In this answer: https://quantumcomputing.stackexchange.com/a/5896/19951 to a question asking how to prove (or disprove) universality, it is stated that a universal gate set "has to be capable of producing any state starting from the all zeros state."
My question: is demonstrating that a gate set is capable of producing any state starting from the zero state a sufficient condition to prove that the gate set is universal? If so, how can you guarantee that this condition ensures that any unitary can be constructed from the gate set?
