Separation of Orbits Under Group Actions with an Application to Quantum Systems

Motivated by the task to decide whether two quantum states are equally entangled we consider the orbits under the action of the group of all one-qubit operations. To investigate the orbit structure of this group of local unitary operations we propose to use methods from classical invariant theory as well as new results.

Two approaches are presented. The first uses the orbit separation property of invariant rings to distinguish among nonequivalent quantum states. In this context we study the Molien series which describes the structure of the invariant ring as a graded ring. We give a closed formula for the Molien series of the group of one-qubit operations.

Our second approach makes use of an equivalence relation, the so-called graph of the action, which relates two points if they are on the same orbit. For finite groups which factor, are synchronous direct sums, or tensor products we analyze the structure of the graph of the action. This yields new algorithms for the computation of the graph of the action.