You can have a key stamped “do not duplicate” but an unscrupulous or non-caring key shop could make a duplicate key.

The “do not duplicate” notice stamped on keys is legally meaningless. Locksmiths are not required to honor this request,

One solution is the Schlage Primus that has a special key blank that has cuts on the side of the key besides cuts on the top for each lock. The key blanks have to be purchased with the coded side cuts. The top cut is done by the locksmith.

In the dorms, we had key cards. Your student ID would let you in the exterior door, but depending on the door, only if you lived in the building, or during "regular" hours. (After a certain time, all entry was limited to the front door only, and going past the staffer at the front desk.) Once inside, your room key card would actually open your door.

Anything that the tenant is in possession of can be duplicated. The only solution is to seek a solution that is difficult enough that they won't go to the trouble.

A standard blank simply stamped "Do not copy" are worthless, everyone will copy them. Most won't even look at the key other than to find the correct key-way and the copy machines don't read.

Restricted key blanks are much harder, machines copying setups won't have the right blank, most locksmiths will not copy them unless you verify you own the restricted lock. Unscrupulous locksmiths will copy them in a heartbeat (or someone with a grinder and some time). These make it more difficult to copy but not perfect.

RFID key locks use a coded card (some use something that looks like a key) each card has to be synced to open a specific lock so there is more management involved. You can also remove a card that has been lost. They can be duplicated but it is successively more difficult than a traditional key. The locks however, tend to be temperamental. You have to change the lock batteries every few months and sometimes they just wig out. Also, most don't like exposure to the elements. This solution makes it very difficult to copy and allows you to add/remove cards as tenants come and go but also a pain to manage.

These are just a few solutions. Others include Mag-stripe readers (like a credit card) which fit somewhere in the middle, more recent blue tooth locks which requires a synced phone to open, and some really exotic solutions that are very good High Security solutions but are beyond the budget of most landlords.

Find out which unit(s) they are going to and give notice to the tenant to correct. Change locks (at tenant's expense) and proceed to move them out if necessary- when (not if) it happens again..