Relative quantum phase, $m$-tangle, and multi-local Lorentz-group invariant

In this paper we establish a relation between quantum relative phase, $m$-tangle, and multi-local Lorentz-group invariant or $SL(2,\mathbb{C}{)}^{\times m}$-invariant $S_{(m)}^2$. Our construction is based on the orthogonal complement of a positive operator valued measure on quantum phase. In particular, we propose a quantity based on the quantum relative phase of a multi-qubit operator that coincides with $m$-tangle, and multi-local Lorentz-group invariant.