Using and reusing coherence to realize quantum processes

María García Díaz; Kun Fang; Xin Wang; Matteo Rosati; Michalis Skotiniotis; John Calsamiglia; Andreas Winter

doi:10.22331/q-2018-10-19-100

Using and reusing coherence to realize quantum processes

María García Díaz¹, Kun Fang², Xin Wang², Matteo Rosati¹, Michalis Skotiniotis¹, John Calsamiglia¹, and Andreas Winter^1,3

¹Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
²Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia
³ICREA-Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, ES-08001 Barcelona, Spain

Published:	2018-10-19, volume 2, page 100
Eprint:	arXiv:1805.04045v3
Doi:	https://doi.org/10.22331/q-2018-10-19-100
Citation:	Quantum 2, 100 (2018).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Coherent superposition is a key feature of quantum mechanics that underlies the advantage of quantum technologies over their classical counterparts. Recently, coherence has been recast as a resource theory in an attempt to identify and quantify it in an operationally well-defined manner. Here we study how the coherence present in a state can be used to implement a quantum channel via incoherent operations and, in turn, to assess its degree of coherence. We introduce the robustness of coherence of a quantum channel-which reduces to the homonymous measure for states when computed on constant-output channels-and prove that: i) it quantifies the minimal rank of a maximally coherent state required to implement the channel; ii) its logarithm quantifies the amortized cost of implementing the channel provided some coherence is recovered at the output; iii) its logarithm also quantifies the zero-error asymptotic cost of implementation of many independent copies of a channel. We also consider the generalized problem of imperfect implementation with arbitrary resource states. Using the robustness of coherence, we find that in general a quantum channel can be implemented without employing a maximally coherent resource state. In fact, we prove that every pure coherent state in dimension larger than $2$, however weakly so, turns out to be a valuable resource to implement some coherent unitary channel. We illustrate our findings for the case of single-qubit unitary channels.

Featured image: Protocol for implementing a quantum channel N using a MIO (maximally incoherent operation) and a coherent resource $\omega$. After the implementation, a degraded resource $\sigma$ is recovered and ready to be recycled.

Popular summary

Static coherence, the degree of superposition present in a state, can be thought of as different from dynamic coherence, which is the ability to generate coherence itself. Here we develop a framework that puts these two types of coherence on an equal footing and allows us to study their interconversion thereby shifting the paradigm of coherence theory from states to processes. In particular, we introduce a measure for dynamic coherence which uniquely quantifies the implementation cost of a channel using static coherence as a resource, and show that coherence can be used and reused in a continuous fashion.

► BibTeX data

@article{Diaz2018usingreusing,
  doi = {10.22331/q-2018-10-19-100},
  url = {https://doi.org/10.22331/q-2018-10-19-100},
  title = {Using and reusing coherence to realize quantum processes},
  author = {D{\'{i}}az, Mar{\'{i}}a Garc{\'{i}}a and Fang, Kun and Wang, Xin and Rosati, Matteo and Skotiniotis, Michalis and Calsamiglia, John and Winter, Andreas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {100},
  month = oct,
  year = {2018}
}

► References

[1] Johan Åberg. Quantifying superposition. arXiv:quant-ph/0612146, 2006.
arXiv:quant-ph/0612146

[2] Dorit Aharonov, Alexei Kitaev, and Noam Nisan. Quantum circuits with mixed states. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 20–30, New York, NY, USA, 1998. ACM. ISBN 0-89791-962-9. 10.1145/276698.276708. URL http://doi.acm.org/10.1145/276698.276708.
https://doi.org/10.1145/276698.276708

[3] Namit Anand and Arun Kumar Pati. Coherence and entanglement monogamy in the discrete analogue of analog Grover search. arXiv[quant-ph]:1611.04542, 2016.

