Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: End-to-end scalar readout budget

Input, preparation, and measurement budgets › Intrinsic and recorded observables · lines 1418–1468 · proof included or linked

Psy02-A@1#thm:readout, Psy02-A@1#eq:complete-budget

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Under the preceding hypotheses, define

B_{j} = q_{j} + p_{j} + η^{o}_{j} + ϵ (4 n_{σ} + 2) η^{v}_{j} + e_{j} + z_{j} + c_{j},

where

q_{j} = ϵ Tr (ρ Φ (v_{j} (p))^{2}) .

Then

W_{2} (ν_{j}^{bio}, δ_{o_{j} (p)}) \leq B_{j} .

In the surrogate-only case set $c_{j} = 0$ and replace the biological law by the surrogate law.

The mean error is at most $B_{j}$ , and

ν_{j}^{bio} {y : \abs y - o_{j} (p) \geq r} \leq \frac{B _{j} ^{2}}{r ^{2}} .

Proof. Use the triangle inequality for the state quadratic norm in (eq:implementation). Definition def:preparation bounds the principal field term; Lemma lem:number bounds the vector-change term. The scalar and bounded-operator contributions are at most $η^{o}_{j}$ and $e_{j}$ .

The spectral distance to the target point mass is the root mean-square error. Minkowski gives

\norm Y_{j} - o_{j} (p)_{L^{2}} \leq \norm X_{j} - o_{j} (p)_{L^{2}} + \norm Z_{j}_{L^{2}} .

Finally apply the Wasserstein triangle inequality to (eq:carrier-W2). Cauchy–Schwarz and Markov's inequality give the last claims.

Displayed formulas in this passage

Psy02-A@1#eq:complete-budget · line 1421
$B_{j} = q_{j} + p_{j} + η^{o}_{j} + ϵ (4 n_{σ} + 2) η^{v}_{j} + e_{j} + z_{j} + c_{j},$
unlabeled · line 1430
$q_{j} = ϵ Tr (ρ Φ (v_{j} (p))^{2}) .$
unlabeled · line 1434
$W_{2} (ν_{j}^{bio}, δ_{o_{j} (p)}) \leq B_{j} .$
unlabeled · line 1442
$ν_{j}^{bio} {y : \abs y - o_{j} (p) \geq r} \leq \frac{B _{j} ^{2}}{r ^{2}} .$
unlabeled · line 1460
$\norm Y_{j} - o_{j} (p)_{L^{2}} \leq \norm X_{j} - o_{j} (p)_{L^{2}} + \norm Z_{j}_{L^{2}} .$

Theorem: End-to-end scalar readout budget

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