Local content of bipartite qubit correlations

One of the last open problems concerning two qubits in a pure state is to find the exact local content of their correlation, in the sense of Elitzur, Popescu, and Rohrlich (EPR2) [A. C. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A162, 25 (1992)]. We propose an EPR2 decomposition that allows us to prove, for a wide range of states $|\psi (\theta )\rangle =\cos \theta |00\rangle +\sin \theta |11\rangle$, that their local content is $\overline{p_L}(\theta )=\cos 2\theta$. We also share reflections on how to possibly extend this result to all two-qubit pure states.

Figure 1

Bounds on $\overline{p_L}(\theta )$ in the EPR2 decomposition for two-qubit pure states, as a function of the state parameter $\theta$. The new lower bound $\overline{p_L}(\theta )⩾\cos 2\theta$, valid for all states such that $\cos 2\theta ⩽0.8$, coincides with the upper bound obtained in [3]. A gap still exists between the lower and upper bound when $\cos 2\theta >0.8$.

Figure 2

Construction of the set ${\Lambda }_b({\mathrm{\lambda ⃗}}_a,c)$ of allowed local variables ${\mathrm{\lambda ⃗}}_b$, given the first variable ${\mathrm{\lambda ⃗}}_a$. Left: two-dimensional cut in the vertical plane that contains ${\mathrm{\lambda ⃗}}_a$; right: three-dimensional representation of ${\Lambda }_b({\mathrm{\lambda ⃗}}_a,c)$. For both figures, ${\vartheta }_a=\frac{\pi }{3}$, $c=0.5$.