[Todos] ATENCIÓN/CAMBIOS en SEMINARIO MARTES 21 DE JUNIO - IFLP/DEPARTAMENTO DE FÍSICA

[tierno]

ESTIMADOS:  

ADJUNTAMOS NUEVO MATERIAL EN LOS SEMINARIOS DE IFLP/DEPARTAMENTO DE
FÍSICA, POR  CAMBIO DE HORARIO Y DE CONTENIDO  

DISCULPEN LAS MOLESTIAS,  

ATTE.  

ADMINISTRACIÓN IFLP  


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SEMINARIOS  DEL IFLP Y DEL DEPARTAMENTO 

MARTES 21 DE JUNIO A LAS 10HS 

AULA CHICA 

1)  QUANTUM PATTERN RECOGNITION 

 EXPONE: GIUSEPPE SERGIOLI (UNIVERSITY OF CAGLIARI) 

We introduce a new framework for describing pattern recognition tasks by
means of the mathematical language of density matrices. 

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. 

The natural motivations which have led to use quantum states for the
purpose of representing patterns are 

i) the possibility to exploit quantum algorithms to boost the
computational intensive parts of the classification process, 

ii) the possibility of using quantum-inspired algorithms for solving
classical problems more effectively. 

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). 

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. 

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. 

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. 

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. 

The behaviors of QC and NMC on different datasets will be shown and
compared. 

2)  FROM SHARP TO UNSHARP QUANTUM LOGIC: A NEW LOOK AT THE EFFECTS OF A
HILBERT SPACE 

 EXPONE: ROBERTO GIUNTINI 

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). 

The set E(H) can be naturally structured ([1],[2]) as a Brouwer-Zadeh
poset 

(BZ-poset) hE(H) ; ; 0 ; ;O; Ii, where 

(i) E F i for any density operator 2 S(H) : Tr(E) Tr(F); 

(ii) E0 = I 

(iii) E_ = PKer(E), where PKer(E) is the projection associated to the
kernel of E; 

(iv) O and I are the null and the identity projections, respectively. 

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]). 

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. 

REFERENCES 

[1] G. Cattaneo and G. Nistic_o, Brouwer-Zadeh posets and three-valued
Lukasiewicz posets", 

Fuzzy Sets and Systems 33, 165{190, 1986. 

[2] M. L. Dalla Chiara, R. Giuntini and R. Greechie, Reasoning in
Quantum Theory, Kluwer, 

Dordrecht, 2004. 

[3] H. F. de Groote, On a canonical lattice structure on the e_ect
algebra of a von Neumann 

algebra", arXiv:math-ph/0410018v2, 2005. 

[4] M. P. Olson, The selfadjoint operators of a von Neumann algebra form
a conditionally 

complete lattice", Proceedings of the American Mathematical Society 28,
537{544, 1971. 

(Giuntini) Dipartimento di Pedagogia, Psicologia, Filosofia Universit_a
di Cagliari, 

Via Is Mirrionis 1, I-09123 Cagliari, Italy 

E-mail address: giuntini en unica.it 
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