Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics

07/11/2017
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The study of positivity properties of trace class operators is essential in the theory of quantum mechanical density matrices; the latter describe the “mixed states” of quantum mechanics and are essential in information theory. While a general theory for these positivity results is still lacking, we present some new results we have recently obtained and which generalize and extend the well-known conditions given in the 1970s by Kastler, Loupias, and Miracle-Sole, generalizing Bochner’s theorem on the Fourier transform of a probability measure. The tools we use are the theory of pseudodi¤erential operators, symplectic geometry, and Gabor frame theory. We also speculate about some consequences of a possibly varying Planck’s constant for the early universe.

Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics

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application/pdf Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics Elena Cordero, Maurice A. de Gosson, Fabio Nicola
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Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics

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Name Your Journal manuscript No. (will be inserted by the editor) Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics Elena Cordero1 , Maurice A. de Gosson2 , Fabio Nicola3 1 Universita di Torino e-mail: elena.cordero@unito.it 2 Universität Wien e-mail: maurice.de.gosson@univie.ac.at 3 Politecnico di Torino e-mail: fabio.nicola@polito.it The date of receipt and acceptance will be inserted by the editor Abstract The study of positivity properties of trace class operators is essential in the theory of quantum mechanical density matrices; the lat- ter describe the “mixed states”of quantum mechanics and are essential in information theory. While a general theory for these positivity results is still lacking, we present some new results we have recently obtained and which generalize and extend the well-known conditions given in the 1970s by Kastler, Loupias, and Miracle-Sole, generalizing Bochner’ s theorem on the Fourier transform of a probability measure. The tools we use are the the- ory of pseudodi¤erential operators, symplectic geometry, and Gabor frame theory. We also speculate about some consequences of a possibly varying Planck’ s constant for the early universe. 1 Introduction The characterization of positivity properties for trace class operators on L2 (Rn ) is important because of its potential applications to quantum me- chanics. It is also a notoriously di¢ cult part of functional analysis which has been tackled by many authors but there have been few decisive advances since the pioneering work of Kastler [6] and Loupias and Miracle-Sole [7,8]. 2 The KLM Conditions Let be a real parameter. The notion of -positivity generalizes the usual notion of positivity. 2 Elena Cordero et al. De…nition 1 Let b 2 S(R2n ). we say that b is of -positive type if for every integer N the N N matrix (N) with entries jk = e i 2 (zj ;zk) b(zj zk) is positive semide…nite for all choices of (z1; z2; :::; zN ) 2 (R2n )N . Here is the standard symplectic form P 1 j n dpj ^ dxj on R2n . In what follows we denote by b A = OpW (a) the -Weyl operator with symbol a: h b A ; i = ha; W ( ; )i (1) for all ; 2 S(Rn ); here W ( ; ) is the -cross-Wigner transform de…ned, for ; 2 L2 (Rn ), by W ( ; )(z) = 1 n (R (z) j )L2 (2) where R (z) is the -parity operator R (z) = T (z)R(0)T (z) 1 (3) (with R(0) (x) = ( x)) and T (z) is Heisenberg’ s -operator T (z0) (x) = e i (p0x 1 2 p0x0) (x x0): (4) Using the symplectic Fourier transform of a 2 S(R2n ) de…ned by a (z) = F a(z) = Z R2n ei (z;z0 ) a(z0 )dz0 : (5) Kastler [6] proved the following result using the theory of C -algebras; we have given in [5] a simpler proof of this result: Theorem 1 Let b A = OpW (a) be a self-adjoint trace-class operator on L2 (Rn ). We have b A 0 if and only the two following conditions hold: (i) a is con- tinuous and (0) = 1; (ii) a is of -positive type. Sketch of the proof: assume that b A 0; we have = P j jW j for a family of normalized functions j 2 L2 (Rn ), the coe¢ cients j being 0. Consider now the expression IN ( ) = X 1 j;k N j ke i 2 (zj ;zk) F ; W (zj zk) 0 (6) where F ; is the -symplectic transform de…ned by F ; a(z) = 1 2 n Z R2n e i (z;z0 ) a(z0 )dz: Title Suppressed Due to Excessive Length 3 We must verify that IN ( ) 0 but this follows from the observation that IN ( ) = 1 2 n jj X 1 j N jT (zj) jj2 L2 : (7) Let us now show that, conversely, the conditions (i) and (ii) imply that ( b A j )L2 0 for all 2 L2 (Rn ); equivalently Z R2n (z)W (z)dz 0 (8) for 2 L2 (Rn ). Setting 0 jk = e i 2 (zj ;zk) a ; (zj zk) to say that a ; is of -positive type means that the matrix 0 = ( 0 jk)1 j;k N is positive semide…nite; choosing zk = 0 and setting zj = z this means that every matrix (a ; (z))1 j;k N is positive semide…nite. Setting jk = e i 2 (zj ;zk) F ; W (zj zk) the matrix (N) = ( jk)1 j;k N is positive semide…nite. Writing Mjk = F ; W (zj zk) ; (zj zk); one shows using Schur’ s theorem on the positivity of Hadamard product of the positive semide…nite matrices that the matrix (Mjk)1 j;k N is also positive semide…nite. One then concludes using Bochner’ s theorem on the Fourier transform of probability measures. 