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A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors.
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What is the stabilizer group of a $|W\rangle$ state?
As stated in the Schuch's answer: the W state is not a stabilizer state.
We can also find some operators which can "stabilizer" $W_{3}$. You can refer to Entanglement detection in the stabilizer form …