Geometry of quantum systems: Density states and entanglement

Journal of Physics A: Mathematical and General

Volumn 38, Issue 47, 2005, Pages 10217-10244

  Grabowski, Janusz ^a   Kuś, Marek ^b   Marmo, Giuseppe ^c  

^a INSTITUTE OF MATHEMATICS   (Poland)

^b CENTER FOR THEORETICAL PHYSICS   (Poland)

^c UNIVERSITY OF NAPLES FEDERICO II   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27844514662     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/47/011     Document Type: Article

References (60)

1. Dirac P A M 1958 The Principles of Quantum Mechanics 4th edn (Oxford: Pergamon)
2. von Neumann J 1971 Foundations of Quantum Mechanics (Princeton, NJ: Princeton University Press)
3. Pauli W 1980 General Principles of Quantum Mechanics (Berlin: Springer) (translated from German by P Achuthan and K Venkatesan)
4. Weyl H 1931 The Theory of Groups and Quantum Mechanics (New York: Dover)
5. Wigner E 1939 On unitary representations of the inhomogeneous Lorentz group Ann. Math. 40 149-204
6. Wigner E P 1959 Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Pure and Applied Physics vol 5) (New York: Academic) (Expanded and improved edn. Translated from German by J J Griffin)
7. Carĩena J F, Grabowski J and Marmo G 2000 Lie-Scheffers Systems: A Geometrical Approach (Napoli: Bibliopolis)
8. Carĩena J F, Marmo G and Nasarre J 1998 The nonlinear superposition principle and the Weyl-Norman method Int. J. Mod. Phys. A 13 3601-362
9. Man'ko V I, Marmo G, Sudarshan E C G and Zaccaria F 2000 Inner composition law of pure states Phys. Lett. A 273 31-6
10. Cirelli R, Gatti M and Maniá A 1999 On the nonlinear extension of quantum superposition and uncertainty principles J. Geom. Phys. 29 54-86
11. Jordan P, von Neumann J and Wigner E 1934 On an algebraic generalization of the quantum mechanical formalism Ann. Math. 35 29-54
12. Cantoni V 1977 The Riemannian structure on the space of quantum-like systems Commun. Math. Phys. 55 189-93
13. Cantoni V 1977 Intrinsic geometry of the quantum-mechanical phase space, Hamiltonian systems and correspondence principle Licei. Rend. Sc. Fis. Mat. e Nat. 62 628-36
14. Cirelli R, Lanzavecchia P and Maniá A 1983 Normal pure states of the von Neumann algebra of bounded operators as Kähler manifold J. Phys. A: Math. Gen. 15 3829-35
15. Cirelli R and Lanzavecchia P 1984 Hamiltonian vector fields in quantum mechanics Nuovo Cimento B 79 271-83
16. Abbati M C, Cirelli R, Lanzavecchia P and Maniá A 1984 Pure states of general quantum-mechanical systems as Kähler bundle Nuovo Cimento B 83 43-60
17. Bloch A 1987 An infinite-dimensional Hamiltonian system on a projective Hilbert space Trans. Am. Math. Soc. 302 787-96
18. Heslot A 1985 Quantum mechanics as a classical theory Phys. Rev. D 31 1341-48
19. Rowe D J, Ryman A and Rosensteel G 1980 Many-body quantum mechanics as a symplectic dynamical system Phys. Rev. A 22 2362-73
20. Field T R and Anandan J S 2004 Geometric phases and coherent states J. Geom. Phys. 50 56-78
21. Brody D C and Hughston L P 2001 Geometric quantum mechanics J. Geom. Phys. 38 19-53
22. Strocchi F 1956 Complex coordinates and quantum mechanics Rev. Mod. Phys. 38 36-40
23. Ashtekar A and Shilling T A 1998 Geometrical formulation of quantum mechanics On Einstein's Path ed A Harvey (Berlin: Springer) (Preprint gr-qc/9706069)
24. Mańko V I, Marmo G, Sudarshan E C G and Zaccaria F 2005 The geometry of density states Rep. Math. Phys. 55 405-22
25. Fano U 1957 Description of states in quantum mechanics by density matrix and operator techniques Rev. Mod. Phys. 29 74-93
26. Kostant B 1970 Quantization and Unitary Representations. Part I. Prequantization (Lecture Notes in Mathematics vol 170) (Berlin: Springer)
