Discussion on the initial states of controlled bidirectional quantum secure direct communication

In many communication scenarios, it is necessary to involve a third party for control and supervision. In the context of controlled bidirectional quantum secure direct communication (CBQSDC) protocols, the transmission of secret messages between two legitimate users is only permitted with the explicit permission of a controller. To address the issue of controlled communication, a CBQSDC protocol (CLYH2015) utilizing Bell states was proposed in the paper (Quant Inf Process 14, 3515–3522, 2015). Bell states have been widely recognized for their significance in the field of quantum secure direct communication. In a subsequent study published in (Quant Inf Process 16, 147, 2017), the research examined whether CLYH2015 protocol strictly requires the initial states to be Bell states. The conclusion drawn from this investigation is that CLYH2015 protocol working properly necessitates the use of Bell states as initial states. To explore alternative possibilities for the initial states in CLYH2015 protocol, a class of CBQSDC protocols employing the generalized Bell states (GBell states), a|00⟩ + b|11⟩, b ̄|00⟩ − a ̄|11⟩, a|01⟩ + b|10⟩, and b ̄|01⟩ − a ̄|10⟩, are designed where a and b are complex numbers with |a| = |b| = 1/√2, a ̄ and b ̄ the conjugate complex numbers of a and b, respectively. The class of designed CBQSDC protocols demonstrates several favorable properties, including resistance against information leakage, intercept-and-resend attacks, measure-resend attacks, as well as robustness against collective attacks. In addition, the unconditional security of the class of designed protocols is proved. Finally, to show the advantages of the class of designed protocols, they are compared with some with some previous closely associated protocols. Interestingly, it is worth noting that the Bell states can be considered a special case of the GBell states when both a and b are real numbers. Consequently, CLYH2015 protocol can be regarded as a particular instance of the designed CBQSDC protocols. This insight implies that the initial states in CLYH2015 protocol can be extended to include the GBell states, rather than being limited solely to the Bell states. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.