electrolysis - Why can't pure water conduct electricity since it can be reduced at cathode and oxidised at anode?

We all know that pure water can't conduct electricity. But during electrolysis, if add a small amount of $\ce{HCl}$ acid inside, water could be decomposed to hydrogen gas and oxygen gas, as it is reduced at the cathode and oxidised at the anode. So why can't pure water just conduct electricity by electrolysis?