Using and reusing coherence to realize quantum processes

María García Díaz1, Kun Fang2, Xin Wang2, Matteo Rosati1, Michalis Skotiniotis1, John Calsamiglia1, and Andreas Winter1,3

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

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.

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

► 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

[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

Cited by

[1] Chung-Yun Hsieh, "Resource Preservability", Quantum 4, 244 (2020).

[2] Masahito Hayashi, Kun Fang, and Kun Wang, "Finite Block Length Analysis on Quantum Coherence Distillation and Incoherent Randomness Extraction", IEEE Transactions on Information Theory 67 6, 3926 (2021).

[3] Hyukjoon Kwon, Kok Chuan Tan, Tyler Volkoff, and Hyunseok Jeong, "Nonclassicality as a Quantifiable Resource for Quantum Metrology", Physical Review Letters 122 4, 040503 (2019).

[4] Giulio Chiribella, Yuxiang Yang, and Renato Renner, "Fundamental Energy Requirement of Reversible Quantum Operations", Physical Review X 11 2, 021014 (2021).

[5] Carlo Marconi, Pau Colomer Saus, María García Díaz, and Anna Sanpera, "The role of coherence theory in attractor quantum neural networks", Quantum 6, 794 (2022).

[6] Ludovico Lami, Bartosz Regula, and Gerardo Adesso, "Generic Bound Coherence under Strictly Incoherent Operations", Physical Review Letters 122 15, 150402 (2019).

[7] Xin Wang, Mark M Wilde, and Yuan Su, "Quantifying the magic of quantum channels", New Journal of Physics 21 10, 103002 (2019).

[8] María García Díaz, Benjamin Desef, Matteo Rosati, Dario Egloff, John Calsamiglia, Andrea Smirne, Michaelis Skotiniotis, and Susana F. Huelga, "Accessible coherence in open quantum system dynamics", Quantum 4, 249 (2020).

[9] Gilad Gour and Mark M. Wilde, 2020 IEEE International Symposium on Information Theory (ISIT) 1903 (2020) ISBN:978-1-7281-6432-8.

[10] Michele Masini, Thomas Theurer, and Martin B. Plenio, "Coherence of operations and interferometry", Physical Review A 103 4, 042426 (2021).

[11] Masaya Takahashi, Swapan Rana, and Alexander Streltsov, "Creating and destroying coherence with quantum channels", Physical Review A 105 6, L060401 (2022).

[12] Yu Luo, Mingfei Ye, and Yongming Li, "Coherence weight of quantum channels", Physica A: Statistical Mechanics and its Applications 599, 127510 (2022).

[13] Ying Wang, Yu Luo, and Zhengjun Xi, "Robustness of purity of quantum channels", Laser Physics Letters 18 6, 065201 (2021).

[14] Kok Chuan Tan, Seongjeon Choi, and Hyunseok Jeong, "Optimizing nontrivial quantum observables using coherence", New Journal of Physics 21 2, 023013 (2019).

[15] Zi-Wen Liu and Andreas Winter, "Many-Body Quantum Magic", PRX Quantum 3 2, 020333 (2022).

[16] Gilad Gour, "Comparison of Quantum Channels by Superchannels", IEEE Transactions on Information Theory 65 9, 5880 (2019).

[17] Lu Li, Kaifeng Bu, and Zi-Wen Liu, "Quantifying the resource content of quantum channels: An operational approach", Physical Review A 101 2, 022335 (2020).

[18] Philippe Faist, Takahiro Sagawa, Kohtaro Kato, Hiroshi Nagaoka, and Fernando G. S. L. Brandão, "Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems", Physical Review Letters 123 25, 250601 (2019).

[19] Mark M. Wilde, Mario Berta, Christoph Hirche, and Eneet Kaur, "Amortized channel divergence for asymptotic quantum channel discrimination", Letters in Mathematical Physics 110 8, 2277 (2020).

