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A Complete Set of Invariants for LU-Equivalence of Density Operators
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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A Complete Set of Invariants for LU-Equivalence of Density Operators
Інший ID:
2010 Mathematics Subject Classification: 20G05; 20G45; 81R05; 20C35; 22E70
DOI:10.3842/SIGMA.2017.028
DOI:10.3842/SIGMA.2017.028
Посилання:
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.
Підтримка:
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.
Дата:
2017
Переглядів:
635
Завантажень:
396
Анотація:
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.
