In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the unitary, but here we haven't taken environment into consideration.
So, we take environment into consideration and assume the state is in a product state with the environment And then apply the unitary on both, and trace out the environment.
Now, the unitary is something we specify, right? I imagine we have a machine where we can decide what unitary we want to apply and give it as input, and it gets applied on the state inside.
When we know that it might be coupled with the environment, do we apply the right unitary assuming the dimensions? Or do we just apply what we would have applied for our system only, and the dimensions get taken care of?
For example, do we just specify I, and I2 gets applied if it is a 2d system, and I4 gets applied if it's a 4d system?
Or do we exactly specify the matrix? Basically how do we apply these unitaries, atleast on a conceptual level?