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Cost of quantum secret key

Quantum Science and Technology (arXiv overlay)

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Researchers Horodecki, Sikorski, Das, and Wilde introduced a resource theory for quantum secret keys, defining "key cost" as the minimum private key needed to approximate a quantum state via local operations. Their central finding proves the regularized key of formation serves as an upper bound on key cost, using a novel "privacy dilution" protocol that converts ideal private states into diluted ones. The study establishes irreversibility in privacy creation-distillation by showing key cost is lower-bounded by regularized relative entropy of entanglement for specific states. For mixed-state analogs of pure quantum states, the team proved multiple entanglement measures become equivalent, mirroring pure-state behavior in privacy contexts. The work reveals a yield-cost relationship between privacy cost and distillable key in single-shot scenarios, with implications for quantum device energy efficiency.

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AbstractIn this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided. Importantly, our results presented here will remain valid even if entangled states with zero distillable key were shown to exist.Popular summaryQuantum mechanics is a surprising physical theory as it is reversible in principle, contrary to our common experience. After an arbitrary quantum operation, if full access to the environment is available, the system's state can be restored to its initial state by a reversal operation. However, in practice, reversibility is usually not possible because the system typically becomes entangled with an inaccessible environment in an irreversible way. This irreversibility can be quantified in different ways in the resource-theoretic framework, which was first introduced and studied for the case of the resource theory of entanglement. While entanglement theory has been thoroughly studied, this article develops the resource theory of quantum secret key, as pure entanglement is not a strict requirement for a quantum-secured communication. To understand irreversibility in the context of a quantum secret key, we define a quantity called key cost as the amount of private key needed for the creation of an arbitrarily good approximation of a bipartite quantum state by LOCC. We study properties of the key cost via the lens of the quantity key of formation, which represents the minimum average amount of key content of a state, where the average is taken over all decompositions into so-called generalized private states. The central theoretical contribution of the paper is the proof that the regularized key of formation is an upper bound to the key cost. To establish this result, we propose a privacy dilution protocol that converts states with ideal privacy into states with diluted privacy. Conversely, the key cost is bounded from below by the regularized relative entropy of entanglement, which implies irreversibility in the privacy creation-distillation process for a specific class of states. We further show that, for mixed-state analogs of pure quantum states, a number of entanglement measures are equal. Ultimately, the work shows that the privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation. The applicability of the introduced quantities goes beyond the resource theory of (device-dependent) private key. They naturally fit the device-independent scenario. Moreover, it is natural to postulate that the strict gap between the key cost and the distillable key is a natural quantifier of the lower bound on the inevitable energy cost of operating the quantum network.► BibTeX data@article{Horodecki2026costofquantumsecret, doi = {10.22331/q-2026-05-06-2098}, url = {https://doi.org/10.22331/q-2026-05-06-2098}, title = {Cost of quantum secret key}, author = {Horodecki, Karol and Sikorski, Leonard and Das, Siddhartha 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 = {2098}, month = may, year = {2026} }► References [1] Eric Chitambar and Gilad Gour. ``Quantum resource theories''. Reviews of Modern Physics 91, 025001 (2019). https://doi.org/10.1103/RevModPhys.91.025001 [2] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters. ``Purification of noisy entanglement and faithful teleportation via noisy channels''.

Physical Review Letters 76, 722–725 (1996). https://doi.org/10.1103/PhysRevLett.76.722 [3] Ueli M. Maurer. ``Secret key agreement by public discussion from common information''. IEEE Transactions on Information Theory 39, 733–742 (1993). https://doi.org/10.1109/18.256484 [4] Renato Renner and Stefan Wolf. ``New bounds in secret-key agreement: The gap between formation and secrecy extraction''.

In Lecture Notes in Computer Science. Pages 562–577.

Springer Berlin Heidelberg (2003). https://doi.org/10.1007/3-540-39200-9_35 [5] Nicolas Gisin and Stefan Wolf. ``Linking classical and quantum key agreement: Is there ``bound information''?''. In Proceedings of Crypto 2000, Lecture Notes in Computer Science. Volume 1880, pages 482–500. (2000). https://doi.org/10.1007/3-540-44598-6_30 [6] Antonio Acín, Juan Ignacio Cirac, and Lluis Masanes. ``Multipartite bound information exists and can be activated''.

