Manuscript Psy03-B@1 · Psy03-B-v1.tex · theorem · 8ca3caf9

Theorem: Physical irreversible feedback

A quantum stochastic network with a scar register › Complete feedback construction · lines 1901–1933 · proof included or linked

Psy03-B@1#thm:hybrid

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The map

F (X) = s, r, α \sum V_{s, r, α} X V_{s, r, α}^{†}

is CPTP on the full matrix space. It discards off-diagonal input scar-label blocks and preserves the classical–quantum state class.

The complete recorded map uses Kraus operators $∣ r ⟩_{R} \otimes V_{s, r, α}$ . Along every nonzero-probability history, the scar label is nondecreasing. In particular,

E \abs S_{t + 1} \geq E \abs S_{t} .

Proof. Kraus completeness is

s, r, α \sum V_{s, r, α}^{†} V_{s, r, α} = s \sum I_{Q} \otimes ∣ s ⟩ ⟨ s ∣ = I_{QS} .

The formula reads diagonal label blocks and writes diagonal label blocks. Adding orthogonal record labels preserves completeness. Condition (eq:monotone) holds branchwise, so cardinality cannot decrease.

Displayed formulas in this passage

unlabeled · line 1904
$F (X) = s, r, α \sum V_{s, r, α} X V_{s, r, α}^{†}$
unlabeled · line 1917
$E \abs S_{t + 1} \geq E \abs S_{t} .$
unlabeled · line 1924
$s, r, α \sum V_{s, r, α}^{†} V_{s, r, α} = s \sum I_{Q} \otimes ∣ s ⟩ ⟨ s ∣ = I_{QS} .$

Theorem: Physical irreversible feedback

Displayed formulas in this passage

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