Manuscript En02-E@2 · En02-E-v2.tex · theorem · c2769986

Theorem: Flagged cutoff and exact Gaussian error

Deterministic flagged Fock approximation · lines 572–602 · proof included or linked

En02-E@2#thm:cutoff

The map

E_{N} (X) = P_{N} X P_{N} + tr (Q_{N} X) ∣ ⊥ ⟩ ⟨ ⊥ ∣

is deterministic and CPTP. Put

ρ_{K, N} = E_{N} (ρ_{K}), t_{N} (K) = tr (Q_{N} ρ_{K}) .

In the common space $F (C^{J}) \oplus C ∣ ⊥ ⟩$ ,

d_{tr} (ρ_{K} \oplus 0, ρ_{K, N}) = t_{N} (K) .

The retained block is unnormalized; the failure flag carries its actual probability.

Proof. The low-block Kraus map is $P_{N}$ followed by inclusion. For a high-block orthonormal basis $e_{α}$ , add Kraus maps $∣ ⊥ ⟩ ⟨ e_{α} ∣$ . Their effects sum strongly to $I$ .

The number of occupation vectors with total at most $N$ is $\sum_{n = 0}^{N} (n J + n - 1) = (N J + N)$ . Since $[ρ_{K}, N_{tot}] = 0$ , no low–high coherence remains. The difference of the two states is a positive high block of trace $t_{N} (K)$ minus an orthogonal flag block of the same trace.

Displayed formulas in this passage

unlabeled · line 575
$E_{N} (X) = P_{N} X P_{N} + tr (Q_{N} X) ∣ ⊥ ⟩ ⟨ ⊥ ∣$
unlabeled · line 579
$ρ_{K, N} = E_{N} (ρ_{K}), t_{N} (K) = tr (Q_{N} ρ_{K}) .$
unlabeled · line 585
$d_{tr} (ρ_{K} \oplus 0, ρ_{K, N}) = t_{N} (K) .$

Nearby in this section

section: Deterministic flagged Fock approximation