Abstract
Quantum uncertainty relations are mathematical inequalities that describe the lower bound of products of standard deviations of observables (i.e., bounded or unbounded self-adjoint operators). By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for k observables, of which the formulation depends on the even or odd quality of k. This universal uncertainty relation is tight at least for the cases k = 2 and k = 3. For two observables, the uncertainty relation is a simpler reformulation of Schrödinger’s uncertainty principle, which is also tighter than Heisenberg’s and Robertson’s uncertainty relations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge, 2006
Berta, M., Christandl, M., Colbeck, R., et al.: The uncertainty principle in the presence of quantum memory. Nature Physics, 6, 659–662 (2010)
Chen, B., Cao, N., Fei, S., et al: Variance-based uncertainty relations for incompatible observables. Quantum Inf. Process, 15, 3909 (2016)
Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett., 50, 631–633 (1983)
Friedland, S., Gheorghiu, V., Gour, G.: Universal uncertainty relations. Phys. Rev. Lett., 111, 230401 (2013)
Furuichi, S.: Schrödinger uncertainty relation with Wigner—Yanase skew information. Phys. Rev. A, 82, 034101 (2010)
Guise, H. D., Maccone, L., Sanders, B. C., et al.: State-independent preparation uncertainty relations. Phys. Rev. A, 98, 042121 (2018)
Gustafson, K. E., Rao, D. K. M.: Numerical Range, Springer-Verlag, New York, 1997
Halmos, P. R.: A Hilbert Space Problem Book, Springer-Verlag, New York, 1982
Heisenberg, W.: Öber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys., 43, 172–198 (1927)
Heisenberg, W.: The Physical Principles of Quantum Theory, University of Chicago Press, Chicago, 1930
Hertz, A., Cerf, N. J.: Continuous-variable entropic uncertainty relations. J. Phys. A: Math. Theor., 52(17), 173001 (2019)
Hou, J. C., He, K.: Non-linear maps on self-adjoint operators preserving numerical radius and numerical range of Lie product. Linear and Multilinear Algebra, 64(1), 36–57 (2016)
Kechrimparis, S., Weigert, S.: Heisenberg uncertainty relation for three canonical observables. Phys. Rev. A, 90, 062118 (2014)
Keeler, D. S., Rodman, L., Spitkovsky, I. M.: The numerical range of 3 × 3 matrices. Linear Algebra Appl., 252, 115–139 (1997)
Li, T., Xiao, Y. L., Ma, T., et al.: Optimal universal uncertainty relations. Sci. Rep., 6, 35735 (2016)
Ma, W. C., Chen, B., Liu, Y., et al.: Experimental demonstration of uncertainty relations for the triple components of angular momentum. Phys. Rev. Lett., 118(18), 180402 (2017)
Maccone, L., Pati, A. K.: Stronger uncertainty relations for all incompatible observables. Phys. Rev. Lett., 113, 260401 (2014)
Pati, A. K., Sahu, P. K.: Sum uncertainty relation in quantum theory. Phys. Lett. A, 367, 177–181 (2007)
Prevedel, R., Hamel, D. R., Colbeck, R., et al.: Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement. Nat. Phys., 7, 757–761 (2011)
Qin, H., Fei, S., Li-Jost, X.: Multi-observable uncertainty relations in product form of variances. Sci. Rep., 6, 31192 (2016)
Robertson, H. P.: The uncertainty principle. Phys. Rev., 34, 163 (1929)
Schrodinger, E. (annotated by Angelow, A., Batoni, M. C.): About Heisenberg uncertainty relation. Bulg. J. Phys., 26(5/6), 193–203 (1999)
Szymański, K., Yczkowski, K.: Geometric and algebraic origins of additive uncertainty relations. J. Phys. A Math. Theor., 53(1), 015302 (2020)
Von Neumann, J.: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1932
Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys., 12, 025009 (2010)
Acknowledgements
We thank referees for helping to improve this work.
Funding
Supported by National Natural Science Foundation of China (Grant Nos. 11771011, 12071336)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, K., Hou, J.C. Applying the Theory of Numerical Radius of Operators to Obtain Multi-observable Quantum Uncertainty Relations. Acta. Math. Sin.-English Ser. 38, 1241–1254 (2022). https://doi.org/10.1007/s10114-022-1474-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-022-1474-y