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.
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.
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.
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.
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 for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p M , with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′ the scalar product b⋅b′ has modulus d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cd when measured in the mutually unbiased bases B_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when d is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension d. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the d-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for d even or odd number, and to recover the known result of existence of d+1 MUB's for d a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.