A Complete Set of Invariants for LU-Equivalence of Density Operators

Symmetry Integrability and Geometry-Methods and Applications

article

Jacob TURNER¹  Jason MORTON² 

[1] Korteweg-de Vries Institute, University of Amsterdam;Department of Mathematics, The Pennsylvania State University, University Park

关键词: quantum entanglement;    local unitary invariants;    SLOCC invariants;    invariant rings;    geometric invariant theory;    complete set of invariants;    density operators;    tensor networks;   

DOI  :  10.3842/SIGMA.2017.028

来源: National Academy of Science of Ukraine

【 摘 要 】

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 ${\rm GL}$ 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.

【 授权许可】

Unknown