3 A New Positivity Test The KLM conditions are not easily computable since they involve the veri- …cation of a non-countable set of inequalities. The following result replaces the KLM conditions by a countable set of conditions: Theorem 2 Let a 2 L2 (R2n ) and G( ; ) a Gabor frame for L2 (Rn ). For (z ; z ) 2 set a ; = Z R2n e i (z;z z ) a(z)W (z 1 2 (z + z ))dz: (9) The operator b A = OpW (a) is positive semide…nite if and only if for every integer N 0 the matrix with entries M ; = e i 2 (z ;z ) a ; , jz j; jz j N (10) is positive semide…nite. 4 Elena Cordero et al. Sketch of the proof: The condition b A 0 is equivalent to Z R2n a(z)W (z)dz 0 (11) for every 2 L2 (Rn ). The numbers a ; de…ned by (9) are the Gabor coe¢ cients with respect to the Gabor system G(Wg; ). Expanding in the frame G( ; ) we get = X z 2 c(z )T(z ) where c = ( jT(z ) )L2 and hence Z R2n a(z)W P z 2 c T(z ) (z)dz 0: (12) The claim follows using the relations W P z 2 c(z )T(z ) = P z 2 c c W(T(z ) ; T(z ) ); and observing that [4] W(T(z ) ; T(z ) ) = e i 2 (z ;z ) e i (z;z z ) W (z 1 2 (z + z )): Let us next show that the KLM conditions can be recaptured as a lim- iting case of the conditions in Theorem 2. Theorem 3 Let a 2 L1 (Rn ) and z ; z 2 R2n . Let 0(x) = ( ) n=4 e jxj2 =2 be the standard Gaussian and = T(z ) 0. Setting M ; = e i 2 (z ;z ) a ; (z z ) M ; = e i 2 (z ;z ) V 2n W a(1 2 (z + z ); J(z z )): we have M ; = lim "!0+ X z 2"Z2n jz j<1=" "2n M ; : Sketch of the proof: Observe that we can write M ; = e i 2 (z ;z ) V 2n a(1 2 (z + z ); J(z z )) where (z) = 1 for all z 2 R2n the result follows from the dominated convergence theorem using the limit lim "!0+ X z 2"Z2n jz j<1=" "2n W (z) = 1 (13) for all z 2 R2n and the bound P z 2"Z2n jz j<1=" "2n W (z) C: (14) valid for all z 2 R2n . Title Suppressed Due to Excessive Length 5 4 Discussion Positivity questions for operators are notoriously di¢ cult to handle. In the case of trace-class operators we consider here not many progresses have been done since the work of Narcowich and his collaborators [11– 14] and Bröcker and Werner [1]; also see Dias and Prata [3]. The interest in these questions come from the quantum mechanical problem of characterizing the so-called “mixed states” , which are statistical mixtures of well-de…ned quantum-mechanical states (the “pure states” ). Mixed states are mathe- matically represented by the quantum density operators: such an operator is a self-adjoint positive semide…nite trace class operators with unit trace (on any Hilbert space). While it is usually a rather trivial matter to ver- ify self-adjointness and the trace property, positivity is very delicate – as exempli…ed by our discussion above – . The importance of this concept has increased since it has been realized by cosmological observations [9,10] that Planck’ s constant = h=2 might very well have been changing it value since the early Universe. If true, this would mean that some quantum states have evolved in classical ones (see our analysis in [5]). Acknowledgement. Maurice de Gosson has been …nanced by the grant P27773-N23 of the Austrian Research Foundation FWF. References 1. T. Bröcker, and R. F. Werner, Mixed states with positive Wigner functions, J. Math. Phys., 36(1) 62– 75 (1995) 2. E. Cordero, M. de Gosson, and F. Nicola, Positivity of Trace-Class Operators and the Cohen Class; Application to Born– Jordan Quantization (in prepara- tion) 3. N. C. Dias, and J. N. Prata, The Narcowich-Wigner spectrum of a pure state, Rep. Math. Phys. 63(1), 43– 54 (2009) 4. M. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhäuser, 2011. 5. M. de Gosson, Quantum Harmonic Analysis of the Density Matrix: Basics, arXiv:1703.00889 [quant-ph] 6. D. Kastler, The C -Algebras of a Free Boson Field, Commun. math. Phys. 1, 14– 48 (1965) 7. G. Loupias and S. Miracle-Sole, C -Algèbres des systèmes canoniques, I, Com- mun. math. Phys., 2, 31– 48 (1966) 8. G. Loupias and S. Miracle-Sole, C -Algèbres des systèmes canoniques, II, Ann. Inst. Henri Poincaré, 6(1), 39– 58 (1967) 9. M. T. Murphy, J. K. Webb, V. V. Flambaum, V. A. Dzuba, C.W. Churchill, J. X. Prochaska, J. D. Barrow, and A. M. Wolfe, Possible evidence for a variable …ne-structure constant from QSO absorption lines: motivations, analysis and results. Mon. Not. R. Astron. Soc. 327, 1208– 1222 (2001) 10. M. T. Murphy, J. K. Webb, and V. V. Flambaum, Further evidence for a variable …ne-structure constant from Keck/HIRES QSO absorption spectra. Mon. Not. R. Astron. Soc. 345(2), 609– 638 (2003) 6 Elena Cordero et al. 11. F. J. Narcowich, Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution, J. Math. Phys., 29(9), 2036– 2041 (1988). 12. F. J. Narcowich, Distributions of -positive type and applications, J. Math. Phys., 30(11), 2565– 2573 (1989). 13. F. J. Narcowich, Geometry and uncertainty, J. Math. Phys., 31(2),354– 364 (1990). 14. F. J. Narcowich and R. F. O’ Connell, Necessary and su¢ cient conditions for a phase-space function to be a Wigner distribution, Phys, Rev. A, 34(1), 1– 6 (1986).