<p>For the sake of completeness, I'll note that my version of the proof appears in</p>

<ul>
<li>NdB. <em>A linearized stabilizer formalism for systems of finite dimension</em>.<br> <a href="http://www.rintonpress.com/journals/qiconline.html#v13n12" rel="nofollow">Quantum Information &amp; Computation <strong>13</strong> (pp.&nbsp;73&ndash;115)</a>, 2013.<br> [<a href="http://arxiv.org/abs/1102.3354v4" rel="nofollow">arXiv:1102.3354v4</a>]</li>
</ul>

<p>appearing as <em>Lemma&nbsp;B.3</em> (page 38) in the published version, and <em>Lemma&nbsp;12</em> (page 23) in the arXiv preprint; in both cases occuring in Appendix B.</p>

<p>If anyone can point to an reference to a proof which is older than this question, I'll accept and reward the earliest such reference which is provided.</p>

CC BY-SA 3.0
2013-01-07T15:31:05.380Z
2013-01-07T15:31:05.380Z
Niel de Beaudrap
<h2>Context.</h2>

<p>I am writing on topics such as the <a href="http://en.wikipedia.org/wiki/Gottesman%E2%80%93Knill_theorem" rel="nofollow">Gottesman-Knill theorem</a>, using Pauli stabilizer groups, but in the case of <em>d</em>-dimensional qudits &mdash; where <em>d</em> 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 <em>d</em> prime or <em>d</em> a prime power, and makes use of finite fields; I'm considering instead the cyclic groups &#8484;<sub><em>d</em></sub>&nbsp;.)</p>

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

<ul>
<li><p>I'm writing about a result which is well-known for <em>d</em>&nbsp;=&nbsp;2 (and easily generalized to <em>d</em> prime):</p>

<blockquote>
  <p>A stabilizer group stabilizes a unique pure state if and only if it is maximal</p>
</blockquote>

<p>where by maximality, I mean that any extension either lies outside the Pauli group, or is non-abelian, or contains operators without +1 eigenvalues.</p></li>
<li><p>Proofs of such results for <em>d</em> prime usually rely on the fact that &#8484;<sub><em>d</em></sub><sup>2n</sup> is a vector space (<em>i.e.</em>&nbsp;that &#8484;<sub><em>d</em></sub> is a field): this does not hold for <em>d</em> composite. There are two recourses: generalize the existing proofs in a way that is robust to the existence of zero divisiors (<em>e.g.</em> using tools such as the <a href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>), or avoid number theory altogether and use ideas such as orthogonality relations of Pauli operators.</p></li>
</ul>

<h2>Problem.</h2>

<p>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). </p>

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

<p><strong>Note</strong> &mdash; the result I'm considering is not one which relates the <em>cardinality</em> 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 <em>which cannot be extended</em> 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 <em>d</em> being prime or &#8484;<sub><em>d</em></sub><sup>2n</sup> being a vector space.</p>

<reference-request><quantum-computing>
Reference request: number-theory-free proof that maximal stabilizer groups determine unique states
Niel de Beaudrap
# Reference request: number-theory-free proof that maximal stabilizer groups determine unique states

(Asked by: username_0 on 2011-04-22)

## Context.

I am writing on topics such as the [Gottesman-Knill theorem](http://en.wikipedia.org/wiki/Gottesman%E2%80%93Knill_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 ℤ*d* .)

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 ℤ*d*2n is a vector space (*i.e.* that ℤ*d* 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](http://en.wikipedia.org/wiki/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]](http://arxiv.org/abs/quant-ph/9608048) and [[quant-ph/9608049]](http://arxiv.org/abs/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]](http://arxiv.org/abs/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 ℤ*d*2n being a vector space.

**Tags:** reference-request, quantum-computing


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## 1 answers

### Answer by username_0 on 2013-01-07. Score: 4 | Accepted answer

For the sake of completeness, I'll note that my version of the proof appears in

* NdB. *A linearized stabilizer formalism for systems of finite dimension*.  
   [Quantum Information & Computation **13** (pp. 73–115)](http://www.rintonpress.com/journals/qiconline.html#v13n12), 2013.  
   [[arXiv:1102.3354v4](http://arxiv.org/abs/1102.3354v4)]

appearing as *Lemma B.3* (page 38) in the published version, and *Lemma 12* (page 23) in the arXiv preprint; in both cases occuring in Appendix B.

If anyone can point to an reference to a proof which is older than this question, I'll accept and reward the earliest such reference which is provided.


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