Teaching the mathematics of quantum entanglement using elementary mathematical tools

A. Gómez-Rodríguez, J.-L. Aragón

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

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REVISTA MEXICANA DE FÍSICA E, year 18, issue 1, January-June 2021. Semiannual Journal published by Sociedad Mexicana de Física, A. C. Departamento de Física, 2º Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Apartado Postal 70-348. Tel. (+52)55-5622-4946, https://rmf.smf.mx/ojs/rmf-e, rmf@ciencias.unam.mx. Chief Editor: José Alejandro Ayala Mercado.  INDAUTOR Certificate of Reserve: 04-2019-080216402500-203, ISSN: 2683-2216 (on line), 1870-3542 (print), both granted by Instituto Nacional del Derecho de Autor.  Responsible for the last update of this issue, Technical Staff of Sociedad Mexicana de Física, A. C., Fís. Efraín Garrido Román, 2º. Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Date of last modification, January 04, 2021.

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