In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the properties of the Pauli matrices. Here, a few classes of such matrices are summarized.
Let be the matrix with 1 in the -th entry and 0 elsewhere. Consider the space of d×dcomplex matrices,, for a fixed d. Define the following matrices, The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension. The symbol ⊕ means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert–Schmidt inner product on. By dimension count, one sees that they span the vector space of complex matrices,. They then provide a Lie-algebra-generator basis acting on the fundamental representation of. In dimensions = 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.
A non-Hermitian generalization of Pauli matrices
The Pauli matrices and satisfy the following: The so-called Walsh–Hadamard conjugation matrix is Like the Pauli matrices, W is both Hermitian and unitary. and W satisfy the relation The goal now is to extend the above to higher dimensions, d, a problem solved by J. J. Sylvester.
Fix the dimension as before. Let , a root of unity. Since and , the sum of all roots annuls: Integer indices may then be cyclically identified mod. Now define, with Sylvester, the shift matrix and the clock matrix, These matrices generalize σ1 and σ3, respectively. Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe Quaternions, Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc. These two matrices are also the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the exponential of position in a "clock" of d hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are representations of the corresponding elements of the Weyl-Heisenberg on a d-dimensional Hilbert space. The following relations echo and generalize those of the Pauli matrices: and the braiding relation, the Weyl formulation of the CCR, and can be rewritten as On the other hand, to generalize the Walsh–Hadamard matrix W, note Define, again with Sylvester, the following analog matrix, still denoted by W in a slight abuse of notation, It is evident that W is no longer Hermitian, but is still unitary. Direct calculation yields which is the desired analog result. Thus, , a Vandermonde matrix, arrays the eigenvectors of , which has the same eigenvalues as . When d = 2k, W * is precisely the matrix of the discrete Fourier transform, converting position coordinates to momentum coordinates and vice versa. The complete family of d2 unitary independent matrices provides Sylvester's well-known trace-orthogonal basis for, known as "nonions", "sedenions", etc... This basis can be systematically connected to the above Hermitian basis. It can further be used to identify , as, with the algebra of Poisson brackets.
