Maximal randomness expansion from steering inequality violations using qudits

TL;DR

This paper demonstrates that maximal violation of certain EPR steering inequalities certifies maximal randomness expansion in a one-sided device-independent setting using qudits, with the amount of certifiable randomness quantified by a semidefinite program.

Contribution

It identifies specific steering inequalities whose maximal violation certifies maximal randomness, and shows all pure partially entangled states can achieve this maximal violation across all dimensions.

Findings

Maximal violation certifies log(d) bits of randomness.

All pure partially entangled full-Schmidt-rank states achieve maximal violation.

The amount of certifiable randomness is characterized by a semidefinite program.

Abstract

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 expansion. We show that in the simplest scenario -- involving only two parties applying two measurements with $d$ outcomes each -- that there exist EPR steering inequalities whose maximal violation certifies the maximal amount of randomness, 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 expansion in the one-sided device-independent setting. More generally, the amount of randomness that can be certified is given by a semidefinite program, which we use to study the behaviour for non-maximal violations of the inequalities.