# A question from Aaronson 2004 paper

In Aaronson's paper about the efficient simulation of a stabilizer circuit (https://journals.aps.org/pra/pdf/10.1103/PhysRevA.70.052328), I have a problem with finding the reason why the following statement holds. Restating the statement in the end of the 4th page of the paper, Let $$\{R_{h+n}\}$$ with $$h=1...n$$ are the Pauli stabilizer of a state $$|{\psi}\rangle$$. If $$Z_a$$ commutes with all of $$\{R_{h+n}\}$$, then $$\sum_{h=1}^n c_hR_{n+h} = \pm Z_a$$ for a unique choice of $$c_1 \dots c_n \in {0,1}$$.

But for $$|\psi\rangle=|11\rangle$$, we have stabilizers of $${II, ZZ}$$ only, and its combination can make neither of single-qubit $$Z$$ gate.

If I have misunderstood, I will be very appreciative if you can correct it. Thank you in advance.

The terms $$II$$ and $$ZZ$$ do not uniquely specify the state $$|11\rangle$$ because you could equally have the state $$|00\rangle$$. Indeed, you should not include the identity term in your stabilizer. Thus, you need to add a second term, which could be either $$-ZI$$ or $$-IZ$$. Either way, you can easily see how to make a product $$-ZI$$ out of your stabilizers.