Elemental and tight monogamy relations in nonsignaling theories

Physical principles constrain the way nonlocal correlations can be distributed among distant parties. These constraints are usually expressed by monogamy relations that bound the amount of Bell inequality violation observed among a set of parties by the violation observed by a different set of parties. We prove here that much stronger monogamy relations are possible for nonsignaling correlations by showing how nonlocal correlations among a set of parties limit any form of correlations, not necessarily nonlocal, shared among other parties. In particular, we provide tight bounds between the violation of a family of Bell inequalities among an arbitrary number of parties and the knowledge an external observer can gain about outcomes of any single measurement performed by the parties. Finally, we show how the obtained monogamy relations offer an improvement over the existing protocols for device-independent quantum key distribution and randomness amplification.

Figure 1

(a) The usual monogamies compare nonlocality (measured by the value of some Bell expression $I)$ between different groups of parties (here between two pairs of parties $AB$ and $AC)$. Instead, (b) our monogamy relations compare nonlocality observed by a group of parties (here $AB)$ to the knowledge, represented by the probability $p(A=C)$, the third party $C$ can have about the outcomes observed by either of the parties. As such, they are qualitatively different, and in fact stronger than those of type (a).

Figure 2

Comparison of the upper bounds on GP: present bound (32) (red line) and (33) (purple line). Our bound is tight—for any value $0\le I_{\mathsf{A}}^{N,M,d}\le d-1$, it provides the maximum attainable value of GP. Instead, the bound (33) is nontrivial only in some restricted range of $I_{\mathsf{A}}^{N,M,d}$, namely, when $I_{\mathsf{A}}^{N,M,d}<4(d-1)/d^2$, which tends to zero for $d\to \infty$.

Figure 3

(a) Guessing probability (and simultaneously the tight analogs of monogamies in Theorem 3) for $i=2$ as a function of $({\tilde{I}}_{AB}^{\alpha }-2\alpha )/2(\sqrt{1+{\alpha }^2}-\alpha )$ for various values of $\alpha$. All curves were found using two methods. First, we maximized the guessing probability for a given value of ${\tilde{I}}_{AB}^{\alpha }$ over two-ququart states and one-ququart dichotomic measurements. Then, we used the hierarchy of Ref. [29] and with its third level we arrived at curves that coincide with those obtained with the first method with precision ${10}^{-7}$. For comparison, (b) presents the corresponding nontight monogamies proven in theorem 3 $\left(i=2\right)$ for $\alpha =1,3$ (the curves for $\alpha =1.5,2$ fall in between these two). The black curve is the same on both plots.

Figure 4

Minimal number of measurements $M$ on a maximally entangled state of local dimension $d$ necessary for the secret-key rate $R$ secure against nonsignaling eavesdroppers to be at least one (dots), ${log}_{2}3$ (squares), and two (triangles) bits, when (32) (green) and (33) (red) are used to bound $R$. Using our bound the parties need to use many fewer measurements to reach the same key rate. Moreover, contrary to what is predicted by the previously known bound, the number of measurements decreases with the dimension.