This method of generalizing the Pauli matrices refers to a generalization from a single 2-level system (qubit) to multiple such systems.
We use the tensor product notation to refer to the operator on
such generalized Pauli matrices if we include the identity
In quantum computation, it is conventional to denote the Pauli matrices with single upper case letters This allows subscripts on Pauli matrices to indicate the qubit index.
For example, in a system with 3 qubits, Multi-qubit Pauli matrices can be written as products of single-qubit Paulis on disjoint qubits.
Alternatively, when it is clear from context, the tensor product symbol
can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product.
For example: The traditional Pauli matrices are the matrix representation of the
This method of generalizing the Pauli matrices refers to a generalization from 2-level systems (Pauli matrices acting on qubits) to 3-level systems (Gell-Mann matrices acting on qutrits) and generic
-level systems (generalized Gell-Mann matrices acting on qudits).
[2][3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum.
By dimension count, one sees that they span the vector space of
They then provide a Lie-algebra-generator basis acting on the fundamental representation of
= 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.
A particularly notable generalization of the Pauli matrices was constructed by James Joseph Sylvester in 1882.
satisfy the following: The so-called Walsh–Hadamard conjugation matrix is Like the Pauli matrices,
satisfy the relation The goal now is to extend the above to higher dimensions,
, the sum of all roots annuls: Integer indices may then be cyclically identified mod d. Now define, with Sylvester, the shift matrix and the clock matrix, These matrices generalize
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[7][8][9] as formulated by Hermann Weyl, and they find routine applications in numerous areas of mathematical physics.
hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum.
They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg group on a
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
, note Define, again with Sylvester, the following analog matrix,[11] still denoted by
Direct calculation yields which is the desired analog result.
is precisely the discrete Fourier transform matrix, converting position coordinates to momentum coordinates and vice versa.
This provides Sylvester's well-known trace-orthogonal basis for
, the Cartan subalgebra, map to linear combinations of the
, Sylvester's generalized Pauli operators are orthogonal and normalized to