I have seen multiple definitions of the depolaring channel on $N$ qubits :
- $\varepsilon(\rho)=(1-p)\rho + \frac{pI}{2^N}Tr(\rho)$ (Depolarizing channel for $n$ qubits: why is there a trace term?)
- $\varepsilon(\rho)=(1-p)\rho + \frac{p}{4^N-1}\sum_i P_i \rho P_i$ where $\{P_i\}$ are all the Pauli strings except identity. (https://quantumai.google/reference/python/cirq/DepolarizingChannel)
- $\varepsilon(\rho)=(1-p)\rho + \frac{p}{3N}\sum_i P_i \rho P_i$ where $\{P_i\}$ are Pauli strings supported on a single qubit ($X_1, Y_1, Z_1, X_2, Y_2 ...$)
- The channel that damp every Pauli string $P$ by a factor $e^{-\gamma w(P)}$ where $w(P)$ is the length of $P$ (number of non identity Pauli matrices). (https://arxiv.org/pdf/2407.12768 below eq 1)
These are not equivalent. In particular, (1) and (2) do not make long Pauli strings decay faster than short ones, therefore are obviously incompatible with (4). How to reconcile these different definitions ?
