SR20 Instructor pilot reported an engine malfunction during initial climb during a training flight. The Instructor took control and returned to the departure airport; landing safely on an opposite direction runway.

Narrative

I was giving instruction in a Cirrus SR20. We got our departure off of runway XX; with a left turnout clearance to a heading of 180. All of our runups went well. On takeoff the student applied full power; no abnormal indications were observed. Upon reaching approximately 500' AGL; we were advised to switch to Departure frequency. As the student was making contact; while in the turn at approximately 700' AGL; or 1500 MSL; I happened to observe a loss of power from around 95% drop to approximately 30% power. I glanced at the students hand to see if he pulled the power lever aft; however it was still in the full forward position. I asked if he reduced power and he said no. I grabbed flight controls from there; cycled the power lever full aft and forward a few times to see if a response was generated however the engine was not responding to any input. From there I [requested assistance]; and decided to land on runway XY. Once I observed I was able to safely land on the runway with the remaining power available; I brought the mixture to the Cutoff position and landed the aircraft safely without power causing no damage. After inspection; the probable cause of the power loss was due to the throttle cable detaching from the engine controls.