[4] 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

[5] Khaled Ben Dana, María García Díaz, Mohamed Mejatty, and Andreas Winter. Resource theory of coherence: Beyond states. Phys. Rev. A, 95: 062327, Jun 2017. 10.1103/PhysRevA.95.062327. URL https://link.aps.org/doi/10.1103/PhysRevA.95.062327.
https://doi.org/10.1103/PhysRevA.95.062327

[6] M. Berta, F. G. S. L. Brandão, M. Christandl, and S. Wehner. Entanglement cost of quantum channels. IEEE Transactions on Information Theory, 59 (10): 6779–6795, Oct 2013. ISSN 0018-9448. 10.1109/TIT.2013.2268533.
https://doi.org/10.1109/TIT.2013.2268533

[7] Tanmoy Biswas, María García Díaz, and Andreas Winter. Interferometric visibility and coherence. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 473 (2203), 2017. ISSN 1364-5021. 10.1098/rspa.2017.0170. URL http://rspa.royalsocietypublishing.org/content/473/2203/20170170.
https://doi.org/10.1098/rspa.2017.0170
http://rspa.royalsocietypublishing.org/content/473/2203/20170170

[8] 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

[9] 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

[10] Fernando Brandão, Michał Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner. The second laws of quantum thermodynamics. Proceedings of the National Academy of Sciences of the United States of America, 112 (11): 3275–3279, 03 2015. 10.1073/pnas.1411728112. URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4372001/.
https://doi.org/10.1073/pnas.1411728112
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4372001/

[11] Daniel Braun and Bertrand Georgeot. Quantitative measure of interference. Phys. Rev. A, 73: 022314, Feb 2006. 10.1103/PhysRevA.73.022314. URL https://link.aps.org/doi/10.1103/PhysRevA.73.022314.
https://doi.org/10.1103/PhysRevA.73.022314

[12] Kaifeng Bu and Chunhe Xiong. A note on cohering power and de-cohering power. Quantum Info. Comput., 17 (13-14): 1206–1220, November 2017. ISSN 1533-7146. 10.26421/QIC17.13-14.
https://doi.org/10.26421/QIC17.13-14

[13] Kaifeng Bu, Asutosh Kumar, Lin Zhang, and Junde Wu. Cohering power of quantum operations. Phys. Lett. A, 381 (19): 1670 – 1676, 2017. ISSN 0375-9601. 10.1016/j.physleta.2017.03.022. URL http://www.sciencedirect.com/science/article/pii/S0375960117302621.
https://doi.org/10.1016/j.physleta.2017.03.022
http://www.sciencedirect.com/science/article/pii/S0375960117302621

[14] Iulia Buluta and Franco Nori. Quantum simulators. Science, 326 (5949): 108–111, 2009. ISSN 0036-8075. 10.1126/science.1177838. URL http://science.sciencemag.org/content/326/5949/108.
https://doi.org/10.1126/science.1177838
http://science.sciencemag.org/content/326/5949/108

[15] A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio. The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigment–protein complexes. Nature Physics, 9: 113 EP –, 01 2013. 10.1038/nphys2515.
https://doi.org/10.1038/nphys2515

[16] G. Chiribella, G. M. D'Ariano, and P. Perinotti. Quantum circuit architecture. Phys. Rev. Lett., 101: 060401, Aug 2008. 10.1103/PhysRevLett.101.060401. URL https://link.aps.org/doi/10.1103/PhysRevLett.101.060401.
https://doi.org/10.1103/PhysRevLett.101.060401

[17] 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 2016. 10.1103/PhysRevLett.117.030401. URL https://link.aps.org/doi/10.1103/PhysRevLett.117.030401.
https://doi.org/10.1103/PhysRevLett.117.030401

[18] Eric Chitambar and Min-Hsiu Hsieh. Relating the resource theories of entanglement and quantum coherence. Phys. Rev. Lett., 117: 020402, Jul 2016. 10.1103/PhysRevLett.117.020402. URL https://link.aps.org/doi/10.1103/PhysRevLett.117.020402.
https://doi.org/10.1103/PhysRevLett.117.020402

[19] C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Rev. Mod. Phys., 89: 035002, Jul 2017. 10.1103/RevModPhys.89.035002. URL https://link.aps.org/doi/10.1103/RevModPhys.89.035002.
https://doi.org/10.1103/RevModPhys.89.035002

[20] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47: 777–780, May 1935. 10.1103/PhysRev.47.777. URL https://link.aps.org/doi/10.1103/PhysRev.47.777.
https://doi.org/10.1103/PhysRev.47.777

[21] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio. Optimal local implementation of nonlocal quantum gates. Phys. Rev. A, 62: 052317, Oct 2000. 10.1103/PhysRevA.62.052317. URL https://link.aps.org/doi/10.1103/PhysRevA.62.052317.
https://doi.org/10.1103/PhysRevA.62.052317

