Separation between quantum Lov\'asz number and entanglement-assisted zero-error classical capacity

Quantum Lov\'asz number is a quantum generalization of the Lov\'asz number in graph theory. It is the best known efficiently computable upper bound of the entanglement-assisted zero-error classical capacity of a quantum channel. However, it remains an intriguing open problem whether quantum entanglement can always enhance the zero-error capacity to achieve the quantum Lov\'asz number. In this paper, by constructing a particular class of qutrit-to-qutrit channels, we show that there exists a strict gap between the entanglement-assisted zero-error capacity and the quantum Lov\'asz number. Interestingly, for this class of quantum channels, the quantum generalization of fractional packing number is strictly larger than the zero-error capacity assisted with feedback or no-signalling correlations, which differs from the case of classical channels.