Manuscript Hawk02-B@2 · Hawk02-B.tex · proof · 95ea744f

Proof

Complete residual leakage · lines 648–687

Proof. For the stated comparison,

C_{N} - T_{τ_{M, N}} = q_{N} Ad_{J_{C}} (id - T_{I / d}) .

Isometric output embedding preserves the diamond norm. Lemma lem:replacement proves the upper bound.

For any alternative $τ_{M}$ , set $P_{ex} = J_{C} J_{C}^{†}$ and $t = Tr (P_{ex} τ_{M})$ . On a Bell input the actual complement output is

q_{N} Φ_{J_{C}} + (1 - q_{N}) ∣ 0 ⟩ ⟨ 0 ∣ \otimes I_{Q} / d .

Pinch the output difference by the projections

Φ_{J_{C}}, P_{ex} \otimes I_{Q} - Φ_{J_{C}}, ∣ 0 ⟩ ⟨ 0 ∣ \otimes I_{Q} .

Their block trace norms sum to

q_{N} - \frac{t}{d ^{2}} + t (1 - d^{- 2}) + ∣ t - q_{N} ∣.

For $t \leq q_{N}$ , half this is $q_{N} - t / d^{2} \geq q_{N} c_{d}$ . For $t \geq q_{N}$ , using $∣ q_{N} - t / d^{2} ∣ \geq q_{N} - t / d^{2}$ gives at least $t c_{d} \geq q_{N} c_{d}$ . Pinching contracts trace norm, so this is a lower bound on the original distance for every $τ_{M}$ .

The reference estimate follows from the diamond bound. For comparison with $∣ 0 ⟩ ⟨ 0 ∣$ , the difference is $q_{N}$ times two channels with orthogonal output supports. Their half-diamond distance is one.

Displayed formulas in this passage

unlabeled · line 650
$C_{N} - T_{τ_{M, N}} = q_{N} Ad_{J_{C}} (id - T_{I / d}) .$
unlabeled · line 661
$q_{N} Φ_{J_{C}} + (1 - q_{N}) ∣ 0 ⟩ ⟨ 0 ∣ \otimes I_{Q} / d .$
unlabeled · line 665
$Φ_{J_{C}}, P_{ex} \otimes I_{Q} - Φ_{J_{C}}, ∣ 0 ⟩ ⟨ 0 ∣ \otimes I_{Q} .$
unlabeled · line 671
$q_{N} - \frac{t}{d ^{2}} + t (1 - d^{- 2}) + ∣ t - q_{N} ∣.$

Proof

Displayed formulas in this passage

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