It really depends on what one considers as a "state" or not. There are those who would consider decentralized communities using militias to defend themselves from attempts to enforce authority and property rights as qualifying as a state. If that is the definition being used, sure, a state could enforce the abolition of private property. I don't personally think that definition of state is what is normally meant when people say the word, and I think people often use this definition of state as a way of justifying their support for more centralized and authoritarian states.

And, if by "state" one means an entity that claims sovereign authority over a certain area, and enforces such authority with violence, then no, such an entity could never abolish private property. I say this because, the state that forms to lead the working class is never made up of the entirety of the working class. It is made up of a political class claiming to act on behalf of the working class. Yet, as they come to manage the resources they take from the private sector (which is a necessity if they are to maintain their sovereign authority) this political class becomes the new ruling and owning class. They become the new bosses and owners that everyone else is coerced into working for, obeying and serving.

Thus the private property that is "communalized" in this manner is not actually communalized at all, but rather becomes the private property of the new ruling political class.

And, again, the reason they have to put this property under state control instead of allowing communities and workers to directly manage it, is that allowing direct management will mean the sovereign authority of the state won't be able to maintain control of the communities and individuals they feel the need and right to rule. So, since political power over an area or group is impossible without economic power over them (and vice versa), the state will always feel the need to control the material resources people depend on in order to maintain control over the people they claim to represent.