BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Drupal iCal API//EN
X-WR-CALNAME:Events items teaser
X-WR-TIMEZONE:America/Toronto
BEGIN:VTIMEZONE
TZID:America/Toronto
X-LIC-LOCATION:America/Toronto
BEGIN:DAYLIGHT
TZNAME:EDT
TZOFFSETFROM:-0500
TZOFFSETTO:-0400
DTSTART:20260308T070000
END:DAYLIGHT
BEGIN:STANDARD
TZNAME:EST
TZOFFSETFROM:-0400
TZOFFSETTO:-0500
DTSTART:20251102T060000
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
UID:6a2c06d824e4c
DTSTART;TZID=America/Toronto:20260611T163000
SEQUENCE:0
TRANSP:TRANSPARENT
DTEND;TZID=America/Toronto:20260611T173000
URL:https://uwaterloo.ca/pure-mathematics/events/quantum-catalyst-seminar
SUMMARY:Quantum Catalyst Seminar
CLASS:PUBLIC
DESCRIPTION:OLIVIER LALONDE\, UNIVERSITY OF WATERLOO\n\n_Quantum chromatic 
 numbers\, orthogonal representations\, and the\nHadamard’conjecture_\n\n
 Cameron\, Montanaro\, Newman\, Severini and Winter gave a construction\nwh
 ich shows that\, for \\(n \\in \\{2\,4\,8\\}\\) any graph _G_ which\na
 dmits a real \\(n\\)-dimensional orthogonal representation\nsatisfies \\(
 \\chi_q(G) \\leq n\\).This result can be recast as the\nstatement that \\
 (\\chi_q(S^{n-1}_\\mathbb{R}) = n\\)  for these values\nof \\(n\\)\, whe
 re \\(S^{n-1}_\\mathbb{F}\\) stands for the orthogonality\ngraph of the 
 unit sphere in \\(\\mathbb{F}^n\\). We investigate possible\nextensions o
 f their construction. We first show that their hypothesis\nthat the orthog
 onal representation be real-valued is required by\nproving that \\(\\chi_
 q(S^{n-1}_\\mathbb{C}) > n\\) for all \\(n \\geq\n3\\). We also exhibit 
 a finite\nsubgraph \\(G_{19}\\) of \\(S^{2}_\\mathbb{C}\\)  and show t
 hat \\(k+4\n= \\chi_q^{(1)}(G_{19} \\vee K_k) > \\xi_{\\mathbb{C}}(G_{19}
  \\vee K_k) =\nk+3\\) for all \\(k\\)\, so that the joins \\(G_{19} \\v
 ee K_k\\) form a\nfamily of finitary witnesses of the aforementioned sepa
 ration for the\nspecial case of rank-one colorings. As a byproduct\, we sh
 ow\nthat \\(\\xi_\\mathbb{R}(G_{19}) = 4\\)\, thereby separating the real
  and\ncomplex orthogonal ranks. For the case of the real sphere\, we show\
 nthat \\(\\chi_q(S^{n-1}_\\mathbb{R}) > n\\) whenever \\(n \\neq\n2\\)
  and \\(n\\) is not a multiple of 4. On the other hand\, we show\nthat
  \\(\\chi_q(S^{n-1}_\\mathbb{R}) = n\\) does does hold whenever a\nHadam
 ard matrix of order \\(n\\) exists. Hence\, assuming the Hadamard\nconjec
 ture\, it follows that the CMNSW construction can be extended to\nreal \\(
 n\\)-dimensional orthogonal representations if and only\nif \\(n=2\\) or
  \\(n\\) is a multiple of 4. Our method of proof\ninvolves showing the e
 quivalence between the existence of such a\nconstruction and the ability t
 o find a maximal code space for\nClifford-algebraic errors given a clean a
 ncilla\, and we believe that\nthe representation-theoretic techniques we u
 se for tackling the latter\nproblem could be of independent interest. It a
 lso follows from this\nequivalence that \\(\\chi^{(1)}_q(S^{n-1}_\\mathbb
 {R}) = n\\) if and\nonly if \\(n \\in \\{2\,4\,8\\}\\)\, thereby settlin
 g a conjecture of Zeng\nand Zhang.\n\nQNC 1201
DTSTAMP:20260612T131712Z
END:VEVENT
END:VCALENDAR