Depolarizing channel on GHZ-state

Question

Consider a GHZ-state $|\psi\rangle =\frac{1}{\sqrt{2}}(|0\rangle^{n}+|1\rangle^n)$, and consider a depolarizing channel that maps a density matrix $$\rho\to(1-\lambda)\rho + \frac{\lambda}{2^d}I.$$

Now we want to see the effect of the depolarizing channel on the GHZ state. What is the natural way to think about this? Two options that came to mind:

• Apply the channel on the state as a whole (with $d=n$)
• Apply the channel on each individual qubit (with $d=1$)

Which of these two (or what other option) best suits depolarizing noise in practice?

Note that I also tried this for a dephasing channel ($\rho\to(1-\lambda)\rho+\lambda Z\rho Z$), in which case the two options above result in the same resulting state.

Answer 1

Either can be considered (except I think you've got your values of $d$ swapped around, but that's easy to check because you want your output to have trace 1 if your input did).

In terms of which is the closest reflection of an actual experiment, you want to apply it to each individual qubit.