9 research outputs found
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
Non-Locality of Experimental Qutrit Pairs
The insight due to John Bell that the joint behavior of individually measured
entangled quantum systems cannot be explained by shared information remains a
mystery to this day. We describe an experiment, and its analysis, displaying
non-locality of entangled qutrit pairs. The non-locality of such systems, as
compared to qubit pairs, is of particular interest since it potentially opens
the door for tests of bipartite non-local behavior independent of probabilistic
Bell inequalities, but of deterministic nature
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
Two new non-equivalent three-qubit CHSH games
In this paper, we generalize to three players the well-known CHSH quantum
game. To do so, we consider all possible 3 variables Boolean functions and
search among them which ones correspond to a game scenario with a quantum
advantage (for a given entangled state). In particular we provide two new three
players quantum games where, in one case, the best quantum strategy is obtained
when the players share a state, while in the other one the players have a
better advantage when they use a state as their quantum resource. To
illustrate our findings we implement our game scenarios on an online quantum
computer and prove experimentally the advantage of the corresponding quantum
resource for each game.Comment: 19 pages, 3 figure
Counterexamples in self-testing
In the recent years self-testing has grown into a rich and active area of
study with applications ranging from practical verification of quantum devices
to deep complexity theoretic results. Self-testing allows a classical verifier
to deduce which quantum measurements and on what state are used, for example,
by provers Alice and Bob in a nonlocal game. Hence, self-testing as well as its
noise-tolerant cousin -- robust self-testing -- are desirable features for a
nonlocal game to have.
Contrary to what one might expect, we have a rather incomplete understanding
of if and how self-testing could fail to hold. In particular, could it be that
every 2-party nonlocal game or Bell inequality with a quantum advantage
certifies the presence of a specific quantum state? Also, is it the case that
every self-testing result can be turned robust with enough ingeniuty and
effort? We answer these questions in the negative by providing simple and fully
explicit counterexamples. To this end, given two nonlocal games
and , we introduce the -game, in which the players get pairs of questions and choose
which game they want to play. The players win if they choose the same game and
win it with the answers they have given. Our counterexamples are based on this
game.Comment: Added references, changed titl
Quantum independence and chromatic numbers
In this thesis we are studying the cases when quantum independence and quantum chromatic numbers coincide with or differ from their classical counterparts. Knowing about the relation of chromatic numbers separation to the projective Kochen-Specker sets, we found an analogous characterisation for the independence numbers case. Additionally, all the graphs that we studied that had known quantum parameters exhibited both the separation between the classical and quantum independence numbers and the separation between the classical and quantum chromatic numbers. This observation and the Kochen-Specker connection suggested the possibility of the chromatic and independence numbers separations occurring simultaneously. We have disproved this idea with a counterexample. Furthermore, we generalised Manĉinska-Roberson’s example of the chromatic numbers separation to an infinite family. We investigate some known instances with strictly larger quantum independence numbers in-depth, find a more general description and generalise Piovesan’s example. Using the Lovász theta bound, we prove that there is no separation between the independence numbers in bipartite and perfect graphs. We also show that there is no separation when the classical independence number is two; and that the cone over a graph has the same quantum independence number as the underlying graph