bibkey: rajchelgasiorowskizyczkowski2018robust
authors: Grzegorz Rajchel; Adam Gąsiorowski; Karol Życzkowski
year: 2018
title: "Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces"
doi: 10.1007/s11786-018-0384-y
url: https://arxiv.org/abs/1804.10715v2
claim: "Two n×n permutation matrices are complementary when P_ij = P_hk = Q_ik = 1 implies Q_hj = 1, and strongly complementary when moreover their nonzero entries lie in different places; the paper asks how the maximal number of pairwise strongly complementary permutation matrices depends on n."
strata_touched:
  - D5/S3/Combinatorics/Permutation/StronglyComplementaryPermutationMaximum
license: citation-only
triage: anchor

Rajchel, Gąsiorowski and Życzkowski, Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces

G. Rajchel, A. Gąsiorowski and K. Życzkowski, Mathematics in Computer Science 12, 473–490 (2018); arXiv:1804.10715v2. Quotations are from the v2 TeX source.

Permutation matrices $P$ and $Q$ are complementary if $P_{ij}=P_{hk}=Q_{ik}=1$ implies $Q_{hj}=1$ for all $i,j,h,k$, and strongly complementary if moreover $P_{ij}=1$ implies $Q_{ij}=0$. With $P=\mathbb{I}$, $Q$ is strongly complementary exactly when it is an involution with zero diagonal. For a set of $k$ pairwise strongly complementary permutation matrices of dimension $n$, the paper notes $k\le n$ and states:

We are not able to determine how the maximal number $k$ of such matrices depends on the dimension $n$, nor whether the bistochastic matrices belonging to the interior of this polytope are unistochastic.

Verified locator

Source: https://arxiv.org/abs/1804.10715v2

• DOI: https://doi.org/10.1007/s11786-018-0384-y (Mathematics in Computer Science 12(4), 473–490; title, authors, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/1804.10715v2. The e-print member unisto9.tex has 1401 lines and md5 bc7893394ee3708e2bb9d4ba277b4a41. Locations:
  • complementary permutation matrices, Definition 4.1, l. 605–611; strong complementarity, Definition 4.3, l. 622–626;
  • the involution reformulation, l. 630–636; the question, l. 678–690.