Quieting existing hardwood floor

I recently purchased a home which has a nice hardwood installed floor on the second floor. However, I’ve noticed that that you can easily hear people on the second floor (talking at a reasonable volume), while standing in the corresponding room on the first floor.

I was wondering what can be done to dramatically reduce the sound, while not throwing away (or covering) the nice hardwood floor.

One idea I had was to remove the hardwood, install underlay then install the same hardwood panels I just removed. I was wondering, how difficult this would be. I am open to to different ideas.