Quantum Colorings of Spheres

Abstract:Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for $n \in \{2,4,8\}$, any graph $G$ which admits a real $n$-dimensional orthogonal representation is quantumly $n$-colorable. This result can be recast as the statement that the real sphere $S^{n-1}$ is quantumly $n$-colorable for these values of $n$. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide candidate finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if $S^{n-1}$ is quantumly $n$-colorable, then either $n=2$ or $n$ is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order $n$ exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the existence of a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that $S^{n-1}$ admits a rank-one quantum $n$-coloring if and only if $n \in \{2,4,8\}$, thereby settling a conjecture of Zeng and Zhang, as does the fact that for all $m \geq 1$, there exists a catalytic zero-error remote state preparation protocol for real $m$-qubit states with $m$ bits of communication and which consumes $m$ ebits.

arXiv:2606.10872 [quant-ph]

(or arXiv:2606.10872v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.10872 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.