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There is a well known $ [[3,1,2]]_3 $ qutrit stabilizer code with stabilizer generators $ XXX $ and $ ZZZ $. This code is related to a $ [[4,0,3]]_3 $ qutrit stabilizer state with stabilizer generators $ XXXI,ZZZI,ZZ^{-1}IZ,XX^{-1}IX $, see appendix A.2 of https://arxiv.org/abs/1503.06237 . And this $ [[4,0,3]]_3 $ stabilizer state seems to be equivalent to $ AME(4,3) $, an absolutely maximally entangled state of $ 4 $ qutrits.

The paper https://arxiv.org/abs/2212.06737 states "However, we prove the surprising result that there is truly only one AME state of four qutrits up to local unitary equivalence."

So, up to local unitary equivalence, are $ AME(4,3) $ and the $ [[4,0,3]]_3 $ stabilizer state unique and equal to each other? Does that imply that the $ [[3,1,2]]_3 $ qutrit stabilize code is also unique (i.e. any distance $ d=2 $ encoding of one logical qutrit into three physical qutrits is equivalent to the $ [[3,1,2]]_3 $ code by a local unitary)?

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    $\begingroup$ Hi Ian, you might find an answer here: arxiv.org/abs/2003.13639; e.g. page 9 Example 8 and table I/II on page 11. They talk of minimal support states, which might or might not (I keep forgetting) is equivalent to the construction from Helwig from MDS codes. $\endgroup$ May 14 at 21:12
  • $\begingroup$ @FelixHuber ah ok so that shows that AME(4,3) i.e. $ ((4,1,3))_3 $ is unique up to local unitaries. Does the correspondence between $ ((4,1,3))_q $ codes and $ ((3,q,2))_q $ codes mentioned in Proposition 7 of your paper have a uniqueness to it? Like can you use that to show that LU uniqueness of $ ((4,1,3))_3 $ implies LU uniqueness of $ ((3,q,2))_3 $? $\endgroup$ May 15 at 16:02
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    $\begingroup$ > Does the correspondence have a uniqueness to it? I do not think so, due to the reason that going from $(\!(n,1,d))$ to $((n-1,q,d-1)\!)$ is done trough a partial trace applied on the code projector. And you are free to apply the partial trace it at the coordinate you want. For codes that are not symmetric under permutation, you might end up with a different LU-inequalivalent code. $\endgroup$ May 15 at 18:08
  • $\begingroup$ Hmm ok maybe something like the techniques from arxiv.org/abs/quant-ph/9704043 that Rains uses to show that the $ [[4,2,2]], [[5,1,3]], [[6,0,4]] $ are all unique up to LU equivalence would be required to show $ [[3,1,2]]_3 , [[4,0,3]]_3 $ are unique up to LU. $\endgroup$ May 15 at 20:52
  • $\begingroup$ So the first step would be that, say, the qutrit $[\![3,1,2]\!]_3$ commutes with the group $\langle XXX, ZZZ\rangle$..? $\endgroup$ May 15 at 21:28

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