Abstract
Heisenberg's uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement. The information concerning incompatibility (non-commutativity) of observables is well included but that concerning correlation is missing. Schrödinger's uncertainty relations remedy this defect by supplementing the correlation in terms of anti-commutators. However, both Heisenberg's uncertainty relations and Schrödinger's uncertainty relations are expressed in terms of variances, which are not good measures of uncertainty in general situations (e.g., when mixed states are involved). By virtue of the Wigner–Yanase skew information, we will establish an uncertainty relation along the spirit of Schrödinger from a statistical inference perspective and propose a conjecture. The result may be interpreted as a quantification of certain aspect of the celebrated Wigner–Araki–Yanase theorem for quantum measurement, which states that observables not commuting with a conserved quantity cannot be measured exactly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
H. Araki and M. M. Yanase, Measurement of quantum mechanical operators, Phys. Rev. 120:622–626 (1960).
I. Bialynicki-Birula and J. Mycielski, Uncertainty relations for information entropy in wave mechanics, Comm. Math. Phys. 44:129–132 (1975).
E. A. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal. 101:194–211 (1991).
E. A. Carlen and A. Soffer, Entropy production by block variable summation and central limit theorems, Comm. Math. Phys. 140:339–371 (1991).
A. Connes and E. Stormer, Homogeneity of the state space of factors of type III1 J. Funct. Anal. 28:187–196 (1978).
H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1974).
R. A. Fisher, Theory of statistical estimation, Proc. Camb. Phil. Soc. 22:700–725 (1925).
G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3:207–236 (1997).
B. R. Frieden, Physics from Fisher Information, A Unification (Cambridge University Press, Cambridge, 1998).
M. J. W. Hall, Exact Heisenberg uncertainty relations, Phys. Rev. A 64:052103(2001).
M. J. W. Hall and M. Reginatto, Schrö;dinger equation from an exact uncertainty principle, J. Phys. A 35:3289–3303 (2002).
W. Heisenberg, Ü;ber Den anschaulichen Inhalt Der quantumtheoretischen Kinematik und Mechanik, Z. Phys. 43:172–198 (1927).
C. W. Helstrom, Quantum Detection and Estimation Theory (AcaDemic Press, New York, 1976).
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North Holland, Amsterdam, 1982).
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd edn. (Butterworth-Heinemann, Oxford, 1977), p. 2.
E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math. 11:267–288 (1973).
S. L. Luo, Quantum Fisher information and uncertainty relations, Lett. Math. Phys. 53:243–251 (2000).
S. L. Luo, Fisher information matrix of Husimi distribution, J. Stat. Phys. 102:1417–1428 (2001).
S. L. Luo, Maximum Shannon entropy, minimum Fisher information, and an elementary game, Found. Phys. 32:1757–1772 (2002).
N. A. Nielsen and I. L. Chung, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
D. Petz, Monotone metrics on matrix spaces, Linear Alg. Appl. 244:81–96 (1996).
H. P. Robertson, The uncertainty principle, Phys. Rev. 34:163–164 (1929).
E. Schrö;dinger, About Heisenberg uncertainty relation, Proc. Prussian Acad. Sci., Phy-Math. Section XIX:293–303 (1930). See also arXiv: quant-ph/9903100 for an English translation.
A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. Control 2:101–112 (1959).
A. D. Sukhanov, Schrö;dinger uncertainty relations and physical features of correlated-coherent states, Theor. Math. Phys. 132:1277–1294 (2002).
J. Summhammer, Inference in quantum physics, Found. Phys. Lett. 1:113–137 (1988).
A. Wehrl, General concept of entropy, Rev. MoDern Phys. 50:221–260 (1978).
R. F. Werner and M. M. Wolf, Bell inequalities and entanglement, quant-ph/0107093.
J. A. Wheeler, Information, physics, quantum: the search for links, in Complexity, Entropy, and Physics of Information, Z. H. Zurek, ed. (Addison-Wesley, MA, 1990), pp. 3–28.
H. Weyl, De Gruppentheorie und Quantenmechanik (Leipzig, 1928), p. 272.
E. P. Wigner, Die messung quantenmechanischer operatoren, Z. Phys. 133:101–118 (1952).
E. P. Wigner and M. M. Yanase, Information contents of distributions, Proc. Nat. Acad. Sci. USA, 49:910–918 (1963).
M. M. Yanase, Optimal measuring apparatus, Phys. Rev. 123:666–668 (1961).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Luo, S., Zhang, Z. An Informational Characterization of Schrödinger's Uncertainty Relations. Journal of Statistical Physics 114, 1557–1576 (2004). https://doi.org/10.1023/B:JOSS.0000013971.75667.c8
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000013971.75667.c8