1Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
2Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
3Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
4Institute for Quantum Optics and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
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Abstract
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover $textit{catalytic replication}$ of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce $textit{consistent resource measures}$ that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.
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► References
[1] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, Apr 2019. 10.1103/RevModPhys.91.025001. URL https://link.aps.org/doi/10.1103/RevModPhys.91.025001.
https://doi.org/10.1103/RevModPhys.91.025001
[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, Jun 2009. 10.1103/RevModPhys.81.865. URL https://link.aps.org/doi/10.1103/RevModPhys.81.865.
https://doi.org/10.1103/RevModPhys.81.865
[3] E. M. Rains. A semidefinite program for distillable entanglement. IEEE Trans. Inf. Theory, 47 (7): 2921–2933, Nov 2001. ISSN 1557-9654. 10.1109/18.959270. URL https://doi.org/10.1109/18.959270.
https://doi.org/10.1109/18.959270
[4] Fernando G. S. L. Brandão and N. Datta. One-shot rates for entanglement manipulation under non-entangling maps. IEEE Trans. Inf. Theory, 57 (3): 1754–1760, March 2011. ISSN 1557-9654. 10.1109/TIT.2011.2104531. URL https://doi.org/10.1109/TIT.2011.2104531.
https://doi.org/10.1109/TIT.2011.2104531
[5] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. Colloquium: Quantum coherence as a resource. Rev. Mod. Phys., 89: 041003, Oct 2017. 10.1103/RevModPhys.89.041003. URL https://link.aps.org/doi/10.1103/RevModPhys.89.041003.
https://doi.org/10.1103/RevModPhys.89.041003
[6] Iman Marvian and Robert W. Spekkens. How to quantify coherence: Distinguishing speakable and unspeakable notions. Phys. Rev. A, 94: 052324, Nov 2016. 10.1103/PhysRevA.94.052324. URL https://link.aps.org/doi/10.1103/PhysRevA.94.052324.
https://doi.org/10.1103/PhysRevA.94.052324
[7] Johan Aberg. Quantifying superposition. arXiv preprint arXiv:quant-ph/0612146, 2006. URL https://arxiv.org/abs/quant-ph/0612146.
arXiv:quant-ph/0612146
[8] T. Baumgratz, M. Cramer, and M. B. Plenio. Quantifying coherence. Phys. Rev. Lett., 113: 140401, Sep 2014. 10.1103/PhysRevLett.113.140401. URL https://link.aps.org/doi/10.1103/PhysRevLett.113.140401.
https://doi.org/10.1103/PhysRevLett.113.140401
[9] Benjamin Yadin, Jiajun Ma, Davide Girolami, Mile Gu, and Vlatko Vedral. Quantum processes which do not use coherence. Phys. Rev. X, 6: 041028, Nov 2016. 10.1103/PhysRevX.6.041028. URL https://link.aps.org/doi/10.1103/PhysRevX.6.041028.
https://doi.org/10.1103/PhysRevX.6.041028
[10] Eric Chitambar and Gilad Gour. Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett., 117: 030401, Jul 2016a. 10.1103/PhysRevLett.117.030401. URL https://link.aps.org/doi/10.1103/PhysRevLett.117.030401.
https://doi.org/10.1103/PhysRevLett.117.030401
[11] Eric Chitambar and Gilad Gour. Comparison of incoherent operations and measures of coherence. Phys. Rev. A, 94: 052336, Nov 2016b. 10.1103/PhysRevA.94.052336. URL https://link.aps.org/doi/10.1103/PhysRevA.94.052336.
https://doi.org/10.1103/PhysRevA.94.052336
[12] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and Th. Beth. Thermodynamic cost of reliability and low temperatures: Tightening Landauer’s principle and the second law. Int. J. Theor. Phys., 39: 2717, 2000. 10.1023/A:1026422630734. URL https://doi.org/10.1023/A:1026422630734.
https://doi.org/10.1023/A:1026422630734
[13] Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett., 111: 250404, Dec 2013. 10.1103/PhysRevLett.111.250404. URL https://link.aps.org/doi/10.1103/PhysRevLett.111.250404.
