A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics

A lower bound for the Wehrl entropy of a single quantum spin is derived. The high-spin asymptotics of this bound coincides with Lieb’s conjecture up to, but not including, terms of first and higher order in the inverse spin quantum number. The result presented here may be seen as complementary to the verification of the conjecture in cases of lowest spin by Schupp [Commun. Math. Phys. 207, 481 (1999)]. The present result for the Wehrl-entropy is obtained from interpolating a sharp norm bound that also implies a sharp lower bound for the so-called Rényi-Wehrl entropy with certain indices that are evenly spaced by half of the inverse spin quantum number.

[1]  William Beckner,et al.  Pitt’s inequality and the uncertainty principle , 1995 .

[2]  E. Carlen.  Some integral identities and inequalities for entire functions and their application to the coherent state transform , 1991 .

[3]  P. Schupp.  On Lieb's Conjecture for the Wehrl Entropy¶of Bloch Coherent States , 1999, math-ph/9902017.

[4]  J. Serrin,et al.  UNIQUENESS OF GROUND STATES FOR QUASILINEAR ELLIPTIC EQUATIONS IN THE EXPONENTIAL CASE , 1998 .

[5]  Lambertus A. Peletier,et al.  Uniqueness of positive solutions of semilinear equations in ℝn , 1983 .

[6]  C. Mueller,et al.  Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere , 1982 .

[7]  Elliott H. Lieb,et al.  Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities , 1983 .

[8]  S. Luo.  A simple proof of Wehrl's conjecture on entropy , 2000 .

[9]  Karol Zyczkowski,et al.  Rényi-Wehrl entropies as measures of localization in phase space , 2001 .

[10]  J. Klauder.  The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers , 1960 .

[11]  L. Gross.  Hypercontractivity over complex manifolds , 1999 .

[12]  J. Serrin,et al.  Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations inRn , 1996 .

[13]  Ayumu Sugita.  LETTER TO THE EDITOR: Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one , 2002 .

[14]  V. Bargmann.  On a Hilbert space of analytic functions and an associated integral transform part I , 1961 .

[15]  J. Eells,et al.  NONLINEAR ANALYSIS ON MANIFOLDS MONGE-AMPÈRE EQUATIONS (Grundlehren der mathematischen Wissenschaften, 252) , 1984 .

[16]  Lambertus A. Peletier,et al.  Uniqueness of non-negative solutions of semilinear equations in Rn , 1986 .

[17]  A. Wehrl.  On the relation between classical and quantum-mechanical entropy , 1979 .

[18]  C. Beck,et al.  Thermodynamics of chaotic systems : an introduction , 1993 .

[19]  E. Lieb.  Proof of an entropy conjecture of Wehrl , 1978 .

[20]  S. Luo.  A harmonic oscillator on the Poincaré disc and hypercontractivity , 1997 .

[21]  William Beckner,et al.  Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality , 1993 .

[22]  F. Arecchi,et al.  Atomic coherent states in quantum optics , 1972 .

[23]  Some inequalities for entire functions , 1980 .

[24]  On the Grushin operator and hyperbolic symmetry , 1999, math/9903126.

[25]  J. Burbea.  Inequalities for holomorphic functions of several complex variables , 1983 .

[26]  A. Perelomov.  Generalized Coherent States and Their Applications , 1986 .

[27]  C. T. Lee.  Wehrl's entropy of spin states and Lieb's conjecture , 1988 .

[28]  B. Bodmann,et al.  A rigorous path integral for quantum spin using flat-space Wiener regularization , 1998, math-ph/9811016.

[29]  Thierry Aubin,et al.  Nonlinear analysis on manifolds, Monge-Ampère equations , 1982 .

[30]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[31]  A. Wehrl.  The many facets of entropy , 1991 .

[32]  J. Serrin,et al.  UNIQUENESS OF GROUND STATES FOR QUASILINEAR ELLIPTIC OPERATORS , 1998 .

[33]  Karol Zyczkowski,et al.  Rényi Extrapolation of Shannon Entropy , 2003, Open Syst. Inf. Dyn..