Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number

The quantum chromatic number of a graph $G$ 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 $\chi_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number $\chi(G)$ and the minimum dimension of orthogonal representations $\xi(G)$. It is known that $\xi(G) \leq \chi_q^{(1)}(G) \leq \chi(G)$. We answer three open questions about these relations: we give a necessary and sufficient condition to have $\xi(G) = \chi_q^{(1)}(G)$, we exhibit a class of graphs such that $\xi(G) < \chi_q^{(1)}(G)$, and we give a necessary and sufficient condition to have $\chi_q^{(1)}(G) < \chi(G)$. 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

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

Necessary and sufficient condition for quantum state-independent contextuality

We solve the problem of whether a set of quantum tests reveals state-independent contextuality and use this result to identify the simplest set of the minimal dimension. We also show that identifying state-independent contextuality graphs [R. Ramanathan and P. Horodecki, Phys. Rev. Lett. 112, 040404 (2014)] is not sufficient for revealing state-independent contextuality.Comment: 5 pages, 3 graph

Quantum state-independent contextuality requires 13 rays

We show that, regardless of the dimension of the Hilbert space, there exists no set of rays revealing state-independent contextuality with less than 13 rays. This implies that the set proposed by Yu and Oh in dimension three [Phys. Rev. Lett. 108, 030402 (2012)] is actually the minimal set in quantum theory. This contrasts with the case of Kochen-Specker sets, where the smallest set occurs in dimension four.Comment: 8 pages, 2 tables, 1 figure, v2: minor change

Kochen-Specker set with seven contexts

The Kochen-Specker (KS) theorem is a central result in quantum theory and has applications in quantum information. Its proof requires several yes-no tests that can be grouped in contexts or subsets of jointly measurable tests. Arguably, the best measure of simplicity of a KS set is the number of contexts. The smaller this number is, the smaller the number of experiments needed to reveal the conflict between quantum theory and noncontextual theories and to get a quantum vs classical outperformance. The original KS set had 132 contexts. Here we introduce a KS set with seven contexts and prove that this is the simplest KS set that admits a symmetric parity proof.Comment: REVTeX4, 7 pages, 1 figur

Systematic construction of quantum contextuality scenarios with rank advantage

A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.Comment: 8+8 page