https://doi.org/10.1103/PhysRevLett.111.250404
[14] Michal Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun., 4: 2059, 2013a. 10.1038/ncomms3059. URL https://doi.org/10.1038/ncomms3059.
https://doi.org/10.1038/ncomms3059
[15] Nicole Yunger Halpern. Toward physical realizations of thermodynamic resource theories. In Information and Interaction, pages 135–166. Springer, 2017. 10.1007/978-3-319-43760-6_8. URL https://doi.org/10.1007/978-3-319-43760-6_8.
https://doi.org/10.1007/978-3-319-43760-6_8
[16] Nicole Yunger Halpern and David T. Limmer. Fundamental limitations on photoisomerization from thermodynamic resource theories. Phys. Rev. A, 101: 042116, Apr 2020. 10.1103/PhysRevA.101.042116. URL https://link.aps.org/doi/10.1103/PhysRevA.101.042116.
https://doi.org/10.1103/PhysRevA.101.042116
[17] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New J. Phys., 16 (1): 013009, jan 2014. 10.1088/1367-2630/16/1/013009. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/16/1/013009
[18] Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett., 118: 090501, Mar 2017. 10.1103/PhysRevLett.118.090501. URL https://link.aps.org/doi/10.1103/PhysRevLett.118.090501.
https://doi.org/10.1103/PhysRevLett.118.090501
[19] Gilad Gour and Robert W Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New J. Phys., 10 (3): 033023, 2008. 10.1088/1367-2630/10/3/033023. URL https://doi.org/10.1088/1367-2630/10/3/033023.
https://doi.org/10.1088/1367-2630/10/3/033023
[20] Gilad Gour, Iman Marvian, and Robert W. Spekkens. Measuring the quality of a quantum reference frame: The relative entropy of frameness. Phys. Rev. A, 80: 012307, Jul 2009. 10.1103/PhysRevA.80.012307. URL https://link.aps.org/doi/10.1103/PhysRevA.80.012307.
https://doi.org/10.1103/PhysRevA.80.012307
[21] Iman Marvian and Robert W Spekkens. The theory of manipulations of pure state asymmetry: I. basic tools, equivalence classes and single copy transformations. New J. Phys., 15 (3): 033001, mar 2013. 10.1088/1367-2630/15/3/033001. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/15/3/033001
[22] Iman Marvian and Robert W. Spekkens. Asymmetry properties of pure quantum states. Phys. Rev. A, 90: 014102, Jul 2014a. 10.1103/PhysRevA.90.014102. URL https://link.aps.org/doi/10.1103/PhysRevA.90.014102.
https://doi.org/10.1103/PhysRevA.90.014102
[23] Iman Marvian and Robert W Spekkens. Extending Noether’s theorem by quantifying the asymmetry of quantum states. Nat. Commun., 5 (1): 1–8, 2014b. 10.1038/ncomms4821. URL https://doi.org/10.1038/ncomms4821.
https://doi.org/10.1038/ncomms4821
[24] Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Yunger Halpern. The resource theory of informational nonequilibrium in thermodynamics. Phys. Rep., 583: 1 – 58, 2015. ISSN 0370-1573. https://doi.org/10.1016/j.physrep.2015.04.003. URL http://www.sciencedirect.com/science/article/pii/S037015731500229X.
https://doi.org/https://doi.org/10.1016/j.physrep.2015.04.003
http://www.sciencedirect.com/science/article/pii/S037015731500229X
[25] Ryuji Takagi and Quntao Zhuang. Convex resource theory of non-Gaussianity. Phys. Rev. A, 97: 062337, Jun 2018. 10.1103/PhysRevA.97.062337. URL https://link.aps.org/doi/10.1103/PhysRevA.97.062337.
https://doi.org/10.1103/PhysRevA.97.062337
[26] Ludovico Lami, Bartosz Regula, Xin Wang, Rosanna Nichols, Andreas Winter, and Gerardo Adesso. Gaussian quantum resource theories. Phys. Rev. A, 98: 022335, Aug 2018. 10.1103/PhysRevA.98.022335. URL https://link.aps.org/doi/10.1103/PhysRevA.98.022335.
