282 research outputs found
Mutually unbiased bases in dimension six: The four most distant bases
We consider the average distance between four bases in dimension six. The
distance between two orthonormal bases vanishes when the bases are the same,
and the distance reaches its maximal value of unity when the bases are
unbiased. We perform a numerical search for the maximum average distance and
find it to be strictly smaller than unity. This is strong evidence that no four
mutually unbiased bases exist in dimension six. We also provide a two-parameter
family of three bases which, together with the canonical basis, reach the
numerically-found maximum of the average distance, and we conduct a detailed
study of the structure of the extremal set of bases.Comment: 10 pages, 2 figures, 1 tabl
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
Unitary reflection groups for quantum fault tolerance
This paper explores the representation of quantum computing in terms of
unitary reflections (unitary transformations that leave invariant a hyperplane
of a vector space). The symmetries of qubit systems are found to be supported
by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive
reflection groups, introduced (but not named) in a recent paper [Planat M and
Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The
automorphisms of multiple qubit systems are found to relate to some Clifford
operations once the corresponding group of reflections is identified. For a
short list, one may point out the Coxeter systems of type and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups and groups No 9 and 31 in
the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford
groups (the Bell groups) are generated by the Hadamard gate, the phase
gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it
New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum
computing, the lattice approach and the geometry of smooth cubic surfaces are
discussed.Comment: new version for the Journal of Computational and Theoretical
Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices
for Quantum Applications
Complex Hadamard matrices of order 6: a four-parameter family
In this paper we construct a new, previously unknown four-parameter family of
complex Hadamard matrices of order 6, the entries of which are described by
algebraic functions of roots of various sextic polynomials. We conjecture that
the new, generic family G together with Karlsson's degenerate family K and
Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of
order 6. Such a complete characterization might finally lead to the solution of
the famous MUB-6 problem.Comment: 17 pages; Contribution to the workshop "Quantum Physics in higher
dimensional Hilbert Spaces", Traunkirchen, Austria, July 201
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