Parisi, Vincenzo (2024) Towards a p-adic model of quantum information theory. [Tesi di dottorato]

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p-Adic quantum mechanics is a branch of modern theoretical physics which arose, at the end of the last century, from Volovich and Vladimirov's conjecture that the existence of a smallest measurable length --- i.e., the so-called Planck length --- entails a non-Archimedean character for the space-time at small distances. Since, up to isomorphisms, the only non-Archimedean field one can construct by completing the field of rational numbers is the field of p-adic numbers Q_p, it is natural to take this field as the right candidate to model space-time coordinates at the Planck regime. There are actually two possible routes one can follow when trying to construct p-adic models of quantum mechanics. The first approach describes physical states of a p-adic quantum system by means of p-adic wave functions, i.e., square integrable functions from Q_p to the field of complex numbers C. The second approach, instead, follows a more ‘radical’ p-adic point of view and postulates a non-Archimedean structure also for the carrier vector space (i.e., the Hilbert space) of physical states. In this dissertation, we aim at extending p-adic quantum mechanics towards quantum information theory, following both the two approaches to the p-adic quantization. In particular, pursuing the first route, we observe that a suitable model of a p-adic qubit is provided by considering two-dimensional irreducible projective representations of the group SO(3,Q_p) --- the special orthogonal group on Q_p. Since this group is compact, according to the celebrated Peter-Weyl theorem, the determination of the Haar measure (or, the Haar integral, viewing such a measure as a functional) can be regarded as a preliminary step for studying its irreducible representations. Hence, as a starting point, we provide a general method for constructing the Haar measure on every p-adic Lie group; in particular, we also argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for standard real Lie groups. We then apply our general results to the special orthogonal groups over Q_p (in dimension 2,3 and 4), hence paving the way to the development of harmonic analysis on these groups, with potential applications in both p-adic quantum mechanics and p-adic quantum information theory. Following the second approach to the p-adic quantization, instead, we first introduce a suitable notion of a p-adic Hilbert space over a quadratic extension of Q_p. Next, we characterize some classes of linear operators acting in it. Resorting to the algebraic definition of quantum states, we define a state for a p-adic quantum system as a suitable functional acting on the (ultrametric) Banach *-algebra of p-adic (bounded) observables. Eventually, to complete the statistical interpretation of the theory, we introduce the notion of a SOVM as a suitable p-adic counterpart to a POVM of the complex quantum theory.

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