Quantum value for a family of $I_{3322}$-like Bell functionals

We introduce a three-parameter family of Bell functionals that extends those studied previously [Kaniewski, Phys. Rev. Res. 2, 033420 (2020)] by including a marginal contribution. An analysis of their largest value achievable by quantum realizations naturally splits the family into two branches, and for the first of them we show that this value is given by a simple function of the parameters defining the functionals. In this case we completely characterize the realizations attaining the optimal value and show that these functionals can be used to self-test any partially entangled state of two qubits. The optimal measurements, however, are not unique and form a one-parameter family of qubit measurements. The second branch, which includes the well-known $I_{3322}$ functional, is studied numerically. We identify the region in the parameter space where the optimal value can be attained with two-dimensional quantum systems and characterize the state and measurements attaining this value. Finally, we show that the set of realizations introduced by Pál and Vértesi [Phys. Rev. A 82, 022116 (2010)] to obtain the maximal violation of the $I_{3322}$ inequality succeeds in approaching the optimal value for a large subset of the functionals in this branch. In these cases we analyze and discuss the main features of the optimal realizations.

Figure 1

Local value for the functionals in Eq. (1) as a function of the parameters ${\alpha }_1$ and ${\alpha }_3$, for ${\alpha }_2=0$. The functionals in the region below the dashed line ${\alpha }_1={\alpha }_3$ satisfy ${\beta }_{\text{NS}}>{\beta }_L$.

Figure 2

Local value for the functionals in Eq. (1) as a function of the parameters ${\alpha }_1$ and ${\alpha }_3$, for ${\alpha }_2=1$. The functionals in the region below the dashed line ${\alpha }_1={\alpha }_3+2$ satisfy ${\beta }_{\mathrm{NS}}>{\beta }_L$.

Figure 3

Depiction of the quantum advantage (${\beta }_Q>{\beta }_L$) region. The boundary of the region is given by the curve ${\alpha }_1=\sqrt{{\alpha }_3^2+1}-1$. As before, the region above the ${\alpha }_1={\alpha }_3$ line satisfies ${\beta }_{\text{NS}}={\beta }_L$.

Figure 4

Level-3 NPA upper bounds ${\beta }_{\text{NPA}}^3$ for the functionals in Eq. (1) with ${\alpha }_2=1$. The quantum advantage (${\beta }_{\text{NPA}}^3-{\beta }_L>0$) region is depicted in white.

Figure 5

Comparison of optimal two-qubit values ${\beta }_{2\times 2}$ and NPA upper bounds ${\beta }_{\text{NPA}}^3$. The light blue (gray) regions show the functionals for which the quantum value is numerically attained with a two-qubit realization. The dashed line corresponds to the ${\alpha }_1=-{\alpha }_3+2$ line, plotted only for reference.

Figure 6

Distance of the optimal value ${\beta }_{\mathrm{PV}}^{\left(n\right)}$, achieved with an $n$-dimensional ${\mathrm{PV}}^{\left(n\right)}$ realization, to the level-3 NPA value as a function of local dimension $n$.

Figure 7

Optimal values ${\beta }_{\text{PV}}^{\left(n\right)}$, attained with ${\mathrm{PV}}^{\left(n\right)}$ realizations for the functionals in the region of interest, compared with their respective level-3 NPA upper bounds ${\beta }_{\text{NPA}}^3$. For the functionals plotted in the upper light blue (light gray) band the gap between two-qubit values and the NPA upper bound is closed already at dimension 3 or 5.

Figure 8

Functionals for which the quantum value can be approached with ${\mathrm{PV}}^{\left(n\right)}$ realizations with local dimension $n>5$. The color bands show the interval containing the lowest dimension for which ${\beta }_{\text{NPA}}^3-{\beta }_{\text{PV}}^{\left(n\right)}<{10}^{-6}$. For clarity, we only include in the plot the functionals labeled as ${\beta }_{\text{NPA}}^3-{\beta }_{\text{PV}}^{\left(n\right)}<{10}^{-6}|6\le n\le 1000$ in Fig. 7.

Figure 9

Angles for Bob's observables in the optimal PV realisation for the $({\alpha }_1=0.75,{\alpha }_3=1.3)$ functional (top) and for the $({\alpha }_1=0.75,{\alpha }_3=1.85)$ functional (bottom).

Figure 10

Schmidt coefficients of the optimal state $|{\mathrm{\Psi }}^{\left(n\right)}\rangle$ for the $({\alpha }_1=0.75,{\alpha }_3=1.3)$ functional (top) and for the $({\alpha }_1=0.75,{\alpha }_3=1.85)$ functional (bottom).

Figure 11

Classification of solutions by their optimal state $|{\mathrm{\Psi }}^{\left(n\right)}\rangle$. The labeled areas show the subregions in which the Schmidt coefficients exhibit a single maximum at the middle point or two maxima placed symmetrically about the middle point. As in Fig. 8, the plot shows only the functionals being classified.

Figure 12

Curve of Schmidt coefficients of the optimal state $|{\mathrm{\Psi }}^{\left(n\right)}\rangle$ for the functional in $({\alpha }_1=1,{\alpha }_3=1.275)$, a point on the boundary between the two regions depicted in Fig. 11.

Figure 13

Curve of Schmidt coefficients of the optimal state $|{\mathrm{\Psi }}^{\left(n\right)}\rangle$ for the functionals in $({\alpha }_1=0.05,{\alpha }_3=1.975)$, $({\alpha }_1=0.075,{\alpha }_3=1.975)$, and $({\alpha }_1=0.125,{\alpha }_3=1.975)$.

Figure 14

Depiction of the quantum advantage (${\beta }_Q>{\beta }_L$) region in the whole parameter space. The solid lines in the plot demarcate the boundary of this region, while the dashed-dotted line indicates the boundary between the region where the local value is ${\beta }_L=4{\alpha }_3$ (${\alpha }_1<{\alpha }_3-1$) and that where ${\beta }_L=4({\alpha }_1+1)$ (${\alpha }_1>{\alpha }_3-1$). As in the previous plots, the region above the ${\alpha }_1={\alpha }_3$ curve satisfies ${\beta }_{\text{NS}}={\beta }_L$