Manuscript I-B@1 · I-B-v1.tex · proposition · 5357688f

Proposition: Exact event matching at a fixed event is deterministic

The dictionary and its calibration limits › Time labels and reading events · lines 585–607 · proof included or linked

I-B@1#prop:fixed-event

Under distinct-label calibration, (eq:event-calibration) holds if and only if there is a unique $j$ with $t_{j} = T (e)$ and

ρ = (P_{j} \otimes I_{X}) ρ (P_{j} \otimes I_{X}) .

The readout then returns $j$ with probability one. If $T (e) \in / {t_{k}}$ , no normalized exactly calibrated input exists.

Proof. Distinctness permits at most one index $j$ satisfying $t_{j} = T (e)$ . Calibration and normalization give $p_{j} = 1$ . Put $Q = I - P_{j} \otimes I_{X}$ . Positivity and $Tr (Qρ) = 0$ imply

0 = Tr (ρ^{1/2} Q ρ^{1/2}) = ∥ Q ρ^{1/2} ∥_{HS}^{2} .

Therefore $Qρ = ρQ = 0$ , proving the support statement. The converse follows immediately from (eq:instrument).

Displayed formulas in this passage

unlabeled · line 589
$ρ = (P_{j} \otimes I_{X}) ρ (P_{j} \otimes I_{X}) .$
unlabeled · line 601
$0 = Tr (ρ^{1/2} Q ρ^{1/2}) = ∥ Q ρ^{1/2} ∥_{HS}^{2} .$

Proposition: Exact event matching at a fixed event is deterministic

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