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

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Convex Analysisの二,三の進展について

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.

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Entanglement in many-body systems

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

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Gaussian quantum information

Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn [2001, Nature (London) 409, 46] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. The original theory and its improvements are reviewed, and a few examples of experimental two-qubit gates are given. The use of realistic components, the errors they induce in the computation, and how these errors can be corrected is discussed.

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Linear optical quantum computing with photonic qubits

We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators We consider the ground state and thermal states of this system, which are Gaussian states The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement

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Entanglement properties of the harmonic chain

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

Evenly distributed unitaries: On the structure of unitary designs

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

https://iopscience.iop.org/article/10.1088/1751-8113/40/28/S18/pdf

A sharp continuity estimate for the von Neumann entropy

The authors present a high-resolution sensor composed of three ultrasonic sensors, one transmitter/receiver and two extra receivers, which allows a significant improvement in the information-extraction process. With this sensor the position (distance and bearing) of all isolated objects in an approximately 25 degrees field of view can be determined using information contained in one single snapshot of a moderately complex scene. Within limits, the sensor system can also discriminate between different types of reflectors, in particular, walls and edges, based on their radius of curvature. These results are all based on the determination of the arrival times of the echoes present at the three receivers. A noise model that accounts for the measured variations of the arrival times is used to derive limits on the resolution of the results provided by the sensor. Based on this model it is shown that, to a large extent, the sensor results are impervious to measurement variations common to all three receivers. Results obtained in a realistic environment are compared with those obtained from a conventional time-of-flight sensor. >

A high-resolution sensor based on tri-aural perception

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.

https://arxiv.org/pdf/quant-ph/0207146.pdf

Koenraad M.R. Audenaert

Papers

Entanglement properties of the harmonic chain

Evenly distributed unitaries: On the structure of unitary designs

A sharp continuity estimate for the von Neumann entropy

A high-resolution sensor based on tri-aural perception

Entanglement cost under positive-partial-transpose-preserving operations.