Dual bounds for the positive definite functions approach to mutually unbiased bases
METADATA ONLY
Loading...
Author / Producer
Date
2022-12
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
Abstract
A long-standing open problem asks if there can exist 7 mutually unbiased bases (MUBs) in C6, or, more generally, d + 1 MUBs in Cd for any d that is not a prime power. Recent work of Proc Am Math Soc 146(3), 1143–1150 (2018) proposed an application of the method of positive definite functions (a relative of Delsarte’s method in coding theory and Lovász’s semidefinite programming relaxation of the independent set problem) as a means of answering this question in the negative. Namely, they ask whether there exists a polynomial of a unitary matrix input satisfying various properties which, through the method of positive definite functions, would show that 7 MUBs cannot exist in C6. Using a convex duality argument, we prove that there is no such polynomial of degree at most 6. We also propose a general dual certificate which we conjecture to certify that this method can never show that there exist strictly fewer than d + 1 MUBs in Cd.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
20 (2)
Pages / Article No.
18
Publisher
Springer
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Mutually unbiased bases; Semidefinite programming; Positive definite functions; Convex optimization
Organisational unit
09679 - Bandeira, Afonso / Bandeira, Afonso