Efficient verification of entangled continuous-variable quantum states with local measurements

Continuous-variable quantum states are of particular importance in various quantum information processing tasks including quantum communication and quantum sensing. However, a bottleneck has emerged with the fast increasing in size of the quantum systems which severely hinders their efficient characterization. In this work, we establish a systematic framework for verifying entangled continuous-variable quantum states by employing local measurements only. Our protocol is able to achieve the unconditionally high verification efficiency which is quadratically better than quantum tomography as well as other nontomographic methods. Specifically, we demonstrate the power of our protocol by showing the efficient verification of entangled two-mode and multimode coherent states with local measurements.

Figure 1

General framework for the verification of two-mode entangled coherent states $|{\psi }_{\alpha ,\beta }^{\text{ECS}}\rangle$ as in Eq. (13). By randomly choosing the displacement operation ${\mathbb{D}}_l$, the outcomes with 1 or 0 of the coincidences of the two detectors are counted. The outcome 1 represents the “pass” instance and 0 for “fail.” With the number of successive “pass” instances larger than $N$, $|{\psi }_{\alpha ,\beta }^{\text{ECS}}\rangle$ is verified.