Abstract:
This Dissertation collects my results on the interpretation, characterization, quantification and application of bipartite and multipartite entanglement in Gaussian states of continuous variable systems.

Abstract:
We investigate the correlation structure of pure N-mode Gaussian resources which can be experimentally generated by means of squeezers and beam splitters, whose entanglement properties are generic. We show that those states are specified (up to local unitaries) by N(N-1)/2 parameters, corresponding to the two-point correlations between any pair of modes. Our construction yields a practical scheme to engineer such generic-entangled N-mode pure Gaussian states by linear optics. We discuss our findings in the framework of Gaussian matrix product states of harmonic lattices, raising connections with entanglement frustration and the entropic area law.

Abstract:
We demonstrate that the entanglement in a class of two-mode non-Gaussian states obtained by subtracting photons from Gaussian twin beams can be bounded from above and from below by functionals of the second moments only. Knowledge of the covariance matrix thus suffices for an entanglement quantification with appreciable precision. The absolute error in the entanglement estimation scales with the non-Gaussianity of the considered states.

Abstract:
The interferometric power of a bipartite quantum state quantifies the precision, measured by quantum Fisher information, that such a state enables for the estimation of a parameter embedded in a unitary dynamics applied to one subsystem only, in the worst-case scenario where a full knowledge of the generator of the dynamics is not available a priori. For finite-dimensional systems, this quantity was proven to be a faithful measure of quantum correlations beyond entanglement. Here we extend the notion of interferometric power to the technologically relevant setting of optical interferometry with continuous-variable probes. By restricting to Gaussian local dynamics, we obtain a closed formula for the interferometric power of all two-mode Gaussian states. We identify separable and entangled Gaussian states which maximize the interferometric power at fixed mean photon number of the probes, and discuss the associated metrological scaling. At fixed entanglement of the probes, highly thermalized states can guarantee considerably larger precision than pure two-mode squeezed states.

Abstract:
This brief article discusses some aspects of quantum theory and their impact on popular culture. The basic features of quantum entanglement between two or more parties are introduced in a language suitable for a general audience, and metaphorically connected to love and faithfulness in human relationships.

Abstract:
We study a system represented by a Bose-Einstein condensate interacting with a cavity field in presence of a strong off-resonant pumping laser. This system can be described by a three-mode Gaussian state, where two are the atomic modes corresponding to atoms populating upper and lower momentum sidebands and the third mode describes the scattered cavity field light. We show that, as a consequence of the collective atomic recoil instability, these modes possess a genuine tripartite entanglement that increases unboundedly with the evolution time and is larger than the bipartite entanglement in any reduced two-mode bipartition. We further show that the state of the system exhibits genuine tripartite nonlocality, which can be revealed by a robust violation of the Svetlichny inequality when performing displaced parity measurements. Our exact results are obtained by exploiting the powerful machinery of phase-space informational measures for Gaussian states, which we briefly review in the opening sections of the paper.

Abstract:
We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of [Adesso & Illuminati, Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows to quantify genuine N-partite entanglement in terms of the "residual contangle" not encoded into 2,...,K,...,(N-1)-partite quantum correlations. The explicit expression of this entanglement measure is derived, by a recursive formula, for a subclass of Gaussian states. These are fully symmetric (permutation-invariant) states multi-partitioned into blocks, each consisting of an arbitrary number of modes. We compute the genuine multipartite entanglement shared by the blocks ("molecules") and investigate its scaling properties with the number and size of the molecules, the total number of modes, the global mixedness of the state, and the squeezed resources. To achieve the exact computation we prove a general result of symplectic analysis: Correlations among K molecules in N-mode multi-symmetric Gaussian states can be transformed by a local unitary operation into correlations shared by K single modes, one per molecule, in effective non-symmetric states. Due to this theorem, the above results extend to the subclass of non-symmetric Gaussian states that are obtained by unitarily localizing the multipartite molecular entanglement of symmetric states. These findings provide evidence that distributed Gaussian entanglement is strongly monogamous even beyond specific symmetry constraints, and that the residual contangle is a bona fide measure of genuine multipartite entanglement for permutation-invariant Gaussian states under any multi-partition of the modes.

Abstract:
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures, and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization, and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states, and its consequences for the characterization of multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non Gaussian states, a field of research that is to a large extent yet to be explored.

Abstract:
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.

Abstract:
We provide a quantum benchmark for teleportation and storage of single-mode squeezed states with zero displacement and a completely unknown degree of squeezing along a given direction. For pure squeezed input states, a fidelity higher than 81.5% has to be attained in order to outperform any classical strategy based on an estimation of the unknown squeezing and repreparation of squeezed states. For squeezed thermal input states, we derive an upper and a lower bound on the classical average fidelity which tighten for moderate degree of mixedness. These results enable a critical discussion of recent experiments with squeezed light.