Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.
Two orthonormal bases A and B are said to be mutually unbiased when, for every pair of vectors |αi〉 ∈ A, |βj〉 ∈ B, |〈αi|βj〉|2 = 1/d, where d is the dimension of the system. We have tried different methods to construct MUB’s in six dimensions. These include numerical searches for orthonormal bases, extension of Wootters and Fields’ method using the concept of field, and trying to partition the tensor product of qubits and qutrits operators to generate MUB’s as their eigenbases as suggested by Lawrence et al. Our studies have yielded no more than three MUB’s, and we are led to the conclusion that either only these three bases exist or the method for constructing such bases is radically different from those used so far. We shall present the difficulties encountered in each of these methods while searching for MUB’s in composite dimensions.
A set of
k orthonormal bases of
Cd is called mutually unbiased if
∣⟨e,f⟩∣2=1/d whenever
e and
f are basis vectors in distinct bases. A natural question is for which pairs
(d,k) there exist~
k mutually unbiased bases in dimension
d. The (well-known) upper bound
k≤d+1 is attained when~
d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain
C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of
Sd and
Sk. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the
Sd≀Sk-module
C([d]×[k])t into irreducible modules. We present numerical results for small
d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist
d+2 mutually unbiased bases in dimensions~
d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than
3 MUBs in dimension
6 requires polynomials of total degree at least~
12.
A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d . The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗ -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk .We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk -module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8 . Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12 .
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $(6\ifmmode\times\else\texttimes\fi{}6)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.