We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$ . It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.

arXiv:1612.09000Dec 28, 20160 citations

On approximate real mutually unbiased bases in square dimension

Ajeet Kumar, Subhamoy Maitra, Chandra Sekhar Mukherjee

Cryptography and CommunicationsJan 11, 20214 citations

On the mathematical foundations of mutually unbiased bases

Koen Thas

Quantum Information ProcessingDec 21, 20173 citations

Mutually unbiased bases and semi-definite programming

Stephen Brierley, Stefan Weigert

A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Gröbner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Gröbner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.

arXiv:1006.0093Journal of Physics Conference SeriesNov 1, 201037 citations

d+1

MUBs in

{\mathbb C}^d

. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.