M-Qubit states embedded in Clifford algebras

Abstract

The quantum theory of a finite quantum system with L degrees of freedom is usually set up by associating it with a Hilbert space of dimension d(L) and identifying observables and states in the matrix algebra . For the case d = 2m, m integer, this algebra can be identified with the Clifford algebra . The case of d = 2m dimensions is simply realized by a system with m dichotomic degrees of freedom, an m-qubit system for instance. The physically relevant new point is the appearance of a new (symmetry-?)group SO(2m). A possible interpretation of the space in which this group operates is proposed. It is shown that the eigenvalues of m-qubit-type states only depend on SO(2m)-invariants. We use this fact to determine state parameter domains (generalized Bloch spheres) for states classified as SO(2m)-tensors. The classification of states and interactions of components of a physical m-qubit system as k-tensors and pseudotensors (0 ⩽ k ⩽ m) leads to rules similar to those found in elementary quantum mechanics. The question of electromagnetic interactions is shortly broached. We sketch, pars pro toto, a graphical interpretation of tensor contractions appearing in perturbative expansions.