https://doi.org/10.1103/PhysRevA.98.022335
[27] Francesco Albarelli, Marco G. Genoni, Matteo G. A. Paris, and Alessandro Ferraro. Resource theory of quantum non-Gaussianity and wigner negativity. Phys. Rev. A, 98: 052350, Nov 2018. 10.1103/PhysRevA.98.052350. URL https://link.aps.org/doi/10.1103/PhysRevA.98.052350.
https://doi.org/10.1103/PhysRevA.98.052350
[28] Hayata Yamasaki, Takaya Matsuura, and Masato Koashi. Cost-reduced all-gaussian universality with the gottesman-kitaev-preskill code: Resource-theoretic approach to cost analysis. Phys. Rev. Research, 2: 023270, Jun 2020. 10.1103/PhysRevResearch.2.023270. URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.023270.
https://doi.org/10.1103/PhysRevResearch.2.023270
[29] Eyuri Wakakuwa. Operational resource theory of non-markovianity. arXiv preprint arXiv:1709.07248 [quant-ph], 2017. URL https://arxiv.org/abs/1709.07248.
arXiv:1709.07248
[30] Eyuri Wakakuwa. Communication cost for non-markovianity of tripartite quantum states: A resource theoretic approach. arXiv preprint arXiv:1904.08852 [quant-ph], 2019. URL https://arxiv.org/abs/1904.08852.
arXiv:1904.08852
[31] Michal Horodecki and Jonathan Oppenheim. (Quantumness in the context of) resource theories. Int. J. Mod. Phys. B, 27 (1-3): 1345019, 2013b. 10.1142/S0217979213450197. URL https://doi.org/10.1142/S0217979213450197.
https://doi.org/10.1142/S0217979213450197
[32] Fernando G. S. L. Brandão and Gilad Gour. Reversible framework for quantum resource theories. Phys. Rev. Lett., 115: 070503, Aug 2015. 10.1103/PhysRevLett.115.070503. URL https://link.aps.org/doi/10.1103/PhysRevLett.115.070503.
https://doi.org/10.1103/PhysRevLett.115.070503
[33] Kamil Korzekwa, Christopher T. Chubb, and Marco Tomamichel. Avoiding irreversibility: Engineering resonant conversions of quantum resources. Phys. Rev. Lett., 122: 110403, Mar 2019. 10.1103/PhysRevLett.122.110403. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.110403.
https://doi.org/10.1103/PhysRevLett.122.110403
[34] C. L. Liu, Xiao-Dong Yu, and D. M. Tong. Flag additivity in quantum resource theories. Phys. Rev. A, 99: 042322, Apr 2019a. 10.1103/PhysRevA.99.042322. URL https://link.aps.org/doi/10.1103/PhysRevA.99.042322.
https://doi.org/10.1103/PhysRevA.99.042322
[35] Zi-Wen Liu, Kaifeng Bu, and Ryuji Takagi. One-shot operational quantum resource theory. Phys. Rev. Lett., 123: 020401, Jul 2019b. 10.1103/PhysRevLett.123.020401. URL https://link.aps.org/doi/10.1103/PhysRevLett.123.020401.
https://doi.org/10.1103/PhysRevLett.123.020401
[36] Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso. Operational advantage of quantum resources in subchannel discrimination. Phys. Rev. Lett., 122: 140402, Apr 2019. 10.1103/PhysRevLett.122.140402. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.140402.
https://doi.org/10.1103/PhysRevLett.122.140402
[37] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks. Phys. Rev. X, 9: 031053, Sep 2019. 10.1103/PhysRevX.9.031053. URL https://link.aps.org/doi/10.1103/PhysRevX.9.031053.
https://doi.org/10.1103/PhysRevX.9.031053
[38] Kun Fang and Zi-Wen Liu. No-go theorems for quantum resource purification. Phys. Rev. Lett., 125: 060405, Aug 2020. 10.1103/PhysRevLett.125.060405. URL https://link.aps.org/doi/10.1103/PhysRevLett.125.060405.
https://doi.org/10.1103/PhysRevLett.125.060405
[39] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu. Benchmarking one-shot distillation in general quantum resource theories. Phys. Rev. A, 101: 062315, Jun 2020. 10.1103/PhysRevA.101.062315. URL https://link.aps.org/doi/10.1103/PhysRevA.101.062315.
