Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well-known that in prime power dimensions N=p^m (with p prime and m a positive integer) there exists a maximal set of N+1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic in odd prime dimensions (J. Phys. A, 14, 3241 (1981)), and Wootters and Fields (Ann. Phys. 191, 363 (1989)) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalised Pauli group in terms of the additive characters of the field and we derive an exact groupal composition law inside the elements of the commuting subsets of the generalised Pauli group, renormalised by a well-chosen phase-factor.
Mutually unbiased bases generalize the X , Y and Z qubit bases. They possess numerous applications in quantum information science. It is well known that in prime power dimensions N = p m (with p prime and m a positive integer), there exists a maximal set of N + 1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic ( 1981 J. Phys. A: Math. Gen. 14 3241 ) in odd prime dimensions, and Wootters and Fields (1989 Ann. Phys. 191 363) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalized Pauli group in terms of the additive characters of the field, and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized Pauli group, renormalized by a well-chosen phase-factor.
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.
Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in 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 complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.