Physical Review Letters 92, 107903 (2004). https://doi.org/10.1103/PhysRevLett.92.107903 [7] Andreas Winter. ``Secret, public and quantum correlation cost of triples of random variables''. In Proceedings of the 2005 International Symposium on Information Theory. Pages 2270–2274. (2005). https://doi.org/10.1109/ISIT.2005.1523752 [8] Karol Horodecki, Michał Horodecki, Pawel Horodecki, and Jonathan Oppenheim. ``Information theories with adversaries, intrinsic information, and entanglement''. Foundations of Physics 35, 2027–2040 (2005). https://doi.org/10.1007/s10701-005-8660-5 [9] Eric Chitambar, Min-Hsiu Hsieh, and Andreas Winter. ``The private and public correlation cost of three random variables with collaboration''. IEEE Transactions on Information Theory 62, 2034–2043 (2016). https://doi.org/10.1109/TIT.2016.2530086 [10] Eric Chitambar, Ben Fortescue, and Min-Hsiu Hsieh. ``Classical analog to entanglement reversibility''.

Physical Review Letters 115, 090501 (2015). https://doi.org/10.1103/PhysRevLett.115.090501 [11] Matthias Christandl, Artur Ekert, Michał Horodecki, Paweł Horodecki, Jonathan Oppenheim, and Renato Renner. ``Unifying classical and quantum key distillation''. In Theory of Cryptography. Pages 456–478.

Springer Berlin Heidelberg (2007). https://doi.org/10.1007/978-3-540-70936-7_25 [12] Maris Ozols, Graeme Smith, and John A. Smolin. ``Bound entangled states with a private key and their classical counterpart''.

Physical Review Letters 112, 110502 (2014). https://doi.org/10.1103/PhysRevLett.112.110502 [13] Eric Chitambar, Ben Fortescue, and Min-Hsiu Hsieh. ``The conditional common information in classical and quantum secret key distillation''. IEEE Transactions on Information Theory 64, 7381–7394 (2018). https://doi.org/10.1109/tit.2018.2851564 [14] Daniel Collins and Sandu Popescu. ``Classical analog of entanglement''. Physical Review A 65, 032321 (2002). https://doi.org/10.1103/PhysRevA.65.032321 [15] Stephanie Wehner, David Elkouss, and Ronald Hanson. ``Quantum internet: A vision for the road ahead''. Science 362, eaam9288 (2018). https://doi.org/10.1126/science.aam9288 [16] Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Jonathan Oppenheim. ``Secure key from bound entanglement''.

Physical Review Letters 94, 160502 (2005). https://doi.org/10.1103/PhysRevLett.94.160502 [17] Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Jonathan Oppenheim. ``General paradigm for distilling classical key from quantum states''. IEEE Transactions on Information Theory 55, 1898–1929 (2009). https://doi.org/10.1109/tit.2008.2009798 [18] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''.

Cambridge University Press. Cambridge, UK (2000). https://doi.org/10.1017/CBO9780511976667 [19] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Mixed-state entanglement and distillation: Is there a "bound" entanglement in nature?''.

Physical Review Letters 80, 5239–5242 (1998). https://doi.org/10.1103/physrevlett.80.5239 [20] Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865–942 (2009). https://doi.org/10.1103/revmodphys.81.865 [21] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Physical Review A 54, 3824–3851 (1996). arXiv:quant-ph/9604024. https://doi.org/10.1103/PhysRevA.54.3824 arXiv:quant-ph/9604024 [22] 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). https://doi.org/10.1016/j.tcs.2014.05.025 [23] Artur K. Ekert. ``Quantum cryptography based on Bell's theorem''.

