Secure simultaneous dense coding using $$\chi $$ -type entangled state

A violation of a Bell inequality can formally be expressed as a witness for quantum nonlocality. A new four-qubit Bell inequality is optimally violated by $$\chi $$ -type entangled state, but not by four-qubit GHZ, W and cluster state. Hence, $$\chi $$ -type entangled state is a good candidate to implement simultaneous dense coding. In this paper, we propose a simultaneous dense coding protocol with genuine four-particle entangled state, $$\chi $$ -type entangled state, in which two receivers can simultaneously obtain their respective classical information sent by a sender. The double controlled-NOT operator, which is used as the locking operator, play a crucial role in our protocol. The security of simultaneous dense coding is analyzed against the intercept-resend attack. The preparation of $$\chi $$ -type entangled state has been studied in various physics systems, and it has been experimentally generated with nearly deterministic scheme and high generation rate. Thus, our protocol is feasible with the current experimental technology.