[22] Gregory S. Engel, Tessa R. Calhoun, Elizabeth L. Read, Tae-Kyu Ahn, Tomáš Mančal, Yuan-Chung Cheng, Robert E. Blankenship, and Graham R. Fleming. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature, 446: 782 EP –, 04 2007. 10.1038/nature05678.
https://doi.org/10.1038/nature05678

[23] Philippe Faist, Frédéric Dupuis, Jonathan Oppenheim, and Renato Renner. The minimal work cost of information processing. Nature Communications, 6: 7669 EP –, 07 2015a. 10.1038/ncomms8669.
https://doi.org/10.1038/ncomms8669

[24] Philippe Faist, Jonathan Oppenheim, and Renato Renner. Gibbs-preserving maps outperform thermal operations in the quantum regime. New J. Phys., 17 (4): 043003, 2015b. 10.1088/1367-2630/17/4/043003.
https://doi.org/10.1088/1367-2630/17/4/043003

[25] R. F. Feynman, R. B. Leighton, and M. Sands. Feynman Physics Lectures, volume 3. Addison-Wesley Publishing Company, 1965. ISBN 0-201-02118-8-P.

[26] A. Galindo and M. A. Martín-Delgado. Information and computation: Classical and quantum aspects. Rev. Mod. Phys., 74: 347–423, May 2002. 10.1103/RevModPhys.74.347. URL https://link.aps.org/doi/10.1103/RevModPhys.74.347.
https://doi.org/10.1103/RevModPhys.74.347

[27] María García-Díaz, Dario Egloff, and Martin B. Plenio. A note on coherence power of n-dimensional unitary operators. Quantum Info. Comput., 16 (15-16): 1282–1294, November 2016. ISSN 1533-7146. 10.26421/QIC16.15-16.
https://doi.org/10.26421/QIC16.15-16

[28] I. M. Georgescu, S. Ashhab, and Franco Nori. Quantum simulation. Rev. Mod. Phys., 86: 153–185, Mar 2014. 10.1103/RevModPhys.86.153. URL https://link.aps.org/doi/10.1103/RevModPhys.86.153.
https://doi.org/10.1103/RevModPhys.86.153

[29] Paolo Giorda and Michele Allegra. Coherence in quantum estimation. Journal of Physics A: Mathematical and Theoretical, 51 (2): 025302, 2018. 10.1088/1751-8121/aa9808.
https://doi.org/10.1088/1751-8121/aa9808

[30] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature Photonics, 5: 222 EP –, 03 2011. 10.1038/nphoton.2011.35.
https://doi.org/10.1038/nphoton.2011.35

[31] Nicolas Gisin and Rob Thew. Quantum communication. Nature Photonics, 1: 165 EP –, 03 2007. 10.1038/nphoton.2007.22.
https://doi.org/10.1038/nphoton.2007.22

[32] Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden. Quantum cryptography. Rev. Mod. Phys., 74: 145–195, Mar 2002. 10.1103/RevModPhys.74.145. URL https://link.aps.org/doi/10.1103/RevModPhys.74.145.
https://doi.org/10.1103/RevModPhys.74.145

[33] 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.
https://doi.org/10.1088/1367-2630/10/3/033023

[34] Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Yunger Halpern. The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583: 1 – 58, 2015. ISSN 0370-1573. https://doi.org/10.1016/j.physrep.2015.04.003.
https://doi.org/10.1016/j.physrep.2015.04.003

[35] Mark Hillery. Coherence as a resource in decision problems: The deutsch-jozsa algorithm and a variation. Phys. Rev. A, 93: 012111, Jan 2016. 10.1103/PhysRevA.93.012111. URL https://link.aps.org/doi/10.1103/PhysRevA.93.012111.
https://doi.org/10.1103/PhysRevA.93.012111

[36] Alexander S. Holevo. Quantum Systems, Channels, Information. De Gruyter, Berlin, Boston, jan 2012. ISBN 9783110273403. 10.1515/9783110273403. URL http://www.degruyter.com/view/books/9783110273403/9783110273403/9783110273403.xml.
https://doi.org/10.1515/9783110273403
http://www.degruyter.com/view/books/9783110273403/9783110273403/9783110273403.xml

[37] Michał Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature Communications, 4: 2059 EP –, 06 2013. 10.1038/ncomms3059.
https://doi.org/10.1038/ncomms3059

[38] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A, 60: 1888–1898, Sep 1999. 10.1103/PhysRevA.60.1888. URL https://link.aps.org/doi/10.1103/PhysRevA.60.1888.
https://doi.org/10.1103/PhysRevA.60.1888

[39] 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

[40] M. L. Hu, X. Hu, J. C. Wang, Y. Peng, Y. R. Zhang, and H. Fan. Quantum coherence and quantum correlations. arXiv[quant-ph]:1703.01852, 2017.

