Why is entropy increased when a bit is erased?

I'm writing a report on Landauers principle, and I can see that compressing the volume of the "merged" state requires some work for the system to be able to continue. But, I cannot accurately formulate exactly why mapping 2 possible states to 1 firm state increase "disorder"? If someone could make it clear to me, that would be greatly appreciated.

• From the physics side, any interaction introduces more microstates and thus increases disorder. Your "work" means interactions. From the information entropy side somebody who knows about it should answer. Commented Apr 9, 2017 at 4:11
• Hello! if you think the answer I posted is satisfactory you can select it as the correct one. Otherwise you can point out why it wasn't helpful. Hope it helped! Commented Apr 23, 2017 at 5:19

The laws of physics are, at a fundamental level, reversible, so "mapping 2 possible states to 1 firm state" is actually imposible. This is easy to see: if you are in one of the two initial state you certainly know where you are going (to the 1 firm state) but if you are in the final state you don't know where did you came from (it could be either of the 2 possible states).

However, bits can be "efectively" erased. Supose you have a spin that can be either up or down (that's the same as a bit that can be 0 or 1) and you want to "erase" the information of that spin. You would want to make that spin up whether it was up or down (a.k.a. mapping two possible states to 1 state).

In order to do so, you have to send the information of the original state somewhere, it just can't dissapear. What you can do is this, take another spin in the enviroment and map your spin state to this one and then "erase" your spin. for example

Initial state: Your spin is down

final state: your spin is up and enviroment spin is down

Now you "erased" the information within your system but you have stored the missing bit by fliping a spin in the enviroment. That information is tecnically lost, so you have indeed erased a bit for your system. Evenmore, in order to do so, you had to increase disorder in the enviroment, what caused an increase in entropy.

if you do the math you'll see that the entropy increase is ln(2) which accounts for the two possible configurations that you have added to the enviroment.

Landauer Principle is very important because it relates information entropy with thermodynamic entropy in a very clear way.

Hope this makes it clear.