Paolo Giorda

Quantum phase transitions and quantum fidelity in free fermion graphs

In this paper we analyze the ground-state phase diagram of a class of fermionic Hamiltonians by looking at the fidelity of ground states corresponding to slightly different Hamiltonian parameters. The Hamiltonians under investigation can be considered as the variable range generalization of the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the

Two-point versus multipartite entanglement in quantum phase transitions

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Experimental Estimation of Entanglement at the Quantum Limit

spin chain in a transverse magnetic field. Under periodic boundary conditions, the matrices of the problem become circulant and the models are exactly solvable. Their free-ends counterparts are instead analyzed numerically. In particular, we focus on the long-range model corresponding to a fully connected directed graph, providing asymptotic results in the thermodynamic limit, as well as the finite-size scaling analysis of the second-order quantum phase transitions of the system. A strict relation between fidelity and single particle spectrum is demonstrated, and a peculiar gapful transition due to the long-range nature of the coupling is found. A comparison between fidelity and another transition marker borrowed from quantum information, i.e., single site entanglement, is also considered.

Bures metric over thermal state manifolds and quantum criticality

We analyze correlations between subsystems for an extended Hubbard model exactly solvable in one dimension, which exhibits a rich structure of quantum phase transitions (QPTs). The T = 0 phase diagram is exactly reproduced by studying singularities of single-site entanglement. It is shown how comparison of the latter quantity and quantum mutual information allows one to recognize whether two-point or shared quantum correlations are responsible for each of the occurring QPTs. The method works in principle for any number D of degrees of freedom per site. As a by-product, we are providing a benchmark for direct measures of bipartite entanglement; in particular, here we discuss the role of negativity at the transition.

Quantum discord and classical correlations in the bond-charge Hubbard model: Quantum phase transitions, off-diagonal long-range order, and violation of the monogamy property for discord

Entanglement is the central resource of quantum information processing and the precise characterization of entangled states is a crucial issue for the development of quantum technologies. This leads to the necessity of a precise, experimental feasible measure of entanglement. Nevertheless, such measurements are limited both from experimental uncertainties and intrinsic quantum bounds. Here we present an experiment where the amount of entanglement of a family of two-qubit mixed photon states is estimated with the ultimate precision allowed by quantum mechanics.

Ground state fidelity and quantum phase transitions in free Fermi systems

We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows one to complement the understanding of the phase diagram including crossover regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.

Optimal estimation of entanglement

We study the quantum discord (QD) and the classical correlations (CC) in a reference model for strongly correlated electrons, the one-dimensional bond-charge extended Hubbard model. We show that the comparison of QD and CC and of their derivatives in the direct and reciprocal lattice allows one to efficiently inspect the structure of two-points driven quantum phase transitions, discriminating those at which off diagonal long-range order (ODLRO) is involved. Moreover, we observe that QD between pair of sites is a monotonic function of ODLRO, thus establishing a direct relation between the latter and two point quantum correlations that differ from the entanglement. The study of the ground-state properties allows us to show that for a whole class of permutation invariant (η-pair) states quantum discord can violate the monogamy property, both in presence and in absence of bipartite entanglement. In the thermodynamic limit, due to the presence of ODLRO, the violation for η-pair states is maximal, while, for the purely fermionic ground state, it is finite. From a general perspective, all our results validate the importance of the concepts of QD and CC for the study of critical condensed-matter systems.

Quantum information-geometry of dissipative quantum phase transitions.

We compute the fidelity of the ground states of general quadratic fermionic Hamiltonians and analyse its connections with quantum phase transitions. Each of these systems is characterized by an L × L real matrix whose polar decomposition, into a non-negative ΛΦ and a unitary T, contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of ΛΦ. This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exemplification by a model of fermions on a totally connected graph.

Quantum discord for Gaussian states with non-Gaussian measurements

Entanglement does not correspond to an observable, and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address optimal estimation of entanglement in the framework of local quantum estimation theory and derive the optimal observable in terms of the symmetric logarithmic derivative. We evaluate the quantum Fisher information and, in turn, the ultimate bound to precision for several families of bipartite states for either for qubits or continuous-variable systems and for different measures of entanglement. We found that for discrete variables, entanglement may be efficiently estimated when it is large, whereas estimation of weakly entangled states is an inherently inefficient procedure. For continuous-variable Gaussian systems the effectiveness of entanglement estimation strongly depends on the chosen entanglement measure. Our analysis makes an important point of principle and may be relevant in the design of quantum information protocols based on the entanglement content of quantum states.

Entanglement in Extended Hubbard models and Quantum Phase Transitions

A general framework for analyzing the recently discovered phase transitions in the steady state of dissipation-driven open quantum systems is still lacking. To fill this gap, we extend the so-called fidelity approach to quantum phase transitions to open systems whose steady state is a Gaussian fermionic state. We endow the manifold of correlation matrices of steady states with a metric tensor g measuring the distinguishability distance between solutions corresponding to a different set of control parameters. The phase diagram can then be mapped out in terms of the scaling behavior of g and connections with the Liouvillean gap and the model correlation functions unveiled. We argue that the fidelity approach, thanks to its differential-geometric and information-theoretic nature, provides insights into dissipative quantum critical phenomena as well as a general and powerful strategy to explore them.