Quantum Lunch: Finding large gaps between tensor rank and border rank via algebras

Speaker: Jeroen Zuiddam

Abstract: In quantum information theory, it is well-known that the GHZ-state 000+111 cannot be (SLOCC-)transformed into the W-state 000+101+011, but the W-state can be approximated with states from the GHZ-orbit. The W-state is thus an example of a tensor of tensor rank 3 that can be approximated by tensors of tensor rank 2. We say that the border rank of W is 2. I will talk about the following open problem. Let S be a 3-qudit state of border rank d. How large can the tensor rank of S be? I will show how one can use the deformation theory of algebras and the theory of the rank of algebras to get states S with high rank.