Eliahu Cohen

Measuring Incompatible Observables by Exploiting Sequential Weak Values

F. Piacentini, M. P. Levi, A. Avella, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, M. Gramegna, G. Brida, I. P. Degiovanni, M. Genovese INRIM, Strada delle Cacce 91, I-10135 Torino, Italy Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129Torino, Italy School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Piazza Leonardo da Vinci 32, 20133 Milano, Italy and INFN, Via P. Giuria 1, I-10125 Torino, Italy

Interaction-Free Effects Between Distant Atoms

A Gedanken experiment is presented where an excited and a ground-state atom are positioned such that, within the former’s half-life time, they exchange a photon with 50% probability. A measurement of their energy state will therefore indicate in 50% of the cases that no photon was exchanged. Yet other measurements would reveal that, by the mere possibility of exchange, the two atoms have become entangled. Consequently, the “no exchange” result, apparently precluding entanglement, is non-locally established between the atoms by this very entanglement. This quantum-mechanical version of the ancient Liar Paradox can be realized with already existing transmission schemes, with the addition of Bell’s theorem applied to the no-exchange cases. Under appropriate probabilities, the initially-excited atom, still excited, can be entangled with additional atoms time and again, or alternatively, exert multipartite nonlocal correlations in an interaction free manner. When densely repeated several times, this result also gives rise to the Quantum Zeno effect, again exerted between distant atoms without photon exchange. We discuss these experiments as variants of interaction-free-measurement, now generalized for both spatial and temporal uncertainties. We next employ weak measurements for elucidating the paradox. Interpretational issues are discussed in the conclusion, and a resolution is offered within the Two-State Vector Formalism and its new Heisenberg framework.

Finally making sense of the double-slit experiment

Significance We put forth a time-symmetric interpretation of quantum mechanics that does not stem from the wave properties of the particle. Rather, it posits corpuscular properties along with nonlocal properties, all of which are deterministic. This change of perspective points to deterministic properties in the Heisenberg picture as primitive instead of the wave function, which remains an ensemble property. This way, within a double-slit experiment, the particle goes only through one of the slits. In addition, a nonlocal property originating from the other distant slit has been affected through the Heisenberg equations of motion. Under the assumption of nonlocality, uncertainty turns out to be crucial to preserve causality. Hence, a (qualitative) uncertainty principle can be derived rather than assumed. Feynman stated that the double-slit experiment “…has in it the heart of quantum mechanics. In reality, it contains the only mystery” and that “nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it” [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures on Physics]. We rise to the challenge with an alternative to the wave function-centered interpretations: instead of a quantum wave passing through both slits, we have a localized particle with nonlocal interactions with the other slit. Key to this explanation is dynamical nonlocality, which naturally appears in the Heisenberg picture as nonlocal equations of motion. This insight led us to develop an approach to quantum mechanics which relies on pre- and postselection, weak measurements, deterministic, and modular variables. We consider those properties of a single particle that are deterministic to be primal. The Heisenberg picture allows us to specify the most complete enumeration of such deterministic properties in contrast to the Schrödinger wave function, which remains an ensemble property. We exercise this approach by analyzing a version of the double-slit experiment augmented with postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric interpretation, where interference appears even when the particle is localized. Although the Heisenberg and Schrödinger pictures are equivalent formulations, nevertheless, the framework presented here has led to insights, intuitions, and experiments that were missed from the old perspective.

Nonlocality of the Aharonov-Bohm effect

Although the Aharonov-Bohm and related effects are familiar in solid-state and high-energy physics, the nonlocality of these effects has been questioned. Here we show that the Aharonov-Bohm effect has two very different aspects. One aspect is instantaneous and nonlocal; the other aspect, which depends on entanglement, unfolds continuously over time. While local, gauge-invariant variables may occasionally suffice for explaining the continuous aspect, we argue that they cannot explain the instantaneous aspect. Thus the Aharonov-Bohm effect is, in general, nonlocal.

1-1=Counterfactual: on the potency and significance of quantum non-events.

We study the unique role played in quantum mechanics by non-events or ‘counterfactuals’. Our earlier analysis of ‘quantum oblivion’ has revealed some subtle stages in the measurement process, which may end up in self-cancellation. To these findings, we now add two insights derived by two time-symmetric interpretations of quantum mechanics. (i) Like all quantum interactions, the non-event is formed by the conjunction of forward-plus-backward-evolving wave functions. (ii) Then, it is another feature of such dual evolutions, namely the involvement of negative masses and energies, that enables Nature to make some events ‘unhappen’ while leaving causal traces.

