You can’t shoot someone for stealing at all. Period! Not in the US anyway. Under some circumstances you can use deadly force to thwart a forcible felony which is a very narrow category of crimes.
You can use deadly force to protect yourself or another from death or great bodily harm.
While you are correct that you can’t use deadly force to stop someone from stealing a paper clip you can’t use deadly force to stop someone from stealing the Hope Diamond either.
An exception would be if the person who wanted to steal your paper clip decided to take it from you by the threat of using force that would put you in reasonable fear of death or great bodily harm.
ARIZONA REVISED STATUTES, from
https://www.azleg.gov/arsDetail/?title=13. Note that
13-411 includes prevention of first- or second-degree burglary.
13-408.
Justification; use of physical force in defense of property
A person is justified in using physical force against another when and to the extent that a reasonable person would believe it necessary to prevent what a reasonable person would believe is an attempt or commission by the other person of theft or criminal damage involving tangible movable property under his possession or control, but such person may use deadly physical force under these circumstances as provided in sections 13-405, 13-406 and 13-411.
13-405.
Justification; use of deadly physical force
A. A person is justified in threatening or using deadly physical force against another:
1. If such person would be justified in threatening or using physical force against the other under section 13-404, and
2. When and to the degree a reasonable person would believe that deadly physical force is immediately necessary to protect himself against the other's use or attempted use of unlawful deadly physical force.
B. A person has no duty to retreat before threatening or using deadly physical force pursuant to this section if the person is in a place where the person may legally be and is not engaged in an unlawful act.
13-406.
Justification; defense of a third person
A person is justified in threatening or using physical force or deadly physical force against another to protect a third person if, under the circumstances as a reasonable person would believe them to be, such person would be justified under section 13-404 or 13-405 in threatening or using physical force or deadly physical force to protect himself against the unlawful physical force or deadly physical force a reasonable person would believe is threatening the third person he seeks to protect.
13-411.
Justification; use of force in crime prevention; applicability
A. A person is justified in threatening or using both physical force and deadly physical force against another if and to the extent the person reasonably believes that physical force or deadly physical force is immediately necessary to prevent the other's commission of arson of an occupied structure under section 13-1704, burglary in the second or first degree under section 13-1507 or 13-1508, kidnapping under section 13-1304, manslaughter under section 13-1103, second or first degree murder under section 13-1104 or 13-1105, sexual conduct with a minor under section 13-1405, sexual assault under section 13-1406, child molestation under section 13-1410, armed robbery under section 13-1904 or aggravated assault under section 13-1204, subsection A, paragraphs 1 and 2.
B. There is no duty to retreat before threatening or using physical force or deadly physical force justified by subsection A of this section.
C. A person is presumed to be acting reasonably for the purposes of this section if the person is acting to prevent what the person reasonably believes is the imminent or actual commission of any of the offenses listed in subsection A of this section.
D. This section includes the use or threatened use of physical force or deadly physical force in a person's home, residence, place of business, land the person owns or leases, conveyance of any kind, or any other place in this state where a person has a right to be.