Demonstrating genuine multipartite entanglement and nonseparability without shared reference frames

Multipartite nonlocality is of great fundamental interest and constitutes a useful resource for many quantum information protocols. However, demonstrating it in practice, by violating a Bell inequality, can be difficult. In particular, standard experimental setups require the alignment of distant parties' reference frames, which can be challenging or impossible in practice. In this work we study the violation of certain Bell inequalities, namely the Mermin, Mermin-Klyshko, and Svetlichny inequalities, without shared reference frames, when parties share a Greenberger-Horne-Zeilinger state. Furthermore, we analyze how these violations demonstrate genuine multipartite features of entanglement and nonlocality. For three, four, and five parties, we show that it is possible to violate these inequalities with high probability, when the parties choose their measurements from the three Pauli operators, defined only with respect to their local frames. Moreover, the probability of violation, and the amount of violation, are increased when each party chooses their measurements from the four operators describing the vertices of a tetrahedron. We also consider how many randomly chosen measurement directions are needed to violate the Bell inequalities with high probability. We see that the obtained levels of violation are sufficient to also demonstrate genuine multipartite entanglement and nonseparability. Finally, we show analytically that choosing from two measurement settings per party is sufficient to demonstrate the maximum degree of genuine multipartite entanglement and nonseparability with certainty when the parties' reference frames are aligned in one direction so that they differ only in rotations around one axis.

Figure 1

The four vertices of the tetrahedron are used to define measurement directions evenly spaced over the Bloch sphere.

Figure 2

Probability distributions of ${\left({\mathcal{I}}_{\mathrm{M}}\right)}_n,{\left({\mathcal{I}}_{\mathrm{MK}}\right)}_n$, and ${\left({\mathcal{I}}_{\mathrm{S}}\right)}_n$ values for $n=3,4,5$ parties when a randomly rotated GHZ state is measured in the Pauli (red, dashed) and tetrahedral (blue, dotted) bases (color online), and in $3,4,5,6,7$ randomly chosen directions (in increasingly dark shades of gray). The vertical lines mark the LHV, and multipartite entanglement and separability bounds. Note that the latter calculation was not possible for all numbers of random measurement operators beyond the three-party case.