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 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 $σ_i$'s, $i=0,1,2,3$, then we have shown that under some assumption on $σ_i$'s, the lower bound for a distillable secret key rate $K_D$ can be expressed in terms of GQD of $\frac{σ_0+σ_1}{2}$ and $\frac{σ_2+σ_3}{2}$. Thus, the distillable key rate depends upon the GQD of $\frac{σ_0+σ_1}{2}$ and $\frac{σ_2+σ_3}{2}$, 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 $\frac{σ_0+σ_1}{2}$ and $\frac{σ_2+σ_3}{2}$, 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 $\frac{σ_0+σ_1}{2}$ and $\frac{σ_2+σ_3}{2}$ increases, decreases or remains constant, with the help of a few examples.

Figures (12)

• Figure 1: Geometric discord for the isotropic separable state $\rho_{\beta}$, where $\frac{-1}{8}\leq \beta \leq \frac{1}{3}$. The $X$ axis represents the value of the state parameter $\beta$ lying between $\frac{-1}{8}$ and $\frac{1}{3}$, and the $Y$ axis represents the corresponding geometric discord $D_G^{(3,3)}(\rho_{\beta})$ given by (\ref{['gd_sep_rho_isotropic']}).
• Figure 2: Geometric discord for the state $\rho_{\alpha}$, where $2\leq \alpha \leq 3$ is plotted. The $X$ axis represents the value of the state parameter $\alpha$ lying in the interval $[2,3]$, and the $Y$ axis represents the corresponding geometric discord $D_G^{(3,3)}(\rho_{\alpha})$ given by (\ref{['gd_sep_rho_alpha']}).
• Figure 3: Geometric discord for the two-qutrit separable state $\rho_1(\gamma)$, defined in (\ref{['sep_rhogamma']}) is plotted. The $X$ axis represents the range of the state parameter $\gamma\in [\frac{1}{5},1]$, and the $Y$ axis represents the corresponding geometric discord given by (\ref{['gd_sep_rhogammai']}).
• Figure 4: Geometric discord for the NPTES isotropic state $\rho_{\beta}$, where $\beta\in [\frac{1}{3},1]$ is plotted. The $X$ axis represents the state parameter $\beta$ lying in the interval $[\frac{1}{3},1]$, and the $Y$ axis represents the corresponding geometric discord $D_G^{(3,3)}(\rho_{\beta})$ given by (\ref{['gd_nptes_rho_isotropic']}).
• Figure 5: Geometric discord of the state $\rho_a$ in the interval $\frac{1}{\sqrt{2}}\leq a\leq 1$, is plotted. The $X$ axis represents the state parameter $a$, and the $Y$ axis represents the corresponding geometric discord $D_G^{(3,3)}(\rho_a)$ given by (\ref{['gd_nptes_rhoa']}).