NCN Research Grant Maestro 7
Uncertainty Relations and Quantum Entanglement
Number 2015/18/A/ST2/00274 realised in 2016-2021

National Science Center Jagiellonian University National Science Center
Hilbert basis for orthogonal arrays...

4) L. Seveso, D. Goyeneche, K. ┼╗yczkowski

Coarse-grained entanglement classification through orthogonal arrays

Hilbert bases for orthogonal arrays OA($r,5,2,k$) composed of five columns and two symbols, as well as for mixed orthogonal arrays MOA($r,2^2\cdot 3,k$) and MOA($r,2\cdot 3^2,k$).

Each Hilbert basis is made up of the arrays generating all other orthogonal arrays having the same parameters. As we show in the paper, any orthogonal array with fixed number of columns over a given alphabet, can be written as a linear composition of the elements of the corresponding Hilbert basis.

Generating arrays for OA($r,5,2,1$): Download

Generating arrays for OA($r,5,2,2$): Download

Generating arrays for OA($r,5,2,3$): Download

Generating arrays for OA($r,5,2,4$): Download

Generating arrays for MOA($r,2^2 \cdot 3,1$): Download

Generating arrays for MOA($r,2^2 \cdot 3,2$): Download

Generating arrays for MOA($r,2 \cdot 3^2,1$): Download

Generating arrays for MOA($r,2 \cdot 3^2,2$): Download

All files zip: Download

Here $r$ denotes the number of rows of the OA. In the files, the arrays are given in "vector notation", see Sect. III of the paper. Each row represents an orthogonal array, with the $j^{\text{th}}$ entry specifying the number of occurrences of the $j^{\text{th}}$ run, with the runs ordered lexicographically. For example, the OA $$ \text{OA($2,2,2,1$)} = \begin{array} 00 & 0 \\ 1 & 1 \end{array} $$ would give $1,\,0,\,0,\,1$.

The generating arrays given above are fundamental to construct a coarse-grained classification of entanglement classes for five qubit systems, as well as for heterogeneous systems of two qubits and one qutrit, and two qutrits and one qubit. See Section VI of our paper.