From Bell’s Theorem to Secure Quantum Key Distribution

The first step in any quantum key distribution (QKD) protocol consists of sequences of measurements that produce correlated classical data. We show that these correlation data must violate some Bell inequality in order to contain distillable secrecy, if not they could be produced by quantum measurements performed on a separable state of larger dimension. We introduce a new QKD protocol and prove its security against any individual attack by an adversary only limited by the no-signaling condition.

Figure 1

Pictorial representation of nonsignaling correlations $P(a,b|x,y)$ for binary inputs and outputs. The closed thick line defines one of the facets of the polytope of local correlations, corresponding to the CHSH inequality. All the extreme points lying on this facet are also extreme points of the more general polytope of nonsignaling correlations. Only one extreme point is on top of the CHSH facet, given by the nonlocal machine. The curved line schematically represents the region of points achievable using quantum states. Isotropic correlations (4) lie along the vertical line starting from the nonlocal machine and entering the local polytope through the center of the CHSH facet. In the optimal eavesdropping attack, Eve simulates Alice and Bob’s correlation, square point, by the mixture (i.e., convex combination) of extreme points of the nonsignaling polytope, circles in the figure.

Figure 2

Key rates using one-way communication against nonsignaling individual attacks. The dashed line corresponds to the situation where no preprocessing is employed, while the thick one to the optimal preprocessing. In the inset, the key rate against a standard quantum eavesdropper is given, compared to the BB84 protocol (dashed line).