How to find density matrix of 3 qubit W state?

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?

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.

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)

• Sir kindly give me the full detailed solution of my state. Nov 21, 2023 at 6:45
• I downvoted this because it doesn't really answer the question. Nov 21, 2023 at 11:54