When you are in Milla's mind, it may be difficult to track down a vault. You might also notice that achievements suggest there is a secret in her mind, but you can't find it. This is how you hit two birds with one stone (is that the right phrase?).
After you enter her mind, make your way to the area where Milla asks you to go through the hoops using levitation. You'll be standing on a flying platform where your classmates or some random NPC's are dancing, so walk around it and look at the walls. It's difficult to spot, but there is a small door on the wall with a very small platform on which you can land. To make sure you reach it, go up to the part where you need to levitate into the currents in order to reach the ladder. Once you are certain you can make the jump, make it and levitate towards the entrance. If you happen to fall, then you can use the springs to quickly get back up and retry.
Now that you are here, you'll notice there is only a vault and some toys. If you are during the training mission, then Milla will beg you to come back to the party and not investigate further. Get the vault instead and you'll get some pretty interesting background-story, but we still don't have that "secret". To get it, punch the chest in the corner and it will open. The orange light might seem suspicious, but just jump in for a quick peek and an achievement (if your system supports it). Once you have seen enough, simply jump into the obvious hole in the middle to return where you left off.