Bell nonlocality

Bell’s 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell’s theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. The main concepts and tools which have been developed to describe and study the nonlocality of quantum theory and which have raised this topic to the status of a full subfield of quantum information science are reviewed.

Figure 1

Sketch of a Bell experiment. A source ($S$) distributes two physical systems to distant observers, Alice and Bob. Upon receiving their systems, each observer performs a measurement on it. The measurement chosen by Alice is labeled $x$ and its outcome $a$. Similarly, Bob chooses measurement $y$ and gets outcome $b$. The experiment is characterized by the joint probability distribution $p(ab|xy)$ of obtaining outcomes $a$ and $b$ when Alice and Bob choose measurements $x$ and $y$.

Figure 2

Sketch of the no-signaling ($\mathcal{N}\mathcal{S}$), quantum ($\mathcal{Q}$), and local ($\mathcal{L}$) sets. Notice the strict inclusions $\mathcal{L}\subset \mathcal{Q}\subset \mathcal{N}\mathcal{S}$. Moreover, $\mathcal{N}\mathcal{S}$ and $\mathcal{L}$ are polytopes, i.e., they can be defined as the convex combination of a finite number of extremal points. The set $\mathcal{Q}$ is convex, but not a polytope. The hyperplanes delimiting the set $\mathcal{L}$ correspond to Bell inequalities.

Figure 3

Hierarchy of sets ${\mathcal{Q}}_{\nu }$ generated by the hierarchy of SDPs defined by [339] (see Sec. 2c1d). Each set in the hierarchy better approximates the set of quantum correlations $\mathcal{Q}$. In the CHSH scenario, the set ${\mathcal{Q}}_1$ already achieves the maximum quantum value of the CHSH inequality, i.e., Tsirelson’s bound.

Figure 4

A two-dimensional section of the no-signaling polytope in the CHSH scenario ($m=\mathrm{\Delta }=2$). The vertical axis represents the CHSH value $S$, while the horizontal axis represents the value of a symmetry of the CHSH expression $S^{\prime }$ (where inputs have been relabeled). Local correlations satisfy $|S|\le 2$ and $|S^{\prime }|\le 2$. The PR box is the no-signaling behavior achieving the maximum CHSH value $S=4$. Tsirelson’s bound corresponds to the point where $S=2\sqrt{2}$, i.e., the maximum CHSH value that a quantum behavior can achieve.

Figure 5

The nonlocality of a quantum state $\rho$ can be revealed in different scenarios. (a) The simplest scenario: Alice and Bob directly perform local measurements on a single copy of $\rho$. (b) The hidden nonlocality scenario: Alice and Bob first apply a filtering to the state; upon successful operation of the filter, they perform the local measurements for the Bell test. (c) Many-copy scenario: Alice and Bob measure collectively many copies of the state $\rho$. (d) Network scenario: several copies of $\rho$ are distributed in a quantum network, where each observer performs local measurements.

Figure 6

Nonlocal properties of the isotropic state (53). The state is separable for $p\le p_s=1/(d+1)$, admits a local model for $p\le p_L$ ([22]), and violates a Bell inequality for $p_L<p_{NL}<p$ ([156]). In the interval $p_L<p<p_{NL}$, it is not known whether the state admits a local model or, on the contrary, violates a Bell inequality. Finally, when several copies of the isotropic state can be measured jointly, nonlocality is obtained whenever a single copy of the state is entangled, that is, if $p>p_s$ ([123]). In the gray region, superactivation of quantum nonlocality occurs ([352]).

Figure 7

A Bell test based on distant entangled atoms. Each atom is entangled with an emitted photon. Upon successful projection of the two photons onto a Bell state, the two atoms become entangled. The scheme is therefore “event ready,” which makes it robust to photon losses in the channel. Moreover, since atomic measurements have an efficiency close to 1, this scheme is free of the detection loophole. This setup has been implemented experimentally with a distance of 20 m between the atoms ([248]), and used for device-independent randomness expansion. From [373].