The quantum internet is predicted to be the next-generation information processing platform, promising secure communication and an exponential speed-up in distributed computation. The distribution of single qubits over large distances via quantum teleportation is a key ingredient for realizing such a global platform. By using quantum teleportation, unknown quantum states can be transferred over arbitrary distances to a party whose location is unknown. Since the first experimental demonstrations of quantum teleportation of independent external qubits, an internal qubit and squeezed states, researchers have progressively extended the communication distance. Usually this occurs without active feed-forward of the classical Bell-state measurement result, which is an essential ingredient in future applications such as communication between quantum computers. The benchmark for a global quantum internet is quantum teleportation of independent qubits over a free-space link whose attenuation corresponds to the path between a satellite and a ground station. Here we report such an experiment, using active feed-forward in real time. The experiment uses two free-space optical links, quantum and classical, over 143 kilometres between the two Canary Islands of La Palma and Tenerife. To achieve this, we combine advanced techniques involving a frequency-uncorrelated polarization-entangled photon pair source, ultra-low-noise single-photon detectors and entanglement-assisted clock synchronization. The average teleported state fidelity is well beyond the classical limit of two-thirds. Furthermore, we confirm the quality of the quantum teleportation procedure without feed-forward by complete quantum process tomography. Our experiment verifies the maturity and applicability of such technologies in real-world scenarios, in particular for future satellite-based quantum teleportation.

We review in detail recent advances in our understanding of the phase structure and the phase transitions of hadronic matter in strong magnetic fields B and zero quark chemical potentials µ f . Many aspects of QCD are described using low-energy effective theories and models such as the MIT bag model, the hadron resonance gas model, chiral perturbation theory, the Nambu-Jona-Lasinio (NJL) model, the quark-meson (QM) model and Polyakov-loop extended versions of the NJL and QM models. We critically examine their properties and applications. This includes mean-field calculations as well as approaches beyond the mean-field approximation such as the functional renormalization group (FRG). Renormalization issues are discussed and the influence of the vacuum fluctuations on the chiral phase transition is pointed out. Magnetic catalysis at T = 0 is covered as well. We discuss recent lattice results for the thermodynamics of nonabelian gauge theories with emphasis on SU (2)c and SU (3)c. In particular, we focus on inverse magnetic catalysis around the transition temperature Tc as a competition between contributions from valence quarks and sea quarks resulting in a decrease of Tc as a function of B. Finally, we discuss recent efforts to modify models in order to reproduce the behavior observed on the lattice. A. B-dependent transition temperature T0 51 B. B-dependent coupling constant 52 XI. Anisotropic pressure and magnetization 55 XII. Conclusions and outlook 57 Acknowledgments 59 A. Notation and conventions 59 B. Sum-integrals 59 C. Small and large-B expansions 61 D. Propagators in a magnetic background 61References 63 1 Another common choice is the symmetric gauge, Aµ = 1 2 (0, By, −Bx, 0).

† These authors contributed equally to this work Quantum simulators are controllable quantum systems that can reproduce the dynamics of the system of interest, which are unfeasible for classical computers. Recent developments in quantum technology enable the precise control of individual quantum particles as required for studying complex quantum systems. Particularly, quantum simulators capable of simulating frustrated Heisenberg spin systems provide platforms for understanding exotic matter such as high-temperature superconductors. Here we report the analog quantum simulation of the ground-state wavefunction to probe arbitrary Heisenberg-type interactions among four spin-1/2 particles . Depending on the interaction strength, frustration within the system emerges such that the ground state evolves from a localized to a resonating valence-bond state. This spin-1/2 tetramer is created using the polarization states of four photons. The single-particle addressability and tunable measurement-induced interactions provide us insights into entanglement dynamics among individual particles. We directly extract ground-state energies and pair-wise quantum correlations to observe the monogamy of entanglement.During the past years, there has been an explosion of interest in quantum-enhanced technologies. The applications are many-fold and reach from quantum metrology[1] to quantum information processing [2]. In particular quantum computation has generated a lot of interest due to the discovery of quantum algorithms [3][4][5] which outperform classical ones. The first proposed application for which quantum computation can give an exponential enhancement over classical computation was suggested by Richard Feynman [6,7]. He considered a universal quantum mechanical simulator, which is a controllable quantum system that can be used to imitate other quantum systems, therefore being able to tackle problems that are intractable on classical computers. Since then the motivation to use a quantum simulator as a powerful tool to address the most important and difficult problems in multidisciplinary science has led to many theoretical proposals [8][9][10][11][12][13]. Vast technological developments allowed for recent realizations of such devices in atoms [14][15][16], trapped ions [17][18][19][20], single photons [21][22][23][24] and NMR [25,26]. The quantum simulation of strongly correlated quantum systems (e.g. frustrated spin systems) is of special interest and would provide new results that cannot be otherwise classically simulated [27].In order to manipulate and measure individual properties of microscopic quantum systems the complete control over all degrees of freedom for each particle is required. Typically, atoms in optical lattices [14] are used for realizing physical systems that can simulate various models in condensed-matter physics. The fact that the experimental addressability of single atoms in optical lattices remains very challenging [28][29][30] leads to the studies of bulk properties of the atomic ensemble (≈ 10 5 at...