Semi-device-independent certification of high-dimensional quantum channels

Certifying high-dimensional quantum channels is essential for ensuring the reliability of quantum communication protocols. Existing certification schemes often rely on fully trusted internal devices, which is difficult to achieve in realistic scenarios. Here, we propose a semi-device-independent framework for certifying channel properties directly from observed statistics, assuming only that the system dimension is known. By explicitly incorporating the full set of structural constraints inherent to Choi states, our approach exploits the Choi-Jamiołkowski isomorphism for rigorous certification of quantum channels. The entanglement dimensionality of quantum channels is first certified by introducing a witness and numerically determining its Schmidt-number-dependent bounds. This certification method reproduces known analytical benchmarks and is applied to dephasing and depolarizing noise channels, thereby confirming its validity. To provide a more complete assessment of channel performance, the entanglement fidelity of quantum channels is also certified using a hierarchy of semidefinite programming relaxations based on localizing matrices. Lower bounds on the entanglement fidelity are obtained that are compatible with either the full set of observed statistics or a single witness value.

💡 Research Summary

**
The paper introduces a semi‑device‑independent (SDI) framework for certifying high‑dimensional quantum channels directly from observed prepare‑and‑measure statistics, assuming only that the Hilbert space dimension d is known. By exploiting the Choi‑Jamiołkowski (CJ) isomorphism, the authors map a quantum channel Λ to its CJ state Φ_Λ = (id ⊗ Λ)(|Φ⁺_d⟩⟨Φ⁺_d|). They emphasize three structural constraints that any CJ state must satisfy: positivity, unit trace, and the partial‑trace condition Tr_B Φ_Λ = I_A/d. The partial‑trace constraint, often omitted in previous high‑dimensional channel certification works, is shown to be essential for obtaining physically meaningful bounds.

Two main figures of merit are considered: (i) the entanglement dimensionality, quantified by the Schmidt number SN(Λ) = SN(Φ_Λ), and (ii) the entanglement fidelity F(Φ_Λ) = max_Ξ ⟨Φ⁺| (id ⊗ Ξ)