992 research outputs found
The conic-gearing image of a complex number and a spinor-born surface geometry
Quaternion (Q-) mathematics formally contains many fragments of physical
laws; in particular, the Hamiltonian for the Pauli equation automatically
emerges in a space with Q-metric. The eigenfunction method shows that any
Q-unit has an interior structure consisting of spinor functions; this helps us
to represent any complex number in an orthogonal form associated with a novel
geometric image (the conic-gearing picture). Fundamental Q-unit-spinor
relations are found, revealing the geometric meaning of spinors as Lam\'e
coefficients (dyads) locally coupling the base and tangent surfaces.Comment: 7 pages, 1 figur
Orthogonality for Quantum Latin Isometry Squares
Goyeneche et al recently proposed a notion of orthogonality for quantum Latin
squares, and showed that orthogonal quantum Latin squares yield quantum codes.
We give a simplified characterization of orthogonality for quantum Latin
squares, which we show is equivalent to the existing notion. We use this
simplified characterization to give an upper bound for the number of mutually
orthogonal quantum Latin squares of a given size, and to give the first
examples of orthogonal quantum Latin squares that do not arise from ordinary
Latin squares. We then discuss quantum Latin isometry squares, generalizations
of quantum Latin squares recently introduced by Benoist and Nechita, and define
a new orthogonality property for these objects, showing that it also allows the
construction of quantum codes. We give a new characterization of unitary error
bases using these structures.Comment: In Proceedings QPL 2018, arXiv:1901.0947
On the structure tensors of almost contact B-metric manifolds
The space of the structure (0,3)-tensors of the covariant derivatives of the
structure endomorphism and the metric on almost contact B-metric manifolds is
considered. A known decomposition of this space in orthogonal and invariant
subspaces with respect to the action of the structure group is used. We
determine the corresponding components of the structure tensor and consider the
case of the lowest dimension 3 of the studied manifolds. Some examples are
commented.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1105.5715 by
other author
Adversary Lower Bound for Element Distinctness with Small Range
The Element Distinctness problem is to decide whether each character of an
input string is unique. The quantum query complexity of Element Distinctness is
known to be ; the polynomial method gives a tight lower bound
for any input alphabet, while a tight adversary construction was only known for
alphabets of size .
We construct a tight adversary lower bound for Element
Distinctness with minimal non-trivial alphabet size, which equals the length of
the input. This result may help to improve lower bounds for other related query
problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos
fixed. v3: minor typos fixe
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