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arXiv:1109.5537 · Europhysics Letters (EPL)

Geometrical view of quantum entanglement

A. Ramšak

Published Nov 1, 2011• 6 citations
arXivHTMLPDF

Abstract

Although a precise description of microscopic physical problems requires a full quantum-mechanical treatment, physical quantities are generally discussed in terms of classical variables. One exception is quantum entanglement which apparently has no classical counterpart. We demonstrate here how quantum entanglement may be within the de Broglie-Bohm interpretation of quantum mechanics visualized in geometrical terms, giving new insight into this mysterious phenomenon and a language to describe it. On the basis of our analysis of the dynamics of a pair of qubits, quantum entanglement is linked to concurrent motion of angular momenta in the Bohmian space of hidden variables and to the average angle between these momenta.

DOI
10.1209/0295-5075/96/40004

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Although a precise description of microscopic physical problems requires a full quantum-mechanical treatment, physical quantities are generally discussed in terms of classical variables. One exception is quantum entanglement which apparently has no classical counterpart. We demonstrate here how quantum entanglement may be within the de Broglie-Bohm interpretation of quantum mechanics visualized in geometrical terms, giving new insight into this mysterious phenomenon and a language to describe it. On the basis of our analysis of the dynamics of a pair of qubits, quantum entanglement is linked to concurrent motion of angular momenta in the Bohmian space of hidden variables and to the average angle between these momenta.
A general entangled qubit pair is analyzed in the de Broglie-Bohm formalism corresponding to two spin-1/2 quantum rotors. Several spin-spin correlators of Bohm's hidden variables are analyzed numerically and a detailed comparison with results obtained by standard quantum mechanics is outlined. In addition to various expectation values the Bohm interpretation allows also a study of the corresponding probability distributions, which enables a novel understanding of entangled qubit dynamics. In particular, it is shown how the angular momenta of two qubits in this formalism can be viewed geometrically and characterized by their relative angles. For perfectly entangled pairs, for example, a compelling picture is given, where the qubits exhibit a unison precession making a constant angle between their angular momenta. It is also demonstrated that the properties of standard quantum mechanical spin-spin correlators responsible for the violation of Bell's inequalities are identical to their counterparts emerging from the probability distributions obtained by the Bohmian approach.
The possibility that quantum mechanics is foundationally the same as classical theories in explaining phenomena in space and time is postulated. Such a view is motivated by interpreting the experimental violation of Bell inequalities as resulting from questions of geometry and algebraic representation of variables, and thereby the structure of space, rather than realism or locality. While time remains Euclidean in the proposed new structure, space is described by Projective geometry. A dual geometry facilitates description of a physically real quantum particle trajectory. Implications for the physical basis of Bohmian mechanics is briefly examined, and found that the hidden variables pilot-wave model is local. Conceptually, the consequence of this proposal is that quantum mechanics has common ground with relativity as ultimately geometrical. This permits the derivation of physically meaningful quantum Lorentz transformations. Departure from classical notions of measurability is discussed.
The cardinal differences in the theoretical description of classical and quantum particles, the contradictions of the probabilistic (Copenhagen School) and deterministic (De Broglie - Bohm theory) interpretation of quantum mechanics, and the features of entangled quantum states are briefly discussed. It is assumed that the strange properties of the entangled pair can be explained by the localization of particles not in 3D physical space, but in the domain of a fractal space arising in the process of purely mathematical derivation of the Schrodinger and Pauli equations.
Quantum entanglement, a cornerstone of quantum mechanics, presents a profound challenge to classical intuition regarding information and space. This paper explores the potential of entanglement as a fundamental geometric entity, proposing a novel framework centered around "纠缠几何" – a model that leverages the non-linearity inherent in entangled states to uncover latent geometric structures within quantum information. We aim to demonstrate how entanglement, when treated as a geometric variable, can lead to unexpected and potentially transformative insights into quantum information storage and transmission. This research seeks to move beyond traditional approaches by explicitly incorporating entanglement as a geometric principle, potentially revolutionizing our understanding of quantum information's structure.
The de Broglie–Bohm interpretation (Bohmian mechanics, hidden-variables theory, causal interpretation or pilot-wave theory) is one of the interpretations of quantum mechanics which opposes to the claim of standard interpretation on the completeness of description of the state of quantummechanical system, that description in Bohmian mechanics is completed by defining positions of the particles of the system i.e. hidden variables and their dynamics. These facts are the basis of motivation for the introduction of a new theory that can describe dubious axiomatic requirements of standard interpretation. The presented theory is deterministic and explicitly nonlocal. Originators of Bohmian mechanics were L. de Broglie (1927.) and D. Bohm (1952.) who presented mathematically more general and conceptual elaborate formulation. Along with the historical overview, a note is given about the main properties of the theory, such as the issue of nonlocality and hidden variables. The aim of this paper is to present and explain the main ideas and properties of the theory as well as to give a systematic overview of present and historical mathematical construction of the theory and most of all to discuss implications of de Broglie–Bohm theory on the ontology of quantum-mechanical picture, showcasing examples of the equivalency of the description of a free particle as a propagation of the plane wave in space or as the ensemble of the trajectories and of the spreading of the Gaussian wave packet. The main example of the obvious conceptual clashing of the interpretations is presented: the double-slit experiment, with a detailed examination of the Bohmian mechanics’ predictions using presented formalism on the two double-slit experiment. The graphs of trajectories for given examples are obtained.