2,110 research outputs found
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
Coarse-grained entanglement classification through orthogonal arrays
Classification of entanglement in multipartite quantum systems is an open
problem solved so far only for bipartite systems and for systems composed of
three and four qubits. We propose here a coarse-grained classification of
entanglement in systems consisting of subsystems with an arbitrary number
of internal levels each, based on properties of orthogonal arrays with
columns. In particular, we investigate in detail a subset of highly entangled
pure states which contains all states defining maximum distance separable
codes. To illustrate the methods presented, we analyze systems of four and five
qubits, as well as heterogeneous tripartite systems consisting of two qubits
and one qutrit or one qubit and two qutrits.Comment: 38 pages, 1 figur
Multi-latin squares
A multi-latin square of order and index is an array of
multisets, each of cardinality , such that each symbol from a fixed set of
size occurs times in each row and times in each column. A
multi-latin square of index is also referred to as a -latin square. A
-latin square is equivalent to a latin square, so a multi-latin square can
be thought of as a generalization of a latin square.
In this note we show that any partially filled-in -latin square of order
embeds in a -latin square of order , for each , thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable -latin squares of order for each . We also show
that for each , there exists some finite value such that for
all , every -latin square of order is separable.
We discuss the connection between -latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and -latin trades. We also enumerate and classify -latin squares of small
orders.Comment: Final version as sent to journa
Some Implications on Amorphic Association Schemes
AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition
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