Manuscript Hawk07-A@1 · Hawk07-A.tex · proposition · f9c0f5da

Proposition: Purely classical return is insufficient

The classical obstruction and the surviving quantum mouth · lines 1470–1487 · proof included or linked

Hawk07-A@1#prop:classical

For a purely classical returned channel on a $d$ -dimensional diary, every deterministic decoder has

d_{⋄} (D M, id_{L}) \geq b_{d} = 1 - \frac{1}{d} .

Equivalently, its full-diamond error is at least $2 b_{d}$ .

Proof. Decoding a classical channel is measure-and-prepare, hence entanglement breaking EntanglementBreaking. Its Bell-input output is separable. A normalized product vector has squared overlap at most $1/ d$ with $∣ Φ_{d} ⟩$ , and convexity extends the bound to separable states. Testing the Bell projector gives error at least $1 - 1/ d$ .

Displayed formulas in this passage

unlabeled · line 1474
$d_{⋄} (D M, id_{L}) \geq b_{d} = 1 - \frac{1}{d} .$

Nearby in this section

section: The classical obstruction and the surviving quantum mouth
example: Example: Enter plus surviving quantum memory: teleportation
section: The classical obstruction and the surviving quantum mouth