Discriminazione di stati tramite risorse locali nella teoria fermionica

The work investigates the problems of quantum state discrimination by means of local operations and classical communication (LOCC) in the Fermionic theory. The Fermi-Dirac statistics describe half-integer spin particles while their applications range from solid state to high energy physics, bringing fermions to one of the most fundamental notions of nature. However, the analysis of Fermionic theory as of information processing is rather recent and is still open to new investigating results. The contemporary literature thoroughly tackles the Feynman problem of simulating fermions with qubits and the universality of Fermionic computation, providing sharp tools to better understand the theory. Thus, we firstly review the Jordan-Wigner transform and the results concerning the entanglement between fermions, along with the operational definitions of LOCC. In the thesis, Fermionic theory is treated as an instance of operational probabilistic theories (OPTs). An OPT is an operational language that expresses the possible connections between events, dressed with a probabilistic structure. In such a novel framework, we are interested in rigorously expressing the conditions for which two states are discriminable using some measurement protocols. It is known that quantum mechanics satisfies local discriminability, namely any pair of composite states can be probabilistically discriminated using only local measurements on the component system. Moreover, orthogonal pure states can be perfectly (with probability one) discriminated using LOCC, whereas pairs of orthogonal pure states can be optimally discriminated by LOCC, i. e. with the optimal probability of discrimination. Finally, under certain hypothesis any pair of quantum states allows for LOCC unambiguous discrimination, thus allowing for inconclusive outcomes. The validity of these results, which represent key features within quantum information and quantum computation, is still unexplored for the Fermionic theory. In this thesis, we analyze the problem in the Fermionic theory by providing conditions under which states are discriminable via LOCC. Then we look into the prerequisites for Fermionic states in order to study optimal discrimination with local operations and classical communication, focusing on their operational interpretation. Furthermore, we show the advantages of having some entanglement resources at disposal to the two parts. We conclude by generalizing the results in connection with some key features of theories in the OPT framework: locality, discriminability and classical information. The long-term aim of this work is indeed to study the properties of the Fermionic theory in the context of wider studies still in progress on quantum cellular automata.