T.F. Havel, C.J.L. Doran and S. Furuta
Density Operators in the Multiparticle Spacetime Algebra
Proc. Royal. Soc. (to appear)

Abstract: Two distinct representations for the density operators of multi-qubit systems are developed and compared within the multiparticle spacetime algebra (MSTA). The two representations arise from two different methods for correlating imaginary structures from different qubit spaces. The first is a direct transliteration of the usual representation by Hermitian matrices, with the complex structure provided by correlated pseudoscalar factors. The second arises from a generalisation of the treatment of spinors as elements of the even subalgebra of the Pauli algebra. In this representation the complex structure for quantum states is provided by correlated bivectors, though the action of this on density operators is more subtle. The advantages of the new representation arise from the ease with which spinor observables are constructed and related to terms in the density operator. These are illustrated by rederiving a well-known expression for the decoherence of a single qubit through its coherent interactions with other qubits in its environment.

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