Device-independent quantum secret sharing in arbitrary even dimensions

We present a device-independent quantum secret sharing scheme in arbitrary even dimension. We propose a $d$-dimensional $N$-partite linear game, utilizing a generic multipartite higher-dimensional Bell inequality, a generalization of Mermin's inequality in a higher dimension. The probability of winning this linear game defines the device-independence test in the proposed scheme. The security is proved under a causal independence assumption on measurement devices and it is based on the polygamy property of entanglement. By defining ${\epsilon }_{\mathrm{cor}}$ correctness and ${\epsilon }^c$ completeness for a quantum secret sharing scheme, we also show that the proposed scheme is ${\epsilon }_{\mathrm{cor}}$ correct and ${\epsilon }^c$ complete.

Figure 1

Schematic of a round corresponding to the ‘testing’ and the ‘secret sharing’ and ‘secret recovery' phases. (a) A round $i$ corresponding to $T_i=1$ (testing phase). (b) A round $i$ corresponding to $T_i=0$ (secret sharing and secret recovery phases).