Theorem: Symmetry anchoring

Assume $(\mathrm{C}{)}^{\star },(\mathrm{NS}{)}^{\star },(\mathrm{N}{)}^{\star },(\mathrm{K}{)}^{\star }$ and (S5). Then diagonal anchoring holds.

Proof. Let $S$ have $n$ qubits. First take a rational diagonal state

Zero-weight labels are omitted.

Take $M$ blocks of $n$ qubits. Let $\mathcal{A}$ consist of sequences $(k_1,\dots ,k_M)$ containing exactly $m_k$ copies of each $k$, and put

Block permutations are tensor-factor permutations. They preserve this vector and act transitively on $\mathcal{A}$. Nullity and equivariance force a uniform full law.

Composability gives the first-block marginal

since

The first-block reduction is exactly ${\rho }_S$. Completions with different first labels have different remaining multisets, hence disjoint computational supports; all off-diagonal first-block entries vanish. The diagonal entries are the same counting ratios.

Locality transfers the law to every purification of ${\rho }_S$. Rational diagonal states are dense, so reduced-state continuity proves the result for all diagonal states. The empty register has a unique outcome.