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.
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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)
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We thank referees for helping to improve this work.
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Supported by National Natural Science Foundation of China (Grant Nos. 11771011, 12071336)
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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
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DOI: https://doi.org/10.1007/s10114-022-1474-y