Hilbert Spaces and Mutually Unbiased Bases

Prabha Mandayam

Rinie N. M. Nasir, Jesni Shamsul Shaari, Stefano Mancini

Rinie N. M. Nasir, Shamsul Shaari, Jesni, Stefano Mancini

Mohamed Mahmoud Youssef Shalaby

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.

TR Thijs Timmerman

Two orthogonal bases A and B of an n-dimensional Hilbert space are called Mutually Unbiased if for ai ∈ A and bj ∈ B holds: |(ai, bj)| = 1 n for each i, j ∈ {1, . . . , n}. A set B of Mutually Unbiased Bases (MUBs) is a set of bases such that each two distinct bases Bi, Bj ∈ B are mutually unbiased. The bounds on the number of MUBs, dependent on the dimension n, will be derived. Also a construction for MUBs in Cp, with p prime, will be showed. Finally, some open problems will be discussed.

M. Revzen

Analysis of state reconstruction both classical and quantum mechanical on equal footing is outlined. The meaning of "mutual unbiased bases" (MUB) of Hilbert spaces is explained in detail. An alternative quantum state reconstruction, that utilizes mutual unbiased bases (MUB), is given. The MUB approach is then used for state reconstruction in a finite, d, dimensional Hilbert spaces.