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**1 - 6**of**6**### Quantum State Local Distinguishability via Convex Optimization

, 2015

"... Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can ..."

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Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of

### Tensor product methods and entanglement optimization for ab initio quantum chemistry

, 2014

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### unknown title

, 2012

"... Physicists have long recognized that the appropriate framework for quantum theory is that of a Hilbert space H and in the simplest case the algebra of observables is contained in B(H). This motivated von Neumann to develop the more general framework of operator algebras and, in particular C ∗- algeb ..."

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Physicists have long recognized that the appropriate framework for quantum theory is that of a Hilbert space H and in the simplest case the algebra of observables is contained in B(H). This motivated von Neumann to develop the more general framework of operator algebras and, in particular C ∗- algebras and W ∗- algebras (with the latter also known as von Neumann algebras). In the 1970’s these were used extensively in the study of quantum statistical mechanics and quantum field theory. Although at the most basic level, quantum information theory (QIT) is expressed using matrix algebras, interactions with the environment play a critical role. This requires the study of open quantum systems in which the effects of noise can be studied. Operator spaces are used implicitly starting with the identification of completely positive (CP) trace-preserving maps (CPT maps) with quantum channels. During the past five years, the role of operator spaces has played an increasingly important and more explicit role. In many situations, even when the underlying Hilbert space is finite dimensional, many interesting questions, particularly those involving channel capacity require asymptotic limits for infinitely many uses of the channel. In many situations, generic properties are important. Random matrices, free probability, and high dimensional convex analysis have all played an important role. The first workshop on operator structures held at BIRS in February 2007, brought together experts in

### Quantum Entanglement in High dimensions

, 2015

"... These lecture notes study some mathematical aspects of the phenomenon of entangle-ment from quantum mechanics. While the questions we consider are motivated by quantum information theory, where entanglement plays a fundamental role, our exposition targets mostly mathematicians who are not assumed to ..."

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These lecture notes study some mathematical aspects of the phenomenon of entangle-ment from quantum mechanics. While the questions we consider are motivated by quantum information theory, where entanglement plays a fundamental role, our exposition targets mostly mathematicians who are not assumed to be familiar with quantum information theory. We look at entanglement through the prism of “Asymptotic Geometric Analysis”, a branch of functional analysis also known as “local theory of Banach spaces ” whose objects of study are the normed spaces of large but finite dimension. Indeed, we especially focus on the case of quantum systems of large dimension, for which numerical approaches are usually doomed by the curse of dimensionality. These notes are organized as follows: in Section 1 we introduce the dichotomy between entangled vs separated states. In Section 2 we explain various approaches to quantify how much entanglement contains a quantum state, notably the “entanglement of formation”. Section 3 explains how to use concentration of measure in the form of Dvoretzky’s theorem to prove that the entanglement of formation is not additive, a major result first obtain by