Theorem: Flagged cutoff and its correct error regimes

The map

is deterministic and CPTP. The retained subspace has dimension

For $t_K=\mathrm{Tr}(\rho Q_K)$,

In the common output space containing the original Fock space and the flag,

Proof. Use $P_K$ as the low-block Kraus operator and $|\perp ⟩⟨e_{\alpha }|$ for an orthonormal basis of the high block. Their effects sum strongly to $1$. The dimension follows from counting occupation vectors.

On $Q_K$, $N\ge K+1$, giving the tail bound. Purify $\rho$ to $|\Psi ⟩$, and set $|{\Psi }_P⟩=(P_K\otimes 1)|\Psi ⟩$. A calculation in their two-dimensional span gives

Partial trace contracts the trace norm. The orthogonal flag contributes $t_K/2$ to the half trace distance, proving (eq:cutoff-general). When $[\rho ,P_K]=0$, the difference is a positive high block of trace $t_K$ and a negative orthogonal flag block of the same trace, proving (eq:cutoff-diagonal).