https://doi.org/10.1103/PhysRevA.101.062315
[40] Madhav Krishnan Vijayan, Eric Chitambar, and Min-Hsiu Hsieh. One-shot distillation in a general resource theory. arXiv preprint arXiv:1906.04959 [quant-ph], 2019. URL https://arxiv.org/abs/1906.04959.
arXiv:1906.04959
[41] J. Eisert and M. M. Wolf. Gaussian Quantum Channels, pages 23–42. 10.1142/9781860948169_0002. URL https://www.worldscientific.com/doi/abs/10.1142/9781860948169_0002.
https://doi.org/10.1142/9781860948169_0002
[42] A. S. Holevo. One-mode quantum Gaussian channels: Structure and quantum capacity. Probl. Inf. Transm., 43, Mar 2007. 10.1134/S0032946007010012. URL https://doi.org/10.1134/S0032946007010012.
https://doi.org/10.1134/S0032946007010012
[43] Anindita Bera, Tamoghna Das, Debasis Sadhukhan, Sudipto Singha Roy, Aditi Sen(De), and Ujjwal Sen. Quantum discord and its allies: a review of recent progress. Rep. Prog. Phys., 81 (2): 024001, dec 2017. 10.1088/1361-6633/aa872f. URL https://doi.org/10.1088.
https://doi.org/10.1088/1361-6633/aa872f
[44] Stanley Gudder. Quantum markov chains. J. Math. Phys., 49 (7): 072105, 2008. 10.1063/1.2953952. URL https://doi.org/10.1063/1.2953952.
https://doi.org/10.1063/1.2953952
[45] Daniel Jonathan and Martin B. Plenio. Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett., 83: 3566–3569, Oct 1999. 10.1103/PhysRevLett.83.3566. URL https://link.aps.org/doi/10.1103/PhysRevLett.83.3566.
https://doi.org/10.1103/PhysRevLett.83.3566
[46] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54: 3824–3851, Nov 1996a. 10.1103/PhysRevA.54.3824. URL https://link.aps.org/doi/10.1103/PhysRevA.54.3824.
https://doi.org/10.1103/PhysRevA.54.3824
[47] Patrick M Hayden, Michal Horodecki, and Barbara M Terhal. The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen., 34 (35): 6891–6898, aug 2001. 10.1088/0305-4470/34/35/314. URL https://doi.org/10.1088.
https://doi.org/10.1088/0305-4470/34/35/314
[48] Andreas Winter and Dong Yang. Operational resource theory of coherence. Phys. Rev. Lett., 116: 120404, Mar 2016. 10.1103/PhysRevLett.116.120404. URL https://link.aps.org/doi/10.1103/PhysRevLett.116.120404.
https://doi.org/10.1103/PhysRevLett.116.120404
[49] Matthew J. Donald, Michał Horodecki, and Oliver Rudolph. The uniqueness theorem for entanglement measures. J. Math. Phys., 43 (9): 4252–4272, 2002. 10.1063/1.1495917. URL https://doi.org/10.1063/1.1495917.
https://doi.org/10.1063/1.1495917
[50] Michael Reed and Barry Simon. Methods of modern mathematical physics. vol. 1. Functional analysis. Academic New York, 1980.
[51] Masamichi Takesaki. Theory of Operator Algebras I. Springer, 2012. ISBN 0387903917;9780387903910;. 10.1007/978-1-4612-6188-9. URL https://doi.org/10.1007/978-1-4612-6188-9.
https://doi.org/10.1007/978-1-4612-6188-9
[52] A Yu Kitaev. Quantum computations: algorithms and error correction. Russ. Math. Surv., 52 (6): 1191–1249, dec 1997. 10.1070/rm1997v052n06abeh002155. URL https://doi.org/10.1070.
https://doi.org/10.1070/rm1997v052n06abeh002155
[53] Fernando G. S. L. Brandão and Martin Plenio. Entanglement theory and the second law of thermodynamics. Nat. Phys., 4: 873–877, 2008. 10.1038/nphys1100. URL https://doi.org/10.1038/nphys1100.
