No quantum solutions to linear constraint systems in odd dimension from Pauli group and diagonal Cliffords

TL;DR

This paper proves that in odd dimensions, certain quantum groups generated by Pauli and diagonal Clifford operators do not produce solutions to linear constraint systems, extending previous results beyond the Pauli group.

Contribution

It generalizes the non-existence of quantum solutions in odd dimensions from the Pauli group to groups generated by Pauli and diagonal Clifford operators.

Findings

No quantum solutions in odd dimension from Pauli and diagonal Clifford groups

Extends previous results beyond the Pauli group

Provides insights into quantum-classical boundaries in odd dimensions

Abstract

Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of non-classical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. In recent work, [J. Phys. A, 53, 385304 (2020)] have shown that unlike in the Boolean case, where the n-qubit Pauli group gives rise to quantum solutions of LCS such as the Mermin-Peres square, the n-qudit Pauli group never gives rise to quantum solutions of a LCS in odd dimension. Here, we generalise this result towards the Clifford hierarchy. More precisely, we consider tensor products of groups generated by (single-qudit) Pauli and diagonal Clifford operators.