Teaching the mathematics of quantum entanglement using elementary mathematical tools

A. Gómez-Rodríguez^a and J.-L. Aragón^b

^a Departamento de Materia Condensada, Instituto de Física, Universidad Nacional Autónoma de México, Coyoacán 04510, Apartado Postal 20-364, 01000, México D.F, 01000 México, e-mail: alfredo@física.unam.mx

^b Departamento de Nanotecnología, Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Apartado Postal 1-1010, Querétaro, 76000, México, e-mail: aragon@fata.unam.mx

Abstract

We address the problem of teaching the mathematics of entanglement using only elementary linear algebra. For this goal, we first discuss tensor products using only matrix multiplication and with this we discuss entanglement for pure bipartite systems of arbitrary dimensions. We show how to assess entanglement using only Gaussian methods, i.e. the row reduced echelon form of the familiar Gauss-Jordan algorithm for solving systems of linear equations. In this way we can present entanglement avoiding the difficulties of tensor products and without the Schmidt decomposition. Some elementary examples are provided together with MATLAB scripts. A Gaussian algorithm for the factorization of unentangled states is given.

Keywords: Entanglement; tensor products; linear algebra; Gaussian methods.

PACS: 03.65.Ud; 02.10.Ud

DESCARGAR ARTÍCULO EN FORMATO PDF

References

1. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press 2000).         [ Links ]

2. A. Aczel, Entanglement, the greatest mystery in Physics (Four Walls Eight Windows, New York, 2002).         [ Links ]

3. C. Isham, Lectures on Quantum Theory: Mathematical and Structure Foundations (Imperial College Press, London, 1995).         [ Links ]

4. S. Awodey, Category Theory (Clarendon Press, Oxford, 2006).         [ Links ]

5. G. Strang, Linear Algebra and its Applications, Fourth edition, (Wellesley-Cambridge Press, Wellesley MA, 2009).         [ Links ]

6. C.F. Van Loan, Journal of Computational and Applied Mathematics 123 (2000) 85-100.         [ Links ]

7. W. Allen, Without Feathers (Ballantine Books, NY, 1983).         [ Links ]