Nonlocal correlations as an information-theoretic resource

It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are nonlocal, in the sense that a Bell inequality is violated. They cannot, however, be used for superluminal signaling. It is also known that it is possible to write down sets of “superquantum” correlations that are more nonlocal than is allowed by quantum mechanics, yet are still nonsignaling. Viewed as an information-theoretic resource, superquantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signaling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signaling polytope in the case that two observers each choose from two possible measurements with $d$ outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.

Figure 1

A schematic representation of the space of nonsignaling correlation boxes. The vertices are labeled L and NL for local and nonlocal. Bell inequalities are the facets represented in dashed lines. The set bounded by these is $\mathcal{L}$. The region accessible to quantum mechanics is $\mathcal{Q}$. A general nonsignaling box $∊\mathcal{P}$.

Figure 2

An example of how two parties that are given two boxes may process locally their inputs and outputs. They result in simulating another type of box with inputs $X,Y$ and outcomes $a,b$. Note that due to the no-signaling condition, the parties can use their two boxes with a different time ordering.

Figure 3

Making a 4-box from two PR boxes. Alice inputs $X$ into the first box and $\alpha X$ into the second, while Bob inputs $Y$ into both boxes. Alice’s output is given by $a=2{\alpha }^{\prime }+\alpha$ and Bob’s by $b=2{\beta }^{\prime }+\beta$.

Figure 4

Making an $\mathrm{X}(\mathrm{Y}+\mathrm{Z})$ box from two PR boxes. Alice outputs $a=a_1\oplus a_2$, Bob outputs $b$ and Charles outputs $c$.

Figure 5

Making a Svetlichny box from three PR boxes. Alice outputs $a=a_1\oplus a_2$, Bob outputs $b=b_1\oplus b_3$, and Charles outputs $c=c_2\oplus c_3$.

Figure 6

Making an $\mathrm{XYZ}$ box from three PR boxes. Alice outputs $a=a_2$, Bob outputs $b=b_3$, and Charles outputs $c=c_2\oplus c_3$.