Physical Review Letters 67, 661–663 (1991). https://doi.org/10.1103/physrevlett.67.661 [24] Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden. ``Quantum cryptography''. Reviews of Modern Physics 74, 145–195 (2002). https://doi.org/10.1103/RevModPhys.74.145 [25] Igor Devetak and Andreas Winter. ``Distillation of secret key and entanglement from quantum states''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, 207–235 (2005). https://doi.org/10.1098/rspa.2004.1372 [26] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. ``Device-independent security of quantum cryptography against collective attacks''.

Physical Review Letters 98, 230501 (2007). https://doi.org/10.1103/PhysRevLett.98.230501 [27] Marcin Pawłowski and Nicolas Brunner. ``Semi-device-independent security of one-way quantum key distribution''. Physical Review A 84, 010302 (2011). https://doi.org/10.1103/PhysRevA.84.010302 [28] Hoi-Kwong Lo, Marcos Curty, and Bing Qi. ``Measurement-device-independent quantum key distribution''.

Physical Review Letters 108, 130503 (2012). https://doi.org/10.1103/PhysRevLett.108.130503 [29] Rotem Arnon-Friedman, Frédéric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. ``Practical device-independent quantum cryptography via entropy accumulation''. Nature Communications 9, 459 (2018). https://doi.org/10.1038/s41467-017-02307-4 [30] 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 [31] Siddhartha Das, Stefan Bäuml, Marek Winczewski, and Karol Horodecki. ``Universal limitations on quantum key distribution over a network''. Physical Review X 11, 041016 (2021). https://doi.org/10.1103/PhysRevX.11.041016 [32] Matthias Christandl. ``The structure of bipartite quantum states - insights from group theory and cryptography''. PhD thesis. University of Cambridge. (2006). https://doi.org/10.48550/arXiv.quant-ph/0604183 arXiv:quant-ph/0604183 [33] Karol Horodecki, Michał Studziński, Ryszard P. Kostecki, Omer Sakarya, and Dong Yang. ``Upper bounds on the leakage of private data and an operational approach to Markovianity''. Physical Review A 104, 052422 (2021). https://doi.org/10.1103/physreva.104.052422 [34] Karol Horodecki. ``General paradigm for distilling classical key from quantum states — On quantum entanglement and security''. PhD thesis. University of Warsaw. (2008). url: https://www.mimuw.edu.pl/media/uploads/doctorates/thesis-karol-horodecki.pdf. https://www.mimuw.edu.pl/media/uploads/doctorates/thesis-karol-horodecki.pdf [35] Patrick M. Hayden, Michal Horodecki, and Barbara M. Terhal. ``The asymptotic entanglement cost of preparing a quantum state''. Journal of Physics A: Mathematical and General 34, 6891–6898 (2001). https://doi.org/10.1088/0305-4470/34/35/314 [36] Hayata Yamasaki, Kohdai Kuroiwa, Patrick Hayden, and Ludovico Lami. ``Entanglement cost for infinite-dimensional physical systems''. Communications in Mathematical Physics 406, 277 (2025). https://doi.org/10.1007/s00220-025-05431-1 [37] Karol Horodecki, Michal Horodecki, Pawel Horodecki, Debbie Leung, and Jonathan Oppenheim. ``Unconditional privacy over channels which cannot convey quantum information''.

Physical Review Letters 100, 110502 (2008). https://doi.org/10.1103/PhysRevLett.100.110502 [38] Karol Horodecki, Michal Horodecki, Pawel Horodecki, Debbie Leung, and Jonathan Oppenheim. ``Quantum key distribution based on private states: Unconditional security over untrusted channels with zero quantum capacity''. IEEE Transactions on Information Theory 54, 2604–2620 (2008). https://doi.org/10.1109/TIT.2008.921870 [39] Graeme Smith and Jon Yard. ``Quantum communication with zero-capacity channels''. Science 321, 1812–1815 (2008). arXiv:0807.4935. https://doi.org/10.1126/science.1162242 arXiv:0807.4935 [40] Ignatius W. Primaatmaja, Koon Tong Goh, Ernest Y.-Z. Tan, John T.-F. Khoo, Shouvik Ghorai, and Charles C.-W. Lim. ``Security of device-independent quantum key distribution protocols: a review''. Quantum 7, 932 (2023). https://doi.org/10.22331/q-2023-03-02-932 [41] Matthias Christandl, Roberto Ferrara, and Karol Horodecki. ``Upper bounds on device-independent quantum key distribution''.

