Uncertainty relations for quantum coherence with respect to mutually unbiased bases

Abstract

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.

Keywords

coherence / complementarity / uncertainty / mutually unbiased bases

References

[1] T.Baumgratz, M.Cramer, and M. B.Plenio, Quantifying coherence, Phys. Rev. Lett. 113(14), 140401 (2014)

[2] A.Streltsov, G.Adesso, and M. B.Plenio, Quantum coherence as a resource, arXiv: 1609.02439 [quant-ph] (2016)

[3] G.Adesso, T. R.Bromley, and M.Cianciaruso, Measures and applications of quantum correlations, J. Phys. A Math. Theor. 49(47), 473001(2016)

[4] W. H.Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Phys. Rev. D24(6), 1516(1981)

[5] P.Ćwikliński, M.Studziński, M.Horodecki, and J.Oppenheim, Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics, Phys. Rev. Lett. 115(21), 210403(2015)

[6] M.Lostaglio, K.Korzekwa,D.Jennings, and T.Rudolph, Quantum coherence, time-translation symmetry, and thermodynamics, Phys. Rev. X5(2), 021001(2015)

[7] V.Narasimhacharand G.Gour, Low-temperature thermodynamics with quantum coherence, Nat. Commun. 6, 7689(2015)

[8] M.Hillery, Coherence as a resource in decision problems: The Deutsch–Jozsa algorithm and a variation, Phys. Rev. A93(1), 012111(2016)

[9] H. L.Shi, S. Y.Liu, X. H.Wang, W. L.Yang, Z. Y.Yang, and H.Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A95(3), 032307(2017)

[10] M. N.Bera, T.Qureshi, M. A.Siddiqui, and A. K.Pati, Duality of quantum coherence and path distinguishability, Phys. Rev. A92(1), 012118(2015)

[11] E.Bagan, J. A.Bergou, S. S.Cottrell, and M.Hillery, Relations between coherence and path information, Phys. Rev. Lett. 116(16), 160406(2016)

[12] S.Chengand M. J. W.Hall, Complementarity relations for quantum coherence, Phys. Rev. A92(4), 042101(2015)

[13] R. F.Werner, Uncertainty relations for general phase spaces, Front. Phys. 11(3), 110305(2016)

[14] A. E.Rastegin, Entropic uncertainty relations for successive measurements of canonically conjugate observables, Ann. Phys. 528(11–12), 835(2016)

[15] A. E.Rastegin, On entropic uncertainty relations in the presence of a minimal length, Ann. Phys. 382, 170(2017)

[16] S.Wehnerand A.Winter, Entropic uncertainty relations-A survey, New J. Phys. 12(2), 025009(2010)

[17] I.Bia lynicki-Birulaand L.Rudnicki, Entropic Uncertainty Relations in Quantum Physics, in: K. D. Sen (Ed.), Statistical Complexity, Berlin: Springer, 2011

[18] P. J.Coles, M.Berta, M.Tomamichel, and S.Wehner, Entropic uncertainty relations and their applications, Rev. Mod. Phys. 89(1), 015002(2017)

[19] H.Maassenand J. B. M.Uffink, Generalized entropic uncertainty relations, Phys. Rev. Lett. 60(12), 1103(1988)

[20] M.Berta, M.Christandl, R.Colbeck, J. M.Renes, and R.Renner, The uncertainty principle in the presence of quantum memory, Nat. Phys. 6(9), 659(2010)

[21] U.Singh, A. K.Pati, and M. N.Bera, Uncertainty relations for quantum coherence, Mathematics4(3), 47(2016)

[22] Y.Peng, Y. R.Zhang, Z.Y.Fan, S.Liu, and H.Fan, Complementary relation of quantum coherence and quantum correlations in multiple measurements, arXiv: 1608.07950 [quant-ph] (2016)

[23] X.Yuan, G.Bai, T.Peng, and X.Ma, Quantum uncertainty relation of coherence, arXiv: 1612.02573 [quantph] (2016)

[24] T.Durt, B. G.Englert, I.Bengtsson, and K.Życzkowski, On mutually unbiased bases, Int. J. Quant. Inf. 08(04), 535(2010)

[25] M. A.Nielsen, and I. L.Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000

[26] V.Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74(1), 197(2002)

[27] D.Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23(1), 57(1986)

[28] A. E.Rastegin, Quantum-coherence quantifiers based the Tsallis relative entropies, Phys. Rev. A93(3), 032136(2016)

[29] E.Chitambarand G.Gour, Comparison of incoherent operations and measures of coherence, Phys. Rev. A94(5), 052336(2016)

[30] L. H.Shao, Y. M.Li, Y.Luo, and Z. J.Xi, Quantum coherence quantifiers based on Rényi α-relative entropy, Commum. Theor. Phys. 67(6), 631(2017)

[31] A.Streltsov, H.Kampermann, S.Wölk, M.Gessner, and D.Bruß, Maximal coherence and the resource theory of purity, arXiv: 1612.07570 [quant-ph] (2016)

[32] A.Uhlmann, The “transition probability” in the state space of a *-algebra, Rep. Math. Phys. 9(2), 273(1976)

[33] R.Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41(12), 2315(1994)

[34] A.Gilchrist, N. K.Langford, and M. A.Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A71(6), 062310(2005)

[35] A. E.Rastegin, Sine distance for quantum states, arXiv: quant-ph/0602112 (2006)

[36] H. J.Zhang, B.Chen, M.Li, S. M.Fei, and G. L.Long, Estimation on geometric measure of quantum coherence, Commum. Theor. Phys. 67(2), 166(2017)

[37] J. A.Miszczak, Z.Puchała, P.Horodecki, A.Uhlmann, and K.Życzkowski, Sub- and super-fidelity as bounds for quantum fidelity, arXiv: 0805.2037 (2008)

[38] S.Wu, S.Yu, and K.Mølmer, Entropic uncertainty relation for mutually unbiased bases, Phys. Rev. A79(2), 022104(2009)

[39] A. E.Rastegin, Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies, Eur. Phys. J. D67(12), 269(2013)

[40] T.Miyaderaand H.Imai, Generalized Landau–Pollak uncertainty relation, Phys. Rev. A76(6), 062108(2007)

[41] G. M.Bosyk, S.Zozor, M.Portesi, T. M.Osán, and P. W.Lamberti, Geometric approach to extend Landau- Pollak uncertainty relations for positive operator-valued measures, Phys. Rev. A90(5), 052114(2014)

[42] S.Liu, L. Z.Mu, and H.Fan, Entropic uncertainty relations for multiple measurements, Phys. Rev. A91(4), 042133(2015)

[43] P. J.Coles, R.Colbeck, L.Yu, and M.Zwolak, Uncertainty relations from simple entropic properties, Phys. Rev. Lett. 108(21), 210405(2012)

[44] A. E.Rastegin, Separability conditions based on local fine-grained uncertainty relations, Quantum Inform. Process. 15(6), 2621(2016)

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag Berlin Heidelberg