Proposition: The associated six-leg perfect state

The normalized state

has maximally mixed reduction on every set of three of its six qubits. Its tensor therefore gives, up to normalization, an isometry across every bipartition with at most three input legs.

Proof. For three physical qubits, their reduction is the reduction of $P/2$. The stabilizer expansion contains no nonidentity term supported on three qubits, so the reduction is $I_8/8$.

For $R$ and two physical qubits, apply (eq:privacy) to the logical half of the Bell state: the reduction is $I_R/2\otimes I_4/4$. These are all kinds of three-qubit subsets. Smaller subsets follow by tracing. The Schmidt coefficients across each three-versus-three cut are equal, which is precisely the stated isometry property.