https://doi.org/10.1038/nphys1100
[54] Patricia Contreras-Tejada, Carlos Palazuelos, and Julio I. de Vicente. Resource theory of entanglement with a unique multipartite maximally entangled state. Phys. Rev. Lett., 122: 120503, Mar 2019. 10.1103/PhysRevLett.122.120503. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.120503.
https://doi.org/10.1103/PhysRevLett.122.120503
[55] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about locc (but were afraid to ask). Commun. Math. Phys., 328 (1): 303–326, May 2014. ISSN 1432-0916. 10.1007/s00220-014-1953-9. URL https://doi.org/10.1007/s00220-014-1953-9.
https://doi.org/10.1007/s00220-014-1953-9
[56] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. 10.1017/CBO9780511976667. URL https://doi.org/10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667
[57] Berry Groisman, Sandu Popescu, and Andreas Winter. Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A, 72: 032317, Sep 2005. 10.1103/PhysRevA.72.032317. URL https://link.aps.org/doi/10.1103/PhysRevA.72.032317.
https://doi.org/10.1103/PhysRevA.72.032317
[58] Anurag Anshu, Min-Hsiu Hsieh, and Rahul Jain. Quantifying resources in general resource theory with catalysts. Phys. Rev. Lett., 121: 190504, Nov 2018. 10.1103/PhysRevLett.121.190504. URL https://link.aps.org/doi/10.1103/PhysRevLett.121.190504.
https://doi.org/10.1103/PhysRevLett.121.190504
[59] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden. Advances in quantum cryptography. arXiv preprint arXiv:1906.01645 [quant-ph], 2019. URL https://arxiv.org/abs/1906.01645.
arXiv:1906.01645
[60] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, Aug 2018. ISSN 2521-327X. 10.22331/q-2018-08-06-79. URL http://dx.doi.org/10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79
[61] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86: 052329, Nov 2012. 10.1103/PhysRevA.86.052329. URL https://link.aps.org/doi/10.1103/PhysRevA.86.052329.
https://doi.org/10.1103/PhysRevA.86.052329
[62] Ulrich Berger, G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. Continuous Lattices and Domains, volume 93. Cambridge University Press, 2003. 10.1017/CBO9780511542725. URL https://doi.org/10.1017/CBO9780511542725.
https://doi.org/10.1017/CBO9780511542725
[63] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64: 012310, Jun 2001. 10.1103/PhysRevA.64.012310. URL https://link.aps.org/doi/10.1103/PhysRevA.64.012310.
https://doi.org/10.1103/PhysRevA.64.012310
[64] Giacomo Pantaleoni, Ben Q. Baragiola, and Nicolas C. Menicucci. Modular bosonic subsystem codes. Phys. Rev. Lett., 125: 040501, Jul 2020. 10.1103/PhysRevLett.125.040501. URL https://link.aps.org/doi/10.1103/PhysRevLett.125.040501.
https://doi.org/10.1103/PhysRevLett.125.040501
[65] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84: 621–669, May 2012. 10.1103/RevModPhys.84.621. URL https://link.aps.org/doi/10.1103/RevModPhys.84.621.
https://doi.org/10.1103/RevModPhys.84.621
[66] Michał Horodecki, Aditi Sen(De), and Ujjwal Sen. Rates of asymptotic entanglement transformations for bipartite mixed states: Maximally entangled states are not special. Phys. Rev. A, 67: 062314, Jun 2003. 10.1103/PhysRevA.67.062314. URL https://link.aps.org/doi/10.1103/PhysRevA.67.062314.
https://doi.org/10.1103/PhysRevA.67.062314
[67] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature? Phys. Rev. Lett., 80: 5239–5242, Jun 1998. 10.1103/PhysRevLett.80.5239. URL https://link.aps.org/doi/10.1103/PhysRevLett.80.5239.
https://doi.org/10.1103/PhysRevLett.80.5239
[68] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher. Concentrating partial entanglement by local operations. Phys. Rev. A, 53: 2046–2052, Apr 1996b. 10.1103/PhysRevA.53.2046. URL https://link.aps.org/doi/10.1103/PhysRevA.53.2046.
