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.
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) https://doi.org/10.1088/1751-8113/49/47/473001
4
W. H.Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Phys. Rev. D24(6), 1516(1981) https://doi.org/10.1103/PhysRevD.24.1516
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) https://doi.org/10.1103/PhysRevLett.115.210403
6
M.Lostaglio, K.Korzekwa,D.Jennings, and T.Rudolph, Quantum coherence, time-translation symmetry, and thermodynamics, Phys. Rev. X5(2), 021001(2015) https://doi.org/10.1103/PhysRevX.5.021001
M.Hillery, Coherence as a resource in decision problems: The Deutsch–Jozsa algorithm and a variation, Phys. Rev. A93(1), 012111(2016) https://doi.org/10.1103/PhysRevA.93.012111
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) https://doi.org/10.1103/PhysRevA.95.032307
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) https://doi.org/10.1103/PhysRevA.92.012118
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) https://doi.org/10.1103/PhysRevLett.116.160406
A. E.Rastegin, Entropic uncertainty relations for successive measurements of canonically conjugate observables, Ann. Phys. 528(11–12), 835(2016) https://doi.org/10.1002/andp.201600130
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) https://doi.org/10.1103/RevModPhys.89.015002
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) https://doi.org/10.3390/math4030047
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)
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) https://doi.org/10.1088/0253-6102/67/6/631
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)
A.Gilchrist, N. K.Langford, and M. A.Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A71(6), 062310(2005) https://doi.org/10.1103/PhysRevA.71.062310
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) https://doi.org/10.1088/0253-6102/67/2/166
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)
A. E.Rastegin, Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies, Eur. Phys. J. D67(12), 269(2013) https://doi.org/10.1140/epjd/e2013-40453-2
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) https://doi.org/10.1103/PhysRevA.90.052114
A. E.Rastegin, Separability conditions based on local fine-grained uncertainty relations, Quantum Inform. Process. 15(6), 2621(2016) https://doi.org/10.1007/s11128-016-1286-z