Physical Review Letters 126, 160501 (2021). https://doi.org/10.1103/PhysRevLett.126.160501 [42] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Reviews of Modern Physics 86, 419–478 (2014). https://doi.org/10.1103/revmodphys.86.419 [43] Marcos Curty, Maciej Lewenstein, and Norbert Lütkenhaus. ``Entanglement as a precondition for secure quantum key distribution''.

Physical Review Letters 92, 217903 (2004). https://doi.org/10.1103/PhysRevLett.92.217903 [44] ``Open quantum problems—IQOQI Vienna''. https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems. https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems [45] Karol Horodecki, Łukasz Pankowski, Michał Horodecki, and Paweł Horodecki. ``Low-dimensional bound entanglement with one-way distillable cryptographic key''. IEEE Transactions on Information Theory 54, 2621–2625 (2008). https://doi.org/10.1109/tit.2008.921709 [46] Karol Horodecki, Michal Horodecki, Pawel Horodecki, and Jonathan Oppenheim. ``Locking entanglement with a single qubit''.

Physical Review Letters 94, 200501 (2005). https://doi.org/10.1103/PhysRevLett.94.200501 [47] Vlatko Vedral, Martin B. Plenio, M. A. Rippin, and Peter L. Knight. ``Quantifying entanglement''.

Physical Review Letters 78, 2275–2279 (1997). https://doi.org/10.1103/PhysRevLett.78.2275 [48] Matthias Christandl, Norbert Schuch, and Andreas Winter. ``Entanglement of the antisymmetric state''. Communications in Mathematical Physics 311, 397–422 (2012). arXiv:0910.4151. https://doi.org/10.1007/s00220-012-1446-7 arXiv:0910.4151 [49] Stefan Bäuml, Matthias Christandl, Karol Horodecki, and Andreas Winter. ``Limitations on quantum key repeaters''. Nature Communications 6, 6908 (2015). arXiv:1402.5927. https://doi.org/10.1038/ncomms7908 arXiv:1402.5927 [50] Mark M. Wilde. ``Squashed entanglement and approximate private states''.

Quantum Information Processing 15, 4563–4580 (2016). arXiv:1606.08028. https://doi.org/10.1007/s11128-016-1432-7 arXiv:1606.08028 [51] Karol Horodecki, Piotr Ć wikliński, Adam Rutkowski, and Michał Studziński. ``On distilling secure key from reducible private states and (non)existence of entangled key-undistillable states''. New Journal of Physics 20, 083021 (2016). arXiv:1612.08938. https://doi.org/10.1088/1367-2630/aad75a arXiv:1612.08938 [52] Matthias Christandl and Roberto Ferrara. ``Private states, quantum data hiding, and the swapping of perfect secrecy''.

Physical Review Letters 119, 220506 (2017). https://doi.org/10.1103/PhysRevLett.119.220506 [53] Mark M. Wilde. ``Second law of entanglement dynamics for the non-asymptotic regime''. In Proceedings of the 2021 IEEE Information Theory Workshop (ITW). Pages 1–6. (2021). https://doi.org/10.1109/ITW48936.2021.9611411 [54] Ryuji Takagi, Bartosz Regula, and Mark M. Wilde. ``One-shot yield-cost relations in general quantum resource theories''. PRX Quantum 3, 010348 (2022). https://doi.org/10.1103/PRXQuantum.3.010348 [55] Dong Yang, Michał Horodecki, Ryszard Horodecki, and Barbara Synak-Radtke. ``Irreversibility for all bound entangled states''.

Physical Review Letters 95, 190501 (2005). https://doi.org/10.1103/PhysRevLett.95.190501 [56] Matthias Christandl and Andreas Winter. ````Squashed entanglement'': An additive entanglement measure''. Journal of Mathematical Physics 45, 829–840 (2004). https://doi.org/10.1063/1.1643788 [57] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Limits for entanglement measures''.

