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In addition to the Kraus operator and Choi matrix, the Pauli transfer matrix (PTM) is another useful representation of a quantum map, its matrix entries are

$(R)_{ij}=\frac{1}{d}tr\{P_i\Lambda(P_j)\}$,

where $d=2^n$ is the Hilbert space dimension, $P_i,i\in\{0,1,2,3\}$ is Pauli matrix, which can be used as basis, and $\Lambda$ is a map. I know that any operator can be expressed as $O=\sum_itr(P_iO)P_i$. But I don't understand the meaning of $\Lambda(P_j)$, why we need a map of Pauli basis?

Refs: https://arxiv.org/abs/1509.02921

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Pauli transfer matrix is usually used in tomography. Quantum channels are often assumed to be a completely positive, trace-preserving (CPTP) map. In the PTM representation of a quantum channel, it only contains real elements and provides a simple way to see whether the quantum operation is trace-preserving or whether the process is unital. For Clifford operation there is exactly one non-zero element in each row and column with unit magnitude. And for a composite map is simply the matrix product of the individual Pauli transfer matrices. They can be used to characterize and mitigate state preparation and measurement (SPAM) errors. It gives a comprehensive knowledge of the errors and provides a convenient way. $O_{ij}=Tr(P_iO(P_j))$

Particularly, for an initial state $\rho$, we denote its vectorization in the Pauli basis by the real column vector $|\rho\rangle\rangle =(\cdots \rho_P \cdots)^T\in \mathbb{R}^{2^n}$, where $\rho_P=Tr(\rho P)$ with $P\in\{I,X,Y,Z\}^{\otimes n}$ being the $n$-qubit Pauli matrices. Similarly, the measurement operator can be represented as a real row vector, namely, $\langle \langle O|=(\cdots O_P \cdots)$ with $O_P=2^{-n}Tr(OP)$. In this regard, the state is mainly measured in Pauli basis. So the states can be written as \begin{equation} |0 \rangle \rangle = \left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \end{array} \right), |1 \rangle \rangle = \left(\begin{array}{l} 1 \\ 0 \\ 0 \\ -1 \end{array} \right), \frac{|0 \rangle \rangle +|1 \rangle \rangle}{\sqrt{2}} = \left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \end{array} \right), , \frac{|0 \rangle \rangle -|1 \rangle \rangle}{\sqrt{2}} = \left(\begin{array}{l} 1 \\ -1 \\ 0 \\ 0 \end{array} \right), \frac{|0 \rangle \rangle + i|1 \rangle \rangle}{\sqrt{2}} = \left(\begin{array}{l} 1 \\ 0 \\ 1 \\ 0 \end{array} \right), \frac{|0 \rangle \rangle - i|1 \rangle \rangle}{\sqrt{2}} = \left(\begin{array}{l} 1 \\ 0 \\ -1 \\ 0 \end{array} \right). \end{equation} Similarly, the Pauli basis can be written as \begin{equation} \langle \langle I|=(1,0,0,0),\langle \langle X|=(0,1,0,0),\langle \langle Y|=(0,0,1,0),\langle \langle Z|=(0,0,0,1). \end{equation} So, why do we need to use this formalism? Usually, when we would like to quantify error or the distance between two states, we employ the metric of fidelity which is simple and clear. Think about this, if we calculate the fidelity between the ground state $|0\rangle $ and the superposition of the ground state and the excited state $|+\rangle$, the fidelity is 0.701. For the case of any pole state $|-\rangle, |i\rangle, |-i\rangle$, we will get the same fidelity. If the pole state here means an error, even though we have the error in different directions, we get exactly the same things, just one number. This makes fidelity not good for error mitigation. It could be very good at quantifying a quantum device, or quantum gates, but not suitable we need detailed error information, that's where other methods come in. With the PTM formalism, it kind of takes pictures for all sides of the black sphere, so you have a comprehensive idea of the error model

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