Persistent Tensors and Multiqudit Entanglement Transformation

TL;DR

This paper introduces persistent tensors, establishes their tensor rank bounds, explores their relation to multiqudit entanglement states, and demonstrates how they can be transformed from GHZ states using asymptotic SLOCC, with implications for tensor rank multiplicativity.

Contribution

The paper constructs tight tensor rank bounds for persistent tensors, links them to multiqudit W and GHZ states, and extends these bounds to block pyramidal tensors, advancing understanding of entanglement transformations.

Findings

Lower bound of tensor rank for persistent tensors is tight.

Persistent tensors generalize multiqubit W states within multiqudit systems.

Multiqudit W states can be obtained from GHZ states via asymptotic SLOCC with rate one.

Abstract

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 $\rm{W}$ states within multiqudit systems and are geometrically in the orbit closure of multiqudit $\rm{GHZ}$ states. Consequently, we show that one can obtain every one of the generalizations of $\rm{W}$ state from a multiqudit $\rm{GHZ}$ 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 $\rm{GHZ}$ tensor.