for example, people have a "negative" right to be free of physical violence. If someone uses physical violence an another, that person forfeits their rights, and the state can use violence against them. It can also take away their freedom of movement and incarcerate them.

Agreed - rights that are guaranteed by prohibiting certain actions, i.e. you have a right to $FOO, meaning that $BAR is prohibited.

Compare with:

> They conceptualize rights "negatively", as things the state shouldn't take away from you.

The formulation seems a little different: you have a right to $FOO, meaning that $BAR is allowed ("the state shouldn't take [$BAR] away from you").

You have a negative right to freedom from the state: the state can't lock you up without a good reason.

Also relevant, you have a negative right to your property: someone can't take your property from you.

Contrast with a postive-right versions of these two things, right to due process and right to property protection by the police.

In the positive rights case, the state extends property protection via the law and police, thereby implicitly granting you property rights. This is a grant, and you are not allowed to defend your own property?

That's my rough reading of ops text.