Physical Review Letters 84, 2014–2017 (2000). https://doi.org/10.1103/PhysRevLett.84.2014 [58] Hoi-Kwong Lo and Sandu Popescu. ``Classical communication cost of entanglement manipulation: Is entanglement an interconvertible resource?''.

Physical Review Letters 83, 1459–1462 (1999). https://doi.org/10.1103/PhysRevLett.83.1459 [59] Andreas Winter and Dong Yang. ``Operational resource theory of coherence''.

Physical Review Letters 116, 120404 (2016). https://doi.org/10.1103/PhysRevLett.116.120404 [60] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states. an introduction to quantum entanglement''.

Cambridge University Press. (2006). https://doi.org/10.1017/CBO9780511535048 [61] John Watrous. ``Theory of quantum information''.

Cambridge University Press. (2018). https://doi.org/10.1017/9781316848142 [62] Armin Uhlmann. ``The ``transition probability'' in the state space of a *-algebra''. Reports on Mathematical Physics 9, 273–279 (1976). https://doi.org/10.1016/0034-4877(76)90060-4 [63] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter. ``Everything you always wanted to know about LOCC (but were afraid to ask)''. Communications in Mathematical Physics 328, 303–326 (2014). https://doi.org/10.1007/s00220-014-1953-9 [64] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters. ``Quantum nonlocality without entanglement''. Physical Review A 59, 1070–1091 (1999). https://doi.org/10.1103/PhysRevA.59.1070 [65] Abbas El Gamal and Young-Han Kim. ``Network information theory''.

Cambridge University Press. (2011). https://doi.org/10.1017/CBO9781139030687 [66] Thomas M. Cover and Joy A. Thomas. ``Elements of information theory''. Wiley Series in Telecommunications and Signal Processing. Wiley-Interscience. (2006). 2nd edition. https://doi.org/10.1002/047174882X [67] Claude E. Shannon. ``A mathematical theory of communication''.

Bell System Technical Journal 27, 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb01338.x [68] Nicolas J. Cerf and Christoph Adami. ``Negative entropy and information in quantum mechanics''.

Physical Review Letters 79, 5194–5197 (1997). https://doi.org/10.1103/PhysRevLett.79.5194 [69] Elliott H. Lieb and Mary Beth Ruskai. ``Proof of the strong subadditivity of quantum-mechanical entropy''. Journal of Mathematical Physics 14, 1938–1941 (1973). https://doi.org/10.1063/1.1666274 [70] Yury Polyanskiy and Sergio Verdú. ``Arimoto channel coding converse and Rényi divergence''. In Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computation. Pages 1327–1333. (2010). https://doi.org/10.1109/ALLERTON.2010.5707067 [71] Mark M. Wilde, Andreas Winter, and Dong Yang. ``Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy''. Communications in Mathematical Physics 331, 593–622 (2014). https://doi.org/10.1007/s00220-014-2122-x [72] Hisaharu Umegaki. ``Conditional expectations in an operator algebra, IV (entropy and information)''.

Kodai Mathematical Seminar Reports 14, 59–85 (1962). https://doi.org/10.2996/kmj/1138844604 [73] Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. ``On quantum Rényi entropies: a new definition and some properties''. Journal of Mathematical Physics 54, 122203 (2013). https://doi.org/10.1063/1.4838856 [74] Rupert L. Frank and Elliott H. Lieb. ``Monotonicity of a relative Rényi entropy''. Journal of Mathematical Physics 54, 122201 (2013). https://doi.org/10.1063/1.4838835 [75] Mark M. Wilde. ``Optimized quantum $f$-divergences and data processing''. Journal of Physics A 51, 374002 (2018). arXiv:1710.10252. https://doi.org/10.1088/1751-8121/aad5a1 arXiv:1710.10252 [76] Nilanjana Datta. ``Min- and max-relative entropies and a new entanglement monotone''. IEEE Transactions on Information Theory 55, 2816–2826 (2009). https://doi.org/10.1109/TIT.2009.2018325 [77] Nilanjana Datta. ``Max-relative entropy of entanglement, alias log robustness''. International Journal of Quantum Information 7, 475–491 (2009). https://doi.org/10.1142/S0219749909005298 [78] Francesco Buscemi and Nilanjana Datta. ``The quantum capacity of channels with arbitrarily correlated noise''. IEEE Transactions on Information Theory 56, 1447–1460 (2010). https://doi.org/10.1109/TIT.2009.2039166 [79] Ligong Wang and Renato Renner. ``One-shot classical-quantum capacity and hypothesis testing''.