https://doi.org/10.1103/PhysRevA.53.2046
[69] Mark M. Wilde. Entanglement Manipulation, page 517–536. Cambridge University Press, 2 edition, 2017. 10.1017/9781316809976.022. URL https://doi.org/10.1017/9781316809976.022.
https://doi.org/10.1017/9781316809976.022
[70] W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299: 802–803, Oct 1982. 10.1038/299802a0. URL https://doi.org/10.1038/299802a0.
https://doi.org/10.1038/299802a0
[71] Xin Wang, Mark M. Wilde, and Yuan Su. Efficiently computable bounds for magic state distillation. Phys. Rev. Lett., 124: 090505, Mar 2020. 10.1103/PhysRevLett.124.090505. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.090505.
https://doi.org/10.1103/PhysRevLett.124.090505
[72] Martin B. Plbnio and Shashank Virmani. An introduction to entanglement measures. Quantum Info. Comput., 7 (1): 1–51, jan 2007. ISSN 1533-7146. 10.5555/2011706.2011707. URL https://dl.acm.org/doi/10.5555/2011706.2011707.
https://doi.org/10.5555/2011706.2011707
[73] Thomas R Bromley, Marco Cianciaruso, Sofoklis Vourekas, Bartosz Regula, and Gerardo Adesso. Accessible bounds for general quantum resources. J. Phys. A: Math. Theor., 51 (32): 325303, jul 2018. 10.1088/1751-8121/aacb4a. URL https://doi.org/10.1088.
https://doi.org/10.1088/1751-8121/aacb4a
[74] Zi-Wen Liu, Xueyuan Hu, and Seth Lloyd. Resource destroying maps. Phys. Rev. Lett., 118: 060502, Feb 2017. 10.1103/PhysRevLett.118.060502. URL https://link.aps.org/doi/10.1103/PhysRevLett.118.060502.
https://doi.org/10.1103/PhysRevLett.118.060502
[75] Bartosz Regula. Convex geometry of quantum resource quantification. J. Phys. A: Math. Theor., 51 (4): 045303, dec 2017. 10.1088/1751-8121/aa9100. URL https://doi.org/10.1088.
https://doi.org/10.1088/1751-8121/aa9100
[76] Barbara Synak-Radtke and Michał Horodecki. On asymptotic continuity of functions of quantum states. J. Phys. A: Math. Gen., 39 (26): L423–L437, jun 2006. 10.1088/0305-4470/39/26/l02. URL https://doi.org/10.1088.
https://doi.org/10.1088/0305-4470/39/26/l02
[77] L. Lami. Completing the grand tour of asymptotic quantum coherence manipulation. IEEE Trans. Inf. Theory, pages 1–1, 2019. ISSN 1557-9654. 10.1109/TIT.2019.2945798. URL https://doi.org/10.1109/TIT.2019.2945798.
https://doi.org/10.1109/TIT.2019.2945798
[78] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight. Quantifying entanglement. Phys. Rev. Lett., 78: 2275–2279, Mar 1997. 10.1103/PhysRevLett.78.2275. URL https://link.aps.org/doi/10.1103/PhysRevLett.78.2275.
https://doi.org/10.1103/PhysRevLett.78.2275
[79] Fernando G. S. L. Brandão and Martin B. Plenio. A generalization of quantum Stein’s lemma. Commun. Math. Phys., 295 (3): 791–828, May 2010. ISSN 1432-0916. 10.1007/s00220-010-1005-z. URL https://doi.org/10.1007/s00220-010-1005-z.
https://doi.org/10.1007/s00220-010-1005-z
[80] Michał Horodecki, Jonathan Oppenheim, and Ryszard Horodecki. Are the laws of entanglement theory thermodynamical? Phys. Rev. Lett., 89: 240403, Nov 2002. 10.1103/PhysRevLett.89.240403. URL https://link.aps.org/doi/10.1103/PhysRevLett.89.240403.
https://doi.org/10.1103/PhysRevLett.89.240403
[81] Stephan Weis and Andreas Knauf. Entropy distance: New quantum phenomena. J. Math. Phys., 53 (10): 102206, Oct 2012. ISSN 1089-7658. 10.1063/1.4757652. URL http://dx.doi.org/10.1063/1.4757652.
