Manuscript XXII-F@1 · XXII-F.tex · section · 67821705

An optional finite full-output estimation certificate

An optional finite full-output estimation certificate · lines 1305–1347

XXII-F@1#eq:normalized-choi

The exact data assignment (eq:data) has no shot count. Finite empirical certification is a different interface. The following proposition gives one explicit finite certificate for complete channels, not for the infinite cone or its exact signalling quotient.

Fix a finite nonempty set of validation tasks $k = 1, \dots, K$ . Task $k$ tests a fixed complete channel $Γ_{0, k}$ from a $d = 2^{b}$ -dimensional message to a declared $o_{k} = 2^{q_{k}}$ -dimensional output, possibly after a fixed padding isometry. All retained records, residuals, and failures intended in the claim are included in that output.

Let $Γ_{⋆, k}$ be the claimed implemented channel on the same interface and $Θ_{k}$ its specified ideal target. Define normalized channel Choi states

χ (Γ) = (Γ \otimes id) (∣ Ω_{d} ⟩ ⟨ Ω_{d} ∣), ∣ Ω_{d} ⟩ = d^{- 1/2} j = 1 \sum^{d} ∣ j ⟩ ∣ j ⟩ .

This convention is distinct from the transposed unnormalized process testers in (eq:choi).

The Choi output dimension is $N_{k} = d o_{k}$ . Let $P_{k, ν}$ , $1 \leq ν \leq N_{k}^{2} - 1$ , be its nonidentity Pauli strings, so

Tr (P_{k, ν} P_{k, λ}) = N_{k} δ_{ν λ}, ∥ P_{k, ν} ∥_{\infty} = 1.

Write

c_{k, ν} = Tr [P_{k, ν} χ (Γ_{0, k})], t_{k, ν} = Tr [P_{k, ν} χ (Θ_{k})] .

Displayed formulas in this passage

XXII-F@1#eq:normalized-choi · line 1323
$χ (Γ) = (Γ \otimes id) (∣ Ω_{d} ⟩ ⟨ Ω_{d} ∣), ∣ Ω_{d} ⟩ = d^{- 1/2} j = 1 \sum^{d} ∣ j ⟩ ∣ j ⟩ .$
unlabeled · line 1336
$Tr (P_{k, ν} P_{k, λ}) = N_{k} δ_{ν λ}, ∥ P_{k, ν} ∥_{\infty} = 1.$
unlabeled · line 1342
$c_{k, ν} = Tr [P_{k, ν} χ (Γ_{0, k})], t_{k, ν} = Tr [P_{k, ν} χ (Θ_{k})] .$

An optional finite full-output estimation certificate

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