arXiv subjects

Chris J. L. Doran

Publications and source records attributed to Chris J. L. Doran.

3 recordsLinked to original sources

arXiv2004

A Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional special unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.

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arXiv2001

Geometric Algebra in Quantum Information Processing

This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.

quant-ph↗

arXiv2001

Interaction and Entanglement in the Multiparticle Spacetime Algebra

The multiparticle spacetime algebra (MSTA) is an extension of Dirac theory to a multiparticle setting, which was first studied by Doran, Gull and Lasenby. The geometric interpretation of this algebra, which it inherits from its one-particle factors, possesses a number of physically compelling features, including simple derivations of the Pauli exclusion principle and other nonlocal effects in quantum physics. Of particular importance here is the fact that all the operations needed in the quantum (statistical) mechanics of spin 1/2 particles can be carried out in the ``even subalgebra'' of the MSTA. This enables us to ``lift'' existing results in quantum information theory regarding entanglement, decoherence and the quantum/classical transition to space-time. The full power of the MSTA and its geometric interpretation can then be used to obtain new insights into these foundational issues in quantum theory. A system of spin 1/2 particles located at fixed positions in space, and interacting with an external magnetic field and/or with one another via their intrinsic magnetic dipoles provides a simple paradigm for the study of these issues. This paradigm can further be easily realized and studied in the laboratory by nuclear magnetic resonance spectroscopy.

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