Foundations of quantum mechanics and quantum computing Prof. Gian Carlo Ghirardi, Full Professor Prof. Tullio Weber, Full Professor Dr Emiliano Ippoliti, PhD student Dr Davide Salvetti, PhD student Dr Luca Marinatto, Postdoctoral fellow (ICTP) Dr Angelo Bassi, Postdoctoral fellow (ICTP) FOUNDATIONS OF QUANTUM MECHANICS: DYNAMICAL REDUCTION MODELS The group is mainly involved in the study of "Dynamical Reduction Models" [1-4], which represent one of the few mathematically consistent and physically sensible solutions to the measurement problem of Quantum Mechanics [5]. The following topics are part of our current research activity: Stochastic differential equations in Hilbert spaces. We analyze the problem of the existence and uniqueness of solutions of particular types of stochastic differential equations in infinite dimensions (i.e. in Banach and Hilbert spaces), namely those reproducing the evolution of a quantum system subject to spontaneous localizations in space. We also study analytically properties like the asymptotic limit of the solutions for large times. Stochastic differential equations in terms of square-integrable martingales. We are interested in generalizing stochastic differential equations (in Hilbert spaces) to stochastic processes representing integrable martingale, instead of "simple" Wiener processes. This generalization, from the physical point of view, allows one to take into account spontaneous localizations driven by non-white noises, and to analyze how the reduction mechanism depends of the statistics of the noise. Stochastic differential equations for relativistic quantum field theories. One of the major problems in the development of a relativistic theory of dynamical reductions is the appearance (when a quantum field is coupled to a white-noise process) of additional infinite divergences which cannot be renormalized in the usual way. We are working for finding a consistent relativistic model, free of intractable divergences: a possibility which will be explored is to replace the white-noises so far used with more general martingales. Numerical simulation of stochastic differential equations. The stochastic differential equations describing dynamical reduction models are non-linear, and as a consequence in many interesting physical situations they cannot be solved analytically. We have recently started a research project aiming at solving numerically such equations: this kind of approach allows to gain information about the behavior of the solutions for finite times, and to give estimates of parameters like the reduction time and the localization length. FOUNDATIONS OF QUANTUM MECHANICS: ENTANGLEMENT The group deals also with a mathematical analysis of the properties of the entanglement, which represents undoubtly one of the most striking features of the quantum formalism and an essential resource both in quantum computation and in quantum information theory. In fact, the possibility of devising clever quantum algorithms for solving efficiently computational problems which are regarded as intractable by classical means, the possibility of performing reliable teleportation processes and to create truly random and provably secure private keys in quantum cryptography, is essentially based on the properties of entangled states. A deep mathematical study of entanglement therefore appears as a necessary prerequisite for understanding and exploiting practically its useful features. The analysis of the group [6,9] mainly focussed on the entanglement for composite quantum systems involving identical particles. It has been shown how the non-factorized mathematical form of the state vectors describing identical constituents does not necessarily imply the presence of entanglement. The cases of boson and fermion systems have been thoroughly analyzed. FOUNDATIONS OF QUANTUM MECHANICS: UNCERTAINTY RELATIONS The uncertainty principle of quantum mechanics is usually expressed as an inequality relating the product of the variances of two given observables with the mean value of their commutator. Such an inequality turns out to be a not too significant relation when at least one of the two observables is a bounded operator. For this reason entropic uncertainty relations has been suggested as optimal measures of the indeterminacy connected with the measurement process of a pair of observables. The group has discussed [13] the simple case of a two-dimensional Hilbert space where an optimal lower bound on the entropic uncertainty has been obtained. The current activity is concerned with the generalization of the above-mentioned results to higher dimensional Hilbert spaces and with the analysis of the properties of a new class of uncertainty relations. QUANTUM COMPUTATION AND INFORMATION In recent years, the research group has become involved in the study of quantum computation: Geometrical quantum computation. In collaboration with the University of Genova, we are studying the effects of noises and decoherence on holonomic quantum gates, implemented by means of semiconductor devices. The most relevant sources of noise are the fluctuations of the lasers (determining the form of the gate) intensity and the vibration of quantum dots (which carry information). Query complexity theory is mainly concerned with the computational cost required to determine some specific property of functions. The cost is measured by the number of queries which can be addressed to a "black box" device (the oracle) that outputs instantaneously an answer to a query. Various results concerning query complexity of Boolean functions, i.e. functions with a finite-size domain which have two possible output values only, exist in the literature (Deutsch-Josza algorithms for determining whether a function is constant or balanced, or Simon's algorithm for period finding). The interest of the group is focussed on the harder problem of determining whether a Boolean function is constant or not [12], and an efficiency comparison between a quantum and a classical querying procedure has been performed. Our results show that quantum querying is almost always better than the classical one for this specific problem. MOST RELEVANT PUBLICATIONS: [1] G.C. Ghirardi, A. Rimini and T. Weber: "Unified dynamics for microscopic and macroscopic systems", Physical Review D 34, p. 470 (1986). [2] G.C. Ghirardi, P. Pearle and A. Rimini: "Markov processes in Hilbert spaces and continuous spontaneous localization of systems of identical particles", Physical Review A 42, p. 78 (1990). [3] G.C. Ghirardi, R. Grassi and F. Benatti: "Describing the macroscopic world: closing the circle within the dynamical reduction program", Foundations of Physics 25, p. 5 (1995) [4] A. Bassi and G.C. Ghirardi: "Dynamical Reduction Models", Physics Reports 379, P. 257 (2003). [5] A. Bassi and G.C. Ghirardi: "A general argument against the universal validity of the superposition principle", Physics Letters A 275, p. 373 (2000). [6] G.C. Ghirardi, L. Marinatto, T. Weber: "Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis", Journ. Stat. Phys. 108, 49 (2002). PUBLICATIONS: YEARS 2002/2003 [7] A. Bassi and G.C. Ghirardi: "Dynamical Reduction Models with General Gaussian Noises", Phys. Rev. A 65, p. 042114 (2002). [8] A. Bassi and G.C. Ghirardi: "A general scheme for ensemble purification", Phys. Lett. A 309, p. 24 (2003). [9] G.C. Ghirardi and L. Marinatto: "Entanglement and properties", Fortsch. Physik 51, 379 (2003). [10] L. Marinatto: "Comment on "Bell's Theorem without Inequalities and without Probabilities for Two Observers", Phys. Rev. Lett. 90, 258901 (2003). [11] A. Bassi: "Stochastic Schroedinger Equations with General Complex Gaussian Noises", Phys. Rev. A 67, p. 062101 (2003). [12] F. Benatti and L. Marinatto: "On deciding whether a Boolean function is constant or not", preprint quant-ph/0304073. To appear in: Int. J. Quant. Inf. [13] G.C. Ghirardi, L. Marinatto and R. Romano: "An optimal entropic uncertainity in a two-dimensional Hilbert space", to appear in Phys. Lett.A.

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