Manuscript Em03-E@1 · Em03-E-v1.tex · proposition · c4b07e6a

Proposition: Collective departure from replacement

E3: controlled semiclassical comparison › The collective channel is not a replacement channel · lines 1275–1319 · proof included or linked

Em03-E@1#prop:fullreplacement, Em03-E@1#eq:fullreplacement

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For the named replacement center,

d_{⋄} (N_{2}, R_{τ_{rad}}) = p max {\frac{1}{2}, \frac{1 + 2 η}{4}} .

For every replacement channel $R_{σ}$ ,

d_{⋄} (N_{2}, R_{σ}) \geq p /2.

Thus exact private single-cell channels can coexist with a nonconstant collective radiation channel, but the collective channel cannot simultaneously be arbitrarily close to a replacement channel at fixed nonzero $p$ .

Proof. In the comparison with $τ_{rad}$ , the failure blocks cancel. The successful key blocks are unitary copies of $Λ_{η} - T$ . Its Pauli coefficient vectors are

(\frac{1 + η}{2}, 0, 0, \frac{1 - η}{2}) and (1/4, 1/4, 1/4, 1/4) .

Lemma lem:Pauli gives $max {1/2, (1 + 2 η) /4}$ . Orthogonal key blocks and the common occupation factor preserve this distance, giving (eq:fullreplacement).

For an arbitrary replacement output, the triangle inequality between the two computational-basis radiation outputs and that common output gives

p \leq d_{tr} (N_{2} (∣ 0 ⟩ ⟨ 0 ∣), σ) + d_{tr} (N_{2} (∣ 1 ⟩ ⟨ 1 ∣), σ) \leq 2 d_{⋄} (N_{2}, R_{σ}) .

Displayed formulas in this passage

Em03-E@1#eq:fullreplacement · line 1278
$d_{⋄} (N_{2}, R_{τ_{rad}}) = p max {\frac{1}{2}, \frac{1 + 2 η}{4}} .$
unlabeled · line 1286
$d_{⋄} (N_{2}, R_{σ}) \geq p /2.$
unlabeled · line 1300
$(\frac{1 + η}{2}, 0, 0, \frac{1 - η}{2}) and (1/4, 1/4, 1/4, 1/4) .$
unlabeled · line 1313
$p \leq d_{tr} (N_{2} (∣ 0 ⟩ ⟨ 0 ∣), σ) + d_{tr} (N_{2} (∣ 1 ⟩ ⟨ 1 ∣), σ) \leq 2 d_{⋄} (N_{2}, R_{σ}) .$

Nearby in this section

section: The collective channel is not a replacement channel