Proposition: Cutoff error and a moment certificate

Embed the original state in the Fock space enlarged by the overflow flag, and put ${\delta }_L=\mathrm{Tr}((I-P_L)\rho )$. Then

This statement holds with an untouched reference. If $[\rho ,P_L]=0$, the distance equals ${\delta }_L$. Moreover,

Proof. For a purification $|\psi ⟩$, write its projected and discarded components with squared norms $1-{\delta }_L$ and ${\delta }_L$. On their two-dimensional span, direct diagonalization gives

Partial trace is contractive, so the same upper bound holds for mixed states and with a reference. The added overflow block has trace norm ${\delta }_L$. The triangle inequality proves (eq:cutofferror).

If the state commutes with $P_L$, there are no off-diagonal low/high blocks. The removed high block and the added overflow block have orthogonal supports and each has trace ${\delta }_L$, so the trace distance is ${\delta }_L$. Finally, $N_{\mathrm{tot}}\ge (L+1)(I-P_L)$, proving (eq:cutoffmoment).