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arXiv:2109.00507

Characterizing the generalized complementarity polytope with extractable information from MUBs

Gautam Sharma

Published Sep 1, 2021• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2109.00507

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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.
A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 - 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N = pk, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N = pk; perhaps they do not exist otherwise. However, the link to the existence of MUBs — if any — remains to be found.
A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = pk, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = pk. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.
The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUBs subsets remains a major open problem. The existing methods not only fail to conclude on the exact classification, but also are severely limited by computational resources and suffer from the numerical precision problem. Here we introduce an operational approach to identify the inequivalence of MUBs subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUBs subsets of kkk elements in any prime dimension, this method yields a universal analytical upper bound for the amount of MUBs equivalence classes. By applying this method through simple iterations, we further obtain tighter classification upper bounds for any prime dimension d≤37d\leq 37d≤37. Crucially, the comparison of these upper bounds with existing lower bounds successfully determines the exact classification for all MUBs subsets in any dimension d≤17d \leq 17d≤17. We further extend this method to the case that the dimension is a power of prime number. This general and scalable framework for the classification of MUBs subsets sheds new light on related applications.
A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d -dimensional Hilbert space, one can construct a complete set of mutually unbiased measurements. Besides their intrinsic link to MUB, we show that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy a non-trivial entropic uncertainty relation similar to MUB.
Mutually unbiased bases (MUB) are an elusive discrete structure in Hilbert spaces. Many (complete sets of) MUB are group covariant, but little is known whether they can be sharply covariant in the sense that the generating groups can have order equal to the total number of basis states, that is, d(d+1)d(d+1)d(d+1) for MUB in dimension ddd. Sharply covariant MUB, if they exist, would be most appealing from both theoretical and practical point of view. Since stabilizer MUB subsume almost all MUB that have ever been constructed, it is of fundamental interest to single out those candidates that are sharply covariant. We show that, quite surprisingly, only two stabilizer MUB are sharply covariant, and the conclusion still holds even if antiunitary transformations are taken into account. Our study provides valuable insight on the symmetry of stabilizer MUB, which may have implications for a number of research topics in quantum information and foundations. In addition, it exposes a sharp contrast between MUB and another elusive discrete structure known as symmetric informationally complete measurements (SICs), all known examples of which are sharply covariant.