27. Kirillov A A 1976 Elements of the Theory of Representations (Berlin: Springer)
28. Souriau J-M 1970 Structure des Systemes Dynamiques (Paris: Dunod)
29. Sommers H-J and Życzkowski K 2003 Bures volume of the set of mixed quantum states J. Phys. A: Math. Gen. 36 10083-100
30. Życzkowski K and Sommers H-J 2003 Hilbert-Schmidt volume of the set of mixed quantum states J. Phys. A: Math. Gen. 36 10115-30
31. Chruściński D and Jamiołkowski A 2004 Geometric Phases in Classical and Quantum Mechanics (Boston, MA: Birkhäuser)
32. Berry M V 1984 Quantal phase factors accompanying adiabatic changes Proc. R. Soc. London A 392 45-57
33. Mukunda N and Simon R 1993 Quantum kinematic approach to the geometric phase: I. General formalism Ann. Phys. 228 205-68
34. Esposito G, Marmo G and Sudarshan E C G 2004 From Classical to Quantum Mechanics (Cambridge: Cambridge University Press)
35. Dittmann J and Rudolph G 1992 On a connection governing parallel transport along 2 × 2 density matrices J. Geom. Phys. 10 93-106
36. Schlienz J and Mahler G 1995 Description of entanglement Phys. Rev. A 52 4396-404
37. Grassl M, Rötteler M and Beth T 1998 Computing local invariants of quantum-bit systems Phys. Rev. A 58 1833-39
38. Makhlin Y 2002 Locally invariant properties of two-qubit gates and mixed states and optimization of quantum computations Quantum Inf. Proc. 1 243-52
39. Briand E, Luque J-G and Thibon J-Y 2003 A complete set of covariants of the four qubit system J. Phys. A: Math. Gen. 36 9915-27
40. Luque J-G and Thibon J-Y 2003 Polynomial invariants of four qubits Phys. Rev. A 67 042303
41. Linden N and Popescu S 1998 On multi-particle entanglement Fortschr. Phys. 46 567-78
42. Linden N, Popescu S and Sudbery A 1999 Non-local parameters of multi-particle density matrices Phys. Rev. Lett. 83 243-7
43. Kuś M and Życzkowski K 2001 Geometry of entangled states Phys. Rev. A 63 032307
44. Sinołeçka M M, Życzkowski K and Kuś M 2002 Manifolds of equal entanglement for composite quantum system Acta Phys. Pol. B 33 2081-95
45. Moseri R and Dandoloff R 2001 Geometry of entangled states, Bloch spheres and Hopf fibrations J. Phys. A: Math. Gen. 34 10243-52
46. Moseri R Two and three qubits geometry and Hopf fibrations Topology in Condensed Matter Series: Springer Series in Solid-State Sciences vol 150 ed M I Monastyrsky at press (Preprint quant-ph/0310053)
47. Lévay P 2004 The geometry of entanglement: metrics, connections and the geometric phase J. Phys. A: Math. Gen. 37 1821-41
48. Heydari H 2005 Segre variety, conifold, Hopf fibration, and separable multi-qubit states Preprint quant-ph/0506043
49. Heydari H and Björk G 2005 Complex multi-projective variety and entanglement J. Phys. A: Math. Gen. 38 3203-11
50. Lévay P 2005 On the geometry of a class of N-qubit entanglement monotones Preprint quant-ph/0507070
51. Heydari H 2005 Entanglement monotone for multi-qubit states based on geometric invariant theory Preprint quant-ph/0507077
52. Lévay P 2005 Geometry of three-qubit entanglement Phys. Rev. A 71 012334
53. Chen H 2001 Quantum entanglement and geometry of determinantal varieties Preprint quant-ph/0110103
54. Shi M and Du J 2001 Boundary of the Set of separable states Preprint quant-ph/0103016
55. Cirelli R, Maniá A and Pizzocchero L 1990 Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure Int. J. Math. Phys. 31 2891-7
56. Cirelli R, Maniá A and Pizzocchero L 1990 J. Math. Phys. 31 2898-903
57. Benvegnu A, Sansonetto N and Spera M 2004 Remarks on geometrical quantum mechanics J. Geom. Phys. 51 229-43
58. Kibble T W 1979 Geometrization of quantum mechanics Commun. Math. Phys. 65 189-201
59. Hartshorne R 1977 Algebraic Geometry (Berlin: Springer) section IV.2
60. Vidal G 2000 Entanglement monotones J. Mod. Opt. 47 355-76