The geometry of density states, positive maps and tomograms

Details The geometry of density states, positive maps and tomograms The geometry of density states, positive maps and tomograms

2007-05-24

arxiv.org/abs/0705.3574v1

Physics

Quantum Physics

52 pages, in: Symmetries in Science XI, Bruno J. Gruber, Giuseppe Marmo Naotaka Yoshinag (Eds), Kluwer Academic Publishers, Do

Scientific paper

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 parameters 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.

Affiliated with

Also associated with

No associations

Rating

The geometry of density states, positive maps and tomograms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The geometry of density states, positive maps and tomograms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The geometry of density states, positive maps and tomograms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-606281