• central spin model
• finite size scaling
• quantum phase transitions

In the context of quantum computation and information, the quantum aspects of correlations in composite systems are a key issue. The interaction between the minimal information unit (qubit) with an environment results in entanglement. It destroys the coherence of the initial state, so the gain of information one wished to obtain, in comparison to that stored by a classical bit, quickly disappears: this is what decoherence refers, i.e. the process through which superpositions of quantum states are irreversibly transformed into statistical mixtures.
This loss of coherence may considerably reduce the efficiency of quantum information protocols.

A platform to investigate the underlying mechanism of the decoherence is the study of quantum phase transition (QPT). As a quantum critical phenomenon, QPT happens at zero temperature, at which the thermal fluctuations vanish, thus they are driven only by quantum fluctuation. It is common that the ground state of the critical system is very sensitive to the varying magnitude of the coupling constant, or that the system experiences a spontaneous symmetry breaking at the critical point. Actually, this kind of critical sensitiveness can be well understood by resorting to quantum information concepts.

During the past decade, the central spin model served as a paradigmatic model to investigate the environment-induced decoherence of a qubit in the vicinity of a quantum critical point (QCP) of the surrounding environment. This stimulates a series of works through which it has been shown that at the critical point, where the environment occurs QPT, the decoherence is enhanced, and the disentanglement process is accelerated by the quantum criticality.
In this thesis, we consider a central qubit homogeneously coupled to an environmental Ising chain with periodic boundary conditions, involved in first and continuous quantum phase transitions. Unlike previous works did, we focused our attention on exploring ground-state properties.
Three different types of interactions have been analyzed, in order to highlight differences and similarities. We have thus paid attention to quantifying and explaining the decoherence arising from different configurations of parameters describing our bipartite system.
In particular conditions of integrability, it has been possible to perform an analytical study of the ground state energy, but extensive numerical simulations in support of the theory have been necessary later. Specifically, we exploited the exact diagonalization technique in order to study the ground state features of the analyzed models.
We have fully exploited the power of methods from the Renormalization Group Theory, extending their application to strategic quantum information concepts, formulating appropriate ansatz to be verified or implementing tricks in order to address the nature of the phase transitions we have dealt with, such as the use of universal curves to characterize a universality class, or in parallel the checking of the expected scaling for the ground state fidelity to verify the order of the transition.

A challenging and intriguing question to be answered was: what do we learn if we became blind in respect of the environment?
We have shown that a particular direction for the qubit magnetization results in a further order parameter for the transition that our system undergoes, assuming a non-zero value in the ordered phase.
Furthermore, a FSS analysis is reported also for such magnetizations.
It is not difficult to understand why such an argument can be very convenient. The Hilbert space dimension of the qubit is $2$, while the environmental Ising chain is described by a $2^L$- dimensional Hilbert space, thus the computational effort in analyzing the qubit subsystem only is surely reduced. In addition, it would be a great advantage to look at the qubit alone to have evidence of an occurring criticality for the global system, because this allows us to move the focus on studying this promising quantum object, which is expected to lead to surprising, unexplored and "magical" new physics in different research areas, above all quantum computing.