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Optics Express

Nonlocal generalized quantum measurement of product observables with mixed entanglement

Lan Jiang, Zhenhua Li, Tao Li

Published Mar 21, 2023• 8 citations
Publisher

Abstract

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.

DOI
10.1364/oe.487883

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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.
Measuring a nonlocal observable on a space-like separated quantum system is a resource-hungry and experimentally challenging task. Several theoretical measurement schemes have already been proposed to increase its feasibility, using a shared maximally-entangled ancilla. We present a new approach to this problem, using the language of generalized quantum measurements, to show that it is actually possible to measure a nonlocal spin product observable without necessarily requiring a maximally-entangled ancilla. This approach opens the door to more economical arbitrary-strength nonlocal measurements, with applications ranging from nonlocal weak values to possible new tests of Bell inequalities. The relation between measurement strength and the amount of ancillary entanglement needed is made explicit, bringing a new perspective on the links that tie quantum nonlocality, entanglement and information transmission together.
Abstract Measuring a nonlocal observable on a space-like separated quantum system is a resource-hungry and experimentally challenging task. Several theoretical measurement schemes have already been proposed to increase its feasibility, using a shared maximally-entangled ancilla. We present a new approach to this problem, using the language of generalized quantum measurements, to show that it is actually possible to measure a nonlocal spin product observable without necessarily requiring a maximally-entangled ancilla. This approach opens the door to more economical arbitrary-strength nonlocal measurements, with applications ranging from nonlocal weak values to possible new tests of Bell inequalities. The relation between measurement strength and the amount of ancillary entanglement needed is made explicit, bringing a new perspective on the links that tie quantum nonlocality, entanglement and information transmission together.
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.
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.
The direct measurement of nonlocal properties of entangled quantum systems has been the subject of several recent experimental investigations. Of particular interest is the implementation of nonlocal measurements via indirect measurement schemes, which allow for greater flexibility in the control of the measurement strength. Building on previous results established in the bipartite case, we present a scheme to implement genuine nonlocal measurements of N-qubit systems with variable strength, using GHZ-like entangled qubit meters. This method can be applied to the joint measurement of commuting product observables, enabling us to distinguish between orthogonal nonlocal states, such as Bell states, with minimal disturbance and arbitrary resolution. An explicit relation between the overall measurement strength and the meter entanglement as quantified by the nnn-tangle is derived, opening the door to a new interpretation of the nnn-tangle as a resource for nonlocal measurements.
Entanglement and the wave function description are two of the core concepts that make quantum mechanics such a unique theory. A method to directly measure the wave function, using weak values, was demonstrated by Lundeen et al. [Nature 474, 188 (2011)]. However, it is not applicable to a scenario of two disjoint systems, where nonlocal entanglement can be a crucial element, since that requires obtaining weak values of nonlocal observables. Here, for the first time, we propose a method to directly measure a nonlocal wave function of a bipartite system, using modular values. The method is experimentally implemented for a photon pair in a hyperentangled state, i.e., entangled both in polarization and momentum degrees of freedom.