Proposition: Informationally complete measurement family

For an ideal register density $\tau$,

The family reconstructs every density on the full binary register:

It is therefore informationally complete on the encoded retained subspace as well.

Proof. The diagonal sign observable before rotation is the tensor product of $Z$'s on the nonidentity positions. Conjugation by $R_P$ gives $P$. Pauli matrices obey $\mathrm{tr}(P{P}^{\prime })=d{\delta }_{P,P^{\prime }}$ and form a basis for register operators, proving (eq:PauliExpansion).