Communication cost of breaking the Bell barrier

Correlations in an Einstein-Podolsky-Rosen-Bohm experiment can be made stronger than quantum correlations by allowing a single bit of classical communication between the two sides of the experiment.

Figure 1

Demonstration of the shift mechanism. Concentric circles represent the measurement directions $\widehat{a}$ and $\widehat{b}$ (outer circle), as well as parameter directions $\widehat{\lambda }$ and ${\widehat{\lambda }}^{\perp }$ (inner circle) and their associated projective sign regions. The four measurement regions spanned by $\widehat{a}$ and $\widehat{b}$ are indicated by “$\pm 1$,” respectively. Positive and negative octants spanned by $\widehat{\lambda }$ and ${\widehat{\lambda }}^{\perp }$ are indiated in the inner circle by “±,” respectively. In this configuration, the shift $\widehat{\lambda }\to \widehat{\lambda }-{\widehat{\lambda }}^{\perp }$ pushes $\widehat{\lambda }$ into a positive region.

Figure 2

(Color online) Classical and stronger-than-quantum correlation functions obtained through the exchange of a single bit in the memoryless regime for values of $\delta ∊\{0,\pi ∕10,\pi ∕5,3\pi ∕10,2\pi ∕5,\pi ∕2\}$ between 0 (straight line) and $\pi ∕2$ [cf. Eq (1)].

Figure 3

(Color online) Domains of the correlation function $E(\theta ,\delta )$. Consecutive sections from left to right cover the domains of Eq. (2) from bottom to top: (1) $0⩽\theta ⩽\delta ∕2$, (2) $\delta ∕2<\theta ⩽\frac{1}{2}(\pi -\delta )$, (3) $\frac{1}{2}(\pi -\delta )<\theta ⩽\frac{1}{2}(\pi +\delta )$, (4) $\frac{1}{2}(\pi +\delta )<\theta ⩽\pi -\delta ∕2$, (5) $\pi -\delta ∕2<\theta ⩽\pi$.

Figure 4

The correlation $E\left(\theta \right)$ of Eq. (4) as a function of $\theta =\mid a-b\mid$ for the memoryless exchange of a single bit per experiment in the planar configuration. Note that, although the shape resembles the quantum cosine law, the function is piecewise quadratic.