Algorithmic construction of local models for entangled quantum states: Optimization for two-qubit states

The correlations of certain entangled quantum states can be fully reproduced via a local model. We discuss in detail the practical implementation of an algorithm for constructing local models for entangled states, recently introduced by Hirsch et al. [Phys. Rev. Lett. 117, 190402 (2016)] and Cavalcanti et al. [Phys. Rev. Lett. 117, 190401 (2016)]. The method allows one to construct both local hidden state (LHS) and local hidden variable (LHV) models, and can be applied to arbitrary entangled states in principle. Here, we develop a systematic implementation of the algorithm, discussing the choice of the free parameters. For the case of two-qubit states, we design a ready-to-use procedure. This allows us to construct LHS models (for projective measurements) that are almost optimal, as we show for Bell diagonal states, for which the optimal model has recently been derived. Finally, we show how to construct fully analytical local models, based on the output of the convex optimization procedure.

Figure 1

Schematic view of the final algorithmic method.

Figure 2

Steering properties of Bell diagonal states with $s_1=s_2$. The states are separable below the dotted line, while they admit a LHS model below the dotted curve; this represents the exact steering limit as shown in Refs. [20, 23]. Using our LHS protocol, we obtain the four solid curves, corresponding to each level. At level four, the difference with respect to the exact steerability limit is less than 1%, illustrating the efficiency of our method in this case.

Figure 3

Steering properties of rank-3 Bell diagonal states. States are entangled in the entire region. They are unsteerable inside the white region, while states in the blue region are steerable. Implementing the protocol from level one to four, we obtain the four curves. Again, we observe that level four is very close to being optimal.

Figure 4

Steering properties of partially entangled states mixed with white noise [see Eq. (34)]. The lower dotted curve is the separability limit: states below this curve are separable. The upper dotted curve is a sufficient condition for steering: any state above this curve is steerable, as found via SDP techniques [17] (using 13 measurements on the Bloch sphere). The four solid curves represent the results of the protocol from level one to four, with growing number of measurements from bottom to top (6, 16, 46, and 136 measurements). The dashed-dotted curve corresponds to a sufficient condition for a state to be unsteerable: states below this curve admit a LHS model [21]. Our results show that this criterion is not tight in general.

Figure 5

Steering properties of states (35). States below the lower dashed line are separable, while the upper dashed black curve is an upper bound for steerability found numerically (via SDP techniques [17] and using 13 measurements on the Bloch sphere). The dashed-dotted curve correspond to the best-known LHS models so far, i.e., given by Eq. (36). The solid curves represent three successive levels of the algorithm (corresponding to fixed shrinking factors ${\eta }_1=0.79, {\eta }_2=0.92$, and ${\eta }_3=0.97$). This shows that the criterion (36) of Ref. [21] is not tight.