[20] Masahito Hayashi, Kun Fang, and Kun Wang, 2021 IEEE International Symposium on Information Theory (ISIT) 1326 (2021) ISBN:978-1-5386-8209-8.

[21] Hiroyasu Tajima, Naoto Shiraishi, and Keiji Saito, "Coherence cost for violating conservation laws", Physical Review Research 2 4, 043374 (2020).

[22] Felix Ahnefeld, Thomas Theurer, Dario Egloff, Juan Mauricio Matera, and Martin B. Plenio, "Coherence as a Resource for Shor’s Algorithm", Physical Review Letters 129 12, 120501 (2022).

[23] Ryuji Takagi, Bartosz Regula, and Mark M. Wilde, "One-Shot Yield-Cost Relations in General Quantum Resource Theories", PRX Quantum 3 1, 010348 (2022).

[24] Qi Zhao, Yunchao Liu, Xiao Yuan, Eric Chitambar, and Andreas Winter, "One-Shot Coherence Distillation: Towards Completing the Picture", IEEE Transactions on Information Theory 65 10, 6441 (2019).

[25] Gaurav Saxena, Eric Chitambar, and Gilad Gour, "Dynamical resource theory of quantum coherence", Physical Review Research 2 2, 023298 (2020).

[26] Kang‐Da Wu, Alexander Streltsov, Bartosz Regula, Guo‐Yong Xiang, Chuan‐Feng Li, and Guang‐Can Guo, "Experimental Progress on Quantum Coherence: Detection, Quantification, and Manipulation", Advanced Quantum Technologies 4 9, 2100040 (2021).

[27] Bartosz Regula and Ryuji Takagi, "One-Shot Manipulation of Dynamical Quantum Resources", Physical Review Letters 127 6, 060402 (2021).

[28] Ryuji Takagi and Hiroyasu Tajima, "Universal limitations on implementing resourceful unitary evolutions", Physical Review A 101 2, 022315 (2020).

[29] Xiaorong Wang, Ting Gao, and Fengli Yan, "On coherence of quantum operations by using Choi–Jamiołkowski isomorphism", Laser Physics Letters 19 3, 035206 (2022).

[30] Jiaqing Jiang, Kun Wang, and Xin Wang, "Physical Implementability of Linear Maps and Its Application in Error Mitigation", Quantum 5, 600 (2021).

[31] Thomas Theurer, Dario Egloff, Lijian Zhang, and Martin B. Plenio, "Quantifying Operations with an Application to Coherence", Physical Review Letters 122 19, 190405 (2019).

[32] Kun Fang, Xin Wang, Marco Tomamichel, and Mario Berta, "Quantum Channel Simulation and the Channel’s Smooth Max-Information", IEEE Transactions on Information Theory 66 4, 2129 (2020).

[33] Bartosz Regula and Ryuji Takagi, "Fundamental limitations on distillation of quantum channel resources", Nature Communications 12 1, 4411 (2021).

[34] Jiaqing Jiang and Xin Wang, "Lower Bound for the T Count Via Unitary Stabilizer Nullity", Physical Review Applied 19 3, 034052 (2023).

[35] Xin Wang and Mark M. Wilde, "Resource theory of asymmetric distinguishability for quantum channels", Physical Review Research 1 3, 033169 (2019).

[36] Pei Zeng, You Zhou, and Zhenhuan Liu, "Quantum gate verification and its application in property testing", Physical Review Research 2 2, 023306 (2020).

[37] John Burniston, Michael Grabowecky, Carlo Maria Scandolo, Giulio Chiribella, and Gilad Gour, "Necessary and sufficient conditions on measurements of quantum channels", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 2236, 20190832 (2020).

[38] Gilad Gour and Mark M. Wilde, "Entropy of a quantum channel", Physical Review Research 3 2, 023096 (2021).

[39] Shuanping Du and Zhaofang Bai, "Incoherent Gaussian equivalence of m -mode Gaussian states", Physical Review A 107 1, 012407 (2023).

