Abstract
We construct a multi-observable uncertainty equality as well as an inequality based on the sum of standard deviations in the qubit system. Firstly, an uncertainty equality as well as an inequality is formulated, and we demonstrate that the new uncertainty inequality is tighter than other recent uncertainty relations. Then, a proof, that the triviality problem of the product form uncertainty relation can be completely fixed by the obtained uncertainty relation, is presented. Finally, we show that the uncertainty equality can be used as a measure of the mixedness of the system which usually is expensive in terms of resources involved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Heisenberg, W.: Über den anschaulichen Inhalt derquantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)
Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163 (1929)
Schrodinger, E.: Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 14, 296 (1930)
Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070 (1987)
Zhang, G.F.: Sudden death of entanglement of a quantum model. Chin. Phys. 16, 1855 (2007)
Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659 (2010)
Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072 (2010)
Li, C.F., Xu, J.S., Li, K., Guo, G.C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752 (2011)
Wehner, S., Winter, A.: Entropic uncertainty relations: a survey. New J. Phys. 12, 025009 (2010)
Tomamichel, M., Renner, R.: Uncertainty relation for smooth entropies. Phys. Rev. Lett. 106, 110506 (2011)
Coles, P.J., Colbeck, R., Yu, L., Zwolak, M.: Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012)
Xu, Z.Y., Yang, W.L., Feng, M.: Quantum-memory-assisted entropic uncertainty relation under noise. Phys. Rev. A 86, 012113 (2012)
Hofmann, H.F., Takeuchi, S.: Violation of local uncertainty relations as a signature of entanglement. Phys. Rev. A 68, 032103 (2003)
Guhne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)
Schwonnek, R., Dammeier, L., Werner, R.F.: State-independent uncertainty relations and entanglement detection in noisy systems. Phys. Rev. Lett. 119, 170404 (2017)
Walls, D.F., Zoller, P.: Reduced quantum fluctuations in resonance fluorescence. Phys. Rev. Lett. 47, 709 (1981)
Wineland, D.J., Bollinger, J.J., Itano, W.M., Moore, F.L., Heinzen, D.J.: Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A 46, R6797 (1992)
Kitagawa, M., Ueda, M.: Squeezed spin states. Phys. Rev. A 47, 5138 (1993)
Ma, J., Wang, X.G., Sun, C.P., Nori, F.: Quantum spin squeezing. Phys. Rep. 509, 89 (2011)
Zhang, G.F., Li, S.S.: Entanglement in a spin-one spin chain. Solid State Commun. 138, 17 (2006)
Giovannetti, V., Lloyd, S., Maccone, L.: Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006)
Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nat. Photon. 5, 222 (2011)
Maccone, L., Pati, A.K.: Stronger uncertainty relations for all incompatible observables. Phys. Rev. Lett. 113, 260401 (2014)
Song, Q.C., Qiao, C.F.: Stronger Schrödinger-like uncertainty relations. Phys. Lett. A 380, 2925 (2016)
Chen, B., Fei, S.M.: Sum uncertainty relations for arbitrary N incompatible observables. Sci. Rep. 5, 14238 (2015)
Zheng, X., Zhang, G.F.: The effects of mixedness and entanglement on the properties of the entropic uncertainty in Heisenberg model with Dzyaloshinski–Moriya interaction. Quantum Inf. Process. 16, 1 (2017)
Zheng, X., Zhang, G.F.: Variance-based uncertainty relation for incompatible observers. Quantum Inf. Process. 16, 167 (2017)
Yao, Y., Xiao, X., Wang, X.G., Sun, C.P.: Implications and applications of the variance-based uncertainty equalities. Phys. Rev. A 91, 062113 (2015)
Mal, S., Pramanik, T., Majumdar, A.S.: Detecting mixedness of qutrit systems using the uncertainty relation. Phys. Rev. A 87, 012105 (2013)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11574022 and 11174024) and the Open Project Program of State Key Laboratory of Low-Dimensional Quantum Physics (Tsinghua University) Grants No. KF201407 and also supported by the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No. Y4KF201CJ1), and Beijing Higher Education (Young Elite Teacher Project) YETP 1141.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Xiao Zheng and Shaoqiang Ma have contributed equally to this work.
Rights and permissions
About this article
Cite this article
Zheng, X., Ma, S. & Zhang, G. Multi-observable uncertainty equality based on the sum of standard deviations in the qubit system. Quantum Inf Process 19, 116 (2020). https://doi.org/10.1007/s11128-020-2609-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-020-2609-7