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Quantifying mixed-state entanglement via partial transpose and realignment moments

Poetri Sonya Tarabunga and Tobias Haug

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AbstractEntanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state $4$-design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.Featured image: Quantitative entanglement witnesses provide experimentally accessible lower bounds on mixed-state entanglement (left), reveal the entanglement phases of random quantum states (top right), and enable efficient tests that distinguish states with low and high entanglement (bottom).Popular summaryEntanglement is one of the defining features of quantum mechanics and has been the focus of intense study for decades. However, standard tools for measuring entanglement, such as the Rényi entropies, work only in the idealized case of perfectly pure systems. But real-world computers are never perfectly isolated: noise and imperfections are unavoidable, making entanglement much harder to study. Until now, there has been no general way to extend traditional entanglement measures into this mixed-state regime, leaving scientists without a clear way to study entanglement in realistic systems. Our work develops a new set of tools to fill this gap. We introduce new quantities that can capture entanglement even in noisy, mixed states which naturally appear in experiments. These tools extend the well-known entanglement Rényi entropies and allow us to study entanglement across a wide range of scenarios, providing new insights into how mixed-state entanglement behaves. Our tools have immense potential to advance our understanding of entanglement in quantum systems. They can be applied to quantify entanglement in mixed many-body systems and benchmark the performance of emerging quantum devices. By opening the door to a systematic study of mixed-state entanglement, our results pave the way for deeper understanding of quantum correlations in the real world and for harnessing them in future quantum technologies.► BibTeX data@article{Tarabunga2026quantifyingmixed, doi = {10.22331/q-2026-08-19-2194}, url = {https://doi.org/10.22331/q-2026-08-19-2194}, title = {Quantifying mixed-state entanglement via partial transpose and realignment moments}, author = {Tarabunga, Poetri Sonya and Haug, Tobias}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2194}, month = aug, year = {2026} }► References [1] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/PhysRevA.70.052328. 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Physical Review Letters, 125 (20): 200502, November 2020. ISSN 1079-7114. 10.1103/physrevlett.125.200502. URL http://dx.doi.org/10.1103/PhysRevLett.125.200502. https://doi.org/10.1103/physrevlett.125.200502 [123] Huangjun Zhu. Multiqubit clifford groups are unitary 3-designs. Physical Review A, 96 (6): 062336, December 2017. ISSN 2469-9934. 10.1103/physreva.96.062336. URL http://dx.doi.org/10.1103/PhysRevA.96.062336. https://doi.org/10.1103/physreva.96.062336 [124] Huangjun Zhu, Richard Kueng, Markus Grassl, and David Gross. The clifford group fails gracefully to be a unitary 4-design. arXiv:1609.08172, 2016. URL https://arxiv.org/abs/1609.08172. arXiv:1609.08172 [125] Karol Zyczkowski and Hans-Jürgen Sommers. Induced measures in the space of mixed quantum states. Journal of Physics A: Mathematical and General, 34 (35): 7111–7125, aug 2001. 10.1088/0305-4470/34/35/335. URL https://doi.org/10.1088. https://doi.org/10.1088/0305-4470/34/35/335Cited byCould not fetch Crossref cited-by data during last attempt 2026-08-19 07:34:34: Could not fetch cited-by data for 10.22331/q-2026-08-19-2194 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-08-19 07:34:42: Cannot retrieve data from ADS due to rate limitations.