The answer to your question, generally, is No.
In People v. Brown 69 Cal. App. 2d 602, a California appellate court stated clearly what had been understood implicitly from common law, namely that a U.S. State has criminal jurisdiction only over its own territory when the crime itself is committed within that jurisdiction, even if (for example) the crime is committed on federal land, so long as the state has not ceded that right explicitly. Since the territory of another state does not fall within the criminal jurisdiction of the first state, the first state could not prosecute crimes where the criminal act was committed in another state.
Jurisdiction can get fuzzy when parts of a crime are committed within different jurisdictions, but even then, as stated in United States v. Anderson - 328 U.S. 699 (1946): “[T]he locus delicti must be determined from the nature of the crime alleged and the location of the act or acts constituting it.”
For example, if Kansas makes purchasing and transporting marijuana illegal, and you were to drive from Kansas to Colorado to purchase marijuana, and then transport it back home to Kansas, Kansas could prosecute you for transporting a controlled substance, entering the state of Kansas with a controlled substance, or even conspiring (in Kansas) to enter Colorado for the purpose of acquiring marijuana, but not on the purchase there per-se.
So, regarding ownership, again the answer is No. An example of jurisdictional claims by a state is Texas Penal Code Chapter 1 Sec 1.04 "This state has jurisdiction over an offense that a person commits ... [when] either the conduct or a result that is an element of the offense occurs inside this state". So, a state can prosecute you for conspiring to leave your state to acquire something illegal, it can prosecute you for intending to use the possession to commit a crime inside your former state, or in the commission of a crime against your state. However, unless an act, in whole or part, is committed inside the borders, the state cannot prosecute you.