Local unitary versus local Clifford equivalence of stabilizer and graph states

The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU-equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest [Phys. Rev. A 71, 062323 (2005)]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence (LU double left right arrow LC) to include all stabilizer states represented by graphs with cycles of length neither 3 nor 4. To compare our result with Van den Nest 's, we show that any stabilizer state of distance delta=2 is beyond their criterion. We then further prove that LU double left right arrow LC holds for a more general class of stabilizer states of delta=2. We also explicitly construct graphs representing delta > 2 stabilizer states which are beyond their criterion: we identify all 58 graphs with up to 11 vertices and construct graphs with 2(m)-1 (m >= 4) vertices using quantum error-correcting codes which have non-Clifford transversal gates.