Abstract
We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for density matrices with full rank. We also define “generalized entanglement monotones” and “genuinely coherence monotones” under special one-way LOCC and genuinely incoherent operations respectively. Any entanglement monotone proposed by Vidal et al. is a generalized entanglement monotone. Any coherence monotone under incoherent operations is a genuinely coherence monotone. Furthermore, we clarify the relationship between generalized entanglement monotones and genuinely coherence monotones. We demonstrate that any generalized entanglement monotone of bipartite pure state is the lower bound of a suitable genuinely coherence monotone; any genuinely coherence monotone of a quantum state is the generalized entanglement monotone of the corresponding maximally correlated state.
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Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)
Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001)
Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (1999)
Aberg, J.: Quantifying superposition, arXiv:quant-ph/0612146(2006)
Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)
Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)
Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)
Hu, M.L., Hu, X.Y., Wang, J.C., Peng, Y., Zhang, Y.R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762–764, 1–100 (2018)
Hillery, M.: Coherence as a resource in decision problems: the Deutsch-Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016)
Lostaglio, M., Jennings, D., Rudolph, T.: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6, 6383 (2015)
Lostaglio, M., Jennings, D., Rudolph, T.: Thermodynamic resource theories, non-commutativity and maximum entropy principles. New J. Phys. 19, 043008 (2017)
Micadei, K., Rowlands, D.A., Pollock, F.A., Céleri, L.C., Serra, R.M., Modi, K.: Coherent measurements in quantum metrology. New J. Phys. 17, 023057 (2015)
Lloyd, S.: Quantum coherence in biological systems. J. Phys.: Conf. Ser. 302, 012037 (2011)
Plenio, M.B., Virmani, S.: An introduction to entanglement measures. Quant. Inf. Comput. 7, 1–51 (2007)
de Vicente, J.I., Streltsov, A.: Genuine quantum coherence. J. Phys. A 50, 045301 (2017)
Brandão, F.G.S.L., Gour, G.: Reversible framework for quantum resource theories. Phys. Rev. Lett. 115, 070503 (2015)
Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)
Qi, X., Gao, T., Yan, F.: Measuring coherence with entanglement concurrence. J. Phys. A: Math. Theor. 50, 285301 (2017)
Zhu, H., Ma, Z., Cao, Z., Fei, S.-M., Vedral, V.: Operational one-to-one mapping between coherence and entanglement measures. Phys. Rev. A 96, 032316 (2017)
Chitambar, E., Streltsov, A., Rana, S., Bera, M.N., Adesso, G., Lewenstein, M.: Assisted distillation of quantum coherence. Phys. Rev. Lett. 116, 070402 (2016)
Chitambar, E., Gour, G.: Comparision of incoherent operations and measures of coherence. Phys. Rev. A 94, 052336 (2016)
Vidal, G.: Entanglement monotones. J. Modern Opt. 47, 355 (2000)
Du, S., Bai, Z., Qi, X.: Coherence measures and optimal conversion for coherent states. Quantum Inf. Comput. 15, 1307 (2015)
Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)
Yuan, X., Zhou, H., Cao, Z., Ma, X.: Intrinsic randomness as a measure of quantum coherence. Phys. Rev. A 92, 022124 (2015)
Gour, G.: Family of concurrence monotones and its applications. Phys. Rev. A 71, 012318 (2005)
Chin, S.: Generalized coherence concurrence and path distinguishability. J. Phys. A: Math. Theor. 50, 475302 (2017)
Acknowledgments
This work is supported by the NSFC No.11775306, and 11701568; the Fundamental Research Funds for the Central Universities Grants No. 17CX02033A and 19CX02050A; the Shandong Provincial Natural Science Foundation No.ZR2016AQ06, and ZR2017BA019.
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Qiao, J., Wang, Z., Wang, J. et al. A Note on the Relationship Between Genuinely Coherence and Generalized Entanglement Monotones. Int J Theor Phys 58, 3998–4007 (2019). https://doi.org/10.1007/s10773-019-04266-6
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DOI: https://doi.org/10.1007/s10773-019-04266-6