Physical Review Letters 108, 200501 (2012). https://doi.org/10.1103/PhysRevLett.108.200501 [80] Xin Wang and Mark M. Wilde. ``Resource theory of asymmetric distinguishability''.

Physical Review Research 1, 033170 (2019). https://doi.org/10.1103/PhysRevResearch.1.033170 [81] Andreas Winter. ``Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints''. Communications in Mathematical Physics 347, 291–313 (2016). https://doi.org/10.1007/s00220-016-2609-8 [82] Jens Eisert and Mark M Wilde. ``A smallest computable entanglement monotone''. In 2022 IEEE International Symposium on Information Theory (ISIT). Pages 2439–2444. IEEE (2022). https://doi.org/10.1109/ISIT50566.2022.9834375 [83] Guifré Vidal. ``Entanglement monotones''. Journal of Modern Optics 47, 355–376 (2000). https://doi.org/10.1080/09500340008244048 [84] Dong Yang, Karol Horodecki, Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, and Wei Song. ``Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof''. IEEE Transactions on Information Theory 55, 3375–3387 (2009). https://doi.org/10.1109/tit.2009.2021373 [85] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. ``Colloquium: Quantum coherence as a resource''. Reviews of Modern Physics 89, 041003 (2017). arXiv:1609.02439. https://doi.org/10.1103/RevModPhys.89.041003 arXiv:1609.02439 [86] Karol Horodecki, Debbie Leung, Hoi-Kwong Lo, and Jonathan Oppenheim. ``Quantum key distribution based on arbitrarily weak distillable entangled states''.

Physical Review Letters 96, 070501 (2006). https://doi.org/10.1103/PhysRevLett.96.070501 [87] Matthias Christandl and Andreas Winter. ``Uncertainty, monogamy, and locking of quantum correlations''. IEEE Transactions on Information Theory 51, 3159–3165 (2005). https://doi.org/10.1109/TIT.2005.853338 [88] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu. ``Benchmarking one-shot distillation in general quantum resource theories''. Physical Review A 101, 062315 (2020). https://doi.org/10.1103/physreva.101.062315 [89] Guifré Vidal and Rolf Tarrach. ``Robustness of entanglement''. Physical Review A 59, 141–155 (1999). https://doi.org/10.1103/PhysRevA.59.141 [90] Berry Groisman, Sandu Popescu, and Andreas Winter. ``Quantum, classical, and total amount of correlations in a quantum state''. Physical Review A 72, 032317 (2005). https://doi.org/10.1103/physreva.72.032317 [91] Karol Horodecki, Marek Winczewski, and Siddhartha Das. ``Fundamental limitations on the device-independent quantum conference key agreement''. Physical Review A 105, 022604 (2022). https://doi.org/10.1103/PhysRevA.105.022604 [92] Karol Horodecki and Gláucia Murta. ``Bounds on quantum nonlocality via partial transposition''. Physical Review A 92, 010301(R) (2015). https://doi.org/10.1103/PhysRevA.92.010301 [93] Eneet Kaur, Karol Horodecki, and Siddhartha Das. ``Upper bounds on device-independent quantum key distribution rates in static and dynamic scenarios''.

Physical Review Applied 18 (2022). https://doi.org/10.1103/physrevapplied.18.054033 [94] Gláucia Murta, Federico Grasselli, Hermann Kampermann, and Dagmar Bruß. ``Quantum conference key agreement: A review''.

