Geometric Discord of any arbitrary dimensional bipartite system and its application in quantum key distribution

Abstract: Entangled quantum states are regarded as a key resource in quantum key distribution (QKD) protocols. However, quantum correlations, other than entanglement can also play a significant role the QKD protocols. In this work, we will focus on one such measure of quantum correlation, known as geometric quantum discord (GQD). Firstly, we derive an analytical expression of GQD for two-qutrit quantum systems and further generalize it for $d_1\otimes d_2$ dimensional systems. Next, we apply the concept of GQD in studying QKD. In particular, we derive the lower bound for a distillable secret key rate $K_D$ in terms of GQD when two communicating parties uses private states for generating a secret key in the presence of an eavesdropper. The lower bound of $K_D$ depends upon the GQD of $\frac{\sigma_0+\sigma_1}{2}$ and $\frac{\sigma_2+\sigma_3}{2}$, where $\sigma_i$'s, $i=0,1,2,3$ are the density matrices. We find that for a certain range of GQD, the successful generation of the secret key is not guaranteed. We further study the behavior of distillable key rate when the geometric discord of $\frac{\sigma_0+\sigma_1}{2}$ and $\frac{\sigma_2+\sigma_3}{2}$ increases, decreases or remains constant, with the help of a few examples. Moreover, we find that even when $\sigma_i$'s are separable or positive partial transpose entangled states, the distillable key can still be generated. %Thus, indicating that entanglement is not strictly necessary for a successful QKD protocol.