I have some very basic questions about stabilizers.
What I understood:
To describe a state $|\psi \rangle$ that lives in an $n$-qubit Hilbert space, we can either give the wavefunction (so the expression of $|\psi\rangle$), either give a set of commuting observable that $|\psi\rangle$ is an eigenvector with $+1$ eigenvalue.
We define a stabilizer $M$ of $|\psi \rangle$ as a tensor product of $n$ Pauli matrices (including the identity) that verifies $M |\psi \rangle = |\psi\rangle$.
And (apparently) we need $n$ stabilizers to fully define a state.
The things I don't understand:
- How can a stabilizer necessarily be a product of Pauli matrices?
With $n=1$, I take $|\psi \rangle = \alpha | 0 \rangle + \beta |1 \rangle$, excepted for specific values of $\alpha$ and $\beta$, this state is only an eigenvector of $I$ (not of the other pauli matrices). But saying $I$ is the stabilizer doesn't give me which state I am working for.
- How can we need only $n$ stabilizers to fully define a state?
With $n$ qubits we have $2^n$ dimensional Hilbert space. I thus expect to have $2^n$ stabilizers, not $n$ to fully describe a state.
I am looking for a simple answer. Preferably an answer based on the same materials as my question, if possible. I am really a beginner in quantum error correction.
I learned these things within a 1-hour tutorial, so I don't have references for which book I learned this from. It is what I understood (maybe badly) from the professor talking.