Advanced Quantum Technologies 3, 2000025 (2020). https://doi.org/10.1002/qute.202000025 [95] Stefan Bäuml. ``On bound key and the use of bound entanglement''. Master's thesis.

Ludwig Maximilians Universität München. (2010).Cited by[1] Himanshu Badhani, S. Dhanuja G., and Siddhartha Das, "Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality", arXiv:2510.12790, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-06 09:17:06). 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 2026-05-06 09:17:05: Could not fetch cited-by data for 10.22331/q-2026-05-06-2098 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. AbstractIn this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided. Importantly, our results presented here will remain valid even if entangled states with zero distillable key were shown to exist.Popular summaryQuantum mechanics is a surprising physical theory as it is reversible in principle, contrary to our common experience. After an arbitrary quantum operation, if full access to the environment is available, the system's state can be restored to its initial state by a reversal operation. However, in practice, reversibility is usually not possible because the system typically becomes entangled with an inaccessible environment in an irreversible way. This irreversibility can be quantified in different ways in the resource-theoretic framework, which was first introduced and studied for the case of the resource theory of entanglement. While entanglement theory has been thoroughly studied, this article develops the resource theory of quantum secret key, as pure entanglement is not a strict requirement for a quantum-secured communication. To understand irreversibility in the context of a quantum secret key, we define a quantity called key cost as the amount of private key needed for the creation of an arbitrarily good approximation of a bipartite quantum state by LOCC. We study properties of the key cost via the lens of the quantity key of formation, which represents the minimum average amount of key content of a state, where the average is taken over all decompositions into so-called generalized private states. The central theoretical contribution of the paper is the proof that the regularized key of formation is an upper bound to the key cost. To establish this result, we propose a privacy dilution protocol that converts states with ideal privacy into states with diluted privacy. Conversely, the key cost is bounded from below by the regularized relative entropy of entanglement, which implies irreversibility in the privacy creation-distillation process for a specific class of states. We further show that, for mixed-state analogs of pure quantum states, a number of entanglement measures are equal. Ultimately, the work shows that the privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation. The applicability of the introduced quantities goes beyond the resource theory of (device-dependent) private key. They naturally fit the device-independent scenario. Moreover, it is natural to postulate that the strict gap between the key cost and the distillable key is a natural quantifier of the lower bound on the inevitable energy cost of operating the quantum network.► BibTeX data@article{Horodecki2026costofquantumsecret, doi = {10.22331/q-2026-05-06-2098}, url = {https://doi.org/10.22331/q-2026-05-06-2098}, title = {Cost of quantum secret key}, author = {Horodecki, Karol and Sikorski, Leonard and Das, Siddhartha 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 = {2098}, month = may, year = {2026} }► References [1] Eric Chitambar and Gilad Gour. ``Quantum resource theories''. Reviews of Modern Physics 91, 025001 (2019). https://doi.org/10.1103/RevModPhys.91.025001 [2] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters. ``Purification of noisy entanglement and faithful teleportation via noisy channels''.

In Lecture Notes in Computer Science. Pages 562–577.

Physical Review Letters 94, 200501 (2005). https://doi.org/10.1103/PhysRevLett.94.200501 [47] Vlatko Vedral, Martin B. Plenio, M. A. Rippin, and Peter L. Knight. ``Quantifying entanglement''.

Physical Review Letters 83, 1459–1462 (1999). https://doi.org/10.1103/PhysRevLett.83.1459 [59] Andreas Winter and Dong Yang. ``Operational resource theory of coherence''.

Cambridge University Press. (2006). https://doi.org/10.1017/CBO9780511535048 [61] John Watrous. ``Theory of quantum information''.

Physical Review Letters 108, 200501 (2012). https://doi.org/10.1103/PhysRevLett.108.200501 [80] Xin Wang and Mark M. Wilde. ``Resource theory of asymmetric distinguishability''.

Advanced Quantum Technologies 3, 2000025 (2020). https://doi.org/10.1002/qute.202000025 [95] Stefan Bäuml. ``On bound key and the use of bound entanglement''. Master's thesis.

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