General Method for Constructing Local Hidden Variable Models for Entangled Quantum States

Entanglement allows for the nonlocality of quantum theory, which is the resource behind device-independent quantum information protocols. However, not all entangled quantum states display nonlocality. A central question is to determine the precise relation between entanglement and nonlocality. Here we present the first general test to decide whether a quantum state is local, and show that the test can be implemented by semidefinite programing. This method can be applied to any given state and for the construction of new examples of states with local hidden variable models for both projective and general measurements. As applications, we provide a lower-bound estimate of the fraction of two-qubit local entangled states and present new explicit examples of such states, including those that arise from physical noise models, Bell-diagonal states, and noisy Greenberger-Horne-Zeilinger and $W$ states.

Figure 1

Diagrammatic representation of the method (restricted to two dimensions for illustrative purposes). The vectors $\pm {\widehat{u}}_x$ are the Bloch vectors corresponding to each of the measurements from the set $\mathcal{M}$. The area enclosed by the dashed lines is the polytope that these measurements form. Any measurement contained inside this polytope can be simulated by appropriately mixing the LHS model that simulates the measurements in $\mathcal{M}$. The shaded circle, of radius $r$, is the largest circle which is completely contained in the convex hull, and contains all noisy projective measurements ${\mathrm{\Pi }}_{a|\widehat{u}}^{(r)}$.