Extended Cahill-Glauber formalism for finite-dimensional spaces: I. Fundamentals

Journal of Physics A: Mathematical and General

Volumn 38, Issue 27, 2005, Pages 6239-6251

  Ruzzi, M ^a   Marchiolli, M A ^b   Galetti, D ^a  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21344471118     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/27/010     Document Type: Article

References (46)

1. Wootters W K 1987 A Wigner-function formulation of finite-state quantum mechanics Ann. Phys., NY 176 1
2. Galetti D and de Toledo Piza A F R 1988 An extended Weyl-Wigner transformation for special finite spaces Physica A 149 267
3. Cohendet O, Combe P, Sirugue M and Sirugue-Collin M 1988 A stochastic treatment of the dynamics of an integer spin J. Phys. A: Math. Gen. 21 2875
4. Galetti D and de Toledo Piza A F R 1992 Discrete quantum phase spaces and the mod(N) invariance Physica A 186 513
5. Opatrný T, Welsch D-G and Bužek V 1996 Parametrized discrete phase-space functions Phys. Rev. A 53 3822
6. Galetti D and Marchiolli M A 1996 Discrete coherent states and probability distributions in finite-dimensional spaces Ann. Phys., NY 249 454
7. Marchiolli M A 2003 Nonclassical statistical properties of finite-coherent states in the framework of the Jaynes-Cummings model Physica A 319 331
8. Luis A and Peřina J 1998 Discrete Wigner function for finite-dimensional systems J. Phys. A: Math. Gen. 31 1423
9. Hakioǧlu T 1998 Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase J. Phys. A: Math. Gen. 31 6975
10. Mukunda N, Chatuverdi S and Simon R 2004 Wigner distributions for non-Abelian finite groups of odd order Phys. Lett. A 321 160
11. Gibbons K S, Hoffman M J and Wootters W K 2004 Discrete phase space based on finite fields Phys. Rev. A 70 062101
12. Vourdas A 2004 Quantum systems with finite Hilbert space Rep. Prog. Phys. 67 267 and references therein
13. Leonhardt U 1995 Quantum-state tomography and discrete Wigner functions Phys. Rev. Lett. 74 4101
14. Leonhardt U 1996 Discrete Wigner function and quantum-state tomography Phys. Rev. A 53 2998
15. Miquel C, Paz J P, Saraceno M, Knill E, Laflamme R and Negrevergne C 2002 Interpretation of tomography and spectroscopy and dual forms of quantum computation Nature 418 59
16. Paz J P and Roncaglia A J 2003 Quantum gate arrays can be programmed to evaluate the expectation value of any operator Phys. Rev. A 68 052316
17. Paz J P, Roncaglia A J and Saraceno M 2004 Quantum algorithms for phase-space tomography Phys. Rev. A 69 032312
18. Koniorczyk M, Bužek V and Janszky J 2001 Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit Phys. Rev. A 64 034301
19. Paz J P 2002 Discrete Wigner functions and the phase-space representation of quantum teleportation Phys. Rev. A 65 062311
20. Miquel C, Paz J P and Saraceno M 2002 Quantum computers in phase space Phys. Rev. A 65 062309
21. Bianucci P, Miquel C, Paz J P and Saraceno M 2002 Discrete Wigner functions and the phase space representation of quantum computers Phys. Lett. A 297 353
22. Bianucci P, Paz J P and Saraceno M 2002 Decoherence for classically chaotic quantum maps Phys. Rev. E 65 046226
23. López C C and Paz J P 2003 Phase-space approach to the study of decoherence in quantum walks Phys. Rev. A 68 052305
24. Aolita M L, García-Mata I and Saraceno M 2004 Noise models for superoperators in the chord representation Phys. Rev. A 70 062301
25. Paz J P, Roncaglia A J and Saraceno M 2004 Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem Preprint quant-ph/0410117
26. Galvão E F 2005 Discrete Wigner functions and quantum computational speedup Phys. Rev. A 71 042302
27. Hillery M, O'Connell R F, Scully M O and Wigner E P 1984 Distribution functions in physics: fundamentals Phys. Rep. 106 121
28. Lee H-W 1995 Theory and application of the quantum phase-space distribution functions Phys. Rep. 259 147
29. Cahill K E and Glauber R J 1969 Ordered expansions in boson amplitude operators Phys. Rev. 177 1857
30. Cahill K E and Glauber R J 1969 Density operators and quasiprobability distributions Phys. Rev. 177 1882
31. Marchiolli M A, Ruzzi M and Galetti D 2005 Extended Cahill-Glauber formalism for finite-dimensional spaces: II. Applications in quantum tomography and quantum teleportation Preprint quant-ph/0504107
32. See, for instance Schleich W P 2001 Quantum Optics in Phase-Space (Berlin: Wiley-VCH)
33. Schwinger J 2000 Advanced Book Classics: Quantum Kinematics and Dynamics (New York: Perseus Publishing)
34. Schwinger J 2001 Quantum Mechanics: Symbolism of Atomic Measurements (New York: Springer)
35. Mehta M L 1987 Eigenvalues and eigenvectors of the finite Fourier transform J. Math. Phys. 28 781
36. Leboeuf P and Voros A 1990 Chaos-revealing multiplicative representation of quantum eigenstates J. Phys. A: Math. Gen. 23 1765
37. Zhang S and Vourdas A 2004 Analytic representation of finite quantum systems J. Phys. A: Math. Gen. 37 8349
38. Ruzzi M and Galetti D 2000 Quantum discrete phase space dynamics and its continuous limit J. Phys. A: Math. Gen. 33 1065
39. Ruzzi M and Galetti D 2002 Schwinger and Pegg-Barnett approaches and a relationship between angular and Cartesian quantum descriptions: II. Phase spaces J. Phys. A: Math. Gen. 35 4633
40. Barker L 2001 Continuum quantum systems as limits of discrete quantum systems: I. State vectors J. Funct. Anal. 186 153
41. Barker L 2001 Continuum quantum systems as limits of discrete quantum systems: II. State functions J. Phys. A: Math. Gen. 34 4673
42. Barker L 2001 Continuum quantum systems as limits of discrete quantum systems: III. Operators J. Math. Phys. 42 4653
43. Vilenkin N J and Klimyk A U 1992 Representation of Lie Groups and Special Functions: Simplest Lie Groups, Special Functions and Integral Transforms (Dordrecht: Kluwer)
44. Bellman R 1961 A Brief Introduction to Theta Functions (New York: Holt, Rinehart and Winston)
45. Magnus W, Oberhettinger F and Soni R P 1966 Formulas and Theorems for the Special Functions of Mathematical Physics (New York: Springer)
46. Whittaker E T and Watson G N 2000 Cambridge Mathematical Library: A Course of Modern Analysis (Cambridge: Cambridge University Press)