Authors: Richard D. Gill
Geometric algebra has been proposed as an alternative framework to the quantum mechanics of interacting qubits in a number of pioneering papers by Chris Doran, Anthony Lasenby and others, building on the foundations laid by David Hestenes. This line of work is summarised in two chapters of the book Doran and Lasenby (2003). Since then however, the approach has been pretty much completely neglected, with one exception: in 2007 Joy Christian published the first of a series of works, culminating in a book Christian (2014), which took off in a completely different direction: he claimed to have refuted Bell's theorem and to have obtained a local realistic model of the famous singlet correlations by taking account of the geometry of space, as expressed through geometric algebra. His geometric algebraic model of the singlet correlations is completely different from that of Doran and Lasenby. One of the aims of the paper is to explore geometric algebra as a tool for quantum information and to explain why it did not live up to its early promise. The short answer is: because the mapping between 3D geometry and the mathematics of one qubit is already thoroughly understood, while the extension to a system of entangled qubits does not bring in new geometric insights but on the contrary merely reproduces the usual complex Hilbert space approach in a slightly clumsy way. We also work through the mathematical core of two of Christian's shortest, least technical, and most accessible works (Christian 2007, 2011), exposing both a conceptual and an algebraic error at their heart.
Comments: 16 Pages. Revised version: added discussion of Doran and Lasenby's attempt to geometrize quantum information; more neutral, less extensive, evaluation of Christian's model
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