Certification contract and complete experiment

All spaces in the primary protocol are finite dimensional. For a Hilbert space $H$, $\mathcal{B}(H)$ denotes its matrix algebra. A channel is completely positive and trace preserving, abbreviated CPTP.

Let $C\cong {\mathbb{C}}^2$ be the logical input, $R$ the accessible memory output, and $C^{\prime }\cong C$ the returned logical output. The identification $C^{\prime }\cong C$, including its Pauli frame, is fixed before validation.

Let

include the declared encoding, storage, correction, and accessible collection operations. Let

be the implemented deterministic decoder. The tested logical channel is

The channel ends before the destructive certification analyzer. The analyzer is not part of the returned quantum-memory task.

We use

For any finite untouched reference $A$, input state ${\omega }_{AC}$, and joint output measurement, $e(\mathcal{E})$ bounds the trace distance, and hence the total variation distance of outcome laws, between