Tensor Cross Interpolation of Purities in Quantum Many-Body Systems
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AbstractA defining feature of quantum many-body systems is the exponential scaling of the Hilbert space with the number of degrees of freedom. This exponential complexity naïvely renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.Featured image: Overview of entanglement feature applications. Left: entanglement pattern map showing various quantum states arranged according to the distance between their entanglement features. Right: learned entanglement feature allows for an efficient recovery of physical index ordering that lowers entanglement profile of a quantum state.Popular summaryEntanglement is key to characterizing complex states encountered in equilibrium and non-equilibrium quantum many-body systems. For pure states, entanglement quantifies the quantum correlations between two complementary regions (bipartition) of the system. Usually entanglement characterization is restricted to simplest bipartitions, where system is split into left/right part via one cut. This characterization is oversimplified and in many cases may not be sufficient. However, more generic characterization is generally not possible due to large resources required to measure entanglement in experiments or even in numerical simulations as number of subregions grows as $2^N$ where $N$ is a number of degrees of freedom. Our work introduces a resource-efficient method to characterize the entanglement structure of a quantum many-body state across all bipartitions, not just contiguous left/right cuts. The $2^N$ subregion purities of an $N$-qubit state can be packaged as the amplitudes of an auxiliary wave function known as entanglement feature in the literature. We show that whenever the entanglement feature admits a low-bond-dimension matrix-product-state representation, it can be reconstructed from only $O(N)$ purity evaluations using the tensor cross interpolation algorithm, resulting in an exponential reduction of complexity. Crucially, we analytically show that entanglement feature is learnable not only for weakly entangled, but also for volume-law entangled states, such as individual instances of Haar random states for large enough systems. We also test our method numerically for a broad range of complex quantum many-body wave functions and demonstrate that it works for many physically interesting states, opening up powerful new capabilities for analyzing and comparing entanglement structures. Some applications that we explore in our work include quantifying distances between entanglement structures, revealing structural differences invisible to standard scaling analyses, and finding optimal spatial ordering for the most efficient tensor network representations of quantum states. We believe our results will interest a broad audience working at the interface of quantum information, simulation, and many-body theory, and provide a scalable framework for entanglement analysis in future quantum technologies.► BibTeX data@article{Kolisnyk2026tensorcross, doi = {10.22331/q-2026-05-22-2114}, url = {https://doi.org/10.22331/q-2026-05-22-2114}, title = {Tensor {C}ross {I}nterpolation of {P}urities in {Q}uantum {M}any-{B}ody {S}ystems}, author = {Kolisnyk, Dmytro and Medina, Raimel A. and Vasseur, Romain and Serbyn, Maksym}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2114}, month = may, year = {2026} }► References [1] A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nat. Phys. 16, 132 (2020). https://doi.org/10.1038/s41567-019-0733-z [2] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. 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[3] Chun-Yue Zhang, Shi-Xin Zhang, and Zi-Xiang Li, "Bond Additivity and Persistent Geometric Imprints of Entanglement in Quantum Thermalization", arXiv:2601.01327, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-22 11:42:47). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-22 11:42:46: Could not fetch cited-by data for 10.22331/q-2026-05-22-2114 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractA defining feature of quantum many-body systems is the exponential scaling of the Hilbert space with the number of degrees of freedom. This exponential complexity naïvely renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.Featured image: Overview of entanglement feature applications. Left: entanglement pattern map showing various quantum states arranged according to the distance between their entanglement features. Right: learned entanglement feature allows for an efficient recovery of physical index ordering that lowers entanglement profile of a quantum state.Popular summaryEntanglement is key to characterizing complex states encountered in equilibrium and non-equilibrium quantum many-body systems. For pure states, entanglement quantifies the quantum correlations between two complementary regions (bipartition) of the system. Usually entanglement characterization is restricted to simplest bipartitions, where system is split into left/right part via one cut. This characterization is oversimplified and in many cases may not be sufficient. However, more generic characterization is generally not possible due to large resources required to measure entanglement in experiments or even in numerical simulations as number of subregions grows as $2^N$ where $N$ is a number of degrees of freedom. Our work introduces a resource-efficient method to characterize the entanglement structure of a quantum many-body state across all bipartitions, not just contiguous left/right cuts. The $2^N$ subregion purities of an $N$-qubit state can be packaged as the amplitudes of an auxiliary wave function known as entanglement feature in the literature. We show that whenever the entanglement feature admits a low-bond-dimension matrix-product-state representation, it can be reconstructed from only $O(N)$ purity evaluations using the tensor cross interpolation algorithm, resulting in an exponential reduction of complexity. Crucially, we analytically show that entanglement feature is learnable not only for weakly entangled, but also for volume-law entangled states, such as individual instances of Haar random states for large enough systems. We also test our method numerically for a broad range of complex quantum many-body wave functions and demonstrate that it works for many physically interesting states, opening up powerful new capabilities for analyzing and comparing entanglement structures. Some applications that we explore in our work include quantifying distances between entanglement structures, revealing structural differences invisible to standard scaling analyses, and finding optimal spatial ordering for the most efficient tensor network representations of quantum states. We believe our results will interest a broad audience working at the interface of quantum information, simulation, and many-body theory, and provide a scalable framework for entanglement analysis in future quantum technologies.► BibTeX data@article{Kolisnyk2026tensorcross, doi = {10.22331/q-2026-05-22-2114}, url = {https://doi.org/10.22331/q-2026-05-22-2114}, title = {Tensor {C}ross {I}nterpolation of {P}urities in {Q}uantum {M}any-{B}ody {S}ystems}, author = {Kolisnyk, Dmytro and Medina, Raimel A. and Vasseur, Romain and Serbyn, Maksym}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2114}, month = may, year = {2026} }► References [1] A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nat. Phys. 16, 132 (2020). https://doi.org/10.1038/s41567-019-0733-z [2] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. 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[3] Chun-Yue Zhang, Shi-Xin Zhang, and Zi-Xiang Li, "Bond Additivity and Persistent Geometric Imprints of Entanglement in Quantum Thermalization", arXiv:2601.01327, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-22 11:42:47). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-22 11:42:46: Could not fetch cited-by data for 10.22331/q-2026-05-22-2114 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.