The coherent superposition of states, in combination with the quantization of observables, represents one of the most fundamental features that mark the departure of quantum mechanics from the classical realm. Quantum coherence in many-body systems embodies the essence of entanglement and is an essential ingredient for a plethora of physical phenomena in quantum optics, quantum information, solid state physics, and nanoscale thermodynamics. In recent years, research on the presence and functional role of quantum coherence in biological systems has also attracted considerable interest. Despite the fundamental importance of quantum coherence, the development of a rigorous theory of quantum coherence as a physical resource has been initiated only recently. This Colloquium discusses and reviews the development of this rapidly growing research field that encompasses the characterization, quantification, manipulation, dynamical evolution, and operational application of quantum coherence.

Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.

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.

We investigate the relationship between mixedness and entanglement for Gaussian states of continuous variable systems. We introduce generalized entropies based on Schatten $p$ norms to quantify the mixedness of a state and derive their explicit expressions in terms of symplectic spectra. We compare the hierarchies of mixedness provided by such measures with the one provided by the purity (defined as $\mathrm{tr}{\varrho }^2$ for the state $\varrho$) for generic $n$-mode states. We then review the analysis proving the existence of both maximally and minimally entangled states at given global and marginal purities, with the entanglement quantified by the logarithmic negativity. Based on these results, we extend such an analysis to generalized entropies, introducing and fully characterizing maximally and minimally entangled states for given global and local generalized entropies. We compare the different roles played by the purity and by the generalized $p$ entropies in quantifying the entanglement and the mixedness of continuous variable systems. We introduce the concept of average logarithmic negativity, showing that it allows a reliable quantitative estimate of continuous variable entanglement by direct measurements of global and marginal generalized $p$ entropies.

Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations (QCs) in composite states. We focus on various approaches to define and quantify general QCs, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.