Manuscript Oai280-C@1 · Oai280-C.tex · proposition · 7d44b8c0

Proposition: Classical-record obstruction

Tasks, references, and claim contracts › Two mandatory task separations · lines 266–300 · proof included or linked

Oai280-C@1#prop:EB

Suppose the only input carrying information to the decoder is a finite classical record, and all other decoder inputs are fixed and independent of the logical system and its reference. Then the resulting logical channel is measure-and-prepare:

Γ (ρ) = m \sum Tr (F_{m} ρ) σ_{m} .

It is entanglement-breaking. For a $d$ -dimensional identity target, $d \geq 2$ ,

d_{⋄} (Γ, id_{d}) \geq 1 - \frac{1}{d} .

Proof. The record probabilities form a POVM ${F_{m}}$ ; conditional decoder outputs are states $σ_{m}$ . For any reference $R$ ,

(Γ \otimes id_{R}) (ρ) = m \sum σ_{m} \otimes Tr_{H} ((F_{m} \otimes I_{R}) ρ),

which is separable.

Let $Φ_{d}$ be the maximally entangled state. A product state $σ \otimes τ$ has overlap $Tr (σ τ^{T}) / d \leq 1/ d$ with $Φ_{d}$ . By convexity the same bound holds for separable states. Measuring ${Φ_{d}, I - Φ_{d}}$ therefore gives trace distance at least $1 - 1/ d$ from $Φ_{d}$ . This is a permissible diamond-norm input.

Displayed formulas in this passage

unlabeled · line 272
$Γ (ρ) = m \sum Tr (F_{m} ρ) σ_{m} .$
unlabeled · line 277
$d_{⋄} (Γ, id_{d}) \geq 1 - \frac{1}{d} .$
unlabeled · line 285
$(Γ \otimes id_{R}) (ρ) = m \sum σ_{m} \otimes Tr_{H} ((F_{m} \otimes I_{R}) ρ),$

Proposition: Classical-record obstruction

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