Self-testing quantum states and measurements in the prepare-and-measure scenario

The goal of self-testing is to characterize an a priori unknown quantum system based solely on measurement statistics, i.e., using an uncharacterized measurement device. Here we develop self-testing methods for quantum prepare-and-measure experiments, thus not necessarily relying on entanglement and/or violation of a Bell inequality. We present noise-robust techniques for self-testing sets of quantum states and measurements, assuming an upper bound on the Hilbert space dimension. We discuss in detail the case of a $2\to 1$ random access code with qubits, for which we provide analytically optimal self-tests. The simplicity and noise robustness of our methods should make them directly applicable to experiments.

Figure 1

Average fidelity $\mathcal{F}$ $\left({\mathcal{F}}^{\prime }\right)$ for prepared states (measurements), as a function of the observed value of ${\mathcal{A}}_2$. The black line is our analytical lower bound of Eq. (13). The blue region is accessible via single qubit strategies without shared randomness, as confirmed by strong numerical evidence (see Appendixes). When allowing for shared randomness between the devices, the accessible region (obtained by taking the convex hull of the blue region) now also includes the gray area, and our analytic lower bound is tight in general.

Figure 2

Black line is the analytic lower bound on the average fidelity $\mathcal{F}$ $\left({\mathcal{F}}^{\prime }\right)$ for prepared states (measurements), as a function of the observed value of ${\mathcal{A}}_2$. To characterize the region accessible via pure qubit strategies (i.e., without shared randomness), we perform numerics generating randomly sets of qubit preparations (blue circles and crosses); here we show the numerical results for the case of states, but similar results are obtained for the case of measurements. In the region $C_2<{\mathcal{A}}_2<Q_2$, we conjecture that the class of strategies given in the text (corresponding to the red curve) are optimal, both for $\mathcal{F}$ and ${\mathcal{F}}^{\prime }$. Finally, the green dashed line is our conjectured upper bound on the average fidelity.

Figure 3

Lower bound on $\mathcal{F}\left({\mathcal{A}}_2\right)$ as obtained by the swap method and by analytical technique.

Figure 4

Lower bound on $\mathcal{F}\left(\mathcal{A}\right)$ as obtained by the swap method.