Characterizing the generalized complementarity polytope with extractable information from MUBs

TL;DR

This paper explores the geometric structure of the complementarity polytope in quantum state space and shows it can be characterized by the total extractable information from mutually unbiased bases, simplifying the understanding of quantum complementarity.

Contribution

It introduces the concept of the generalized complementarity polytope characterized by extractable information from a subset of MUBs, extending previous geometric insights.

Findings

Complementarity polytope can be characterized by extractable information from N+1 MUBs.

A generalized polytope exists for t MUBs in t(N-1) dimensional space.

Total extractable information from MUBs can define these polytopes.

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.