This is a fairly common way to make a locked room mystery work: the door was locked, there is but one key, and occasionally someone will say there was no sign of forced entry or of the lock being picked. I should note that this doesn't bother me, but I do think it's interesting: just how much evidence does lockpicking leave behind?
The thing to understand about picking locks is it's very unlikely to be useful to someone who is committing a crime unless that person needs it to appear as if no crime was committed at all. If someone were to steal your TV, for instance, there's no reason to pick the lock when they could more quickly use a destructive method--the crime will be obvious. Given that it is (generally) time-consuming to pick a lock, is highly suspicious if you are caught in the act, and requires the development of a specialized skill, the only reason you might consider using lockpicks in a crime is if you don't intend for anyone to know that the crime happened in the first place--something like espionage--because under normal circumstances, lockpicking doesn't leave behind any evidence that the lock's normal users will be able to detect.
However, while it doesn't leave behind obvious signs that the lock was tampered with, it (along with most forms of bypassing a lock) actually does leave behind some evidence that a forensic locksmith can identify (see this 2009 paper on the subject by datagram), but these require a specialist to identify, and would (probably) need the lock to be disassembled and viewed under high magnification*. And perhaps most critically, these marks only tell us that the lock was picked; it can't give us the precise time. In many cases it may not even be able to tell us the lock was successfully opened, only that an attempt was made.
As the paper I linked above notes, it's far more likely that the key system was compromised than that someone picked it; that is, someone probably simply obtained a key who should not have. That is more or less what happened in the case from which I got this screenshot, where someone social engineered their way into the office by tricking the guard into using their master key to get in there. And the forensic marks a key used by an unauthorized user leaves behind should be more or less identical to the marks of an authorized key user--that is to say, it effectively leaves no evidence at all. (Some techniques for copying keys might leave residue behind on the original, though.)
(As an aside, master keys make a lock less secure, both because there are now more keys that can open the door, and because a master keyed lock is generally easier to pick.)
And now you know enough that, when you write a locked-door murder mystery and need to dismiss the possibility that the criminal used lockpicks to open the door, you can have your detective say that forensics didn't find any sign that the lock was picked, rather than that the guard didn't find them, and you will make me and the six other locksporters who also play your game smile.
*I know there exist some imaging cameras that are used by locksmiths to inspect the inside of locks without disassembling them; I am assuming here that they are inadequate to the task of doing forensic analysis of the lock, but that is only an assumption. I've never interacted with one.
