Univerza na Primorskem Fakulteta za matematiko, naravoslovje in informacijske tehnologije
Več informacij o projektu / More info about the project
Vsebina projekta / Project content
(SI) Ohranjevalci spektra na neomejenih operatorjih. V matematični formulaciji kvantno-mehanske opazljivke (tj. stvari, ki jih lahko opazujemo in merimo) modeliramo z (neomejenimi) sebi-adjungiranimi operatorji, ki delujejo na kompleksnem separabilnem Hilbertovem prostoru. V okviru projekta tako nameravamo klasificirati vse aditivne transformacije, ki (neomejene) sebi-adjungirane operatorje slikajo same vase in ohranjajo njihov spekter. Izometrije pri kompleksificiranih normah. Vsak realen vektorski prostor lahko kompleksificiramo, tj. naredimo za kompleksen vektorski prostor; procedura je enolična. Problemi ohranjevalcev v kvantno-informacijskih znanostih. Tenzorski produkti igrajo ključno vlogo v kvantni mehaniki in teoriji kvantne informacije: V matematični terminologiji kvantno stanje opiše pozitivno-semidefinitna matrika s sledjo ena; tenzorski produkt dveh (ali več) stanj pa opisuje skupni bipartitni (ali multipartitni) sistem. V kvantni mehaniki imamo pogosto dostop do majhnega nabora stanj (kot so npr. tenzorska stanja). Preučevali bomo tudi preslikave, ki ohranjajo določene lastnosti, npr. spekter, norma, meritve (numerični zaklad) na manjših podmnožicah, kot so vsa tenzorska stanja ali algebre efektov, in določili strukturo njihovih ohranjevalcev.
(EN) Spectrum preservers on unbounded operators. In a mathematical formulation, quantum-mechanical observables (i.e., things that can be observed and measured) are modelled with (unbounded) self-adjoint operators acting on a complex separable Hilbert space. As part of the project, we intend to classify all additive transformations that map (unbounded) self-adjoint operators into themselves and preserve their spectrum. Isometries of complexified norms. Any real vector space can be complexified, ie. made into a complex vector space by extending the scalars; the procedure is unique. In contrast to this, for a given norm on a real space, we can define several different complex norms on the complexified space, all of which extend the initial real norm. Preserver problems in quantum information sciences. Tensor products play a key role in quantum mechanics and quantum information theory: In mathematical terminology, a quantum state is described by a positive-semidefinite matrix of trace-one; the tensor product of two (or more) states describes a joint bipartite (or multipartite) system.
(EN) Spectrum preservers on unbounded operators. In a mathematical formulation, quantum-mechanical observables (i.e., things that can be observed and measured) are modelled with (unbounded) self-adjoint operators acting on a complex separable Hilbert space. As part of the project, we intend to classify all additive transformations that map (unbounded) self-adjoint operators into themselves and preserve their spectrum. Isometries of complexified norms. Any real vector space can be complexified, ie. made into a complex vector space by extending the scalars; the procedure is unique. In contrast to this, for a given norm on a real space, we can define several different complex norms on the complexified space, all of which extend the initial real norm. Preserver problems in quantum information sciences. Tensor products play a key role in quantum mechanics and quantum information theory: In mathematical terminology, a quantum state is described by a positive-semidefinite matrix of trace-one; the tensor product of two (or more) states describes a joint bipartite (or multipartite) system.