Custom Bell inequalities from formal sums of squares

TL;DR

This paper introduces a method to construct Bell inequalities tailored to arbitrary quantum states by leveraging sums of squares and nullifiers, enabling improved self-testing and insights into quantum correlation sets.

Contribution

The authors develop a novel approach to design Bell inequalities for any quantum state using nullifiers, overcoming previous measurement setting restrictions.

Findings

Constructed Bell inequalities for multipartite GHZ and qutrit states.

Achieved self-testing of target states with the new inequalities.

Demonstrated multiple Bell inequalities can self-test the same statistics.

Abstract

Bell inequalities play a key role in certifying quantum properties for device-independent quantum information protocols. It is still a major challenge, however, to devise Bell inequalities tailored for an arbitrary given quantum state. Existing approaches based on sums of squares provide results in this direction, but they are restricted by the necessity of first choosing measurement settings suited to the state. Here, we show how the sum of square property can be enforced for an arbitrary target state by making an appropriate choice of nullifiers, which is made possible by leaving freedom in the choice of measurement. Using our method, we construct simple Bell inequalities for several families of quantum states, including partially entangled multipartite GHZ states and qutrit states. In most cases we are able to prove that the constructed Bell inequalities achieve self-testing of the target state. We also use the freedom in the choice of measurement to self-test partially entangled two-qubit states with a family of settings with two parameters. Finally, we show that some statistics can be self-tested with distinct Bell inequalities, hence obtaining new insight on the shape of the set of quantum correlations.