For the purposes of this answer, Player 1 is the one with the Ax hand. Player 2 is the player with the non-Ace hand. I'm also assuming that Player 2 has a random hand - in other words, sometimes their starting hand will also include an Ace.
This is an interesting question. My conclusion is that Player 2, knowing that Player 1 has an Ace, has the advantage, but it won't be easy to win!
Furthermore, the advantage comes from the one card knowledge, as opposed to something else like position.
If Player 2 knows the opponent has an Ace, then it would be imperative to narrow down their range drastically and only play things like Broadway cards with an Ace in the hand. Anything else and they are taking too many risks.
However, Player 2 narrowing down the range in this manner gives Player 1 a lot of information too! Player 2 needs to avoid a scenario where Player 1 always knows that they are on an Ace. So Player 2 would have to do some range balancing in order to throw Player 1 off their cards.
And that's really where the advantage comes in - Player 2 can balance their range. But Player 1, with the Ace, is basically "locked-in" to their range! This is the angle that needs to be exploited in order for Player 2 to win.
So, it's not easy, but in order to win, Player 2 would need to be willing to showdown relatively frequently with trashy hands and be prepared to lose with some of those hands. They would need to keep a tight range that keeps all the Broadways with an Ace and occasionally also show down with J2o and T4s
Player 2 would also consider playing other things that flop well and completely miss the range for Player 1. For example something like 8♥ 9♥ wouldn't be a terrible hand if the flop gives Player 2 multiple draws and/or a pair. If you can get to a board like that, then the hands are worthwhile to consider.
Depending on skill, Player 2 can also bluff at a lot of boards. The board texture is going to be either very good or very scary for Player 1. So, as stated above, a board with good draws, that miss the Ax range, are good to make a move at and represent those hands. All your low straights and majority of flushes have the potential to be hands where Player 2 can push Player 1 off their Ace. So Player 2 could consider playing all monochrome boards and connected boards that provide multiple straight possibilities. Again, Player 1 is somewhat locked-in to the range (albeit a high one!) but that means that the low stuff is going to miss them too! That's the part that Player 2 can exploit and potentially win over the long run.
Having said all that, in the process of writing all this, I've also come up with countermoves for Player 1. Basically, they would need to play very aggressive pre-flop. You would need to make it very, very expensive for Player 2 to enter the pot. You need to avoid those board textures that give a lot of draws; best way to do that is raise, raise, raise pre-flop! Player 1 would NEVER just check or call pre-flop. In this process, they would win by blinding-off their opponent and giving Player 2 no cheap flops. The Ace is most valuable pre-flop. After that, the equity is more likely to drop than anything else.
