Orthogonality relations in Quantum Tomography

Quantum estimation of the operators of a system is investigated by analyzing its Liouville space of operators. In this way it is possible to easily derive some general characterization for the sets of observables (i.e. the possible quorums) that are measured for the quantum estimation. In particular we analyze the reconstruction of operators of spin systems.Comment: 10 pages, 2 figure

Added noise in homodyne measurement of field-observables

Homodyne tomography provides a way for measuring generic field-operators. Here we analyze the determination of the most relevant quantities: intensity, field, amplitude and phase. We show that tomographic measurements are affected by additional noise in comparison with the direct detection of each observable by itself. The case of of coherent states has been analyzed in details and earlier estimations of tomographic precision are critically discussed.Comment: Two figures. Submitted to Phys. Lett.

Adaptive Bayesian and frequentist data processing for quantum tomography

The outcome statistics of an informationally complete quantum measurement for a system in a given state can be used to evaluate the ensemble expectation of any linear operator in the same state, by averaging a function of the outcomes that depends on the specific operator. Here we introduce two novel data-processing strategies, non-linear in the frequencies, which lead to faster convergence to theoretical expectations.Comment: 12 pages, 2 figures, revised versio

On the realization of Bell observables

We show how Bell observables on a bipartite quantum system can be obtained by local observables via a controlled-unitary transformation. For continuous variables this result holds for the Bell observable corresponding to the non-conventional heterodyne measurement on two radiation modes, which is connected through a 50-50 beam-splitter to two local observables given by single-mode homodyne measurements. A simple scheme for a controlled-unitary transformation of continuous variables is also presented, which needs only two squeezers, a parametric downconverter and two beam splitters.Comment: 9 pages, elsart, 1 figur

How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.Comment: Work presented at the conference "On the Present Status of Quantum Mechanics" held on 7-9 September 2005, Mali Losinj, Croatia, in celebration of the 70th birthday of Gian Carlo Ghirardi. The dimensionality theorems along with all results already published in quant-ph/0506034 have been presented at the conference, whereas the operational definition of the real adjoint and the Hilbert spaces derivations have been presented as work in progress, and were completed two months late

On the "principle of the quantumness", the quantumness of Relativity, and the computational grand-unification

After reviewing recently suggested operational "principles of the quantumness", I address the problem on whether Quantum Theory (QT) and Special Relativity (SR) are unrelated theories, or instead, if the one implies the other. I show how SR can be indeed derived from causality of QT, within the computational paradigm "the universe is a huge quantum computer", reformulating QFT as a Quantum-Computational Field Theory (QCFT). In QCFT SR emerges from the fabric of the computational network, which also naturally embeds gauge invariance. In this scheme even the quantization rule and the Planck constant can in principle be derived as emergent from the underlying causal tapestry of space-time. In this way QT remains the only theory operating the huge computer of the universe. Is QCFT only a speculative tautology (theory as simulation of reality), or does it have a scientific value? The answer will come from Occam's razor, depending on the mathematical simplicity of QCFT. Here I will just start scratching the surface of QCFT, analyzing simple field theories, including Dirac's. The number of problems and unmotivated recipes that plague QFT strongly motivates us to undertake the QCFT project, since QCFT makes all such problems manifest, and forces a re-foundation of QFT.Comment: To be published on AIP Proceedings of Vaxjo conference. The ideas on Quantum-Circuit Field Theory are more recent. V4 Largely improved, with new interesting results and concepts. Dirac equation solve

Operational Axioms for Quantum Mechanics

The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.Comment: New improvements have been made. Work presented at the conference "Foundations of Probability and Physics-4, Quantum Theory: Reconsideration of Foundations-3" held on 4-9 June at the International Centre for Mathematical Modelling in Physics, Engineering and Cognitive Sciences, Vaxjo University, Sweden. Also contains an errata to "How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms", quant-ph/060301