[41] S.F. Huelga and M.B. Plenio. Vibrations, quanta and biology. Contemporary Physics, 54 (4): 181–207, 2013. 10.1080/00405000.2013.829687. URL https://doi.org/10.1080/00405000.2013.829687.
https://doi.org/10.1080/00405000.2013.829687

[42] Akihito Ishizaki and Graham R. Fleming. Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature. Proceedings of the National Academy of Sciences, 106 (41): 17255–17260, 2009. ISSN 0027-8424. 10.1073/pnas.0908989106. URL http://www.pnas.org/content/106/41/17255.
https://doi.org/10.1073/pnas.0908989106
http://www.pnas.org/content/106/41/17255

[43] N. Killoran, F. E. S. Steinhoff, and M. B. Plenio. Converting nonclassicality into entanglement. Phys. Rev. Lett., 116: 080402, Feb 2016. 10.1103/PhysRevLett.116.080402. URL https://link.aps.org/doi/10.1103/PhysRevLett.116.080402.
https://doi.org/10.1103/PhysRevLett.116.080402

[44] Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, and David Jennings. The extraction of work from quantum coherence. New Journal of Physics, 18 (2): 023045, 2016. 10.1088/1367-2630/18/2/023045.
https://doi.org/10.1088/1367-2630/18/2/023045

[45] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien. Quantum computers. Nature, 464: 45 EP –, 03 2010. 10.1038/nature08812.
https://doi.org/10.1038/nature08812

[46] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nature Communications, 6: 6383 EP –, 03 2015. 10.1038/ncomms7383.
https://doi.org/10.1038/ncomms7383

[47] Matteo Lostaglio, David Jennings, and Terry Rudolph. Thermodynamic resource theories, non-commutativity and maximum entropy principles. New Journal of Physics, 19 (4): 043008, 2017. 1367-2630/19/i=4/a=043008.

[48] Andrew D. Ludlow, Martin M. Boyd, Jun Ye, E. Peik, and P. O. Schmidt. Optical atomic clocks. Rev. Mod. Phys., 87: 637–701, Jun 2015. 10.1103/RevModPhys.87.637. URL https://link.aps.org/doi/10.1103/RevModPhys.87.637.
https://doi.org/10.1103/RevModPhys.87.637

[49] A. L. Malvezzi, G. Karpat, B. Çakmak, F. F. Fanchini, T. Debarba, and R. O. Vianna. Quantum correlations and coherence in spin-1 heisenberg chains. Phys. Rev. B, 93: 184428, May 2016. 10.1103/PhysRevB.93.184428. URL https://link.aps.org/doi/10.1103/PhysRevB.93.184428.
https://doi.org/10.1103/PhysRevB.93.184428

[50] Azam Mani and Vahid Karimipour. Cohering and decohering power of quantum channels. Phys. Rev. A, 92: 032331, Sep 2015. 10.1103/PhysRevA.92.032331. URL https://link.aps.org/doi/10.1103/PhysRevA.92.032331.
https://doi.org/10.1103/PhysRevA.92.032331

[51] Iman Marvian and Robert W Spekkens. Extending noether's theorem by quantifying the asymmetry of quantum states. Nature Communications, 5: 3821 EP –, 05 2014. 10.1038/ncomms4821.
https://doi.org/10.1038/ncomms4821

[52] J M Matera, D Egloff, N Killoran, and M B Plenio. Coherent control of quantum systems as a resource theory. Quantum Science and Technology, 1 (1): 01LT01, 2016. 10.1088/2058-9565/1/1/01LT01.
https://doi.org/10.1088/2058-9565/1/1/01LT01

[53] Avijit Misra, Uttam Singh, Samyadeb Bhattacharya, and Arun Kumar Pati. Energy cost of creating quantum coherence. Phys. Rev. A, 93: 052335, May 2016. 10.1103/PhysRevA.93.052335. URL https://link.aps.org/doi/10.1103/PhysRevA.93.052335.
https://doi.org/10.1103/PhysRevA.93.052335

