Persistent Tensors and Multiqudit Entanglement Transformation

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

19 pages. Your comments are more than welcome

References in corpus (6)

• Qudits and high-dimensional quantum computing
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Tripartite entanglement transformations and tensor rank
• Universal qudit gate synthesis for transmons
• The Simple Criteria of SLOCC Equivalence Classes
• An Algebraic-Geometric Characterization of Tripartite Entanglement

Cited by in corpus (1)

• Classifying Entanglement by Algebraic Geometry