On entropy production in the Madelung fluid and the role of Bohm's potential in classical diffusion

The Madelung equations map the non-relativistic time-dependent Schrödinger equation into hydrodynamic equations of a virtual fluid. While the von Neumann entropy remains constant, we demonstrate that an increase of the Shannon entropy, associated with this Madelung fluid, is proportional to the expectation value of its velocity divergence. Hence, the Shannon entropy may grow (or decrease) due to an expansion (or compression) of the Madelung fluid. These effects result from the interference between solutions of the Schrödinger equation. Growth of the Shannon entropy due to expansion is common in diffusive processes. However, in the latter the process is irreversible while the processes in the Madelung fluid are always reversible. The relations between interference, compressibility and variation of the Shannon entropy are then examined in several simple examples. Furthermore, we demonstrate that for classical diffusive processes, the “force” accelerating diffusion has the form of the positive gradient of the quantum Bohm potential. Expressing then the diffusion coefficient in terms of the Planck constant reveals the lower bound given by the Heisenberg uncertainty principle in terms of the product between the gas mean free path and the Brownian momentum.

Past of a particle in an entangled state

Vaidman has proposed a controversial criterion for determining the past of a single quantum particle based on the weak trace it leaves. We here consider more general examples of entangled systems and analyze the past of single, as well as pairs of entangled pre- and postselected particles. Systems with non-trivial time evolution are also analyzed. We argue that in these cases, examining only the single-particle weak trace provides information which is insufficient for understanding the system as a whole. We therefore suggest to examine, alongside with the past of single particles, also the past of pairs, triplets and eventually the entire system, including higher-order, multipartite traces in the analysis. This resonates with a recently proposed top-down approach by Aharonov, Cohen and Tollaksen for understanding the structure of correlations in pre- and postselected systems.

What Weak Measurements and Weak Values Really Mean: Reply to Kastner

Despite their important applications in metrology and in spite of numerous experimental demonstrations, weak measurements are still confusing for part of the community. This sometimes leads to unjustified criticism. Recent papers have experimentally clarified the meaning and practical significance of weak measurements, yet in Kastner (Found Phys 47:697–707, 2017), Kastner seems to take us many years backwards in the the debate, casting doubt on the very term “weak value” and the meaning of weak measurements. Kastner appears to ignore both the basics and frontiers of weak measurements and misinterprets the weak measurement process and its outcomes. In addition, she accuses the authors of Aharonov et al. (Ann Phys 355:258–268, 2015) in statements completely opposite to the ones they have actually made. There are many points of disagreement between Kastner and us, but in this short reply I will leave aside the ontology (which is indeed interpretational and far more complex than that described by Kastner) and focus mainly on the injustice in her criticism. I shall add some general comments regarding the broader theory of weak measurements and the two-state-vector formalism, as well as supporting experimental results. Finally, I will point out some recent promising results, which can be proven by (strong) projective measurements, without the need of employing weak measurements.

A scheme for performing strong and weak sequential measurements of non-commuting observables

Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state. In general, the possible paths will depend on the future and past boundary conditions, as well as the system’s dynamics. We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously, each with probability one, depending on which measurement was performed. This experiment involves the measurement of observables that do not correspond to Hermitian operators. Our main result is a scheme for measuring these operators. The scheme is based on the erasure protocol [Brodutch and Cohen (Phys. Rev. Lett. 116:070404, 2016)] and allows a wide range of sequential measurements at both the weak and strong limits. At the weak limit the back action of the measurement cannot be used to account for the surprising behaviour and the resulting weak values provide a consistent yet strange account of the system’s past.

Unveiling the curtain of superposition: Recent gedanken and laboratory experiments

What is the true meaning of quantum superposition? Can a particle genuinely reside in several places simultaneously? These questions lie at the heart of this paper which presents an updated survey of some important stages in the evolution of the three-boxes paradox, as well as novel conclusions drawn from it. We begin with the original thought experiment of Aharonov and Vaidman, and proceed to its non-counterfactual version. The latter was recently realized by Okamoto and Takeuchi using a quantum router. We then outline a dynamic version of this experiment, where a particle is shown to disappear and re-appear during the time evolution of the system. This surprising prediction based on self-cancellation of weak values is directly related to our notion of Quantum Oblivion. Finally, we present the non-counterfactual version of this disappearing-reappearing experiment. Within the near future, this last version of the experiment is likely to be realized in the lab, proving the existence of exotic hitherto unknown forms of superposition. With the aid of Bells theorem, we prove the inherent nonlocality and nontemporality underlying such pre- and post-selected systems, rendering anomalous weak values ontologically real.