[54] Carmine Napoli, Thomas R. Bromley, Marco Cianciaruso, Marco Piani, Nathaniel Johnston, and Gerardo Adesso. Robustness of coherence: An operational and observable measure of quantum coherence. Phys. Rev. Lett., 116: 150502, Apr 2016. 10.1103/PhysRevLett.116.150502. URL https://link.aps.org/doi/10.1103/PhysRevLett.116.150502.
https://doi.org/10.1103/PhysRevLett.116.150502

[55] Varun Narasimhachar and Gilad Gour. Low-temperature thermodynamics with quantum coherence. Nature Communications, 6: 7689 EP –, 07 2015. 10.1038/ncomms8689.
https://doi.org/10.1038/ncomms8689

[56] Martin B. Plenio and Shashank Virmani. An introduction to entanglement measures. Quantum Info. Comput., 7 (1): 1–51, January 2007. ISSN 1533-7146. 10.26421/QIC7.1-2.
https://doi.org/10.26421/QIC7.1-2

[57] Massimiliano F. Sacchi. Optimal discrimination of quantum operations. Phys. Rev. A, 71: 062340, Jun 2005. 10.1103/PhysRevA.71.062340. URL https://link.aps.org/doi/10.1103/PhysRevA.71.062340.
https://doi.org/10.1103/PhysRevA.71.062340

[58] Lian He Shao, Zhengjun Xi, Heng Fan, and Yongming Li. Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A, 91 (4), 2015. ISSN 10941622. 10.1103/PhysRevA.91.042120. URL https://journals.aps.org/pra/pdf/10.1103/PhysRevA.91.042120.
https://doi.org/10.1103/PhysRevA.91.042120

[59] Dan Stahlke and Robert B. Griffiths. Entanglement requirements for implementing bipartite unitary operations. Phys. Rev. A, 84: 032316, Sep 2011. 10.1103/PhysRevA.84.032316. URL https://link.aps.org/doi/10.1103/PhysRevA.84.032316.
https://doi.org/10.1103/PhysRevA.84.032316

[60] Alexander Streltsov, Uttam Singh, Himadri Shekhar Dhar, Manabendra Nath Bera, and Gerardo Adesso. Measuring Quantum Coherence with Entanglement. Physical Review Letters, 115 (2), feb 2015. ISSN 10797114. 10.1103/PhysRevLett.115.020403. URL https://arxiv.org/abs/1502.05876.
https://doi.org/10.1103/PhysRevLett.115.020403
arXiv:1502.05876

[61] 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

[62] Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information science perspective. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424006, 2014. 10.1088/1751-8113/47/42/424006.
https://doi.org/10.1088/1751-8113/47/42/424006

[63] V. Vedral and M. B. Plenio. Entanglement measures and purification procedures. Phys. Rev. A, 57: 1619–1633, Mar 1998. 10.1103/PhysRevA.57.1619. URL https://link.aps.org/doi/10.1103/PhysRevA.57.1619.
https://doi.org/10.1103/PhysRevA.57.1619

[64] John Watrous. Semidefinite Programs for Completely Bounded Norms. Theory of Computing, 5: 217–238, 2009. 10.4086/toc.2009.v005a011.
https://doi.org/10.4086/toc.2009.v005a011

[65] 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

[66] W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299: 802 EP –, 10 1982. 10.1038/299802a0.
https://doi.org/10.1038/299802a0

[67] Qi Zhao, Yunchao Liu, Xiao Yuan, Eric Chitambar, and Xiongfeng Ma. One-shot coherence dilution. Phys. Rev. Lett., 120: 070403, Feb 2018. 10.1103/PhysRevLett.120.070403. URL https://link.aps.org/doi/10.1103/PhysRevLett.120.070403.
https://doi.org/10.1103/PhysRevLett.120.070403

[68] Huangjun Zhu, Masahito Hayashi, and Lin Chen. Coherence and entanglement measures based on rényi relative entropies. Journal of Physics A: Mathematical and Theoretical, 50 (47): 475303, 2017. 10.1088/1751-8121/aa8ffc.
https://doi.org/10.1088/1751-8121/aa8ffc

[69] Barış Çakmak, Göktuğ Karpat, and Felipe F. Fanchini. Factorization and criticality in the anisotropic xy chain via correlations. Entropy, 17 (2): 790–817, 2015. ISSN 1099-4300. 10.3390/e17020790. URL http://www.mdpi.com/1099-4300/17/2/790.
https://doi.org/10.3390/e17020790
http://www.mdpi.com/1099-4300/17/2/790