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Journal of Physics A Mathematical and Theoretical

Partial ordering of weak mutually unbiased bases

Semiu Oladipupo Oladejo, Chao Lei, Apostol Vourdas

Published Nov 11, 2014• 3 citations
Publisher

Abstract

A quantum system with variables in , where (with prime numbers), is considered. The non-near-linear geometry of the phase space , is studied. The lines through the origin are factorized in terms of 'prime factor lines' in . Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the 'prime factor Hilbert spaces' , are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of , isomorphic to the partial order in the set of subsystems of .

DOI
10.1088/1751-8113/47/48/485204

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A quantum system with variables in , where (with prime numbers), is considered. The non-near-linear geometry of the phase space , is studied. The lines through the origin are factorized in terms of 'prime factor lines' in . Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the 'prime factor Hilbert spaces' , are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of , isomorphic to the partial order in the set of subsystems of .
There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its \nwide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: \n(i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. \n(ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. \n(iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. \nWe conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.
A phase-space factorization of lines in finite geometry G(m) with variables in Zm and its correspondence in finite Hilbert space H(m) for m a non-prime was discussed. Using the method of Good [15], lines in G(m) were factorized as products of lines G(mi) where mi is a prime divisor of m. A lattice was formed between the non trivial sublines of G(m) and lines of G(mi) and between a subspace of H(m) and bases of H(mi) and existence of a link between lines in phase space finite geometry and bases in Hilbert space of finite quantum systems was discussed.
Quantum systems with variables in are considered. The properties of lines in the phase space of these systems are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d −1/2 or alternatively to one of the d −1/2 i , 0 (where d i is a divisor of d apart from d , 1). They are designed for the geometry of the phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime d , there are no divisors of d apart from 1, d and the weak mutually unbiased bases are mutually unbiased bases.
Mutually unbiased bases is an important topic in the recent quantum system \nresearches. Although there is much work in this area, many problems \nrelated to mutually unbiased bases are still open. For example, constructing \na complete set of mutually unbiased bases in the Hilbert spaces with composite \ndimensions has not been achieved yet. This thesis defines a weaker \nconcept than mutually unbiased bases in the Hilbert spaces with composite \ndimensions. We call this concept, weak mutually unbiased bases. There is \na duality between such bases and the geometry of the phase space Zd × Zd, \nwhere d is the phase space dimension. To show this duality we study the \nproperties of lines through the origin in Zd × Zd, then we explain the correspondence \nbetween the properties of these lines and the properties of the \nweak mutually unbiased bases. We give an explicit construction of a complete \nset of weak mutually unbiased bases in the Hilbert space Hd, where \nd is odd and d = p1p2; p1, p2 are prime numbers. We apply the concept of \nweak mutually unbiased bases in the context of quantum tomography and \nquantum cryptography.
Quantum systems in the d-dimensional Hilbert space are considered. The mutually unbiased bases is a deep problem in this area. The problem of finding all mutually unbiased bases for higher (non-prime) dimension is still open. We derive an alternate approach to mutually unbiased bases by studying a weaker concept which we call weak mutually unbiased bases. We then compare three rather different structures. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is 1/√k (where k∣d) or 0. The second is maximal lines through the origin in the Z(d) × Z(d) phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. The analytic representation of the weak mutually unbiased bases is defined with the zeros examined. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. We give an explicit breakdown of this triality.
We describe sets of mutually unbiased bases (MUBs) for quantum states defined over the p-adic numbers Q_p, i.e. the states that can be described as elements of the (rigged) Hilbert space L2(Q_p). We find that for every prime p>2 there are at least p+1 MUBs, which is in contrast with the situation for quantum states defined over the real line R for which only 3 MUBs are known. We comment on the possible reason for the difference regarding MUBs between these two infinite dimensional Hilbert spaces.