Geometry and Structure of Quantum Phase Space
Tóm tắt
The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an important role in foundations of quantum mechanics and quantum information. In this work we discuss a geometric framework for mixed quantum states represented by density matrices, where the quantum phase space of density matrices is equipped with a symplectic structure, an almost complex structure, and a compatible Riemannian metric. This compatible triple allow us to investigate arbitrary quantum systems. We will also discuss some applications of the geometric framework.
Tài liệu tham khảo
Günther, C.: Prequantum bundles and projective Hilbert geometries. Int. J. Theoret. Phys. 16, 447–464 (1977)
Kibble, T.W.B.: Geometrization of quantum mechanics. Commun. Math. Phys. 65, 189–201 (1979)
Ashtekar, A., Schilling, T.A.: Geometrical formulation of quantum mechanics. In: Harvey, A. (ed.) On Einstein’s Path, pp. 23–65. Springer, New York (1998)
Brody, D.C., Hughston, L.P.: Geometrization of statistical mechanics. Proc. Math. Phys. Eng. Sci. 455(1985), 1683–1715 (1999)
Andersson, O., Heydari, H.: Operational geometric phase for mixed quantum states. New J. Phys. 15(5), 053006 (2013)
Andersson, O., Heydari, H.: Motion in bundles of purifications over spaces of isospectral density matrices. AIP Conf. Proc. 1508(1), 350–353 (2012). GP, MB. DD, GQE, GUR, QSL
Andersson, O., Heydari, H.: Dynamic distance measure on spaces of isospectral mixed quantum states. Entropy 15(9), 3688–3697 (2013)
Andersson, O., Heydari, H.: Geometry of quantum evolution for mixed quantum states. Phys. Scripta 2014(T160), 014004 (2014)
Andersson, O., Heydari, H.: Geometric uncertainty relation for mixed quantum states. J. Math. Phys. 55(4), 042110 (2014)
Andersson, O., Heydari, H.: Quantum speed limits and optimal hamiltonians for driven systems in mixed states. J. Phys. A 47(21), 215301 (2014)
Heydari, H.: A geometric framework for mixed quantum states based on a Kahler structure. arXiv:1409.6891
Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)