The work titled "The Product Structure of Matrix Product States under Permutations" was co-authored by Marta Florido-Llinàs, Álvaro M. Alhambra, Rahul Trivedi, Norbert Schuch, David Pérez-García, and J. Ignacio Cirac and has recently been published in PRX Quantum [PRX Quantum 6, 040338 (2025)].
For more information on this paper, take a look at the popular summary below and at the open-access article (DOI: 10.1103/8sbs-t24w).
Popular summary:
The study of complex systems is at the heart of many diverse fields, ranging from quantum many-body physics to high-dimensional data analysis. They all share a common challenge: they are often too intricate and require more straightforward and manageable representations. Designing optimal representations (or Ansätze) to make these problems tractable is a key question. Our work addresses this issue in scenarios where entanglement across any arbitrary partition is limited. We find that such systems can be well approximated by a remarkably simple Ansatz: a superposition of just a few product states.
To arrive at this result, we define a class of quantum states called “MPS-under-permutations,” which captures the above property and can naturally arise in contexts such as quantum chemistry, machine learning, and solving high-dimensional differential equations. These states are rooted in the widely used matrix product state (MPS) framework, a powerful tool for analyzing quantum many-body systems. By leveraging the well-developed analytical toolbox of MPS, we demonstrate in a mathematically rigorous way that MPS-under-permutations exhibit a very simple product structure.
Therefore, our results advance the fundamental understanding of the entanglement properties of physically relevant many-body states and pave the way for future generalizations to broader classes of states beyond the MPS considered here. Furthermore, they represent a step toward the overarching goal of fully characterizing the structure of systems under specific physical constraints, which is an essential task for tackling complex challenges in quantum physics and beyond.
This work has received support through the ERC grant SEQUAM as well as the Austrian Science Fund FWF (grant DOIs 10.55776/COE1, 10.55776/P36305, 10.55776/F71, and 10.55776/J4796), and the European Union (NextGenerationEU).
