Maximal Randomness Generation from Steering Inequality Violations Using Qudits

We consider the generation of randomness based upon the observed violation of an Einstein-Podolsky-Rosen (EPR) steering inequality, known as one-sided device-independent randomness generation. We show that in the simplest scenario—involving only two parties and two measurements for the uncharacterised party with $d$ outcomes—that there exist EPR steering inequalities whose maximal violation certifies maximal randomness generation, equal to $\log (d)$ bits. We further show that all pure partially entangled full-Schmidt-rank states in all dimensions can achieve maximal violation of these inequalities, and thus lead to maximal randomness generation in the one-sided device-independent setting. More generally, the amount of randomness that can be generated is given by a semidefinite program, which we use to study the behavior for nonmaximal violations of the inequalities.

Figure 1

Randomness generated $H_{\min }(x=1)$, as a function of the observed violation of the steering inequality, ${\beta }^{\text{obs}}$, for dimension $d=4$, 8, 16. Inset: comparison to DIRG. We consider local randomness generation using an isotropic state ${\rho }_w=w|{\mathrm{\Phi }}^{+}⟩⟨{\mathrm{\Phi }}^{+}|+(1-w)\mathbb{1}/d^2$ and optimal measurement settings for violating the Collins-Gisin-Linden-Massar-Popescu (CGLMP) [31] and Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio (SATWAP) [32] Bell inequalities for $d=8$. The code used to generate this figure is available at [33].