Bipartite and Multipartite Entanglement of Gaussian States

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.Comment: 21 pages, 4 figures, WS style. Published as Chapter 1 in the book "Quantum Information with Continuous Variables of Atoms and Light" (Imperial College Press, 2007), edited by N. Cerf, G. Leuchs, and E. Polzik. Details of the book available at http://www.icpress.co.uk/physics/p489.html . For recent follow-ups see quant-ph/070122

A Stochastic Approach to Thermal Fluctuations during a First Order Electroweak Phase Transition

We investigate the role played by subcritical bubbles at the onset of the electroweak phase transition. Treating the configuration modelling the thermal fluctuations around the homogeneous zero configuration of the Higgs field as a stochastic variable, we describe its dynamics by a phenomenological Langevin equation. This approach allows to properly take into account both the effects of the thermal bath on the system: a systematic dyssipative force, which tends to erase out any initial subcritical configuration, and a random stochastic force responsible for the fluctuations. We show that the contribution to the variance \lgh\phi^2(t)\rg_V in a given volume $V$ from any initial subcritical configuration is quickly damped away and that, in the limit of long times, \lgh\phi^2(t)\rg_V approaches its equilibrium value provided by the stochastic force and independent from the viscosity coefficient, as predicted by the fluctuation-dissipation theorem. In agreement with some recent claims, we conclude that thermal fluctuations do not affect the nucleation of critical bubbles at the onset of the electroweak phase transition making electroweak baryogenesis scenarios still a viable possibility to explain the primordial baryon asymmetry in the Universe.Comment: Two figures: fig1.metafile and fig2.metafile. Just print them as usual file.p

Measurement of damping and temperature: Precision bounds in Gaussian dissipative channels

We present a comprehensive analysis of the performance of different classes of Gaussian states in the estimation of Gaussian phase-insensitive dissipative channels. In particular, we investigate the optimal estimation of the damping constant and reservoir temperature. We show that, for two-mode squeezed vacuum probe states, the quantum-limited accuracy of both parameters can be achieved simultaneously. Moreover, we show that for both parameters two-mode squeezed vacuum states are more efficient than either coherent, thermal or single-mode squeezed states. This suggests that at high energy regimes two-mode squeezed vacuum states are optimal within the Gaussian setup. This optimality result indicates a stronger form of compatibility for the estimation of the two parameters. Indeed, not only the minimum variance can be achieved at fixed probe states, but also the optimal state is common to both parameters. Additionally, we explore numerically the performance of non-Gaussian states for particular parameter values to find that maximally entangled states within D-dimensional cutoff subspaces perform better than any randomly sampled states with similar energy. However, we also find that states with very similar performance and energy exist with much less entanglement than the maximally entangled ones.Comment: 14 pages, 6 figure

High-Temperature Atomic Superfluidity in Lattice Boson-Fermion Mixtures

We consider atomic Bose-Fermi mixtures in optical lattices and study the superfluidity of fermionic atoms due to s-wave pairing induced by boson-fermion interactions. We prove that the induced fermion-fermion coupling is always {\it attractive} if the boson-boson on site interaction is repulsive, and predict the existence of an enhanced BEC--BCS crossover as the strength of the lattice potential is varied. We show that for direct on-site fermion-fermion {\it repulsion}, the induced attraction can give rise to superfluidity via s-wave pairing, at striking variance with the case of pure systems of fermionic atoms with direct repulsive interactions.Comment: 4 pages, 2 figures, final versio

Stochastic Variational Approach to Minimum Uncertainty States

We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schr\"{o}dinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials.Comment: 11 pages, latex, 12pt A4wide, no figure

Frustration, Entanglement, and Correlations in Quantum Many Body Systems

We derive an exact lower bound to a universal measure of frustration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide.Comment: 9 pages, 1 figur

Discord of response

The presence of quantum correlations in a quantum state is related to the state response to local unitary perturbations. Such response is quantified by the distance between the unperturbed and perturbed states, minimized with respect to suitably identified sets of local unitary operations. In order to be a bona fide measure of quantum correlations, the distance function must be chosen among those that are contractive under completely positive and trace preserving maps. The most relevant instances of such physically well behaved metrics include the trace, the Bures, and the Hellinger distance. To each of these metrics one can associate the corresponding discord of response, namely the trace, or Hellinger, or Bures minimum distance from the set of unitarily perturbed states. All these three discords of response satisfy the basic axioms for a proper measure of quantum correlations. In the present work we focus in particular on the Bures distance, which enjoys the unique property of being both Riemannian and contractive under completely positive and trace preserving maps, and admits important operational interpretations in terms of state distinguishability. We compute analytically the Bures discord of response for two-qubit states with maximally mixed marginals and we compare it with the corresponding Bures geometric discord, namely the geometric measure of quantum correlations defined as the Bures distance from the set of classically correlated quantum states. Finally, we investigate and identify the maximally quantum correlated two-qubit states according to the Bures discord of response. These states exhibit a remarkable nonlinear dependence on the global state purity.Comment: 10 pages, 2 figures. Improved and expanded version, to be published in J. Phys. A: Math. Ge

Theory of ground state factorization in quantum cooperative systems

We introduce a general analytic approach to the study of factorization points and factorized ground states in quantum cooperative systems. The method allows to determine rigorously existence, location, and exact form of separable ground states in a large variety of, generally non-exactly solvable, spin models belonging to different universality classes. The theory applies to translationally invariant systems, irrespective of spatial dimensionality, and for spin-spin interactions of arbitrary range.Comment: 4 pages, 1 figur

Extended Bose Hubbard model of interacting bosonic atoms in optical lattices: from superfluidity to density waves

For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the {\it lattice attenuation factor}. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on site occupation numbers. In mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is in principle possible, but completely suppressed at lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.Comment: 18 pages, 7 figures, accepted for publication in Phys. Rev.

Surface Entanglement in Quantum Spin Networks

We study the ground-state entanglement in systems of spins forming the boundary of a quantum spin network in arbitrary geometries and dimensionality. We show that as long as they are weakly coupled to the bulk of the network, the surface spins are strongly entangled, even when distant and non directly interacting, thereby generalizing the phenomenon of long-distance entanglement occurring in quantum spin chains. Depending on the structure of the couplings between surface and bulk spins, we discuss in detail how the patterns of surface entanglement can range from multi-pair bipartite to fully multipartite. In the context of quantum information and communication, these results find immediate application to the implementation of quantum routers, that is devices able to distribute quantum correlations on demand among multiple network nodes.Comment: 8 pages, 8 figure