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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?

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  • $\begingroup$ books.google.com/… $\endgroup$ – AHusain Jan 14 at 4:59
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    $\begingroup$ it's not clear what you are asking. What you mean by "apply"? If you mean experimentally, then you do not have access to the global unitary which you can use to express the map. You might be able to do some more or less crude modelling of the system-environment interaction, but how exactly depends entirely on the context. You will often only know the net (non-unitary) effect on the system. $\endgroup$ – glS Jan 14 at 18:24
  • $\begingroup$ Yeah, I mean experimentally. So, if I want to transform some state A to B, how do I do it then, if I don't have access to the global unitary (in terms of this way of modelling the system) $\endgroup$ – Mahathi Vempati Jan 16 at 3:09

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