We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. Topics covered include: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; the distillable entanglement; the relative entropic measures; the squashed entanglement; log-negativity; the robustness monotones; the greatest cross-norm; uniqueness and extremality theorems. Infinite dimensional systems and multi-party settings will be discussed briefly.

We outline the basic questions that are being studied in the theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional quantum systems such as qubits, we will consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for it are applied to a number of problems.

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The currently available techniques for molecular imaging capable of reaching atomic resolution are limited to low temperatures, vacuum conditions, or large amounts of sample. Quantum sensors based on the spin-dependent photoluminescence of nitrogen-vacancy (NV) centers in diamond offer great potential to achieve single-molecule detection with atomic resolution under ambient conditions. Diamond nanoparticles could also be prepared with implanted NV centers, thereby generating unique nanosensors that are able to traffic into living biological systems. Therefore, this technique might provide unprecedented access and insight into the structure and function of individual biomolecules under physiological conditions as well as observation of biological processes down to the quantum level with atomic resolution. The theory of diamond quantum sensors and the current developments from their preparation to sensing techniques have been critically discussed in this Minireview.