Quantifying parallelism of vectors is the quantification of distributed n-party entanglement

Abstract

The three way distributive entanglement is shown to be related to the parallelism of vectors. Using a measurement based approach we form a set of 2-dimensional vectors, representing the post measurement states of one of the parties. These vectors originate at the same point and have an angular distance between them. The area spanned by a pair of such vectors is a measure of the entanglement of formation. This leads to a geometrical manifestation of the 3-tangle in 2-dimensions, from an inequality in area which generalizes for n-qubits to reveal that the n-tangle also has a planar structure. Quantifying the genuine n-party entanglement in every bi-partition, we show that the genuine n-way entanglement does not manifest in n-tangle. A new quantity geometrically similar to 3-tangle is then introduced that represent the genuine n-way entanglement. Extending our formalism to 3-qutrits, we show that the non locality without entanglement arises from a condition under which the post measurement state vectors of a separable state show parallelism. A connection to non trivial sum uncertainty relation analogous to Maccone and Pati uncertainty relation (Maccone and Pati 2014 Phys. Rev. Lett. 113 260401) is then shown using decomposition of post measurement state vectors along parallel and perpendicular direction of the pre-measurement state vectors.