https://doi.org/10.1063/1.4757652
[82] Miguel Herrero-Collantes and Juan Carlos Garcia-Escartin. Quantum random number generators. Rev. Mod. Phys., 89: 015004, Feb 2017. 10.1103/RevModPhys.89.015004. URL https://link.aps.org/doi/10.1103/RevModPhys.89.015004.
https://doi.org/10.1103/RevModPhys.89.015004
[83] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A, 80: 012304, Jul 2009. 10.1103/PhysRevA.80.012304. URL https://link.aps.org/doi/10.1103/PhysRevA.80.012304.
https://doi.org/10.1103/PhysRevA.80.012304
[84] Thomas Theurer, Dario Egloff, Lijian Zhang, and Martin B. Plenio. Quantifying operations with an application to coherence. Phys. Rev. Lett., 122: 190405, May 2019. 10.1103/PhysRevLett.122.190405. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.190405.
https://doi.org/10.1103/PhysRevLett.122.190405
[85] Gilad Gour and Andreas Winter. How to quantify a dynamical quantum resource. Phys. Rev. Lett., 123: 150401, Oct 2019. 10.1103/PhysRevLett.123.150401. URL https://link.aps.org/doi/10.1103/PhysRevLett.123.150401.
https://doi.org/10.1103/PhysRevLett.123.150401
[86] Yunchao Liu and Xiao Yuan. Operational resource theory of quantum channels. Phys. Rev. Research, 2: 012035, Feb 2020. 10.1103/PhysRevResearch.2.012035. URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.012035.
https://doi.org/10.1103/PhysRevResearch.2.012035
[87] Gilad Gour and Mark M. Wilde. Entropy of a quantum channel. arXiv preprint arXiv:1808.06980 [quant-ph], 2018. URL https://arxiv.org/abs/1808.06980.
arXiv:1808.06980
[88] Lu Li, Kaifeng Bu, and Zi-Wen Liu. Quantifying the resource content of quantum channels: An operational approach. Phys. Rev. A, 101: 022335, Feb 2020. 10.1103/PhysRevA.101.022335. URL https://link.aps.org/doi/10.1103/PhysRevA.101.022335.
https://doi.org/10.1103/PhysRevA.101.022335
[89] Zi-Wen Liu and Andreas Winter. Resource theories of quantum channels and the universal role of resource erasure. arXiv preprint arXiv:1904.04201 [quant-ph], 2019. URL https://arxiv.org/abs/1904.04201.
arXiv:1904.04201
[90] Ryuji Takagi, Kun Wang, and Masahito Hayashi. Application of the resource theory of channels to communication scenarios. Phys. Rev. Lett., 124: 120502, Mar 2020. 10.1103/PhysRevLett.124.120502. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.120502.
https://doi.org/10.1103/PhysRevLett.124.120502
[91] Nelson Dunford and Jacob T Schwartz. Linear operators. Part I, General theory. New York : Interscience., 1958.
[92] G. F. Dell’Antonio. On the limits of sequences of normal states. Commun. Pure Appl. Math., 20 (2): 413–429, 1967. 10.1002/cpa.3160200209. URL https://doi.org/10.1002/cpa.3160200209.
https://doi.org/10.1002/cpa.3160200209
Cited by
[1] Bartosz Regula, Ludovico Lami, Giovanni Ferrari, and Ryuji Takagi, “Operational quantification of continuous-variable quantum resources”, arXiv:2009.11302.
[2] Ludovico Lami, Bartosz Regula, Ryuji Takagi, and Giovanni Ferrari, “Taming the infinite: framework for resource quantification in infinite-dimensional general probabilistic theories”, arXiv:2009.11313.
[3] Hayata Yamasaki, Madhav Krishnan Vijayan, and Min-Hsiu Hsieh, “Hierarchy of quantum operations in manipulating coherence and entanglement”, arXiv:1912.11049.
[4] Zi-Wen Liu and Andreas Winter, “Many-body quantum magic”, arXiv:2010.13817.
The above citations are from SAO/NASA ADS (last updated successfully 2020-11-03 11:08:55). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2020-11-03 11:08:53).
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Source: https://quantum-journal.org/papers/q-2020-11-01-355/
