  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.
