All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory} But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding However, it appeared that this new resource is
very complex and difficult to detect Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon A basic role of entanglement
witnesses in detection of entanglement is emphasized

Quantum entanglement

Quantum networks provide opportunities and challenges across a range of intellectual and technical frontiers, including quantum computation, communication and metrology. The realization of quantum networks composed of many nodes and channels requires new scientific capabilities for generating and characterizing quantum coherence and entanglement. Fundamental to this endeavour are quantum interconnects, which convert quantum states from one physical system to those of another in a reversible manner. Such quantum connectivity in networks can be achieved by the optical interactions of single photons and atoms, allowing the distribution of entanglement across the network and the teleportation of quantum states between nodes.

The quantum internet

Coherent preparation by laser light of quantum states of atoms and molecules can lead to quantum interference in the amplitudes of optical transitions. In this way the optical properties of a medium can be dramatically modified, leading to electromagnetically induced transparency and related effects, which have placed gas-phase systems at the center of recent advances in the development of media with radically new optical properties. This article reviews these advances and the new possibilities they offer for nonlinear optics and quantum information science. As a basis for the theory of electromagnetically induced transparency the authors consider the atomic dynamics and the optical response of the medium to a continuous-wave laser. They then discuss pulse propagation and the adiabatic evolution of field-coupled states and show how coherently prepared media can be used to improve frequency conversion in nonlinear optical mixing experiments. The extension of these concepts to very weak optical fields in the few-photon limit is then examined. The review concludes with a discussion of future prospects and potential new applications.

/pdf/electromagnetically-induced-transparency-optics-in-coherent-2dhges92td.pdf

Electromagnetically induced transparency : Optics in coherent media

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

/pdf/entanglement-in-many-body-systems-sqpatuxt14.pdf

Entanglement in many-body systems

The statistical error in any estimation can be reduced by repeating the measurement and averaging the results. The central limit theorem implies that the reduction is proportional to the square root of the number of repetitions. Quantum metrology is the use of quantum techniques such as entanglement to yield higher statistical precision than purely classical approaches. In this Review, we analyse some of the most promising recent developments of this research field and point out some of the new experiments. We then look at one of the major new trends of the field: analyses of the effects of noise and experimental imperfections.

Advances in quantum metrology

Quantum communication holds a promise for absolutely secure transmission of secret messages and faithful transfer of unknown quantum states. Photonic channels appear to be very attractive for physical implementation of quantum communication. However, due to losses and decoherence in the channel, the communication fidelity decreases exponentially with the channel length. We describe a scheme that allows to implement robust quantum communication over long lossy channels. The scheme involves laser manipulation of atomic ensembles, beam splitters, and single-photon detectors with moderate efficiencies, and therefore well fits the status of the current experimental technology. We show that the communication efficiency scale polynomially with the channel length thereby facilitating scalability to very long distances.

Long-distance quantum communication with atomic ensembles and linear optics

We propose a method to produce entangled states of several particles starting from a Bose-Einstein condensate. In the proposal, a single fast $\pi/2$ pulse is applied to the atoms and due to the collisional interaction, the subsequent free time evolution creates an entangled state involving all atoms in the condensate. The created entangled state is a spin-squeezed state which could be used to improve the sensitivity of atomic clocks.

https://arxiv.org/pdf/quant-ph/0006111.pdf

Many-particle entanglement with Bose--Einstein condensates

PEPS as ground states: Degeneracy and topology

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N $(Ml~N),$ and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}g~0.$ For rank $r(\ensuremath{\varrho})l~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})l~2MN\ensuremath{-}M\ensuremath{-}N+2.$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.

https://espace.library.uq.edu.au/view/UQ:250367/UQ250367_OA.pdf

Ignacio Cirac

Papers

Long-distance quantum communication with atomic ensembles and linear optics

Many-particle entanglement with Bose--Einstein condensates

PEPS as ground states: Degeneracy and topology

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Operational criterion and constructive checks for the separability of low-rank density matrices