# Reference request: number-theory-free proof that maximal stabilizer groups determine unique states

## Context.

I am writing on topics such as the Gottesman-Knill theorem, using Pauli stabilizer groups, but in the case of d-dimensional qudits — where d may have more than one prime factor. (I emphasize this because the vast majority of literature on stabilizer formalism in "higher dimensions" involves the cases of d prime or d a prime power, and makes use of finite fields; I'm considering instead the cyclic groups ℤd .)

For any dimension, I characterize a (Pauli) stabilizer group as an abelian subgroup of the Pauli group, in which every operator has a +1 eigenspace.

• I'm writing about a result which is well-known for d = 2 (and easily generalized to d prime):

A stabilizer group stabilizes a unique pure state if and only if it is maximal

where by maximality, I mean that any extension either lies outside the Pauli group, or is non-abelian, or contains operators without +1 eigenvalues.

• Proofs of such results for d prime usually rely on the fact that ℤd2n is a vector space (i.e. that ℤd is a field): this does not hold for d composite. There are two recourses: generalize the existing proofs in a way that is robust to the existence of zero divisiors (e.g. using tools such as the Smith normal form), or avoid number theory altogether and use ideas such as orthogonality relations of Pauli operators.

## Problem.

I actually do have a concise proof of this result already, essentially using no more than orthogonality relations of Pauli operators. But I suspect that I've seen something like it before, and I would like to refer to the prior art if I can (not to mention see if there are better techniques than the one I used, which while not onerous felt less than perfect).

Certainly Knill's papers [quant-ph/9608048] and [quant-ph/9608049] consider similar subjects and use similar techniques; but I couldn't find the result I was looking for there, or in Gottesman's [quant-ph/9802007]. I'm hoping that someone can point me to where such a proof might have been published before.

Note — the result I'm considering is not one which relates the cardinality of the group to the dimension of the stabilized space (which is nice, but trivial both to prove and to find references to); I'm concerned specifically with showing that any stabilizer group which cannot be extended stabilizes a unique state, and vice versa. A reference to a proof that any maximal stabilizer group has the same cardinality would be fine; but again, it must not rely on d being prime or ℤd2n being a vector space.