Manuscript Em03-C@1 · Em03-C-v1.tex · theorem · 47dd71ec

Theorem: Constant radiation channel cannot recover entanglement

C4: input information and reference limitations · lines 2412–2443 · proof included or linked

Em03-C@1#thm:noreference

Let a finite microscopic message space have dimension $D_{M}$ . If a channel $N$ maps every message density to the same radiation density $ρ_{rad}$ , then it is a replacement channel:

N (X) = tr (X) ρ_{rad} .

For a maximally entangled message–reference input, every decoder acting only on radiation has entanglement fidelity exactly $D_{M}^{- 2}$ , and its output has trace distance at least $1 - D_{M}^{- 2}$ from the maximally entangled target.

Proof. Linearity and decomposition of Hermitian matrices into scalar and traceless parts show that a channel constant on all densities vanishes on traceless matrices. Complex linearity proves the replacement formula.

The referenced radiation output is $I_{R} / D_{M} \otimes ρ_{rad}$ . A decoder produces $I_{R} / D_{M} \otimes σ_{M}$ . Its overlap with the maximally entangled projector is

⟨ Φ ∣ (I_{R} / D_{M} \otimes σ_{M}) ∣ Φ ⟩ = \frac{tr σ _{M}}{D _{M} ^{2}} = D_{M}^{- 2} .

Testing that projector gives the trace-distance lower bound.

Displayed formulas in this passage

unlabeled · line 2418
$N (X) = tr (X) ρ_{rad} .$
unlabeled · line 2438
$⟨ Φ ∣ (I_{R} / D_{M} \otimes σ_{M}) ∣ Φ ⟩ = \frac{tr σ _{M}}{D _{M} ^{2}} = D_{M}^{- 2} .$

Theorem: Constant radiation channel cannot recover entanglement

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