7 research outputs found
A Kochen-Specker system has at least 22 vectors (extended abstract)
At the heart of the Conway-Kochen Free Will theorem and Kochen and Specker's
argument against non-contextual hidden variable theories is the existence of a
Kochen-Specker (KS) system: a set of points on the sphere that has no
0,1-coloring such that at most one of two orthogonal points are colored 1 and
of three pairwise orthogonal points exactly one is colored 1. In public
lectures, Conway encouraged the search for small KS systems. At the time of
writing, the smallest known KS system has 31 vectors. Arends, Ouaknine and
Wampler have shown that a KS system has at least 18 vectors, by reducing the
problem to the existence of graphs with a topological embeddability and
non-colorability property. The bottleneck in their search proved to be the
sheer number of graphs on more than 17 vertices and deciding embeddability.
Continuing their effort, we prove a restriction on the class of graphs we
need to consider and develop a more practical decision procedure for
embeddability to improve the lower bound to 22.Comment: In Proceedings QPL 2014, arXiv:1412.810
On the Quantum Chromatic Numbers of Small Graphs
We make two contributions pertaining to the study of the quantum chromatic
numbers of small graphs. Firstly, in an elegant paper, Man\v{c}inska and
Roberson [\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016]
gave an example of a graph on 14 vertices with quantum chromatic
number 4 and classical chromatic number 5, and conjectured that this is the
smallest graph exhibiting a separation between the two parameters. We describe
a computer-assisted proof of this conjecture, thereby resolving a longstanding
open problem in quantum graph theory. Our second contribution pertains to the
study of the rank- quantum chromatic numbers. While it can now be shown that
for every , and are distinct, few small examples of
separations between these parameters are known. We give the smallest known
example of such a separation in the form of a graph on 21 vertices
with and . The previous record was held by a
graph on 57 vertices that was first considered in the aforementioned
paper of Man\v{c}inska and Roberson and which satisfies
and . In addition, provides the first
provable separation between the parameters and .
We believe that our techniques for constructing and lower bounding its
orthogonal rank could be of independent interest
Dagger and Dilation in the Category of Von Neumann algebras
This doctoral thesis is a mathematical study of quantum computing,
concentrating on two related, but independent topics. First up are dilations,
covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the
second topic: effectus theory. Both chapters, or rather parts, can be read
separately and feature a comprehensive introduction of their own
Foundations of Quantum Theory: From Classical Concepts to Operator Algebras
Quantum physics; Mathematical physics; Matrix theory; Algebr