Manuscript Oai265-B@2 · Oai265-B-v2.tex · section · 94b6f42f

Common-state preparation and reference model

A proved calibration budget › Common-state preparation and reference model · lines 543–587

Oai265-B@2#eq:statecal, Oai265-B@2#eq:meascal

Assume a fixed ancillary state independent of the logical-reference input, and let the actual preparation obey

d_{tr} (ρ_{p}, P_{2}) \leq δ_{p} .

Let the reference-storage channel satisfy

d_{⋄} (R_{Q}, id_{Q}) \leq δ_{Q} .

For the actual joint pre-analyzer map $G$ , assume

d_{⋄} (G, R_{Q} \otimes E) \leq δ_{joint} .

The last bound can charge cross-talk, but is not implied by local reference measurements.

Set

σ = G (ρ_{p}), δ = min {1, δ_{p} + δ_{Q} + δ_{joint}} .

Contraction and the triangle inequality give

d_{tr} (σ, J_{E}) \leq δ .

For each setting, let ${M_{j, o}}_{o}$ be the actual complete POVM and $z_{j} (o) \in [- 1, 1]$ its predeclared score. The score observable

B_{j}^{obs} = o \sum z_{j} (o) M_{j, o}

has operator norm at most one. Assume

B_{j}^{obs} - A_{j}_{\infty} \leq μ_{j}, m_{j} = Tr (B_{j}^{obs} σ) .

Displayed formulas in this passage

Oai265-B@2#eq:statecal · line 568
$d_{tr} (σ, J_{E}) \leq δ .$
Oai265-B@2#eq:meascal · line 581
$B_{j}^{obs} - A_{j}_{\infty} \leq μ_{j}, m_{j} = Tr (B_{j}^{obs} σ) .$
unlabeled · line 547
$d_{tr} (ρ_{p}, P_{2}) \leq δ_{p} .$
unlabeled · line 551
$d_{⋄} (R_{Q}, id_{Q}) \leq δ_{Q} .$
unlabeled · line 555
$d_{⋄} (G, R_{Q} \otimes E) \leq δ_{joint} .$
unlabeled · line 563
$σ = G (ρ_{p}), δ = min {1, δ_{p} + δ_{Q} + δ_{joint}} .$
unlabeled · line 576
$B_{j}^{obs} = o \sum z_{j} (o) M_{j, o}$