are satisfied forj 6 =ksince then one will get in the tensor product factors
[σ 3 ,(
0 0
1 0
)
]+= 0 or [σ 3 ,(
0 1
0 0
)
]+= 0
While this sort of tensor product construction is useful for discussing the physics
of multiple qubits, in general it is easier to not work with large tensor products,
and the Clifford algebra formalism we will describe in chapter 28 avoids this.
The number operators will be
NFj=aF†jaFjThese will commute with each other, so can be simultaneously diagonalized,
with eigenvaluesnj= 0,1. One can take as a basis ofHFthe 2dstates
|n 1 ,n 2 ,···,nd〉which are the natural basis states for (C^2 )⊗dgiven bydchoices of either| 0 〉or
| 1 〉.
As an example, for the cased= 3 the picture