<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN">
<html><body style='font-family: Verdana,Geneva,sans-serif'>
<p> </p>
<blockquote type="cite" style="padding-left:5px; border-left:#1010ff 2px solid; margin-left:5px">
<p align="center"><span style="font-size: large;"><strong><span style="text-decoration: underline;">SEMINARIOS DEL IFLP y DEL DEPARTAMENTO</span></strong></span></p>
<p align="center"><span style="font-size: large;"><strong><span style="text-decoration: underline;">MARTES 21</span></strong><strong><span style="text-decoration: underline;"> de JUNIO a las 10hs</span></strong><strong><span style="text-decoration: underline;"></span></strong></span></p>
<p align="center"><span style="font-size: large;"><strong><span style="text-decoration: underline;">AULA CHICA</span></strong></span></p>
<p><br /><span style="font-size: medium;"> <strong></strong></span></p>
<p><span style="text-decoration: underline;"><span style="font-size: medium;"><strong>1) </strong><strong>QUANTUM PATTERN RECOGNITION </strong></span></span></p>
<p><span style="text-decoration: underline;"><span style="font-size: medium;"><strong> </strong></span><strong style="font-size: medium;">EXPONE: GIUSEPPE SERGIOLI (University of Cagliari)</strong></span></p>
<p style="text-align: justify;">We introduce a new framework for describing pattern recognition tasks by means of the mathematical language of density matrices.</p>
<p style="text-align: justify;">In recent years, many efforts to apply the quantum formalism to non-microscopic contexts have been made and, in this direction, important contributions in the areas of pattern recognition and image understanding have been provided. Even if these results seem to suggest some possible computational advantages of an approach of this sort, an extensive and universally recognized treatment of the topic is still missing.</p>
<p style="text-align: justify;">The natural motivations which have led to use quantum states for the purpose of representing patterns are</p>
<p style="text-align: justify;">i) the possibility to exploit quantum algorithms to boost the computational intensive parts of the classification process,</p>
<p style="text-align: justify;">ii) the possibility of using quantum-inspired algorithms for solving classical problems more effectively.</p>
<p style="text-align: justify;">In our work, firstly we provide a one-to-one correspondence between patterns, expressed as n-dimensional feature vectors (according to the standard pattern recognition approach), and pure density operators (i.e. points in the n-dimensional Bloch hypersphere) called "density patterns". By using this correspondence, we give a representation of the well-known Nearest Mean classifier (NMC) in terms of quantum objects by defining an "ad hoc" Normalized Trace Distance (which coincides with the Euclidean distance between patterns in the real space).</p>
<p style="text-align: justify;">Consequently, we have found a quantum version of the discriminant function by means of Pauli components, represented as a plane which intersects the Bloch sphere.</p>
<p style="text-align: justify;">This first result suggests future potential developments, which consist in finding a quantum algorithm able to implement the normalized trace distance between density patterns with a consequent significative reduction of the computational complexity of the process.</p>
<p style="text-align: justify;">But the main result we show consists in introducing a purely quantum classifier (QC), which has not any kind of classical counterpart, through a new definition of "quantum centroid". The convenience of using this quantum centroid lies in the fact that it seems to be more informative than the classical one because it takes into account also information about the distribution of the patterns.</p>
<p style="text-align: justify;">As a consequence, the main implementative result consists in showing how this quantum classifier performs a significative reduction of the error and an improvement of the accuracy and precision of the algorithm with respect to the NMC (and also to other commonly used classifiers) on a classical computer.</p>
<p style="text-align: justify;">The behaviors of QC and NMC on different datasets will be shown and compared.</p>
<p> </p>
<p><span style="text-decoration: underline;"><span style="font-size: medium;"><strong>2) </strong><strong>FROM SHARP TO UNSHARP QUANTUM LOGIC: </strong><strong>A NEW LOOK AT THE EFFECTS OF A HILBERT SPACE</strong></span></span></p>