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. AbstractEntanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state $4$-design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.Featured image: Quantitative entanglement witnesses provide experimentally accessible lower bounds on mixed-state entanglement (left), reveal the entanglement phases of random quantum states (top right), and enable efficient tests that distinguish states with low and high entanglement (bottom).Popular summaryEntanglement is one of the defining features of quantum mechanics and has been the focus of intense study for decades. However, standard tools for measuring entanglement, such as the Rényi entropies, work only in the idealized case of perfectly pure systems. But real-world computers are never perfectly isolated: noise and imperfections are unavoidable, making entanglement much harder to study. Until now, there has been no general way to extend traditional entanglement measures into this mixed-state regime, leaving scientists without a clear way to study entanglement in realistic systems. Our work develops a new set of tools to fill this gap. We introduce new quantities that can capture entanglement even in noisy, mixed states which naturally appear in experiments. These tools extend the well-known entanglement Rényi entropies and allow us to study entanglement across a wide range of scenarios, providing new insights into how mixed-state entanglement behaves. Our tools have immense potential to advance our understanding of entanglement in quantum systems. They can be applied to quantify entanglement in mixed many-body systems and benchmark the performance of emerging quantum devices. By opening the door to a systematic study of mixed-state entanglement, our results pave the way for deeper understanding of quantum correlations in the real world and for harnessing them in future quantum technologies.► BibTeX data@article{Tarabunga2026quantifyingmixed, doi = {10.22331/q-2026-08-19-2194}, url = {https://doi.org/10.22331/q-2026-08-19-2194}, title = {Quantifying mixed-state entanglement via partial transpose and realignment moments}, author = {Tarabunga, Poetri Sonya and Haug, Tobias}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2194}, month = aug, year = {2026} }► References [1] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/PhysRevA.70.052328. URL https://doi.org/10.1103/PhysRevA.70.052328. https://doi.org/10.1103/PhysRevA.70.052328 [2] Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Quantum pseudoentanglement. arXiv preprint arXiv:2211.00747, 2022. URL https://doi.org/10.48550/arXiv.2211.00747. https://doi.org/10.48550/arXiv.2211.00747 arXiv:2211.00747 [3] Shruti Aggarwal, Satyabrata Adhikari, and A. S. Majumdar. Entanglement detection in arbitrary-dimensional bipartite quantum systems through partial realigned moments. Phys. Rev. A, 109: 012404, Jan 2024. 10.1103/PhysRevA.109.012404. URL https://doi.org/10.1103/PhysRevA.109.012404. https://doi.org/10.1103/PhysRevA.109.012404 [4] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 517–576, May 2008. 10.1103/RevModPhys.80.517. URL https://doi.org/10.1103/RevModPhys.80.517. https://doi.org/10.1103/RevModPhys.80.517 [5] GUILLAUME AUBRUN. Partial transposition of random states and non-centered semicircular distributions. Random Matrices: Theory and Applications, 01 (02): 1250001, April 2012. ISSN 2010-3271. 10.1142/s2010326312500013. URL http://dx.doi.org/10.1142/S2010326312500013. https://doi.org/10.1142/s2010326312500013 [6] Guillaume Aubrun and Ion Nechita. Realigning random states. Journal of Mathematical Physics, 53 (10): 4759115, October 2012. ISSN 1089-7658. 10.1063/1.4759115. URL http://dx.doi.org/10.1063/1.4759115. https://doi.org/10.1063/1.4759115 [7] Guillaume Aubrun, Stanisław J. Szarek, and Deping Ye. Phase transitions for random states and a semicircle law for the partial transpose. Physical Review A, 85 (3): 030302, March 2012. ISSN 1094-1622. 10.1103/physreva.85.030302. URL http://dx.doi.org/10.1103/PhysRevA.85.030302. https://doi.org/10.1103/physreva.85.030302 [8] Guillaume Aubrun, Stanislaw J. Szarek, and Deping Ye. Entanglement thresholds for random induced states. Communications on Pure and Applied Mathematics, 67 (1): 129–171, May 2013. ISSN 1097-0312. 10.1002/cpa.21460. URL http://dx.doi.org/10.1002/cpa.21460. https://doi.org/10.1002/cpa.21460 [9] Joonwoo Bae, Dariusz Chruściński, and Beatrix C. Hiesmayr. Mirrored entanglement witnesses. npj Quantum Information, 6 (1): 15, February 2020. ISSN 2056-6387. 10.1038/s41534-020-0242-z. URL http://dx.doi.org/10.1038/s41534-020-0242-z. https://doi.org/10.1038/s41534-020-0242-z [10] Nikhil Bansal, Wai-Keong Mok, Kishor Bharti, Dax Enshan Koh, and Tobias Haug. Pseudorandom density matrices. PRX Quantum, 6: 020322, May 2025. 10.1103/PRXQuantum.6.020322. URL https://doi.org/10.1103/PRXQuantum.6.020322. https://doi.org/10.1103/PRXQuantum.6.020322 [11] Bruno Bertini, Katja Klobas, and Tsung-Cheng Lu. Entanglement negativity and mutual information after a quantum quench: Exact link from space-time duality.

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