[Paper Review] Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

Koen Thas|arXiv (Cornell University)|Jul 10, 2014

Algebraic structures and combinatorial models4 citations

TL;DR

This paper establishes a geometric framework linking generalized Pauli operators in quantum systems to symplectic polar spaces, using finite geometry to analyze unextendible mutually unbiased bases (UMUBs). It provides short, conceptual proofs of prior results and resolves Conjecture 1.2 by showing it holds only for dimensions $ d^N = 4 $ and $ 8 $, i.e., when $ d=2 $, $ N=2 $ or $ N=3 $, and introduces 'Galois MUBs' as optimal unextendible constructions.

ABSTRACT

We consider questions posed in a recent paper of Mandayam, Bandyopadhyay, Grassl and Wootters [10] on the nature of "unextendible mutually unbiased bases." We describe a conceptual framework to study these questions, using a connection proved by the author in [19] between the set of nonidentity generalized Pauli operators on the Hilbert space of $N$ $d$-level quantum systems, $d$ a prime, and the geometry of non-degenerate alternating bilinear forms of rank $N$ over finite fields $\mathbb{F}_d$. We then supply alternative and short proofs of results obtained in [10], as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of [10], and speculate on variations of this conjecture.

Motivation & Objective

• To provide a geometric interpretation of unextendible mutually unbiased bases (UMUBs) via finite geometry.
• To offer concise, alternative proofs of results from Mandayam et al. [10] on UMUBs in $ \mathbb{C}^4 $ and $ \mathbb{C}^8 $.
• To resolve Conjecture 1.2 from [10] regarding the existence of a unique additional maximal commuting class in $ \ell = 2^n $-dimensional systems.
• To introduce and characterize 'Galois MUBs'—a class of weakly unextendible MUBs that achieve optimal bounds in unextendibility.
• To develop new construction techniques for weakly unextendible sets of MUBs using symplectic polar spaces and maximal partial spreads.

Proposed method

• Utilizes the established correspondence between generalized Pauli operators on $ N $-qudit systems and points in the symplectic polar space $ \mathcal{W}_{2N-1}(d) $, where $ d $ is prime.
• Applies properties of symplectic polar spaces—particularly the structure of generators, orthogonal complements, and partial spreads—to analyze commuting classes of Pauli operators.
• Employs duality and incidence geometry in $ \mathcal{W}_{3}(d) $ to show that for a line $ X $ not in a classical spread $ \mathcal{S} $, there exists a unique dual line $ Y $ such that $ \mathcal{S}_X = \mathcal{S}_Y $, implying a unique extension in certain cases.
• Translates geometric configurations in $ \mathcal{W}_{2N-1}(d) $ into quantum operator structures, mapping maximal commuting classes to generators and eigenbases to geometric lines.
• Uses the fact that maximal sets of mutually unbiased bases correspond to spreads in $ \mathcal{W}_{2N-1}(d) $, and unextendible sets to partial spreads.
• Applies results from finite geometry—such as the number of generators through a fixed $ (N-2) $-space and the bijection between points and hyperplanes in disjoint generators—to derive bounds and existence conditions.

Experimental results

Research questions

• RQ1 Under what conditions does a set of $ \ell/2 + 1 $ maximal commuting Pauli classes in a $ \ell = 2^n $-dimensional system admit a unique additional commuting class, forming an unextendible set of Pauli classes?
• RQ2 Is Conjecture 1.2 from Mandayam et al. [10] valid for all $ n $, or only for specific values of $ n $? Specifically, does it hold for $ n=2 $ and $ n=8 $ only?
• RQ3 Can new constructions of weakly unextendible sets of MUBs be systematically derived from geometric structures such as maximal partial spreads in symplectic polar spaces?
• RQ4 What is the role of 'Galois MUBs' in achieving optimal unextendibility bounds, and how do they differ from standard UMUBs?
• RQ5 Are there alternative formulations of Conjecture 1.2 that could be true in broader settings, such as for odd prime dimensions or higher-rank polar spaces?

Key findings

• Conjecture 1.2 from Mandayam et al. [10] is proven false in general: it holds only when $ d=2 $ and $ N=2 $ or $ N=3 $, corresponding to dimensions $ \ell = 4 $ and $ 8 $, respectively.
• The paper provides a short, geometric proof of Theorem 1.1 from [10], which establishes the unextendibility of a MUB set in $ \mathbb{C}^4 $, using the structure of $ \mathcal{W}_3(2) $.
• A new class of maximal partial spreads in $ \mathcal{W}_3(\ell) $ is constructed for any odd prime power $ \ell $, which corresponds to weakly unextendible sets of MUBs and achieves new bounds in every case.
• The existence of a unique additional maximal commuting class in $ \ell = 2^n $ systems is shown to be equivalent to the existence of a unique line $ Y $ such that $ \mathcal{S}_X = \mathcal{S}_Y $, a condition satisfied only for $ n=2 $ and $ n=8 $.
• The concept of 'Galois MUBs' is introduced as a special class of weakly unextendible MUBs that achieve an optimal bound in terms of unextendibility, suggesting a geometrically maximal structure.
• The paper demonstrates that the geometric duality between $ \mathcal{W}_{2N-1}(d) $ and $ \mathcal{Q}(4,d) $ enables a clean translation of quantum operator problems into incidence geometry, yielding new insights and proofs.