<p style="text-align: justify;"><span style="text-decoration: underline;"><span style="font-size: medium;"><strong> </strong><strong>EXPONE: ROBERTO GIUNTINI</strong></span></span></p>
<p style="text-align: justify;"><span style="text-align: justify;">The starting point of the unsharp approach to quantum mechanics (QM) ([2]) is deeply connected with a general problem that naturally arises in the frame- work of Hilbert space quantum theory. Let us consider an event-state system ((H) ; S(H)), where (H) is the set of projections, while S(H) is the set of all density operators of the Hilbert space H (associated to the physical system under investigation). Do the sets (H) and S(H) correspond to an optimal possible choice of adequate mathematical representatives for the intuitive notions of event and of state, respectively? Once (H) is xed, Gleason's Theorem guarantees that S(H) corresponds to an optimal notion of state: for, any probability measure de-ned on (H) is determined by a density operator of H (provided the dimension of H is greater than or equal to 3). On the contrary, (H) does not represent the largest set of operators assigned a probability-value since there are bounded linear operators E of H that are not projections and that satisfy the Born's rule: for any density operator ; Tr(E) 2 [0; 1]: In the unsharp approach to QM, the notion of quantum event is liberalized and the set (H) is replaced by the set of all effects of H (denoted by E(H)), where an effect of H is a bounded linear operator E that satises the following condition, for any density operator : Tr(E) 2 [0; 1]: Clearly, E(H) properly includes (H).</span></p>
<p style="text-align: justify;">The set E(H) can be naturally structured ([1],[2]) as a Brouwer-Zadeh poset</p>
<p style="text-align: justify;">(BZ-poset) hE(H) ; ; 0 ; ;O; Ii, where</p>
<p style="text-align: justify;">(i) E F i for any density operator 2 S(H) : Tr(E) Tr(F);</p>
<p style="text-align: justify;">(ii) E0 = I</p>
<p style="text-align: justify;">(iii) E_ = PKer(E), where PKer(E) is the projection associated to the kernel of E;</p>
<p style="text-align: justify;">(iv) O and I are the null and the identity projections, respectively.</p>
<p style="text-align: justify;">The BZ-poset E(H) turns out to be properly fuzzy since the noncontradiction principle is violated (E ^ E0 6= O). Further, the BZ-poset E(H) fails to be a lattice ([2]). In a quite neglected paper, however, Olson ([4]) proved that E(H) can be equipped with a natural partial order _s (called spectral order) in such a way that hE(H);_si turns out to be a complete lattice. In this paper, we will investigate the algebraic properties of the structure hE(H) ; _s ; 0 ; _ ;O; Ii and we will introduce a new class of BZ-lattices (called BZ_ lattices) that represents a quite faithful abstraction of the concrete model based on E(H) (see also [3]).</p>
<p style="text-align: justify;">Interestingly enough, in the framework of BZ_-lattices di_erent abstract notions of \unsharpness" collapse into the one and the same concept, similarly to what happens in the concrete BZ_-lattices of all effects.</p>
<p style="text-align: justify;"> </p>
<p style="text-align: justify;"><strong>References</strong></p>
<p style="text-align: justify;">[1] G. Cattaneo and G. Nistic_o, \Brouwer-Zadeh posets and three-valued Lukasiewicz posets",</p>
<p style="text-align: justify;">Fuzzy Sets and Systems 33, 165{190, 1986.</p>
<p style="text-align: justify;">[2] M. L. Dalla Chiara, R. Giuntini and R. Greechie, Reasoning in Quantum Theory, Kluwer,</p>
<p style="text-align: justify;">Dordrecht, 2004.</p>
<p style="text-align: justify;">[3] H. F. de Groote, \On a canonical lattice structure on the e_ect algebra of a von Neumann</p>
<p style="text-align: justify;">algebra", arXiv:math-ph/0410018v2, 2005.</p>
<p style="text-align: justify;">[4] M. P. Olson, \The selfadjoint operators of a von Neumann algebra form a conditionally</p>
<p style="text-align: justify;">complete lattice", Proceedings of the American Mathematical Society 28, 537{544, 1971.</p>
<p style="text-align: justify;">(Giuntini) Dipartimento di Pedagogia, Psicologia, Filosofia Universit_a di Cagliari,</p>
<p style="text-align: justify;">Via Is Mirrionis 1, I-09123 Cagliari, Italy</p>
<p style="text-align: justify;"> </p>
<p style="text-align: justify;">E-mail address: giuntini@unica.it</p>
</blockquote>
<p> </p>
<div> </div>
</body></html>