$\ell$-Multiranks of Multipartite Quantum States via Tensor Flattening: A Mathematica Codebase

TL;DR

The paper addresses efficient entanglement characterization in multipartite multiqudit systems using $\ell$-multiranks derived from tensor flattening. It introduces a Mathematica codebase that enumerates all bipartition-induced matricizations $\mathcal{M}_{I}[\psi]$ for $\binom{n}{\ell}$ partitions and computes their ranks to build the full $\ell$-multirank profile. It provides two complementary implementations (SparseArray and TensorProduct) enabling amplitude-tensor and product-basis representations, with automatic generation of all matricizations and extraction of ranks. This tool offers a practical, scalable method for detecting Genuine Multipartite Entanglement (GME) and exploring entanglement structure in high-dimensional quantum states, bridging algebraic geometry and tensor methods.

Abstract

We present a Mathematica codebase for computing $\ell$-multilinear ranks ($\ell$-multiranks) of multiqudit quantum states using tensor-flattening techniques. By calculating the ranks of all bipartition-induced matricizations, the method provides an efficient criterion for detecting Genuine Multipartite Entangled (GME) states in systems with local dimension $d$. The code automatically generates all required tensor reshapes and outputs the full $\ell$-multirank profile, offering a practical tool for characterizing entanglement in high-dimensional multiqudit systems.