Does the $|W\rangle$ state have a stabilizer?

Question

I've been practicing trying to find stabilizer groups for various multi-qubit states $|\psi\rangle$.

So far the only method I know is to start with the state $|0\rangle^n$ which is stabilized by the generating set $\lbrace Z_k\rbrace_{k=1}^n$, (where $Z_k$ is the pauli-Z acting on the kth qubit), and then finding a circuit that constructs $|\psi\rangle$, and evolving the stabilizer generators $S_i \to US_i U^{-1}$ for each gate $U$ in the circuit.

However, as the answer to this post hints, the $W$-state

\begin{align} |W_3\rangle = \frac{|100\rangle + |010\rangle + |001\rangle}{\sqrt{3}} \end{align}

requires non-Clifford gates in its construction, so it seems like you would end up with stabilizers that aren't a subgroup of the Pauli group (even though by definition Stabilizers must be subgroups of the Pauli group).

So I guess my questions are

• Do there exist stabilizers for $|W_3\rangle$ (or in general, any arbitrary state?)

• If not, how can you tell if a given state can or cannot be stabilized?

• Given an arbitrary state $|\psi\rangle$, is the best way to find the generating stabilizers by following through a circuit that constructs $|\psi\rangle$, or is there a more straightforward way?

Answer 1

The W state is not a stabilizer state - for a stabilizer state, the 1-site reduced density matrices must be maximally mixed or pure, which they aren't.

More generally, you can count how many complete set of stabilizers there are. In particular, there are finitely many (though exponential in the number of qubits), so there are also only finitely many stabilizer states. A random state (choosen e.g. according to the Haar measure) will therefore almost never be a stabilizer state.

You can find some more discussion how to identify stabilizer states at https://quantumcomputing.stackexchange.com/questions/3861/stabilizer-state-verification-and-specification-from-state-vector, however, it is not clear to me whether there is a method which is smarter than trying all possible stabilizers brute force.