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CRC Press eBooks

Quantum Entanglement Probability Amplitude for n = N = 21, 22, 23,…, 2 r

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

The generalized probability amplitude, or state, for quantum entanglement, for n = N = 2 1 , 2 2 , 2 3 , … 2 r 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 the generalized probability amplitudes in quantum entanglement, via the | X 〉 = | a 〉 1 , | b 〉 2 , | c 〉 3 …| g 〉 n Dirac identity.

DOI
10.1201/9781003398707-16

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The generalized probability amplitude, or state, for quantum entanglement, for n = N = 2 1 , 2 2 , 2 3 , … 2 r 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 the generalized probability amplitudes in quantum entanglement, via the | X 〉 = | a 〉 1 , | b 〉 2 , | c 〉 3 …| g 〉 n Dirac identity.
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.
Twelve mathematical statements that define the quantum mechanics of interference and entanglement are listed in the chapter. The importance of superposition probability amplitudes, via the Dirac–Feynman principle, in quantum mechanics is highlighted. Particular attention is given to the proper description of experimental configurations via probability amplitudes. The refinement of spatial accuracy, only limited by Nature’s availability of shorter and shorter wavelengths, in N -slit quantum interferometers is described. The transition from superposition probability amplitudes to quantum probabilities is discussed. In all this, the essential importance of the nonlocality of the photon is emphasized. The importance of the probability amplitude of quantum entanglement at the foundations of quantum physics is discussed relative to quantum interference. The intrinsic link between these two principles is highlighted via a series of transparent mathematical statements.
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.