Comment on "Matrix Pencils and Entanglement Classification"

Understand

In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent.

• In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils.
• Thus, two states are equivalent if and only if they have the same canonical form.
• As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.