Geometric analysis of entangled qubit pairs

Abstract

Abstract. Two entangled electron spins, or qubits, are analysed in terms of ordinary three-dimensional space geometric properties, as are the angles between their angular momenta. This formulation allows concurrence, a measure of quantum entanglement, to be expressed as expectation values of trigonometric functions of the azimuthal angle between the two angular momenta.

Spin-spin correlations of entangled qubit pairs in the Bohm interpretation of quantum mechanics

“…This means that for perfectly entangled states, ϑ = π 2 , all pairs of momenta in the ensemble form an equal angle φ = ϕ. In general, a higher degree of entanglement is directly related to a pronounced unison motion of angular momenta, i.e., with suppressed relative angle fluctuations in agreement with the results of a similar treatment in the framework of standard quantum mechanics [42].…”

“…6(a) as a function of ϑ and for various ϕ. For ϑ = π/2 we find exact relation cos Φ = 1 3 (2 cos ϕ − 1), identical to the standard quantum mechanics expression [42]. As expected, perfect anti-parallel alignment is found for the singlet state (ϕ = π, red line), while momenta for the triplet state, ϕ = 0, are only partially aligned, cos Φ = 1 3 .…”

“…In Fig. 5(b) is with full lines presented the correlator M 1 • M 2 for singlet and triplet-type of qubit pair configuration in comparison with quantum mechanical value given by Ψ|S 1 • S 2 |Ψ = 1 4 (2 sin ϑ cos ϕ − 1) [42]. For unentangled state all correlators are equal to − 1 4 while for ϑ = π/2 the Bohmian correlators for any ϕ exhibit anticipated relation…”

“…In standard quantum mechanics a visualization of a relative motion of two angular momenta of an entangled qubit pair is not given directly, although by appropriately defined angle operators some imagery can be given [42]. On the other hand, in de Broglie-Bohm interpretation of quantum mechanics an analysis of geometrical quantities as are, for example, relative angles between the momenta in the space of hidden variables is simple and directly feasible.…”

Geometrical view of quantum entanglement

“…Note that sin(φ − ϕ) B = 0. Similar formulae can be derived also by appropriately defined cosine and sine operators in standard quantum mechanics formalism [33]. A higher degree of entanglement can thus be visualized as a highly correlated distribution of angular momenta making azimuthal angles difference close to ϕ, with suppressed fluctuations for progressively increasing entanglement.…”