Full Network Nonlocality Based Security In Quantum Key Distribution

Abstract

In the last decade research of quantum nonlocality has moved beyond the regime of standard Bell nonlocality to consider network-based experimental set-ups involving multiple independent sources. Notion of full network nonlocality has emerged as some truly network phenomena that cannot be realized in traditional Bell experiments. Present work manifests utility of such form of truly network non-classicality in designing a four partite network-based entanglement assisted quantum key distribution protocol. To be more precise, security of the protocol relies upon full network nonlocality detection via violation of some suitable trilocal inequality. Based on the quantum bit error rate and violation of trilocal inequality, arbitrary two qubit entangled states are characterized in accordance with their utility in successfully executing the protocol. Intuitively, owing to connected structure of entangled sources, any genuine form of network nonlocality may offer advantage over standard Bell nonlocality for designing secure key distribution protocols. To establish that as a fact, another QKD protocol relying upon Bell-CHSH nonlocality detection in all pairs of sender and a receiver party is designed. The former turns out to be more secure compared to the latter. Importantly, while the quantum bit error rate can be less than 14.6% exploiting Bell-CHSH nonlocality, it can be reduced below 13.7% by exploiting full network nonlocality.

Figures (10)

• Figure 1: Schematic Diagram of $n$-local star network
• Figure 2: In sub-figure.(i) the entire region square($\mathbf{S}$) represents the possible subspace formed by largest two singular values($t_1,t_2$) of correlation tensor($T$) of an arbitrary two-qubit state. An enlarged view of the upper right corner of the graph in sub-figure.(i) is provided in sub-figure.(ii). Corresponding to any point $P$ lying inside part of the positive quadrant $\mathbf{C}_{+}$ of the circle $\mathbf{C}$(Eq.(\ref{['cr10']})), $\varrho_P$ is an useless state($\mathcal{N}_4$ fails first security check). Again state corresponding to any point $P$ lying outside $\mathbf{C}_{+}$ but below tangent line $\mathbf{T}$ in $\mathbf{S},$ is an useless state($\mathcal{N}_4$ fails second security check). Only for any point $P$ lying above $\mathbf{T}$ in $\mathbf{S},$$\varrho_P$ can be used to execute $\mathcal{N}_4$ successfully. Point $\mathbf{Q}_{Min}$ represents the minima($2^{-\frac{1}{6}}$,$2^{-\frac{1}{6}}$) of QBER($\mathbf{Q}$) under assumption of no violation. Clearly the region outside $\mathbf{C}_+$ and below $\mathbf{T}$ give $(t_1,t_2)$ for which corresponding sate is FNN but not useful in $\mathcal{N}_4.$
• Figure 3: Shaded region forms a part of three-dimensional space formed by $2^{nd}$ largest singular value $t_{i,2}$ of correlation tensor $T_i$ of $\rho_i(i$$=$$1,2,3)$ specified by Eq. (\ref{['ext1']}). $\rho_1,\rho_2,\rho_3$ corresponding to any point in the shaded region can be used for running $\mathcal{N}_4.$
• Figure 4: Shaded region is a subspace formed by $2^{nd}$ largest singular values of $\rho_1,\rho_2,\rho_3$ specified by $(t_{1,1},t_{2,1},t_{3,1})$$=$$(0.91,0.94,0.93).$ Any such states generate detectable full network nonlocality but cannot be used fo running $\mathcal{N}_4.$
• Figure 5: The figure provides a subspace in two-dimensional space formed by largest two singular values of correlation tensor of arbitrary two-qubit state $\varrho.$ For any point $(t_1,t_2)$ lying in the shaded region, corresponding noisy entangled state satisfies both $\mathcal{C}_{\mathcal{N},1}$ and $\mathcal{C}_{\mathcal{N},2^{'}}$ but violates $\mathcal{C}_{\mathcal{N},2}.$ Consequently, any such state will be rejected as useless state if one uses $\mathcal{C}_{\mathcal{N},2}$ as a second security criterion in $\mathcal{N}_4.$

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 3.1