We report the first implementation of the von Neumann instantaneous measurements of nonlocal variables, which becomes possible due to technological achievements in creating hyperentangled photons. Tests of reliability and of the nondemolition property of the measurements have been performed with high precision, showing the suitability of the scheme as a basic ingredient of numerous quantum information protocols. The method allows us to demonstrate for the first time with strong measurements a special feature of pre- and postselected quantum systems: the failure of the product rule. It has been verified experimentally that for a particular pre- and postselected pair of particles, a single measurement on particle A yields with certainty σ_{x}^{A}=-1, a single measurement on particle B yields with certainty σ_{y}^{B}=-1, and a single nonlocal measurement on particles A and B yields with certainty σ_{x}^{A}σ_{y}^{B}=-1.
We report the first implementation of the von Neumann instantaneous measurements of nonlocal variables, which becomes possible due to technological achievements in creating hyperentangled photons. Tests of reliability and of the nondemolition property of the measurements have been performed with high precision, showing the suitability of the scheme as a basic ingredient of numerous quantum information protocols. The method allows us to demonstrate for the first time with strong measurements a special feature of pre- and postselected quantum systems: the failure of the product rule. It has been verified experimentally that for a particular pre- and postselected pair of particles, a single measurement on particle A yields with certainty σ_{x}^{A}=-1, a single measurement on particle B yields with certainty σ_{y}^{B}=-1, and a single nonlocal measurement on particles A and B yields with certainty σ_{x}^{A}σ_{y}^{B}=-1.
Nonlocal observables play an important role in quantum theory, from Bell inequalities and various postselection paradoxes to quantum error correction codes. Instantaneous measurement of these observables is known to be a difficult problem, especially when the measurements are projective. The standard von Neumann Hamiltonian used to model projective measurements cannot be implemented directly in a nonlocal scenario and can, in some cases, violate causality. We present a scheme for effectively generating the von Neumann Hamiltonian for nonlocal observables without the need to communicate and adapt. The protocol can be used to perform weak and strong (projective) measurements, as well as measurements at any intermediate strength. It can also be used in practical situations beyond nonlocal measurements. We show how the protocol can be used to probe a version of Hardy's paradox with both weak and strong measurements. The outcomes of these measurements provide a nonintuitive picture of the pre- and postselected system. Our results shed new light on the interplay between quantum measurements, uncertainty, nonlocality, causality, and determinism.
Weak measurement is a new technique which allows one to describe the evolution of postselected quantum systems. It appears to be useful for resolving a variety of thorny quantum paradoxes, particularly when used to study properties of pairs of particles. Unfortunately, such nonlocal or joint observables often prove difficult to measure directly in practice (for instance, in optics-a common testing ground for this technique-strong photon-photon interactions would be needed to implement an appropriate von Neumann interaction). Here we derive a general, experimentally feasible, method for extracting these joint weak values from correlations between single-particle observables.
Nonlocal measurement, or instantaneous measurement of nonlocal observables, is a considerably difficult task even for a simple form of product observable since relativistic causality prohibits interaction between spacelike separate subsystems. Following a recent proposal for effectively creating the von Neumann measurement Hamiltonian of nonlocal observables [Brodutch and Cohen, Phys. Rev. Lett. 116, 070404 (2016)], here we report a proof-of-principle demonstration of nonlocally measuring a product observable using linear optics without the violation of relativistic causality. Our scheme provides a feasible approach to perform nonlocal measurements via quantum erasure with linear optics.
We experimentally demonstrate an alternative method for measuring nonlocal weak values in linear optics, avoiding the use of second-order interaction. The method is based on the concept of modular values. The paths of two photons, initialized in hyperentangled states, are adopted as the meter with the polarization acting as the system. The modular values are read out through the reconstructed final states of the meter. The weak value of nonlocal observables is given through its connection to the modular value. Comparing the weak values of local and nonlocal observables, we demonstrate the failure of product rules for an entangled system. Our results significantly simplify the task of measuring nonlocal weak values and will play an important role in the application of weak measurement.
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].
This article reviews six historically important papers in the development of the theory of measurement for nonlocal variables in quantum mechanics, with special emphasis the non violation of relativistic causality. Spanning more than seventy years, we chart the major developments in the field from the declaration, by Landau and Peierls in 1931, that measurement of nonlocal variables was impossible in the relativistic regime to the demonstration, by Vaidman in 2003, that all such variables \emph{can} be measured instantaneously without violation of causality through an appropriate act of "measurement", albeit not of a standard projective (Von Neumann) type.