Local-hidden-state models for Bell diagonal states and beyond

For a bipartite entangled state shared by two observers, Alice and Bob, Alice can affect the postmeasured states left to Bob by choosing different measurements on her half. Alice can convince Bob that she has such an ability if and only if the unnormalized postmeasured states cannot be described by a local-hidden-state (LHS) model. In this case, the state is termed steerable from Alice to Bob. By converting the problem to construct LHS models for two-qubit Bell diagonal states to the one for Werner states, we obtain the optimal models given by Jevtic et al. [J. Opt. Soc. Am. B 32, A40 (2015)], which are developed by using the steering ellipsoid formalism. Such conversion also enables us to derive a sufficient criterion for unsteerability of any two-qubit state.

Figure 1

Maximal lengths of $\vec{a}$, (a) along the $z$ axis, as a function of $|T_z|$ when $|T_z|\le |T_x|=1/2$ and (b) along the $x$ axis, as a function of $|T_x|$ when $|T_z|=1/2$, allowed by the conditions; the dashed blue lines are for (25) and the solid red lines are for (29). When the results from these conditions exceed the physical region, the maximum is given by the positive-semidefinite condition for a state and is shown by the dotted green line.