Local and nonlocal content of bipartite qubit and qutrit correlations

The local and nonlocal contents of nonlocal probability distributions are studied using the approach of Elitzur, Popescu, and Rohrlich [A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A 162, 25 (1992)]. This work focuses on distributions that can be obtained by single-copy von Neumann measurements on bipartite quantum systems. For pure two-qubit states $|\Psi \left(\theta \right)⟩=\text{cos}\theta |00⟩+\text{sin}\theta |11⟩$ (with $\text{cos}\theta \ge \text{sin}\theta$), the local content of the corresponding probability distribution is found to lie between $1-\text{sin}2\theta$ and $\text{cos}2\theta$. For the family $|\Psi \left(\gamma \right)⟩=\left(|00⟩+|11⟩+\gamma |22⟩\right)/\sqrt{2+{\gamma }^2}$ of two-qutrit states, nonzero local content is found for $\gamma >2$.

Figure 1

Bounds on $p_L^{\text{max}}\left(\theta \right)$ as a function of $\theta$ for pure two-qubit states (7)—in particular, $\theta =0$ is the separable state, $\theta =\frac{\pi }{4}$ is the maximally entangled state. Thin full line: the EPR2 lower bound $\frac{1}{4}\left(1-\text{sin}2\theta \right)$. Thick full line: the lower bound (10) obtained in this paper. Dashed line: the numerical upper bound (12).

Figure 2

Present knowledge about $p_L^{\text{max}}\left(\gamma \right)$ for the family (13) of pure two-qutrit states—in particular, $\gamma =0$ is the maximally entangled state of two qubits, $\gamma =1$ is the maximally entangled state of two qutrits, and the state becomes separable in the limit $\gamma \to \infty$. The black dots represent the proved result $p_L^{\text{max}}\left(\gamma =0\right)=p_L^{\text{max}}\left(\gamma =1\right)=0$. The line is a lower bound on $p_L^{\text{max}}\left(\gamma \right)$ obtained for the choice of $P_L$ described in the text (numerical study, the precision on the bound is approximately ${10}^{-5}$).

Figure 3

(Color online) Optimal measurements settings ${\alpha }_k$ $\left(k=1,\dots ,30\right)$ for the chained inequality $I^{\left(30\right)}$ and different values of $\theta$.