Quantumness of correlations in finite dimensional systems - Tiago Debarba (2014)
Some phenomena are exclusive of quantum systems, in other words, there no exist counterpart in classical mechanics. Two examples extensively discussed in recent years are the quantum entanglement and quantum non locality, both are related to the existence of non separable states. The superposition of quantum states is another characteristic which deserves attention. Considering two distinct events in superposition, it implies the nonexistence of a measurement process capable of discriminating them. The superposition and the local measurement process together result in a new class of correlations, without counterpart in classical world, and go beyond quantum entanglement. These quantum correlations are named quantumness of correlations, and they are the main issue of this thesis. In the thesis we study three different approaches for the quantumness of correlations. Firstly, we define a geometrical measure of quantumness of correlations via the Schatten-p norm, which contain in its definition the trace norm, the Hilbert-Schmidt norm and the operator norm. We demonstrate that it is limited below by the quantum entanglement, calculated via entanglement witness. The second approach for the quantumness of correlations is in the context of accessible information and the discrimination of quantum states. It is known that quantum states only can be distinguished if they are orthogonal one each other. Then there exists a maximal amount of information which can be extracted from an ensemble of quantum states, performing measurements. The accessible information is limited by the Holevo¿s quantity, and the bound is attained only for orthogonal states. This limit in the amount of information that can be extracted from a quantum ensemble is related to the incapacity to distinguish quantum states by measurement process. Our study consists to investigate the capacity in to extract information, as well to distinguish the states of a given ensemble, when we are restricted to perform projective measurements. The restriction to projective measurements, as well the generalization to POVMs, can be approached via the Naimark¿s theorem, which state that a given POVM can be approached as a projective measurement in a embedded space. The embedding process can be performed, for example, coupling a pure ancilla on the state. Therefore this process cannot create any correlation between the system and the ancilla. The main goal is approach the quantumness of correlations in this context, to understand how they are affected by the embedding process, once that this process generalizes the measurement to be performed on the system. Finally we study quantum correlations in the context of indistinguishable particles. In our approach we obtained an entanglement measure for fermionic systems, it is a fermionic version of the generalized robustness of entanglement.We also introduced the concept of quantumness of correlations for indistinguishable particles. We calculated who are the states without quantumness of correlations from the activation protocol, in this context. As these states are a subset of separable states, we can attest what states are not entangled, once they are the states without quantumness of correlation. We also calculated a measure of quantumness of correlations for fermionic and bosonic systems.