Sharing preparation contextuality in a Bell experiment by an arbitrary pair of sequential observers

Based on the quantum violation of the bipartite Bell inequality, it has been established that the sharing of nonlocality can be demonstrated for at most two sequential observers at one end and at most one pair of observers at both ends. In this work we study the sharing of nonlocality and preparation contextuality based on a bipartite Bell inequality, involving arbitrary $n$ measurements by one party and $2^{n-1}$ measurements by another party. Such a Bell inequality has two bounds, the local bound and the preparation noncontextual bound, which is smaller than the local bound. We show that while nonlocality can be shared only by the first pair of the sequential observers, the preparation contextuality can be shared by an arbitrary pair of independent sequential observers at both ends.

Figure 1

Local bound (blue line with circles) and maximum quantum value obtained by the ${\mathrm{Alice}}_1\text{−}{\mathrm{Bob}}_1$ pair (red dashed line with squares) and the ${\mathrm{Alice}}_2\text{−}{\mathrm{Bob}}_2$ pair (orange dashed line with diamonds) up to $n=10$ bits.

Figure 2

Preparation noncontextuality bound (blue solid line with circles) and the maximum quantum value obtained by the ${\mathrm{Alice}}_1\text{−}{\mathrm{Bob}}_1$ pair (red dashed line with squares), the ${\mathrm{Alice}}_2\text{−}{\mathrm{Bob}}_2$ pair (orange dashed line with diamonds), and the ${\mathrm{Alice}}_3\text{−}{\mathrm{Bob}}_3$ pair (brown dashed line with triangles) up to $n=10$ bits.

Figure 3

Minimum value of the sharpness parameter of ten pairs of Alice and Bob required for violating the preparation noncontextual bound for the family of Bell expressions for $n=100$.