On the meaning and interpretation of Tomography in abstract Hilbert spaces

The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity decompositions in the Hilbert space of hermitian linear operators, with trace formula as scalar product of operators. Decompositions of identity are considered with respect to over-complete families of projectors labeled by extra parameters and containing a measure, depending on these parameters. It plays the role of a Gram-Schmidt orthonormalization kernel. When the measure is equal to one, the decomposition of identity coincides with a positive operator valued measure (POVM) decomposition. Examples of spin tomography, photon number tomography and symplectic tomography are reconsidered in this new framework.Comment: Submitted to Phys. Lett.

Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics

The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.Comment: 8 page

Bell's inequalities in the tomographic representation

The tomographic approach to quantum mechanics is revisited as a direct tool to investigate violation of Bell-like inequalities. Since quantum tomograms are well defined probability distributions, the tomographic approach is emphasized to be the most natural one to compare the predictions of classical and quantum theory. Examples of inequalities for two qubits an two qutrits are considered in the tomographic probability representation of spin states.Comment: 11 pages, comments and references adde

Introduction to Quantum Mechanics and the Quantum-Classical transition

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.Comment: Survey paper based on the lectures delivered at the XV International Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11-16, 2006. To appear in Publ. de la RSM

Numerical Bayesian state assignment for a three-level quantum system. I. Absolute-frequency data; constant and Gaussian-like priors

This paper offers examples of concrete numerical applications of Bayesian quantum-state-assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in absolute frequencies of the outcomes of N identical von Neumann projective measurements performed on N identically prepared three-level systems. Various small values of N as well as the large-N limit are considered. Two kinds of prior knowledge are used: one represented by a plausibility distribution constant in respect of the convex structure of the set of statistical operators; the other represented by a Gaussian-like distribution centred on a pure statistical operator, and thus reflecting a situation in which one has useful prior knowledge about the likely preparation of the system. In a companion paper the case of measurement data consisting in average values, and an additional prior studied by Slater, are considered.Comment: 23 pages, 14 figures. V2: Added an important note concerning cylindrical algebraic decomposition and thanks to P B Slater, corrected some typos, added reference

Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.Comment: AmsLaTeX, 37 pages, 8 figures. This paper is an expanded version of some lectures delivered by one of us (G. M.) at the ``Advanced Winter School on the Mathematical Foundation of Quantum Control and Quantum Information'' which took place at Castro Urdiales (Spain), February 11-15, 200

Quantum Tomography and the Quantum Radon Transform

A general framework in the setting of $C^*$-algebras for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, is presented. Given a $C^*$-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on $C^*$-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform. The abstract theory is realized by using dynamical systems, that is, groups represented on $C^*$-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory