Clauser–Horne–Shimony–Holt versus three-party pseudo-telepathy: on the optimal number of samples in device-independent quantum private query

In device-independent (DI) paradigm, the trustful assumptions over the devices are removed and CHSH test is performed to check the functionality of the devices toward certifying the security of the protocol. The existing DI protocols consider infinite number of samples from theoretical point of view, though this is not practically implementable. For finite sample analysis of the existing DI protocols, we may also consider strategies for checking device independence other than the CHSH test. In this direction, here we present a comparative analysis between CHSH and three-party Pseudo-telepathy game for the quantum private query protocol in DI paradigm that appeared in Maitra et al. (Phys Rev A 95:042344, 2017) very recently.