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arXiv:quant-ph/0602228 · Open Systems & Information Dynamics

Relations Between Quantum Maps and Quantum States

M. Asorey, Andrzej Kossakowski, Giuseppe Marmo, E. C. G. Sudarshan

Published Nov 11, 2005• 33 citations
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Abstract

The relation between completely positive maps and compound states is investigated in terms of the notion of quantum conditional probability.
DOI
10.1007/s11080-005-4482-3

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Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.
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.