Tensor Product Structure Geometry under Unitary Channels

Faidon Andreadakis; Paolo Zanardi

doi:10.22331/q-2025-03-25-1668

Tensor Product Structure Geometry under Unitary Channels

Faidon Andreadakis¹ and Paolo Zanardi^1,2

¹Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA
²Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532, USA

Published:	2025-03-25, volume 9, page 1668
Editor:	Aaron Goldberg
Eprint:	arXiv:2410.02911v2
Doi:	https://doi.org/10.22331/q-2025-03-25-1668
Citation:	Quantum 9, 1668 (2025).

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Abstract

In quantum many-body systems, complex dynamics delocalize the physical degrees of freedom. This spreading of information throughout the system has been extensively studied in relation to quantum thermalization, scrambling, and chaos. Locality is typically defined with respect to a tensor product structure (TPS) which identifies the local subsystems of the quantum system. In this paper, we investigate a simple geometric measure of operator spreading by quantifying the distance of the space of local operators from itself evolved under a unitary channel. We show that this TPS distance is related to the scrambling properties of the dynamics between the local subsystems and coincides with the entangling power of the dynamics in the case of a symmetric bipartition. Additionally, we provide sufficient conditions for the maximization of the TPS distance and show that the class of 2-unitaries provides examples of dynamics that achieve this maximal value. For Hamiltonian evolutions at short times, the characteristic timescale of the TPS distance depends on scrambling rates determined by the strength of interactions between the local subsystems. Beyond this short-time regime, the behavior of the TPS distance is explored through numerical simulations of prototypical models exhibiting distinct ergodic properties, ranging from quantum chaos and integrability to Hilbert space fragmentation and localization.

Featured image: A unitary evolution $\mathcal{U}$ maps an initial tensor product structure (TPS) $\mathcal{H}^{\otimes N}$ to a new one $\mathcal{U}(\mathcal{H}^{\otimes N})$. In this work, we introduce a geometric measure $\Phi(\mathscr{A},\mathcal{U})$ that quantifies this deviation, where $\mathscr{A}$ encodes the information about the initial TPS.

Popular summary

Locality is a fundamental concept in physics. In quantum systems, it is mathematically described by a tensor product structure (TPS), which encodes the decomposition of the total system into smaller subsystems. However, quantum dynamics can delocalize initially local degrees of freedom, mapping individual subsystem properties into shared properties of the whole system.

In the case of unitary dynamics, the initial TPS is mapped to a new one. How can we quantify this deviation? In our work, we focus on degrees of freedom that represent collections of individual properties—specifically, those expressible as sums of local operators (e.g., the total spin in the z-direction of a spin chain). We show that the deviation of these degrees of freedom under unitary evolution serves as a geometric measure of the distance between the initial and final TPS.

Our work introduces a straightforward geometric framework for quantifying delocalization in quantum many-body systems, while also paving the way for connecting delocalization effects to entanglement generation and the ergodic properties of local Hamiltonian models.

► BibTeX data

@article{Andreadakis2025tensorproduct,
  doi = {10.22331/q-2025-03-25-1668},
  url = {https://doi.org/10.22331/q-2025-03-25-1668},
  title = {Tensor {P}roduct {S}tructure {G}eometry under {U}nitary {C}hannels},
  author = {Andreadakis, Faidon and Zanardi, Paolo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1668},
  month = mar,
  year = {2025}
}

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[1] Antoine Soulas, "Disentangling tensor product structures", International Journal of Quantum Information 23 07, 2550027 (2025).

[2] Michele Arzano, Goffredo Chirco, and Jerzy Kowalski-Glikman, "Bias in Local Spin Measurements from Deformed Symmetries", arXiv:2603.08618, (2026).

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