Fermionic number operator

Under the Jordan-Wigner transform, the fermionic creation/annihilation operators for mode $i$ are

$c^\dagger_i = \Big(\prod_{k<i} Z_k\Big)\sigma^-_i, \qquad c_i = \Big(\prod_{k<i} Z_k\Big)\sigma^+_i, \qquad \sigma^{\pm}_i = \tfrac12(X_i \mp iY_i),$

and the number operator $n_i = c^\dagger_i c_i$ is

$n_i = \Big(\prod_{k<i} Z_k\Big)\sigma^-_i \Big(\prod_{k<i} Z_k\Big)\sigma^+_i = \sigma^-_i\sigma^+_i = \frac{I - Z_i}{2},$

since the two $Z$-strings square to identity and cancel — the number operator, unlike creation or annihilation individually, never needs the nonlocal Jordan-Wigner string. This is why $n_i$ is diagonal and single-qubit local even though $c_i$ and $c^\dagger_i$ individually act nonlocally in the qubit register.

Spectrum. $(I-Z_i)/2$ has eigenvalue $0$ on $|0\rangle$ and $1$ on $|1\rangle$ exactly, matching Fock-space occupation number one-to-one. The total number operator $N = \sum_i n_i$ has eigenvalue on any basis string equal to that string's Hamming weight, and commutes with any Hamiltonian built only from hopping and density-density terms (both are number-conserving), which is exactly the symmetry exploited to block-diagonalize the Fermi-Hubbard operator elsewhere in this catalog.

Role as a building block. Every density-density interaction (Hubbard's $U n_{i\uparrow}n_{i\downarrow}$), every chemical-potential term ($-\mu\sum_i n_i$), and every measurement of "how many particles are in this orbital" in quantum chemistry / condensed-matter simulations is expressed directly in terms of this operator. Because it is diagonal, expectation values $\langle n_i \rangle$ can be read out with a single computational-basis measurement per mode — no ancilla or phase-estimation circuit is required, unlike for off-diagonal observables.