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

Proposition: A shared-reservoir obstruction

Access controls, shared reservoirs, and records › Shared-reservoir dark codes · lines 1979–2003 · proof included or linked

Hawk02-C@2#prop:dark

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Take $m \geq d + 1$ degenerate source states and a common reservoir coupling

G (u_{1}, \dots, u_{m}) = g j = 1 \sum^{m} u_{j} .

Every $d$ -dimensional code in $ker G$ remains in the source up to phase $e^{- i ω_{0} T}$ . Its field channel is constant and its recovery error is $c_{d}$ for every $T$ .

Proof. The one-excitation Hamiltonian is

(ω_{0} I_{m} G G^{†} M_{ν}) .

For $u \in ker G$ , $H (u, 0) = (ω_{0} u, 0)$ . The field therefore remains vacuum. Apply Lemma lem:replacement.

Displayed formulas in this passage

unlabeled · line 1983
$G (u_{1}, \dots, u_{m}) = g j = 1 \sum^{m} u_{j} .$
unlabeled · line 1994
$(ω_{0} I_{m} G G^{†} M_{ν}) .$

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section: Shared-reservoir dark codes