Persistent Tensors and Multiqudit Entanglement Transformation

by   Masoud Gharahi, et al.

We construct a lower bound of the tensor rank for a new class of tensors, which we call persistent tensors. We present three specific families of persistent tensors, of which the lower bound is tight. We show that there is a chain of degenerations between these three families of minimal-rank persistent tensors that can be used to study the entanglement transformation between them. In addition, we show that these three families of persistent tensors are indeed different generalizations of multiqubit W states within multiqudit systems and are geometrically in the orbit closure of multiqudit GHZ states. Consequently, we show that one can obtain every one of the generalizations of W state from a multiqudit GHZ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with rate one. Finally, we extend the obtained lower bound of the tensor rank to direct sums with persistent summands and to even more general combinations of tensors, which we call block pyramidal tensors. As a result, we show that the tensor rank is multiplicative under the Kronecker and tensor products of minimal-rank persistent tensors with the GHZ tensor.


page 1

page 2

page 3

page 4


On the tensor rank of 3× 3 permanent and determinant

The tensor rank and border rank of the 3 × 3 determinant tensor is known...

A geometric study of Strassen's asymptotic rank conjecture and its variants

We establish basic information about the set of tight tensors, the tenso...

Maximal Border Subrank Tensors

We prove a lower bound on the dimension of the set of maximal border sub...

Cannikin's Law in Tensor Modeling: A Rank Study for Entanglement and Separability in Tensor Complexity and Model Capacity

This study clarifies the proper criteria to assess the modeling capacity...

Families of Perfect Tensors

Perfect tensors are the tensors corresponding to the absolutely maximall...

On tensor rank and commuting matrices

Obtaining superlinear lower bounds on tensor rank is a major open proble...

The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the k-partite k-uni...

Please sign up or login with your details

Forgot password? Click here to reset