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

Sphere Packings, Lattices and Groups

This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

The Theory of Quantum Information

Quantum mechanics, information theory, and relativity theory are the basic foundations of theoretical physics The acquisition of information from a quantum system is the interface of classical and quantum physics Essential tools for its description are Kraus matrices and positive operator valued measures (POVMs) Special relativity imposes severe restrictions on the transfer of information between distant systems Quantum entropy is not a Lorentz covariant concept Lorentz transformations of reduced density matrices for entangled systems may not be completely positive maps Quantum field theory, which is necessary for a consistent description of interactions, implies a fundamental trade-off between detector reliability and localizability General relativity produces new, counterintuitive effects, in particular when black holes (or more generally, event horizons) are involved Most of the current concepts in quantum information theory may then require a reassessment

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Quantum Information and Relativity Theory

Quantum state tomography--deducing quantum states from measured data--is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. One of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

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Efficient quantum state tomography

We consider one copy of a quantum system prepared in one of two orthogonal pure states, entangled or otherwise, and distributed between any number of parties. We demonstrate that it is possible to identify which of these two states the system is in by means of local operations and classical communication alone. The protocol we outline is both completely reliable and completely general; it will correctly distinguish any two orthogonal states $100%$ of the time.

https://arxiv.org/pdf/quant-ph/0007098.pdf

Local distinguishability of multipartite orthogonal quantum states

Entanglement is a useful resource because some global operations cannot be locally implemented using classical communication. We prove a number of results about what is and what is not locally possible. We focus on orthogonal states, which can always be globally distinguished. We establish the necessary and sufficient conditions for a general set of $2\ifmmode\times\else\texttimes\fi{}2$ quantum states to be locally distinguishable, and for a general set of $2\ifmmode\times\else\texttimes\fi{}n$ quantum states to be distinguished given an initial measurement of the qubit. These results reveal a fundamental asymmetry to nonlocality, which is the origin of ``nonlocality without entanglement,'' and we present a very simple proof of this phenomenon.

https://arxiv.org/pdf/quant-ph/0202034.pdf

Nonlocality, asymmetry, and distinguishing bipartite states.

A subspace of a multipartite Hilbert space is completely entangled if it contains no product states. Such subspaces can be large with a known maximum size, smax, approaching the full dimension of the system, D. We show that almost all subspaces with dimension s ≤ smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n ≤ D − smax. This condition holds for almost all sets of states of all multipartite systems and reveals something surprising. The criterion is identical for separable and nonseparable states: entanglement makes no difference.

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Generic local distinguishability and completely entangled subspaces

We prove that any three linearly independent pure quantum states can always be locally distinguished with nonzero probability regardless of their dimension, entanglement or multipartite structure. Almost always, all three states can be unambiguously identified. The only exceptional case, where one state is locally knowable but the other two are not, is found among multi-qubit states.

Local Distinguishability of Any Three Quantum States

The anthropic principle, that the universe exists in some sense for life, has persisted in recent religious and scientific thought because it derives from cosmological fact. It has been unsuccessful in furthering our understanding of the world because its advocates tend to impose final metaphysical solutions onto what is a physical problem. We begin by outlining the weak and strong versions of the anthropic principle and reviewing the discoveries that have led to their formulation. We present the reasons some have given for ignoring the anthropic implications of these discoveries and find these reasons wanting—a real phenomenon demands real investigation. Theological and scientific solutions of the problem are then considered and criticized; these solutions provide dead ends for explanation. Finally, we pursue the path that explanation must follow and look at the physical details of the problem. It seems clear that the anthropic principle has been poorly framed. Removing the ambiguities surrounding the meaning of “life” may lead to more profitable investigations.

Jonathan Walgate

Papers

Local distinguishability of multipartite orthogonal quantum states

Nonlocality, asymmetry, and distinguishing bipartite states.

Generic local distinguishability and completely entangled subspaces

Local Distinguishability of Any Three Quantum States

The Anthropic Principle: Life in the Universe