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arXiv:1908.04515 · Optica

Experimental nonlocal measurement of a product observable

Yuan Li, Han-Sen Zhong, Yi-Han Luo, Li-Chao Peng, Chao-Yang Lu, Nai-Le Liu, Jun Zhang, Li Li, Jian-Wei Pan

Published Sep 10, 2019• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1364/optica.6.001199

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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.
Due to the restrictions on quantum measurement imposed by relativistic causality, it is not easy to perform a nonlocal measurement on spacelike separated systems instantaneously. Here we design an experiment in an optical system to perform the nonlocal measurement of a product observable on a two-qubit system, where a maximally-entangled two-qubit state, acting as an ancillary meter, and three identical Kerr media are used to couple the system under consideration and the target meter. It is shown that the nonlocal measurement of the product observable, σzA⊗σzB, which has two degenerate eigenvalues, could be implemented in a deterministic way without violating relativistic causality.
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.
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.
Abstract Relativity theory severely restricts the ability to perform nonlocal measurements in quantum mechanics. Studying such nonlocal schemes may thus reveal insights regarding the relations between these two fundamental theories. Therefore, for the last several decades, nonlocal measurements have stimulated considerable interest. However, the experimental implementation of nonlocal measurements imposes profound restrictions because the interaction Hamiltonian cannot contain, in general, nonlocal observables such as the product of local observables belonging to different particles at spacelike‐separated regions. In this work, we experimentally realize a scheme for nonlocal measurements with the aid of probabilistic quantum erasure. We apply this scheme to the tasks of performing high‐accuracy nonlocal measurements of the parity, as well as measurements in the Bell basis, which do not necessitate classical communication between the parties. Unlike other techniques, the nonlocal measurement outcomes are available locally (upon successful postselection). The state reconstructed via performing quantum tomography on the system after the nonlocal measurement indicates the success of the scheme in retrieving nonlocal information while erasing any local data previously acquired by the parties. This measurement scheme allows to realize any controlled‐controlled‐gate with any coupling strength. Hence, our results are expected to have conceptual and practical applications to quantum communication and quantum computation.
Nonlocal observables of spacelike separated quantum systems in combination with their measurements contribute greatly to quantum theory and its applications. We present a nonlocal generalized quantum measurement protocol for measuring product observables, assisted by a meter in a mixed entangled state rather than maximally or partially entangled pure states. By tuning the entanglement of the meter, measurement strength of arbitrary values can be achieved for nonlocal product observables, since measurement strength equals the concurrence of the meter. Furthermore, we present a specific scheme to measure the polarization of two nonlocal photons using linear optics. We refer to the polarization and spatial-mode degrees of freedom of the same photon pair as the system and the meter, respectively, which significantly simplifies the interaction between the system and the meter. This protocol can be useful for applications involving nonlocal product observables and nonlocal weak values, and for tests of quantum foundations in nonlocal scenarios.
Relativistic causality has dramatic consequences on the measurability of nonlocal variables and poses the fundamental question of whether it is physically meaningful to speak about the value of nonlocal variables at a particular time. Recent work has shown that by weakening the role of the measurement in preparing eigenstates of the variable, it is in fact possible to measure all nonlocal observables instantaneously by exploiting entanglement. However, for these measurement schemes to succeed with certainty, an infinite amount of entanglement must be distributed initially and all this entanglement is necessarily consumed. In this work, we sharpen the characterization of instantaneous nonlocal measurements by explicitly devising schemes in which only a finite amount of the initially distributed entanglement is ever utilized. This enables us to determine an upper bound to the average consumption for the most general cases of nonlocal measurements. This includes the tasks of state verification, where the measurement verifies if the system is in a given state, and verification measurements of a general set of eigenstates of an observable. Despite its finiteness, the growth of entanglement consumption is found to display an extremely unfavourable exponential of an exponential scaling with either the number of qubits needed to contain the Schmidt rank of the target state or the total number of qubits in the system for an operator measurement. This scaling is seen to be a consequence of the combination of the generic exponential scaling of unitary decompositions combined with the highly recursive structure of our scheme required to overcome the no-signalling constraint of relativistic causality.