How to find density matrix of 3 qubit W state?

Question

Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?

Answer 1

We can get the density matrix $\rho$ for a pure state $|\psi\rangle$ using

$$\rho=|\psi\rangle\langle\psi|$$

And to write $\rho$ using the Pauli basis $\left\{ P_i \right\}$,

$$\rho = \sum_{i=0}^N \rho_i P_i$$

we use the fact that $\rho_i = \text{tr} \left( \rho P_i \right)$

See the answers of this question for more details.

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Note: we can easily represent a quantum state in Pauli basis using Qiskit as follows:

from qiskit.quantum_info import DensityMatrix, Statevector, SparsePauliOp
psi = Statevector([0, 1, 1, 0, 1, 0, 0, 0])
rho = DensityMatrix(psi)
pauli_op = SparsePauliOp.from_operator(rho)
print(pauli_op)