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

TL;DR

The paper develops a semi-device-independent framework to certify high-dimensional quantum channels from prepare-and-measure statistics under the sole assumption of a known dimension $d$, leveraging the Choimiolkowski isomorphism to enforce full CJ-state constraints. It introduces a Schmidt-number witness based on a RAC-based average success probability $\alpha_{n,d}$ and uses alternating convex search with generalized DPS and GRM relaxations to certify the entanglement dimensionality, including concrete results for dephasing and depolarizing channels. For a complete performance assessment, it then certifies entanglement fidelity via a dual SDP and a localizing-m Matrix hierarchy, yielding lower bounds compatible with observed data or a witness value $\alpha_{n,d}$. The approach provides a principled, scalable, and practically relevant method for characterizing high-dimensional quantum channels in realistic, less-than-fully-trusted settings.

Abstract

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.

Figures (4)

• Figure 1: Diagram of the SDI-P$\&$M scenario. $\mathcal{A}$ uses an unspecified preparation device that emits a quantum state $\rho_x$ depending on the classical input $x$. This state is transmitted through a quantum channel $\Lambda$. $\mathcal{B}$ receives the transmitted state and performs a measurement using an unspecified measurement device, producing an output $b$ conditioned on the classical input $y$. In this scenario, only the system dimension $d$ is assumed to be known.
• Figure 2: Numerical bounds $\tilde{\beta}_{2,d,r}^{\mathcal{Q}}$ for $d=3,\dots,7$. Panels (a)–(c) correspond to $d=3,4,5$, respectively. Blue squares denote the values computed by enforcing the Schmidt number constraint through the generalized DPS hierarchy (evaluated at the lowest level of the hierarchy), while red circles indicate the values obtained using the GRM criterion. Panel (d) shows the results for $d=6$ and $7$, where the Schmidt number constraint is imposed via the GRM criterion.
• Figure 3: Numerical values $\tilde{\beta}_{2,5,\nu}^{\mathrm{chan}}$ for the optimal ASP under noisy channels as a function of the visibility $\nu$. The blue curve corresponds to the dephasing channel and the red curve to the depolarizing channel. For comparison, horizontal dashed lines indicate the previously computed Schmidt-number-dependent bounds $\tilde{\beta}_{2,5,r}^{\mathcal{Q}}$ for $r=1,\dots,5$.
• Figure 4: Numerical lower bounds on the entanglement fidelity $\mathcal{F}(\Phi_\Lambda)$ are shown as a function of the observed ASP $\alpha_{2,2}$, with $\alpha_{2,2} \in [\beta_{2,2}^{\mathcal{L}}, \beta_{2,2}^{\mathcal{Q}}]$. The red and blue curves correspond to hierarchy levels $\ell=2$ and $\ell=3$, respectively, with the latter already close to convergence and providing a tight lower bound on the entanglement fidelity.