1,282 research outputs found
Constructing Mutually Unbiased Bases in Dimension Six
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 6x6 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.Comment: As published version. Added discussion of the impact of numerical
approximations and corrected the number of triples existing for non-affine
families (cf Table 3
Mubs and Hadamards of Order Six
We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We
find only triplets of MUBs, and thus do not come close to the theoretical upper
bound 7. However, we point out that the natural habitat for sets of MUBs is the
set of all complex Hadamard matrices of the given order, and we introduce a
natural notion of distance between bases in Hilbert space. This allows us to
draw a detailed map of where in the landscape the MUB triplets are situated. We
use available tools, such as the theory of the discrete Fourier transform, to
organise our results. Finally we present some evidence for the conjecture that
there exists a four dimensional family of complex Hadamard matrices of order 6.
If this conjecture is true the landscape in which one may search for MUBs is
much larger than previously thought.Comment: 33 pages, 3 figures. References added in v
Systems of mutually unbiased Hadamard matrices containing real and complex matrices
We use combinatorial and Fourier analytic arguments
to prove various non-existence results on systems of real and com-
plex unbiased Hadamard matrices. In particular, we prove that
a complete system of complex mutually unbiased Hadamard ma-
trices (MUHs) in any dimension cannot contain more than one
real Hadamard matrix. We also give new proofs of several known
structural results in low dimensions
On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six
The complete classification of all 6x6 complex Hadamard matrices is an open
problem. The 3-parameter Karlsson family encapsulates all Hadamards that have
been parametrised explicitly. We prove that such matrices satisfy a non-trivial
constraint conjectured to hold for (almost) all 6x6 Hadamard matrices. Our
result imposes additional conditions in the linear programming approach to the
mutually unbiased bases problem recently proposed by Matolcsi et al.
Unfortunately running the linear programs we were unable to conclude that a
complete set of mutually unbiased bases cannot be constructed from Karlsson
Hadamards alone.Comment: As published versio
The problem of mutually unbiased bases in dimension 6
We outline a discretization approach to determine the
maximal number of mutually unbiased bases in dimension 6. We
describe the basic ideas and introduce the most important definitions
to tackle this famous open problem which has been open for
the last 10 years. Some preliminary results are also listed
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