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No-go theorem for heralded exact one-way key distillation

Quantum Journal

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Researchers Vishal Singh and Mark M. Wilde proved a fundamental limitation in quantum key distillation, showing that "super two-extendible" states—including erased and full-rank states—cannot generate perfect secret keys using one-way communication and local operations. The study reveals a stark contrast between exact and approximate key distillation, with approximate methods yielding far higher rates despite both using the same noisy bipartite states. This no-go theorem extends to heralded exact one-way entanglement distillation, demonstrating that even probabilistic methods fail for most practical quantum states under these constraints. The findings challenge assumptions about noise resilience in quantum networks, as common mixed states—even those allowing near-perfect approximate keys—cannot produce error-free keys under one-way protocols. The work builds on prior research by Horodecki et al., reinforcing that entanglement alone cannot quantify distillable secret keys, particularly in constrained communication scenarios.

No-go theorem for heralded exact one-way key distillation

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AbstractThe heralded exact one-way distillable secret key is equal to the largest expected rate at which perfect secret key bits can be probabilistically distilled from a bipartite state by means of local operations and one-way classical communication. Here we define the set of super two-extendible states and prove that an arbitrary state in this set cannot be used for heralded exact one-way secret-key distillation. This broad class of states includes both erased states and all full-rank states. Comparing the heralded exact one-way distillable secret key with the more commonly studied approximate one-way distillable secret key, our results demonstrate an extreme gap between them for many states of interest, with the approximate one-way distillable secret key being much larger. Our findings naturally extend to heralded exact one-way entanglement distillation, with similar conclusions.Popular summaryQuantum key distribution is one of the most promising applications of quantum technologies in information processing systems. In particular, if two distant parties each hold one share of an Einstein-Podolsky-Rosen pair, they can obtain a pair of perfectly correlated bits that are perfectly secure from any eavesdropping. This unique feature of quantum entanglement has motivated efforts to understand the limits of “key distillation” in more practical settings, for example, when the bipartite state shared between the two parties is affected by noise. Remarkably, Horodecki et. al. showed that despite entanglement being necessary for key distillation, one can extract keys from bound entangled states, demonstrating that entanglement theory is insufficient to quantify the amount of key that can be distilled from a bipartite state. We study heralded exact one-way key distillation. In this task, Alice and Bob share a bipartite state and wish to distill a perfectly secure key by performing local measurements on their respective systems. Additionally, Alice can publicly announce some classical data after performing her measurement to help Bob pick an ideal measurement for his system. We find that nearly all mixed states are useless for this task despite the relaxation to the probabilistic setting. This includes common models for noisy states in quantum networks, such as erased states and all full-rank states, many of which can be used to extract a secret key with an arbitrarily small but non-zero error. Our findings show that for most states of practical interest, ideal secret keys cannot be distilled even probabilistically when restricting to local measurements assisted by one-way public communication.► BibTeX data@article{Singh2026nogotheoremheralded, doi = {10.22331/q-2026-03-10-2020}, url = {https://doi.org/10.22331/q-2026-03-10-2020}, title = {No-go theorem for heralded exact one-way key distillation}, author = {Singh, Vishal and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2020}, month = mar, year = {2026} }► References [1] Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan. ``Secure quantum key distribution with realistic devices''. Reviews of Modern Physics 92, 025002 (2020). https://doi.org/10.1103/RevModPhys.92.025002 [2] Christopher Portmann and Renato Renner. ``Security in quantum cryptography''. Reviews of Modern Physics 94, 025008 (2022). https://doi.org/10.1103/RevModPhys.94.025008 [3] Víctor Zapatero, Tim van Leent, Rotem Arnon-Friedman, Wen-Zhao Liu, Qiang Zhang, Harald Weinfurter, and Marcos Curty. ``Advances in device-independent quantum key distribution''. npj Quantum Information 9, 10 (2023). https://doi.org/10.1038/s41534-023-00684-x [4] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. In Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing. Page 175. India (1984). [5] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''.

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Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927). url: http://eudml.org/doc/59231. http://eudml.org/doc/59231Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-10 14:17:47: Could not fetch cited-by data for 10.22331/q-2026-03-10-2020 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-10 14:17:47: 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. AbstractThe heralded exact one-way distillable secret key is equal to the largest expected rate at which perfect secret key bits can be probabilistically distilled from a bipartite state by means of local operations and one-way classical communication. Here we define the set of super two-extendible states and prove that an arbitrary state in this set cannot be used for heralded exact one-way secret-key distillation. This broad class of states includes both erased states and all full-rank states. Comparing the heralded exact one-way distillable secret key with the more commonly studied approximate one-way distillable secret key, our results demonstrate an extreme gap between them for many states of interest, with the approximate one-way distillable secret key being much larger. Our findings naturally extend to heralded exact one-way entanglement distillation, with similar conclusions.Popular summaryQuantum key distribution is one of the most promising applications of quantum technologies in information processing systems. In particular, if two distant parties each hold one share of an Einstein-Podolsky-Rosen pair, they can obtain a pair of perfectly correlated bits that are perfectly secure from any eavesdropping. This unique feature of quantum entanglement has motivated efforts to understand the limits of “key distillation” in more practical settings, for example, when the bipartite state shared between the two parties is affected by noise. Remarkably, Horodecki et. al. showed that despite entanglement being necessary for key distillation, one can extract keys from bound entangled states, demonstrating that entanglement theory is insufficient to quantify the amount of key that can be distilled from a bipartite state. We study heralded exact one-way key distillation. In this task, Alice and Bob share a bipartite state and wish to distill a perfectly secure key by performing local measurements on their respective systems. Additionally, Alice can publicly announce some classical data after performing her measurement to help Bob pick an ideal measurement for his system. We find that nearly all mixed states are useless for this task despite the relaxation to the probabilistic setting. This includes common models for noisy states in quantum networks, such as erased states and all full-rank states, many of which can be used to extract a secret key with an arbitrarily small but non-zero error. Our findings show that for most states of practical interest, ideal secret keys cannot be distilled even probabilistically when restricting to local measurements assisted by one-way public communication.► BibTeX data@article{Singh2026nogotheoremheralded, doi = {10.22331/q-2026-03-10-2020}, url = {https://doi.org/10.22331/q-2026-03-10-2020}, title = {No-go theorem for heralded exact one-way key distillation}, author = {Singh, Vishal and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2020}, month = mar, year = {2026} }► References [1] Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan. ``Secure quantum key distribution with realistic devices''. Reviews of Modern Physics 92, 025002 (2020). https://doi.org/10.1103/RevModPhys.92.025002 [2] Christopher Portmann and Renato Renner. ``Security in quantum cryptography''. Reviews of Modern Physics 94, 025008 (2022). https://doi.org/10.1103/RevModPhys.94.025008 [3] Víctor Zapatero, Tim van Leent, Rotem Arnon-Friedman, Wen-Zhao Liu, Qiang Zhang, Harald Weinfurter, and Marcos Curty. ``Advances in device-independent quantum key distribution''. npj Quantum Information 9, 10 (2023). https://doi.org/10.1038/s41534-023-00684-x [4] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. In Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing. Page 175. India (1984). [5] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''.

Theoretical Computer Science 560, 7–11 (2014). https://doi.org/10.1016/j.tcs.2014.05.025 [6] Artur K. Ekert. ``Quantum cryptography based on Bell's theorem''.

Physical Review Letters 81, 2839–2841 (1998). https://doi.org/10.1103/PhysRevLett.81.2839 [33] Kun Fang and Zi-Wen Liu. ``No-go theorems for quantum resource purification''.

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No-go theorem for heralded exact one-way key distillation

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