Tight detection efficiency bounds of Bell tests in no-signaling theories

No-signaling theories, which can contain nonlocal correlations stronger than quantum correlations but limited by the no-signaling condition, have deepened our understanding of the quantum theory. In practice, Bell tests are powerful tools to certify nonlocality, but their effectiveness is limited by the detector efficiency. In this work, we provide almost tight detection efficiency bounds for showing the nonlocality of no-signaling theories, by introducing a general class of Bell inequalities. In particular, we provide a tight bound for the bipartite case and an asymptotic tight bound for the multipartite case. The tightness of these bounds shows that they well characterize the structure of no-signaling theories. The bounds also imply that the detector efficiency requirement can be made arbitrarily low in both bipartite and multipartite cases by increasing the number of measurement settings. Furthermore, our work sheds light on the detector efficiency requirement for showing the nonlocality of the quantum theory.

Figure 1

Geometric interpretation of the Bell inequalities $I_{M_A{M}_B(P+1)(P+1)}$ and $I_{P+1}^{\text{multi}}$. (a) In $I_{M_A{M}_B(P+1)(P+1)}$, which is a linear combination of $p(a,b|x,y)$, the coefficients of which are first divided into blocks according to $x\text{and}y$ and then divided into entries according to $a\text{and}b$. The red, green, and white entries are one, $-P^4$, and zero, respectively. Each blue arrow pointing to a column or a row is a term $-p\left(a\right|x)$ or $-p\left(b\right|y)$ in $I_{M_A{M}_B(P+1)(P+1)}$. (b) $I_{P+1}^{\text{multi}}$ is similarly divided into higher-dimensional blocks and entries. The red entries still maintain the property that there is one and only one red entry on every line within a block according to the construction.

Figure 2

(a)–(d) vary $M$ from 4 to 256 and for each $M$ the efficiency upper bound and lower bound are shown for $N=2,3,...,200$. When $M$ increases, the ratio between the efficiency bound of $N=2$ and 200 gradually increases but is always smaller than 2.