dorsal/arxiv
View SchemaLocal unitary versus local Clifford equivalence of stabilizer and graph states
| Authors | Bei Zeng, Hyeyoun Chung, Andrew W. Cross, Isaac L. Chuang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611214 |
| URL | https://arxiv.org/abs/quant-ph/0611214 |
| DOI | 10.1103/PhysRevA.75.032325 |
Abstract
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 et al. [Phys. Rev. \textbf{A71}, 062323]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence ($LU\Leftrightarrow LC$) to include all stabilizer states represented by graphs with neither cycles of length 3 nor 4. To compare our result with Van den Nest et al.'s, we show that any stabilizer state of distance $\delta=2$ is beyond their criterion. We then further prove that $LU\Leftrightarrow LC$ 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 $2^m-1$ ($m\geq 4$) vertices using quantum error correcting codes which have non-Clifford transversal gates.
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"abstract": "The equivalence of stabilizer states under local transformations is of\nfundamental interest in understanding properties and uses of entanglement. Two\nstabilizer states are equivalent under the usual stochastic local operations\nand classical communication criterion if and only if they are equivalent under\nlocal unitary (LU) operations. More surprisingly, under certain conditions, two\nLU equivalent stabilizer states are also equivalent under local Clifford (LC)\noperations, as was shown by Van den Nest et al. [Phys. Rev. \\textbf{A71},\n062323]. Here, we broaden the class of stabilizer states for which LU\nequivalence implies LC equivalence ($LU\\Leftrightarrow LC$) to include all\nstabilizer states represented by graphs with neither cycles of length 3 nor 4.\nTo compare our result with Van den Nest et al.\u0027s, we show that any stabilizer\nstate of distance $\\delta=2$ is beyond their criterion. We then further prove\nthat $LU\\Leftrightarrow LC$ holds for a more general class of stabilizer states\nof $\\delta=2$. We also explicitly construct graphs representing $\\delta\u003e2$\nstabilizer states which are beyond their criterion: we identify all 58 graphs\nwith up to 11 vertices and construct graphs with $2^m-1$ ($m\\geq 4$) vertices\nusing quantum error correcting codes which have non-Clifford transversal gates.",
"arxiv_id": "quant-ph/0611214",
"authors": [
"Bei Zeng",
"Hyeyoun Chung",
"Andrew W. Cross",
"Isaac L. Chuang"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.75.032325",
"title": "Local unitary versus local Clifford equivalence of stabilizer and graph states",
"url": "https://arxiv.org/abs/quant-ph/0611214"
},
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