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

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 in 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, if the shared resource state is an entangled state constructed with the linear combination of the tensor product of the Bell pair and the state $\sigma_i$'s, $i=0,1,2,3$, then we have shown that under some assumption on $\sigma_i$'s, the lower bound for a distillable secret key rate $K_D$ can be expressed in terms of GQD of $\frac{\sigma_0+\sigma_1}{2}$ and $\frac{\sigma_2+\sigma_3}{2}$. Thus, the distillable key rate depends upon the GQD of $\frac{\sigma_0+\sigma_1}{2}$ and $\frac{\sigma_2+\sigma_3}{2}$, when the communicating parties uses private states for generating a secret key in presence of an eavesdropper. Further, for a certain range of GQD of $\frac{\sigma_0+\sigma_1}{2}$ and $\frac{\sigma_2+\sigma_3}{2}$, we find that there exists some NPT entangled resource state for which the successful generation of the secret key may not be guaranteed. We, moreover 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.