Monogamy of Bell’s Inequality Violations in Nonsignaling Theories

We derive monogamy relations (tradeoffs) between strengths of violations of Bell’s inequalities from the nonsignaling condition. Our result applies to general Bell inequalities with an arbitrary large number of partners, outcomes, and measurement settings. The method is simple, efficient, and does not require linear programing. The results are used to derive optimal fidelity for asymmetric cloning in nonsignaling theories.

Figure 1

Diagram of measurements involved in the Bell expression $\mathcal{B}(A,B^{(1)})$ (top) and ${\mathcal{B}}_1$ (bottom). The choices of the measurement settings are marked in red (online) and by positions of the pointers (print). In the setup for $\mathcal{B}(A,B^{(1)})$, both parties have a number of measurements to (freely [21]) choose from. In the setup for ${\mathcal{B}}_1$ only Alice has such a choice whereas each $B^{(y)}$, $y\in \{1,\dots ,n\}$ always performs the same measurement $y$.

Figure 2

Diagram of measurements involved in a direct Bell test between Alice and Bob (top) or the one between Alice and $n$ Bobs after the cloning procedure (bottom). While Alice chooses between an arbitrary number of measurement settings, Bobs choose between $n$ of them. The nonsignaling condition gives an upper bound of the shrinking factor for a general asymmetric cloning procedure employed by Bobs.