Measuring Concurrence in Qubit Werner States Without an Aligned Reference Frame

The genuine concurrence is a standard quantifier of multipartite entanglement, detection, and quantification of which still remains a difficult problem from both the theoretical and experimental points of view. Although many efforts have been devoted to the detection of multipartite entanglement (e.g., using entanglement witnesses), measuring the degree of multipartite entanglement, in general, requires some knowledge about the exact shape of a density matrix of the quantum state. An experimental reconstruction of such a density matrix can be done by full state tomography, which amounts to having the distant parties share a common reference frame and well-calibrated devices. Although this assumption is typically made implicitly in theoretical works, establishing a common reference frame, as well as aligning and calibrating measurement devices in experimental situations, are never trivial tasks. It is therefore an interesting and important question whether the requirements of having a shared reference frame and calibrated devices can be relaxed. In this work, we study both theoretically and experimentally the genuine concurrence for the generalized Greenberger-Horne-Zeilinger states under randomly chosen measurements on individual qubits without a shared frame of reference and calibrated devices. We present the relation between genuine concurrence and the so-called nonlocal volume, a recently introduced indicator of nonlocality.