[40] Shuanping Du and Zhaofang Bai, "Strictly incoherent operations for one-qubit systems", Physics Letters A 394, 127203 (2021).

[41] Bartosz Regula, Ryuji Takagi, and Mile Gu, "Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps", Quantum 5, 522 (2021).

[42] Ryuji Takagi, Kun Wang, and Masahito Hayashi, "Application of the Resource Theory of Channels to Communication Scenarios", Physical Review Letters 124 12, 120502 (2020).

[43] Ho-Joon Kim and Soojoon Lee, "Relation between quantum coherence and quantum entanglement in quantum measurements", Physical Review A 106 2, 022401 (2022).

[44] Maolin Luo, Xiaoqian Zhang, and Xiaoqi Zhou, "Proof-of-principle experimental demonstration of quantum gate verification", Physical Review A 105 1, 012614 (2022).

[45] Ho-Joon Kim and Soojoon Lee, "One-shot static entanglement cost of bipartite quantum channels", Physical Review A 103 6, 062415 (2021).

[46] Xin Wang and Mark M. Wilde, "Resource theory of asymmetric distinguishability", Physical Review Research 1 3, 033170 (2019).

[47] Chung-Yun Hsieh, "Communication, Dynamical Resource Theory, and Thermodynamics", PRX Quantum 2 2, 020318 (2021).

[48] Thomas Theurer, Saipriya Satyajit, and Martin B. Plenio, "Quantifying Dynamical Coherence with Dynamical Entanglement", Physical Review Letters 125 13, 130401 (2020).

[49] Bartosz Regula, Ludovico Lami, Giovanni Ferrari, and Ryuji Takagi, "Operational Quantification of Continuous-Variable Quantum Resources", Physical Review Letters 126 11, 110403 (2021).

[50] Ryuji Takagi and Bartosz Regula, "General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks", Physical Review X 9 3, 031053 (2019).

[51] Yunchao Liu and Xiao Yuan, "Operational resource theory of quantum channels", Physical Review Research 2 1, 012035 (2020).

[52] Zhan Yu, Xuanqiang Zhao, Benchi Zhao, and Xin Wang, "Optimal Quantum Dataset for Learning a Unitary Transformation", Physical Review Applied 19 3, 034017 (2023).

[53] Kun Fang, Jingtian Zhao, Xiufan Li, Yifei Li, and Runyao Duan, "Quantum NETwork: from theory to practice", Science China Information Sciences 66 8, 180509 (2023).

[54] Eric Chitambar, Gilad Gour, Kuntal Sengupta, and Rana Zibakhsh, "Quantum Bell nonlocality as a form of entanglement", Physical Review A 104 5, 052208 (2021).

[55] Zi-Wen Liu and Andreas Winter, "Resource theories of quantum channels and the universal role of resource erasure", arXiv:1904.04201, (2019).

[56] Kun Fang, Xin Wang, Ludovico Lami, Bartosz Regula, and Gerardo Adesso, "Probabilistic Distillation of Quantum Coherence", Physical Review Letters 121 7, 070404 (2018).

[57] Stefan Bäuml, Siddhartha Das, Xin Wang, and Mark M. Wilde, "Resource theory of entanglement for bipartite quantum channels", arXiv:1907.04181, (2019).

[58] Mark M. Wilde, "Entanglement cost and quantum channel simulation", Physical Review A 98 4, 042338 (2018).

[59] Gilad Gour and Mark M. Wilde, "Entropy of a quantum channel", arXiv:1808.06980, (2018).

[60] Xin Wang, Mark M. Wilde, and Yuan Su, "Quantifying the magic of quantum channels", arXiv:1903.04483, (2019).

[61] Lu Li, Kaifeng Bu, and Zi-Wen Liu, "Quantifying the resource content of quantum channels: An operational approach", arXiv:1812.02572, (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2023-10-04 03:05:37) and SAO/NASA ADS (last updated successfully 2023-10-04 03:05:38). The list may be incomplete as not all publishers provide suitable and complete citation data.