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Consider one dimensional quantum codes $[[ n,k=0]] $. One way to describe them is using stabilizer framework with $n$ independent Pauli matrices. Usually, one considers them in the graph state model as well. If you use the distance definition by treating them as stabilizer codes, then as $N(S)=S$, it would detect all errors. Is there a notion of distance of such graph states? I have seen distance being defined after restriction that they are non-degenerate with a mention that graph states are traditionally treated as non-degenerate. Are there any applications where this distance (or some other notion of distance of graph states) plays an important role?

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    $\begingroup$ A one-dimensional stabiliser code is just a stabiliser state; the distance of a state is ill-defined. But yeah, if you encode a state, and you know what state, you'll always be able to correct for an error. $\endgroup$
    – JSdJ
    Aug 23 at 9:24
  • $\begingroup$ BTW - a one dimensional quantum error correcting code is not necessarily a cluster state (in the usual sense of the word). Cluster states are a particular type of graph states, which are themselves a subset of the set of stabiliser states. Every graph state is local-clifford equivalent to a graph state, though. $\endgroup$
    – JSdJ
    Aug 23 at 9:26
  • $\begingroup$ see previous post : quantumcomputing.stackexchange.com/questions/13354/… I think d=infinity makes sense $\endgroup$
    – unknown
    Aug 23 at 15:35

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The distance of an $[[n, 0]]$ code is defined to be the smallest non-zero weight of any stabilizer.

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