Characterizing the generalized complementarity polytope with extractable information from MUBs

Abstract

Complementarity polytope is a geometric structure that exists in N2-1 dimensional space for an N dimensional Hilbert space. The existence of N + 1 mutually unbiased bases(MUBs) is possible, if such a polytope can be shown to be a subset of density matrices, which is a very difficult task. With the hope of simplifying this task, we have shown in this work that, the complementarity polytope can be characterized by the total extractable information from N+1 MUBs. We also demonstrate that t less than (N+1) number of MUBs also form a polytope that exists in t(N-1) dimensional space, which we refer to as generalized complementarity polytope. The generalized complementarity polytope can also be characterized by total extractable information from t MUBs.