One simple answer: private property cannot be maintained without authority to maintain it, because people would immediately appropriate what they need, and the force of law, police, etc would not be there to stop them. It is through these forms of state power that owners are currently able to combat activities such as theft, squatting/trespassing, etc, thereby keeping these activities relatively in check, ie. maintaining their property.
Of course there is the issue of privately-owned security forces, police, armies, prisons, etc. "Anarcho"-capitalists feel entitled to call themselves such because they don't consider these to be forms of government. (They also have a funny definition of capitalism.) To my mind these examples just demonstrate a different form of governmental power in which it is more transparent that the rich have hired mercenaries (a condition that the liberal form of government obscures to some extent).
There is another interesting way of approaching the question. Private property itself functions as a form of authority in that there is an authority held over individuals by the sanctity of property. In this approach, one might view the forms taken by society to enforce property as a social/material actuation of this ideological system. This helps explain the existence of the moral systems in which people believe it is wrong to infringe on property rights and so on--what we experience is not simply a world full of private property that we cannot access because it is protected by armed guards (as some anarchists portray it). This is true, but it is also a world in which most people truly believe in the existing system and in a whole lot of unquestioned abstractions which they hold to be irreproachable, and without these beliefs the armed guards would be nothing.
As for how private property can be abolished: The first paragraph might make it seems as if the abolition of the state would then necessarily lead to the abolition of private property through appropriation. However, just as anarchists reject the idea of using authoritarian measures to abolish private property, we also reject the idea that what we want is simply a matter of abolishing the government, that "everything else" will follow from there. Anarchists are, after all, opposed to all forms of authority, and generally do not believe in confronting them in separation from one another. Most anarchists would probably agree that private property can be abolished through the insurrection of self-actualized individuals and collectivities who self-organize without authority between each other nor between themselves and any higher powers (state, god, property, etc) to free their lives from the systems that have dominated them. This effort of making our lives our own (of appropriating them) is from a certain viewpoint the abolition of private property, although it may be much more as well. It may involve a lot of willpower, but by no means requires authority--in fact, I'd argue that authority as I define it can only be a fetter to this effort.
[edited for italics]
