Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 028, 20 pages      arXiv:1507.03350      https://doi.org/10.3842/SIGMA.2017.028

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

Jacob Turner a and Jason Morton b
a) Korteweg-de Vries Institute, University of Amsterdam, 1098 XG Amsterdam, The Netherlands
b) Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received November 26, 2016, in final form April 28, 2017; Published online May 02, 2017

Abstract
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.

Key words: quantum entanglement; local unitary invariants; SLOCC invariants; invariant rings; geometric invariant theory; complete set of invariants; density operators; tensor networks.

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