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CRC Press eBooks

Quantum Entanglement Probability Amplitude for n = N = 2

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

The probability amplitude, or state, for quantum entanglement in its most basic form, a photon pair and two channels of propagation ( n = N = 2), is derived in a clear and transparent approach from the Dirac–Feynman N -slit superposition probability amplitude. In other words, from probability amplitudes in quantum interference to probability amplitudes in quantum entanglement, via the | X 〉 = | a 〉 1 , | b 〉 2 , | c 〉 3 …| g 〉 n Dirac identity.

DOI
10.1201/9781003398707-15

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The probability amplitude, or state, for quantum entanglement in its most basic form, a photon pair and two channels of propagation ( n = N = 2), is derived in a clear and transparent approach from the Dirac–Feynman N -slit superposition probability amplitude. In other words, from probability amplitudes in quantum interference to probability amplitudes in quantum entanglement, via the | X 〉 = | a 〉 1 , | b 〉 2 , | c 〉 3 …| g 〉 n Dirac identity.
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Quantum mechanics runs on probability amplitudes. Probability amplitudes must accurately describe the physics prior to calculating the quantum probability. Symbolical probability amplitudes are mathematically represented by complex wave functions. Many physicists in the field don’t even mention the words probability amplitude and prefer to refer to wave function, or state, instead. When they do that they miss out on the subtleties of the physics at hand. In this brief chapter we emphasize probability amplitudes à la Dirac since that is the way Feynman and Ward preferred to describe quantum mechanics. The Dirac notation is also ideally suited to describe quantum interference and quantum entanglement.
The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.