Characterizing high-dimensional multipartite entanglement beyond Greenberger-Horne-Zeilinger fidelities
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AbstractCharacterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.Featured image: Fig. (a) In multipartite high-dimensional entanglement, different pure-state components in a decomposition may involve distinct numbers of entangled levels for each particle. For the three particles shown, yellow and blue represent two pure-state components of a mixed state, where one particle contributes 3 effective levels and the other two each contribute 2 effective levels. This corresponds to an entanglement-dimensionality vector of (322). The dots indicate that this definition generalizes to arbitrary particle numbers. (b) This figure illustrates the structure of all possible Schmidt number vectors in a $4\times 3\times 2$ state space, which exhibits only partial nesting. A vector whose last entry is greater than or equal to 2 indicates genuine multipartite entanglement.Popular summaryHigh-dimensional multipartite entanglement, where each particle possesses multiple accessible levels and the entanglement genuinely involves all particles, serves as an important resource for applications like quantum teleportation, computation and communication. However, simultaneously certifying the genuine high-dimensional and multipartite nature of the entanglement of a quantum state is the most challenging task, especially in experiments. The most common approach in current experiments is to prove entanglement by measuring the fidelity with respect to an ideal GHZ state. This strategy, however, typically relies on alignment with a specific target state, close to an ideal highly entangled state such as the Greenberger-Horne-Zeilinger state. Moreover, this only provides a lower bound on the lowest entanglement dimensionality across all bipartitions. As a result, this approach misses potentially crucial properties of the quantum state, in particular for more complex high-dimensional multipartite entangled states that do not closely resemble the GHZ structure. In this work, we complement existing methods by deriving a criterion to witness high-dimensional multipartite entanglement based on measurement data of similar complexity to a GHZ fidelity. Moreover, our approach simultaneously imposes constraints on the entanglement dimensionality across all possible bipartitions of the system, thereby providing a more comprehensive characterization of the multipartite entanglement structure, as compared to the minimal entanglement dimensionality across all bipartitions. We also show that our method often outperforms current existing ones: while not requiring more complex numerical optimizations or analysis, it provides finer entanglement-dimensionality characterization in a wider set of states, in particular those of paradigmatic interest. We therefore provide a broadly applicable toolbox that could trigger further investigation into the applicability of high-dimensional multipartite entanglement, both as a fundamental characterization of quantum states and as a resource, e.g., for quantum networks and related applications, especially under realistic non-ideal conditions.► BibTeX data@article{Liu2026characterizinghigh, doi = {10.22331/q-2026-02-03-1995}, url = {https://doi.org/10.22331/q-2026-02-03-1995}, title = {Characterizing high-dimensional multipartite entanglement beyond {G}reenberger-{H}orne-{Z}eilinger fidelities}, author = {Liu, Shuheng and He, Qiongyi and Huber, Marcus and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1995}, month = feb, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. 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Phys. 12, 053002 (2010). https://doi.org/10.1088/1367-2630/12/5/053002Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-03 12:48:29: Could not fetch cited-by data for 10.22331/q-2026-02-03-1995 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-03 12:48:30: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractCharacterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.Featured image: Fig. (a) In multipartite high-dimensional entanglement, different pure-state components in a decomposition may involve distinct numbers of entangled levels for each particle. For the three particles shown, yellow and blue represent two pure-state components of a mixed state, where one particle contributes 3 effective levels and the other two each contribute 2 effective levels. This corresponds to an entanglement-dimensionality vector of (322). The dots indicate that this definition generalizes to arbitrary particle numbers. (b) This figure illustrates the structure of all possible Schmidt number vectors in a $4\times 3\times 2$ state space, which exhibits only partial nesting. A vector whose last entry is greater than or equal to 2 indicates genuine multipartite entanglement.Popular summaryHigh-dimensional multipartite entanglement, where each particle possesses multiple accessible levels and the entanglement genuinely involves all particles, serves as an important resource for applications like quantum teleportation, computation and communication. However, simultaneously certifying the genuine high-dimensional and multipartite nature of the entanglement of a quantum state is the most challenging task, especially in experiments. The most common approach in current experiments is to prove entanglement by measuring the fidelity with respect to an ideal GHZ state. This strategy, however, typically relies on alignment with a specific target state, close to an ideal highly entangled state such as the Greenberger-Horne-Zeilinger state. Moreover, this only provides a lower bound on the lowest entanglement dimensionality across all bipartitions. As a result, this approach misses potentially crucial properties of the quantum state, in particular for more complex high-dimensional multipartite entangled states that do not closely resemble the GHZ structure. In this work, we complement existing methods by deriving a criterion to witness high-dimensional multipartite entanglement based on measurement data of similar complexity to a GHZ fidelity. Moreover, our approach simultaneously imposes constraints on the entanglement dimensionality across all possible bipartitions of the system, thereby providing a more comprehensive characterization of the multipartite entanglement structure, as compared to the minimal entanglement dimensionality across all bipartitions. We also show that our method often outperforms current existing ones: while not requiring more complex numerical optimizations or analysis, it provides finer entanglement-dimensionality characterization in a wider set of states, in particular those of paradigmatic interest. We therefore provide a broadly applicable toolbox that could trigger further investigation into the applicability of high-dimensional multipartite entanglement, both as a fundamental characterization of quantum states and as a resource, e.g., for quantum networks and related applications, especially under realistic non-ideal conditions.► BibTeX data@article{Liu2026characterizinghigh, doi = {10.22331/q-2026-02-03-1995}, url = {https://doi.org/10.22331/q-2026-02-03-1995}, title = {Characterizing high-dimensional multipartite entanglement beyond {G}reenberger-{H}orne-{Z}eilinger fidelities}, author = {Liu, Shuheng and He, Qiongyi and Huber, Marcus and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1995}, month = feb, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. 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