Detecting two-party quantum correlations in quantum-key-distribution protocols

A necessary precondition for secure quantum key distribution is that sender and receiver can prove the presence of entanglement in a quantum state that is effectively distributed between them. In order to deliver this entanglement proof one can use the class of entanglement witness (EW) operators that can be constructed from the available measurements results. This class of EWs can be used to provide a necessary and sufficient condition for the existence of quantum correlations even when a quantum state cannot be completely reconstructed. The set of optimal EWs for two well-known entanglement-based (EB) schemes, the six-state and the four-state EB protocols, has been obtained recently [M. Curty et al., Phys. Rev. Lett. 92, 217903 (2004).] Here we complete these results, now showing specifically the analysis for the case of prepare and measure (PM) schemes. For this, we investigate the signal states and detection methods of the four-state and the two-state PM schemes. For each of these protocols we obtain a reduced set of EWs. More importantly, each set of EWs can be used to derive a necessary and sufficient condition to prove that quantum correlations are present in these protocols.

Figure 1

Illustration of several regimes of the parameters $\varphi$, $\psi$, and $\theta$ leading to negative expectation values of the operators ${\mathrm{OEW}}_4^{\mathit{EB}}=(1∕2)(\mid {\varphi }_e⟩⟨{\varphi }_e\mid +\mid {\varphi }_e⟩{⟨{\varphi }_e\mid }^{T_P})$, with $\mid {\varphi }_e⟩=\cos \varphi \mid 00⟩+\sin \varphi [\cos \psi \mid 01⟩+\sin \psi (\cos \theta \mid 10⟩+\sin \theta \mid 11⟩)]$, when they are evaluated on the joint probability distribution $P(A,B)$ given in Table . The angles $\varphi$, $\psi$, and $\theta$ are represented in radians.