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The Geometry of Density States, Positive Maps and Tomograms

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Symmetries in Science XI

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Abstract

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group paramete rs and semigroup of positive map parameters. The function is constructed as sum of moduli of U(N)- tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.

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Author information

Authors and Affiliations

  1. P.N. Lebedev Physical Institute, Leninskii Prospect, 53, Moscow, 119991, Russia

    V. I. Man’ko

  2. Dipartimento di Scienze Fisiche, Università “Federico II” di Napoli, Italy

    G. Marmo & F. Zaccaria

  3. Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Via Cintia, 1-80126, Napoli, Italy

    G. Marmo & F. Zaccaria

  4. Physics Department, Center for Particle Physics, University of Texas, 78712, Austin, Texas, USA

    E. C. G. Sudarshan

Authors
  1. V. I. Man’ko
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  2. G. Marmo
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  3. E. C. G. Sudarshan
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  4. F. Zaccaria
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Editor information

Editors and Affiliations

  1. College of Science, Southern Illinois University at Carbondale, Carbondale, Illinois, USA

    Bruno J. Gruber

  2. Instituto Nazionale di Fisica Nucleare, Università di Napoli “Federico II”, Napoli, Italy

    Giuseppe Marmo

  3. Saitama University, Saitama, Japan

    Naotaka Yoshinaga

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© 2004 Kluwer Academic Publishers

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Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F. (2004). The Geometry of Density States, Positive Maps and Tomograms. In: Gruber, B.J., Marmo, G., Yoshinaga, N. (eds) Symmetries in Science XI. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2634-X_19

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