I wrote two pieces arguing for a typology of property rights. Short story shorter. Because property is a relation among men with respect to an object, we can classify property according to the relationship the property owner has with other people. The other people can be family/friends, anonymous strangers, and government. Property can also be distinguished by contracts and personal property which has the threat of violence and theft. These distinctions result in this chart.
Below is a quote from Further typologies of property rights.
The second part is more interesting, how to enforce contracts among anonymous strangers, 3 in the table, and how to ensure there is no violence and theft against anonymous strangers 4. Most economists are unclear on the distinction between the two options, however it is an important one. There is much evidence, international trade being the primary one, that state enforcement is unnecessary to protect contracts among anonymous strangers. However, as Gurri pointed out, the state is likely necessary to protect against violent expropriation from anonymous strangers.
As such, the state exists less to protect private property per se, and more to protect against a specific type of encroachment on private property. In fact, given that many major American cities did not have police departments until the mid 19th century, it seems state exist primary to prevent large scale violence.
I would like to use this framework to critique academic anarchists. They tend to focus on 3 in the table, whether the state is necessary to enforce contracts among anonymous strangers. Some academic anarchists also investigate whether stable rules can emerge in chaotic situations. However, both research agendas miss the hard question 4, whether a non-monopoly of force can prevent theft and violence by anonymous strangers in a modern city like environment or larger.
To the extent anarchism is a normative project, whether it is a desirable alternative to modern first world governments is an important question. This requires a mechanism to protect against anonymous third party theft and violence. David Friedman provided the theoretical mechanism, as well as an important case study. Unfortunately, there has been little focus on this question since.