This isn't true. Air can be compressed to exceptionally high pressures with very, very little energy. A car tire holds approximately 2 atm of pressure in it, air rises at 9.81 m/sec^2, 2atm of water pressure is 20.6 meters deep, so ballparking it the air would be moving at 20 meters per second, or 45 miles per hour.
TL;DR: With the energy required to inflate a car tire, you can cause air to exit the top of the water at 45 miles per hour. This force would be much more than the energy required to compress air to double atmospheric pressure.
Holy stuff you are like pooping out random numbers that accomplish nothing but show that you don't know what you are talking about. You cannot use pressure directly in a kinematics problem like that.
It is a fundamental property of gas laws that the change in internal energy is constant for any full cycle (going back to where it started) and that all energy is conserved. If you compress air, you do a certain amount of work in compressing it. If it then decompresses, it does an equal amount of work back (assuming it returns to the same temperature). I don't know why you are maintaining that air can be compressed with "very, very little energy" because any compression done with a low amount of energy will yield an equally low amount of energy. I can tell you exactly how much energy it will take to compress any gas if you tell me the final and initial pressure and volume. If you think you can handle the calculations yourself: knock yourself out:
Internal Energy = 1.5*(Number of molecules)*(Boltztmann's constant)*(Temperature in K)
Pressure*Volume = (Number of molecules)*(Boltztmann's constant)*(Temperature in K)
Work out = Change in Internal Energy + Heat Added
Just assume one mole of molecules to keep it simple.
