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Partial ordering of weak mutually unbiased bases in finite quantum systems

Semiu Oladipupo Oladejo

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

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.

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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 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 .
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 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.
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.
In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry were expressed as products of lines in near-linear finite geometry (where p is a prime). An existence of lattice between any pair of near-linear finite geometry of is confirmed. For q|d, a one-to-one correspondence between the set of subgeometry of and finite geometry from the subsets of the set {D(d)} of divisors of d (where each divisor represents a finite geometry) and set of subsystems {∏(q)} (with variables in Zq) of a finite quantum system ∏(d) with variables in Zd and a finite system from the subsets of the set of divisors of d is established.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.