Quantifying parallelism of vectors is the quantification of distributed n -party entanglement
Shreya Banerjee, Prasanta Kumar Panigrahi · Journal of Physics A Mathematical and Theoretical · 2020
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.