Manuscript En02-C@1 · En02-C-v1.tex · section · 7549c6df

Sequential models and recovery criteria

Sequential models and recovery criteria · lines 231–277

En02-C@1#eq:Nt

All Hilbert spaces are finite dimensional. Channels are completely positive and trace preserving, abbreviated CPTP.

Let $M$ be the message, $R$ an arbitrary external reference, $B$ the initial internal register, and $E$ accessible early radiation. The initial state $σ_{B E}$ is independent of the message-reference input.

An injection unitary identifies $M B$ with $C_{0}$ . Thereafter,

W_{j} : H_{C_{j - 1}} ⟶ H_{C_{j}} \otimes H_{d_{j}}

is a specified dimension-preserving unitary identification, possibly preceded by gates on $C_{j - 1}$ . Emitted carriers never interact again with the internal evolution.

Set

D_{\leq t} = d_{1} \dots d_{t} .

The consistent evolution is

U_{t} : M B ⟶ C_{t} D_{\leq t},

obtained by composing injection and the first $t$ emission maps. The accessible channel is

N_{t} (ρ) = Tr_{C_{t}} [(U_{t} \otimes I_{E}) (ρ \otimes σ_{B E}) (U_{t}^{†} \otimes I_{E})] .

The observer receives $E D_{\leq t}$ , not $C_{t}$ . We identify the decoder's output register $M$ with $M$ through a fixed computational basis.

Displayed formulas in this passage

En02-C@1#eq:Nt · line 262
$N_{t} (ρ) = Tr_{C_{t}} [(U_{t} \otimes I_{E}) (ρ \otimes σ_{B E}) (U_{t}^{†} \otimes I_{E})] .$
unlabeled · line 243
$W_{j} : H_{C_{j - 1}} ⟶ H_{C_{j}} \otimes H_{d_{j}}$
unlabeled · line 253
$D_{\leq t} = d_{1} \dots d_{t} .$
unlabeled · line 257
$U_{t} : M B ⟶ C_{t} D_{\leq t},$

Sequential models and recovery criteria

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