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.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.

Jan 1, 20090 citations

Numerical evidence for the maximum number of mutually unbiased bases in dimension six

Paul Butterley, William Hall

arXiv:quant-ph/0701122Physics Letters AApr 20, 200766 citations

The Mutually Unbiased Bases Revisited

Monique Combescure

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ 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 $ψ\in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$ . 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.The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ 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 $ψ\in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. 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.

arXiv:quant-ph/0605090May 10, 20064 citations

Mean king’s problem with mutually unbiased bases and orthogonal Latin squares

A. Hayashi, M. Horibe, T. Hashimoto

The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension $d$ is investigated. It is shown that a solution of the problem exists if and only if the maximal number $(d+1)$ of orthogonal Latin squares exists. This implies that there is no solution in $d=6$ or $d=10$ dimensions even if the maximal number of MUB's exists in these dimensions.The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension $d$ is investigated. It is shown that a solution of the problem exists if and only if the maximal number $(d+1)$ of orthogonal Latin squares exists. This implies that there is no solution in $d=6$ or $d=10$ dimensions even if the maximal number of MUB's exists in these dimensions.

arXiv:quant-ph/0502092Physical Review AMay 31, 200549 citations

Mutually unbiased bases in dimension six containing a product-vector basis

Lin Chen, Li Zhi Yu

arXiv:1706.00311Quantum Information ProcessingJun 28, 201817 citations

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in

\mathbb C^d, B {and} B'

are said mutually unbiased if

\forall b\in B, b'\in B'

the scalar product

b\cdot 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\in \mathbb C^d

when measured in the mutually unbiased bases

B\_{j} {of} \mathbb C^d

. 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.

The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension

d

is investigated. It is shown that a solution of the problem exists if and only if the maximal number

(d+1)

of orthogonal Latin squares exists. This implies that there is no solution in

d=6

d=10

dimensions even if the maximal number of MUB's exists in these dimensions.