An Ideal Approach to the Invariant Ring of the Tensor Product

J. Müller-Quade


The problem of deciding if two quantum states $\vert\psi\rangle$ and $\vert\phi\rangle$ have the same entanglement motivates the study of the group $SU_{2}\otimes \dots \otimes SU_{2}$. One is interested in classifying the orbits of $SU_{2}\otimes \dots \otimes SU_{2}$ which resemble the equivalence classes of states with identical entanglement. To classify the orbits of a group G one can use the generators of the ring C[X]^G of invariants of G. This relates to the old question: ``Is there a correspondence between C[X]^Gand $C[X]^{G \otimes G}$?'' We answer (avoid) this question by looking at what we call the orbit relation of G: $\{(x,gx)\vert x\in\mbox{ Vector space },g\in G\}$. The defining equations of the orbit relation of G can be computed from the defining equations of G. Furthermore the invariant ring can be computed from the defining equations of the orbit relation (Derksen's Algorithm). The orbit relation reflects the (direct) product of groups by the relation product and additionally allows to compute the orbit relation of $G\otimes 1$ from the orbit relation of G. As the tensor product can be written as a product of direct sums: $G_{1} \otimes G_{2}=(G_{1}\otimes 1)(1 \otimes G_{2})$ we can conclude that the orbit relation of $G_{1}\otimes G_{2}$ can be computed from the orbit relations of G₁ and G₂.