Thermodynamical formulation of entanglement theory and quantum simulations of many-body systems

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Entanglement is a key concept both in the foundations of quantum theory and, as a resource, in quantum information science. It is thus a central goal of quantum information theory to developed a quantitative theory of entanglement, considering its several manifestations. It has been established that entanglement transformations, in the limit of an arbitrarily large number of identical copies of the state, shares remarkable similarities with thermodynamics. However, the question whether there is a setting for entanglement manipulation which is formally equivalent to thermodynamics has remained elusive to date. In this respect, my research will address the following points: - Reversible transformations: I will analyse several classes of quantum operations for which a reversible conversion of entangled resources and hence a formally connection with thermodynamics could hold. I will try to (dis)prove reversibility for some of these classes.- Thermodynamics analogies: Following the findings in the first item, I will identify counterparts in entanglement theory of advanced concepts in thermodynamics, such as temperature and heat, and explore the new insights that these would bring to the understanding of quantum correlations. - Implications: Finally, I will investigate the implications of such a connection both to entanglement theory and to thermodynamics. Particularly, I will analyse how the new insights gained from thermodynamics can help in the solution of open problems in entanglement theory, e.g. additivities questions and equivalence of entanglement measures in the asymptotic limit. The second strand of my research will be concerned with the use of well controlled quantum systems as quantum simulators of complex quantum many-body dynamics. I will analyse the rich possibilities offered by arrays of coupled micro-cavities, which have been recently proposed as a promising new type of quantum simulator with the distinguishing advantage of allowing the addressability of single sites. Moreover, from a more fundamental perspective, I will investigate issues concerning the complexity of simulating many-body Hamiltonians both on a quantum computer and by classical resources. The research I plan to conduct will address the following points: - Spin Hamiltonians: I will investigate experimentally feasible ways of engineering anisotropic spin Hamiltonians in arrays of coupled micro-cavities, considering the particulars of promising systems such as Cooper paix boxes coupled to rf cavities, atoms in toroidal microcavities, and quantum dots in photonic crystal micro-cavities. - Topologically protected quantum memories: Based on the previous item, I will analyse the feasibility of using these many-body Hamiltonians for the creation of topologically protected quantum memories. The full local addressability of arrays of coupled cavities will also be explored to the realization of active error correction, initialization and quantum processing of the protected qubit. - Complexity of local Hamiltonians versus spectral gap: I will address the computational complexity of calculating the expectation value of local observables in the ground state of local Hamiltonians in one and more dimensions. It will be analysed in particular the dependence of this complexity with the spectral gap of the system.