Abstract
A description of quantum entanglement and its applications. Bell’s inequalities and Bell’s theorem are described, along with their implications for local reality and hidden variables. Other topics: applications using electron spin and photon polarization, Aspect’s experiments, decoherence of quantum states.
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Notes
- 1.
The condition \( c_{11} c_{22} = c_{12} c_{21}\) is also sufficient for the vector \(| \varPsi \rangle \) to be a product state, assuming that \(c_{11} + c_{22} \ne 0\). The product state
$$\begin{aligned} \Big ( |A\uparrow \rangle + \frac{c_{21}}{c_{11}} |A\downarrow \rangle \Big ) \otimes \Big (c_{11} |B\uparrow \rangle + c_{12} |B\downarrow \rangle \Big ) = c_{11} |\uparrow \uparrow \rangle \ + \ c_{12} |\uparrow \downarrow \rangle \ + \ c_{21} |\downarrow \uparrow \rangle \ + \ \frac{ c_{21} c_{12} }{c_{11} } | \downarrow \downarrow \rangle \end{aligned}$$coincides with the state (5.6).
- 2.
EP Wigner (1970), Am J Phys 38:1005–1009.
- 3.
These are, in fact, the \(3^\mathrm{rd}\) and \(4^\mathrm{th}\) of Maxwell’s equation in free space. The \(1^\mathrm{st}\) and \(2^\mathrm{nd}\) are \(\nabla \cdot \mathbf {E} = 0 \) and \(\nabla \cdot \mathbf {B} = 0 \).
- 4.
The distance 2L was increased up to 400 m by Weihs et al. (1998) Phys Rev Lett 81:5039–5043. Entangled photons were transmitted over 144 km by Zeilinger et al. (2007) Proc of SPIE 6780:67800B.
- 5.
Aspect (1999) Bell’s inequality test: more ideal than ever. Nature 398:189–190.
- 6.
Scrupulous analysis of these tests of Bell’s theorem identified two possible “loopholes” to their universal validity. The locality loophole arises from the possibility that, if the measurements are too slow, the photon detectors might be communicating with one another in some way (by a yet unknown mechanism). The detection loophole is the possibility that the observed events might represent a skewed sample of all the emitted photon pairs, since detectors are less than 100% efficient. However, in late 2015, three research groups (at Delft University of Technology, University of Vienna and NIST) have succeeded in carrying out “loophole-free” Bell tests.
- 7.
I denotes the identity in the Hilbert space of the measuring apparatus.
- 8.
Taking the expectation values on \( | E \uparrow \rangle \) and \(| E \downarrow \rangle \) and summing the results.
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Fano, G., Blinder, S.M. (2017). Quantum Entanglement and Bell’s Theorem. In: Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-58732-5_5
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DOI: https://doi.org/10.1007/978-3-319-58732-5_5
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