Proposition: Perfect-tensor property

Every reduction of $|T⟩⟨T|$ to at most three of its six qubits is maximally mixed. Thus every balanced three-input/three-output reshaping of $\sqrt{8}T$ is unitary.

Proof. A triple containing $R$ contains at most two physical qubits. Equation (eq:privacy) applied to the Bell construction makes this reduction maximally mixed. A triple containing only physical qubits is a reduction of $V{V}^{†}/2=P/2$. By Lemma lem:stabilizer, no nonidentity stabilizer is supported on three qubits, so this reduction is $I_8/8$. Smaller reductions follow by tracing. The balanced reshaping statement is precisely the matrix form of a maximally mixed three-qubit reduction.