Conference Proceeding

The algebraic structure of Mutually Unbiased Bases

01/2009; DOI:10.1109/ISITA.2008.4895426 In proceeding of: Information Theory and Its Applications, 2008. ISITA 2008. International Symposium on
Source: IEEE Xplore

ABSTRACT Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in Copfd are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that sets of complete MUBs only exist in Copfd if a projective plane of size d also exists. We investigate the structure of MUBs using two algebraic tools: relation algebras and group rings. We construct two relation algebras from MUBs and compare these to relation algebras constructed from projective planes. We show several examples of complete sets of MUBs in Copfd, that when considered as elements of a group ring form a commutative monoid. We conjecture that complete sets of MUBs will always form a monoid if the appropriate group ring is chosen.

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Keywords

algebraic tools
 
complete MUBs
 
complete sets
 
elements
 
group ring form
 
group rings
 
MUBs
 
Mutually unbiased bases
 
non-prime power dimensions
 
projective plane
 
projective planes
 
quantum information theory
 
relation algebras
 
size d
 

J L Hall