Abstract
Entanglement describes a situation in which two entities share a common attribute. The Einstein-Podolsky-Rosen (EPR) paradox concerns the behavior of entangled states. Their 1935 publication in Physical Review considered the measurements of properties on two systems which are first allowed to interact, and therefore become entangled, and which are subsequently separated so that they are no longer interacting with each other. In proposing this condition, EPR supposed locality. Bell’s theorem analyzes the outcomes of measurements at two different locations, A and B under the assumption of locality. Bell’s theorem allows for the presence of a “local hidden variable” that accounts for some or all of the apparent statistical nature of the measurements at locations A or B. The analysis places upper limits on the correlations that can be measured between results recorded at locations A and B, regardless of the nature or form of the hidden variable. For more than 2 decades, careful measurements of entangled photon pairs show significant violations of Bell’s limit. This result is widely accepted as demonstration that measurement of an entangled quantum does not depend entirely on local conditions where the measurement is made. The reality of the non-local behavior of entangled photon pairs is a foundation stone for quantum cryptography. This technology of ultra-secure communication is based on the distribution to authorized receivers of a quantum key that enables reading of encrypted messages. The quantum key is determined by the detection of photons correlated by entanglement at two different locations.
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Notes
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For example, in 2015, a group led by R. Hensen at the Delft University of Technology achieved an elegant demonstration using a combination of entangled electron spins and photon polarisations (2015). This is an area of active research and discovery.
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This section is adapted from, “The Naive View of an Experimentalist” by Aspect (2002).
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Excerpt from the interview on BBC radio, conducted by Paul Davies. A transcript can be found in the book Davies and Brown (1993).
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Exercises
Exercises
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6.1
Single photon detection can be achieved using a photomultiplier or a silicon avalanche photodiode, both of which work best for visible and near-ir photons. The quantum efficiency \( \eta \) of these detectors is always less than unity.
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(a)
Show that the expression for the Bell function is modified to:
$$ 2 - \frac{4}{\eta } \le E\left( {{\mathbf{a}},{\mathbf{b}}} \right) - E\left( {{\mathbf{a}},{\mathbf{c}}} \right) + E\left( {{\mathbf{d}},{\mathbf{c}}} \right) + E\left( {{\mathbf{d}},{\mathbf{b}}} \right) \le \frac{4}{\eta } - 2 $$ -
(b)
Presuming that an experiment is carried out using entangled photons with polarizers oriented at the optimum settings of \( \frac{\pi }{8},\frac{\pi }{8},\frac{\pi }{8},\frac{3\pi }{8} \). At what level of quantum efficiency will it no longer be possible to detect a violation of Bell’s inequality?
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(a)
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Pearsall, T.P. (2017). Entanglement and Non-locality of Quantum Photonics. In: Quantum Photonics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55144-9_6
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