Self-Testing Entangled Measurements in Quantum Networks

Self-testing refers to the possibility of characterizing an unknown quantum device based only on the observed statistics. Here we develop methods for self-testing entangled quantum measurements, a key element for quantum networks. Our approach is based on the natural assumption that separated physical sources in a network should be considered independent. This provides a natural formulation of the problem of certifying entangled measurements. Considering the setup of entanglement swapping, we derive a robust self-test for the Bell-state measurement, tolerating noise levels up to $\sim 5\%$. We also discuss generalizations to other entangled measurements.

Figure 1

We consider an entanglement swapping scenario for self-testing the Bell-state measurement (BSM). It is well known that, by performing the BSM, Bob can maximally entangle the two systems of Alice and Charlie, that were initially uncorrelated. Hence, Alice and Charlie can observe maximal violation of the CHSH Bell inequality. Here, we prove the converse statement: namely, that the observation of ${\mathrm{CHSH}}_b=2\sqrt{2}$ for all $b$ (see text) necessarily implies that Bob’s measurement is equivalent to the BSM. In the second part of the Letter, we show that this result can be made robust to noise.

Figure 2

Lower bound on the quality of the unknown measurement proven in Theorem 2. The average CHSH violation of ${\beta }_{\mathrm{ave}}\gtrsim 2.689$ certifies that the measurement is entangled.