A Complete Set of Invariants for LU-Equivalence of Density Operators
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unit...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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| Sprache: | English |
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Інститут математики НАН України
2017
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| Zitieren: | A Complete Set of Invariants for LU-Equivalence of Density Operators / J. Turner, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 52 назв. — англ. |
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Turner, J. Morton, J. 2019-02-18T16:42:50Z 2019-02-18T16:42:50Z 2017 A Complete Set of Invariants for LU-Equivalence of Density Operators / J. Turner, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 52 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20G05; 20G45; 81R05; 20C35; 22E70 DOI:10.3842/SIGMA.2017.028 https://nasplib.isofts.kiev.ua/handle/123456789/148619 We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC. The authors would like to acknowledge the helpful comments of the reviewers which greatly improved and strengthened this paper. J. Turner would like to thank Llu´ıs Vena for helpful discussions. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No 339109. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Complete Set of Invariants for LU-Equivalence of Density Operators Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Complete Set of Invariants for LU-Equivalence of Density Operators |
| spellingShingle |
A Complete Set of Invariants for LU-Equivalence of Density Operators Turner, J. Morton, J. |
| title_short |
A Complete Set of Invariants for LU-Equivalence of Density Operators |
| title_full |
A Complete Set of Invariants for LU-Equivalence of Density Operators |
| title_fullStr |
A Complete Set of Invariants for LU-Equivalence of Density Operators |
| title_full_unstemmed |
A Complete Set of Invariants for LU-Equivalence of Density Operators |
| title_sort |
complete set of invariants for lu-equivalence of density operators |
| author |
Turner, J. Morton, J. |
| author_facet |
Turner, J. Morton, J. |
| publishDate |
2017 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148619 |
| citation_txt |
A Complete Set of Invariants for LU-Equivalence of Density Operators / J. Turner, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 52 назв. — англ. |
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2025-12-07T13:36:24Z |
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