel we had better keep to ourselves. We have all – even those of us who never had the fortune to work directly with him – been inspired and motivated by Wheeler's creativity and open-mindedness. For all of our apparent understanding of quantum mechanics, our ability to calculate remarkable things using this theory, and the regularity with which experiment has borne out these predictions, at the turn of the twenty-first century it seems there are as many puzzles on the road to a true understanding of quantum theory as there were at the start of the previous century. Then, at least, one could hope to be guided by the mysteries of unexplained experiment. Now, by contrast, we may seem to have lost our way, as even though our experiments are all “explained” (in some narrow sense which can only be deemed satisfactory out of fear to leap beyond the comfortable realm of formalism), the theory itself is mysterious. Further explorations, without the anchor of experiment, certainly run the risk of becoming mere flights of metaphysical fancy, giving rise to factions characterized less by intellectual rigor than by fundamentalist zeal. Yet it would be premature to give up the journey before at least trying to establish a foothold on the terrain ahead.
Abstract Quantum mechanics, in its orthodox version, imposes severe limits on what can be known, or even said, about the condition of a quantum system between two observations. A relatively new approach, based on so‐called “weak measurements”, suggests that such forbidden knowledge can be gained by studying the system's response to an inaccurate weakly perturbing measuring device. It goes further to propose revising the whole concept of physics variables, and offers various examples of counterintuitive quantum behavior. Both views go to the very heart of quantum theory, and yet are rarely compared directly. A new technique must either transcend the orthodox limits, or just prove that these limits are indeed necessary. Both possibilities are studied and orthodoxy is vindicated.
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].
A history of the discovery of `new' quantum mechanics and the paradoxes of its probabilistic interpretation are briefly reviewed from the modern point of view of quantum probability and information. Modern quantum theory, which has been developed during the last 20 years for the treatment of quantum open systems including quantum noise, decoherence, quantum diffusions and spontaneous jumps occurring under continuous in time observation, is not yet a part of the standard curriculum of quantum physics. It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements, but they have all received an adequate mathematical treatment in quantum stochastics of open systems. Moreover, the only reasonable probabilistic interpretation of quantum mechanics put forward by Max Born was, in fact, in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrödinger. They are reconsidered in this paper from the modern point of view of quantum stochastics and information. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or Bell's be ables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the actual histories with the potential future. This rule, together with the self-compatibility of the measurements ensuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a dynamical solution, in the form of the quantum stochastic filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. It has been recently proved that the quantum stochastic model for the continuous in time measurements is equivalent to a Dirac type boundary-value problem for the secondary quantized input `offer waves from future' in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the `trajectories of the output particles'. This supports the corpuscular-wave duality in the form of the thesis that everything in the future are quantized waves, while everything in the past are trajectories of the recorded particles.
The quantum theory of measurement has been with us since quantum mechanics was invented. It has recently been invigorated, partly due to the increasing interest in quantum information science. In this partly pedagogical review I attempt to give a self-contained overview of non-relativistic quantum theory of measurement expressed in density matrix formalism. I will not dwell on the applications in quantum information theory; it is well covered by several books in that field. The focus is instead on applications to the theory of weak measurement, as developed by Aharonov and collaborators. Their development of weak measurement combined with what they call post-selection - judiciously choosing not only the initial state of a system (pre-selection) but also its final state - has received much attention recently. Not the least has it opened up new, fruitful experimental vistas, like novel approaches to amplification. But the approach has also attached to it some air of mystery. I will attempt to demystify it by showing that (almost) all results can be derived in a straight-forward way from conventional quantum mechanics. Among other things, I develop the formalism not only to first order but also to second order in the weak interaction responsible for the measurement. I apply it to the so called Leggett-Garg inequalities, also known as Bell inequalities in time. I also give an outline, even if rough, of some of the ingenious experiments that the work by Aharonov and collaborators has inspired. As an application of weak measurement, not related to the approach by Aharonov and collaborators, the formalism also allows me to derive the master equation for the density matrix of an open system in interaction with an environment. An issue that remains in the weak measurement plus post-selection approach is the interpretation of the so called weak value of an observable. Is it a bona fide property of the system considered? I have no definite answer to this question; I shall only exhibit the consequences of the proposed interpretation. Quanta 2013; 2: 18–49.
This essay highlights the scientific vision of John Archibald Wheeler, a giant of 20th century physics. His ideas provided the important insight that humanity may be in the very center of disclosure and manifestation of the evolution of our universe. This on the basis of scientific reasoning derived from quantum physics rather than simple anthropocentrism or scale chauvinism. Wheeler, the Joseph Henry Professor of Physics Emeritus at Princeton University, became 96. Over a long, productive scientific life, he was known for his drive to address big, overarching questions in physics, subjects which he liked to say merged with philosophical questions about the origin of matter, information and the universe. His interests ranged far and wide and were characterized by true fearlessness. Wheeler's work was not only in gravity and nuclear physics. In the 1950s Wheeler grew increasingly intrigued by the philosophical implications of quantum physics. According to Wheeler, there was no universe until the rise of consciousness to perceive it. In fact, Wheeler was one of the first prominent physicists seriously to propose that reality might not be a wholly physical phenomenon. In some sense, Wheeler suggested, reality grows out of the act of observation, and thus consciousness itself: it is participatory. He also stated that information is the most fundamental building block of reality, and that the universe should be seen as a self-synthesized information system: a self-excited circuit that is developing through a (closed loop) cycle. His cosmic variant of the delayed choice experiment led to the idea that human observers may not only determine the present, but also may influence the past. According to Wheeler, ultimate mutability is the central feature of physics, and the meaning of reality can only be established if there is a universal knowledge field, that transcend physical past, present and future.