Manuscript Oai280-C@1 · Oai280-C.tex · section · 7d44b8c0

A general certificate for mixed targets

Finite-confidence certification › A general certificate for mixed targets · lines 1422–1454

Oai280-C@1#eq:hoeffding-radius

Put $D = k d$ , the dimension of the output-reference space. Choose an orthonormal Hermitian basis ${G_{ν}}_{ν = 1}^{D^{2}}$ satisfying $∥ G_{ν} ∥_{\infty} \leq 1$ . For example, use the diagonal matrix units and, for $i < j$ ,

\frac{∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣}{2}, \frac{- i ∣ i ⟩ ⟨ j ∣ + i ∣ j ⟩ ⟨ i ∣}{2} .

Measure the spectral observable $G_{ν}$ on fresh Choi outputs. Let $θ_{ν}$ be the mean of $n_{ν}$ outcomes in $[- 1, 1]$ . Allocate $δ_{ν} > 0$ and set

t_{ν} = \frac{2}{n _{ν}} lo g \frac{2}{δ _{ν}} .

Suppose calibration supplies

E θ_{ν} - Tr (G_{ν} J (Λ_{0})) \leq γ_{ν}

on the good calibration event. Define

J r a = ν \sum θ_{ν} G_{ν}, = D (ν \sum (t_{ν} + γ_{ν})^{2})^{1/2}, = J - J (T)_{1} .

No positivity projection of $J$ is required.

Displayed formulas in this passage

Oai280-C@1#eq:hoeffding-radius · line 1437
$t_{ν} = \frac{2}{n _{ν}} lo g \frac{2}{δ _{ν}} .$
unlabeled · line 1447
$J r a = ν \sum θ_{ν} G_{ν}, = D (ν \sum (t_{ν} + γ_{ν})^{2})^{1/2}, = J - J (T)_{1} .$
unlabeled · line 1429
$\frac{∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣}{2}, \frac{- i ∣ i ⟩ ⟨ j ∣ + i ∣ j ⟩ ⟨ i ∣}{2} .$
unlabeled · line 1442
$E θ_{ν} - Tr (G_{ν} J (Λ_{0})) \leq γ_{ν}$

A general certificate for mixed targets

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