Manuscript Hawk02-C@2 · Hawk02-C.tex · theorem · bb332d66

Theorem: Exact finite-access recovery and packet comparison

Prescribed packet access and the complete residual › Accessible and residual spaces · lines 890–998 · proof included or linked

Hawk02-C@2#thm:C4, Hawk02-C@2#eq:Wcol, Hawk02-C@2#eq:packeterrors, Hawk02-C@2#eq:packetdecoder, Hawk02-C@2#eq:packetcomplete, Hawk02-C@2#eq:residualtau

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With outputs ordered as $A_{N} \otimes B_{N}$ ,

W_{N} = L_{N}^{col} \otimes ∣ b_{0} ⟩ + ∣ a_{0} ⟩ \otimes R_{N} .

Its accessible channel and complete complement are

N_{N, T} (X) = L_{N}^{col} X (L_{N}^{col})^{†} + r_{N} Tr (X) ∣ a_{0} ⟩ ⟨ a_{0} ∣,

C_{N, T} (X) = R_{N} X R_{N}^{†} + p_{N} Tr (X) ∣ b_{0} ⟩ ⟨ b_{0} ∣ .

Hence

ε_{rec} (N_{N, T}) = δ_{leak} (C_{N, T}) = c_{d} [q (T) + ℓ_{N} (T)] .

For $p_{N} > 0$ , put $J_{N, T} = p_{N}^{- 1/2} L_{N}^{col}$ . An optimal decoder is

D_{N, T} (Y) = J_{N, T}^{†} Y J_{N, T} + Tr [(I - J_{N, T} J_{N, T}^{†}) Y] I_{C} / d .

For $p_{N} = 0$ , use $T_{I_{C} / d}$ .

There is an identified comparison isometry $W_{N}$ on the same complete spaces that retains $A (T)$ , places all emitted weight $p (T)$ in a collected mode, and leaves missed modes vacant. Writing

h_{N} = q + p p_{N},

its complete distance is

d_{⋄} (Ad_{W_{N}}, Ad_{W_{N}}) = 1 - h_{N}^{2} \leq min {1, 2 ℓ_{N} (T)} .

The collected-channel distance is exactly $ℓ_{N} (T)$ .

Proof. Factorization gives (eq:Wcol). The two branches are orthogonal on both output sides, and

(L_{N}^{col})^{†} L_{N}^{col} = p_{N} I_{C}, R_{N}^{†} R_{N} = r_{N} I_{C} .

The residual embedding is orthogonal to $b_{0}$ . Partial traces give the channels. Lemma lem:erasure proves (eq:packeterrors) and the decoder formula. An optimal residual comparison state is

τ_{B_{N}} = R_{N} (I_{C} / d) R_{N}^{†} + p_{N} ∣ b_{0} ⟩ ⟨ b_{0} ∣ .

If $p_{N} > 0$ , choose

f_{T} = p / p_{N} P_{N} f_{T} .

If $p_{N} = 0 < p$ , choose any normalized collected mode and multiply it by $p$ . If $p = 0$ , set $f_{T} = 0$ . Use (eq:WSF) with this wavefunction. The resulting $W_{N}$ is an isometry, and

W_{N}^{†} W_{N} = (q + p p_{N}) I_{C} = h_{N} I_{C} .

Theorem thm:C1 proves the exact complete distance. Also,

W_{N} - W_{N}_{\infty}^{2} = 2 (1 - h_{N}) = 2 [p - p (p - ℓ_{N})] \leq 2 ℓ_{N},

because $p (p - ℓ_{N}) \geq p - ℓ_{N}$ .

The comparison collected channel minus the actual collected channel is

ℓ_{N} (Ad_{J_{N, T}} - T_{∣ a_{0} ⟩ ⟨ a_{0} ∣}),

with the comparison embedding used at a zero-success endpoint. The two output supports are orthogonal, so the half-diamond distance is $ℓ_{N}$ .

Displayed formulas in this passage

Hawk02-C@2#eq:Wcol · line 893
$W_{N} = L_{N}^{col} \otimes ∣ b_{0} ⟩ + ∣ a_{0} ⟩ \otimes R_{N} .$
Hawk02-C@2#eq:packeterrors · line 910
$ε_{rec} (N_{N, T}) = δ_{leak} (C_{N, T}) = c_{d} [q (T) + ℓ_{N} (T)] .$
Hawk02-C@2#eq:packetdecoder · line 921
$D_{N, T} (Y) = J_{N, T}^{†} Y J_{N, T} + Tr [(I - J_{N, T} J_{N, T}^{†}) Y] I_{C} / d .$
Hawk02-C@2#eq:packetcomplete · line 938
$d_{⋄} (Ad_{W_{N}}, Ad_{W_{N}}) = 1 - h_{N}^{2} \leq min {1, 2 ℓ_{N} (T)} .$
Hawk02-C@2#eq:residualtau · line 961
$τ_{B_{N}} = R_{N} (I_{C} / d) R_{N}^{†} + p_{N} ∣ b_{0} ⟩ ⟨ b_{0} ∣ .$
unlabeled · line 900
$N_{N, T} (X) = L_{N}^{col} X (L_{N}^{col})^{†} + r_{N} Tr (X) ∣ a_{0} ⟩ ⟨ a_{0} ∣,$
unlabeled · line 905
$C_{N, T} (X) = R_{N} X R_{N}^{†} + p_{N} Tr (X) ∣ b_{0} ⟩ ⟨ b_{0} ∣ .$
unlabeled · line 934
$h_{N} = q + p p_{N},$
unlabeled · line 952
$(L_{N}^{col})^{†} L_{N}^{col} = p_{N} I_{C}, R_{N}^{†} R_{N} = r_{N} I_{C} .$
unlabeled · line 968
$f_{T} = p / p_{N} P_{N} f_{T} .$
unlabeled · line 976
$W_{N}^{†} W_{N} = (q + p p_{N}) I_{C} = h_{N} I_{C} .$
unlabeled · line 982
$W_{N} - W_{N}_{\infty}^{2} = 2 (1 - h_{N}) = 2 [p - p (p - ℓ_{N})] \leq 2 ℓ_{N},$
unlabeled · line 991
$ℓ_{N} (Ad_{J_{N, T}} - T_{∣ a_{0} ⟩ ⟨ a_{0} ∣}),$

Theorem: Exact finite-access recovery and packet comparison

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