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Journal of Modern Optics

Beyond the heisenberg uncertainty

Yanhua H. Shih

Published Nov 1, 2004• 0 citations
Publisher

Abstract

An entangled EPR two-particle system is characterized by Δ(p 1 + p 2) = 0 and Δ(x 1 - x 2) = 0, simultaneously, with the momentum and position of each individual particle completely undefined. Statistically, however, the uncertainty of the correlation measurements on the two individual particles must obey Δ(p1 + p 2) > Δp 1,2 and Δ(x 1 - x 2) > Δx 1,2. While one sees the measurements of p 1 + p 2 and x 1 - x 2 of the two individual particles satisfying the EPR δ-function, and believes the classical statistical inequality, one might easily be trapped into concluding either there is a violation of the uncertainty principle or there exists action-at-a-distance. Through the analysis of three typical EPR experiments, we show that the measurement of p 1 + p 2 and x 1 - x 2 for an entangled system can indeed violate the classical statistical inequality. Based on the experimental results, we emphasize, again, the surprising physics of quantum coherent two-particle superposition. Although questions regarding fundamental issues of quantum theory still exist, quantum entanglement has brought us a novel concept in non-local timing and positioning measurements with ultra-high accuracy, even beyond the classical limit.

DOI
10.1080/09500340408231796

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An entangled EPR two-particle system is characterized by Δ(p 1 + p 2) = 0 and Δ(x 1 - x 2) = 0, simultaneously, with the momentum and position of each individual particle completely undefined. Statistically, however, the uncertainty of the correlation measurements on the two individual particles must obey Δ(p1 + p 2) > Δp 1,2 and Δ(x 1 - x 2) > Δx 1,2. While one sees the measurements of p 1 + p 2 and x 1 - x 2 of the two individual particles satisfying the EPR δ-function, and believes the classical statistical inequality, one might easily be trapped into concluding either there is a violation of the uncertainty principle or there exists action-at-a-distance. Through the analysis of three typical EPR experiments, we show that the measurement of p 1 + p 2 and x 1 - x 2 for an entangled system can indeed violate the classical statistical inequality. Based on the experimental results, we emphasize, again, the surprising physics of quantum coherent two-particle superposition. Although questions regarding fundamental issues of quantum theory still exist, quantum entanglement has brought us a novel concept in non-local timing and positioning measurements with ultra-high accuracy, even beyond the classical limit.
Abstract Entangled EPR two‐particle system is characterized by Δ(p1+p2) = 0 and Δ (x1‐ x2) = 0, simultaneous. Even the momentum and position of each individual particle are completely undefined: Δ pj ∼ ∞, Δ xj ∼∞. This remarkable feature does not violate the uncertainty principle of quantum theory. It is the result of the surprising two‐particle quantum coherent superposition. Although questions regarding fundamental issues of quantum theory still exist, quantum entanglement has brought about a novel concept in nonlocal timing and positioning measurements with ultra‐high accuracy, even beyond the classical limit.
Following Einstein, Podolsky, and Rosen (EPR) original argument, we derive a pair of inequalities for the uncertainties in the sum of the momenta and in the difference of the positions of an entangled two-particle system. We prove that only entangled systems can satisfy both EPR-inequalities, in violation of the limits imposed by classical statistics. We propose an operational approach that allows applying the EPR inequalities to two-photon systems, as well. In particular, we demonstrate a scheme that allows implementing the EPR gedanken-experiment and verifying the EPR inequalities on systems of pairs of photons. We report the experimental results obtained in this kind of quantum interference and imaging experiment: SPDC two-photon system satisfies both EPR inequalities. The experimental verification of the EPR inequalities represents a practical way to experimentally distinguish entanglement from classical correlation in momentum and/or position variables for systems of two particles/photons. We emphasize the practical consequences of the EPR inequalities: only entanglement allows one to go beyond the limitations imposed by Heisenberg uncertainty on systems of classically correlated particles/photons. In this context we review a recent experiment of two-photon diffraction and quantum lithography.
Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.
EPR showed that two particles emitted from a source can be entangled by a shared wavefunction where two non-commuting observables (position, momentum) can be simultaneously real, leading to a contradiction with quantum mechanics (two non-commuting variables can not be simultaneously real). John Bell derived an inequality where any local hidden variables prediction is bounded and quantum mechanics can violate the inequality. Bell tests on correlated photon pairs showed a clear violation of the Bell inequality and agreement with quantum mechanics. This study revealed that the Bell inequality holds at any individual instances in a given system but it does not hold over the entire population. The Bell inequality is not an incompatibility criterion for local hidden variables vs quantum mechanics rather it is a criterion for an individual nature vs population dynamics. The nonlocality prediction given by wavefunction is a system approximation and local causality (photon polarization) provides a complete description of the Bell tests. The paper concludes with great satisfaction that the measurement of two correlated photons at a distance was a local cause (photon-filter interaction did not violate the special theory of relativity).
One of the most surprising consequences of quantum mechanics is the entanglement of two or more distant particles. In an entangled EPR two-particle system, the value of the momentum (position) for neither single subsystem is determined. However, if one of the subsystems is measured to have a certain momentum (position), the other subsystem is determined to have a unique corresponding value, despite the distance between them. This peculiar behavior of an entangled quantum system has surprisingly been observed experimentally in two-photon temporal and spatial correlation measurements, such as ghost interference and ghost imaging. This article addresses the fundamental concerns behind these experimental observations and to explore the nonclassical nature of two-photon superposition by emphasizing the physics of 2 is not 1 + 1.