Persistent Tensors and Multiqudit Entanglement Transformation
Masoud Gharahi, Vladimir Lysikov
TL;DR
This work introduces persistent tensors as a new structural tool for multipartite entanglement analysis, establishing a concrete, general lower bound on tensor rank and identifying three minimal-rank, persistent families $\mathcal{L}(d,n)$, $\mathcal{M}(d,n)$, and $\mathcal{N}(d,n)$ that generalize the $n$-qubit $\mathrm{W}$ state to qudits. It proves these families are in the orbit closure of GHZ states and demonstrates rate-one asymptotic SLOCC transformations from GHZ to each family, linking degenerations to feasible interconversions. The paper further extends the rank bound to direct sums and block pyramidal tensors, proves rank multiplicativity with GHZ in these constructions, and provides explicit degeneration chains and exact rank values for the L/M/N families. These results deepen the understanding of entanglement transformations in multipartite, high-dimensional systems and suggest practical avenues for qudit-based quantum information processing. Overall, the framework connects algebraic geometry, tensor rank theory, and asymptotic SLOCC in a way that yields concrete tools for characterizing and converting complex entangled states.
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.