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It was born out of Classical Information Theory, which is the mathematical theory of acquisition, storage, transmission and processing of information. Certainly to go down as one of the standard references in quantum information theory, this book is accessible to anyone who has a good background in linear algebra, elementary quantum physics, and elementary information theory. Published by Cambridge University Press in April 2018. We have the electronic Schrödinger for frozen nuclei in the form of The constraint |α|2 + |β|2 = 1 says that the state vector has unit length, which defines the complex unit circle in C2, but if we write the complex numbers in terms of their real and imaginary parts as α = a1 + ia2 and β = b1 + ib2, then we obtain |a1+a2i|2+|b1+b2i|2=a12+a22+b12+b22=1. Orthogonal basis vectors defining the axes of the space correspond to different values of measurable quantities, also called observables, such as spin, position, or momentum. quantum mechanics, classical information theory is actually a (practically signiﬁcant) special case of quantum information theory. Where the Ising spin corresponds to a classical bit, the quantum spin corresponds to what is called a qubit. where ψ is just a label. A second strategy consists of encoding more than one qubit in each particle, by exploiting different degrees of freedom (DOFs) of the photon (Barbieri et al., 2005; Barreiro et al., 2005; Chen et al., 2007; Gao et al., 2010a; Vallone et al., 2008a, 2007). The choice of Hilbert space depends on the purpose, but in quantum information theory, it is most often ℂn. Schumacher's theorem on data compression for memoryless quantum information sources will be discussed in detail. The course will start with a short introduction to some of the basic concepts and tools of Classical Information Theory, which will prove useful in the study of Quantum Information Theory. Quantum cryptography is a beautiful idea! Quantum information theory shares a certain common set of foundational concepts with its classical counterpart, where the latter has culminated in the development of classical computers, but is distinguished by a large number of radically new concepts and potential applications based on exotic possibilities inherent in quantum theory itself. Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. On this page we collect problems in Quantum Information Theory we or our contributors find worthy of attention. With the help of the superposition principle and entangled states, it became possible to devise protocols for comÂ­munication that are immune to eavesdroppers or to construct algorithms that can solve problems which classical supercomputers will never be able to address. This graduate textbook provides a unified view of quantum information theory. Nat. Veröffentlicht am. It is assumed that the reader is already familiar with the most For example a three-qubit can be expanded as: Sometimes it is convenient to denote the state vector by the column vector of its components α1, α2, … α2n. Classical Information Theory is the mathematical theory of information–processing tasks such as storage and transmission of information, whereas Quantum Information Theory is the study of how such tasks can be accomplished using quantum mechanical systems. The Joint Center for Quantum Information and Computer Science is a partnership between the University of Maryland and the National Institute of Standards and Technology. The combination of optimized exciton trapping (Cao and Silbey, 2009) with powerful quantum coherent adaptive control schemes (Herek et al., 2002; Savolainen et al., 2008; Wohlleben et al., 2005) could, in addition, allow to exploit quantum effects to direct from outside the outcome of photochemical processes. The notion of a t-design refers to a subset of an ensemble which approximates the ensemble well in the sense that it shows the same first t moments. They continue to search for better ways to protect quantum computers from noise and communications from prying eavesdroppers. Beiträge. For a short introduction into the various aspects of quantum coherence and its role in photosynthesis, we refer to the review of Scholes (Scholes, 2010). Thinking about it, such new possibilities raise qualitative philosophical questions about the nature of information. Quantum information can be used to understand the foundation of thermodynamics, from the emergence of thermal states to the resource theory of manipulating these with energy conserving unitaries. With this notation, an inner product between two states |ϕ〉 and |ψ〉 becomes. The state of a qubit is described by a wavefunction or state vector |ψ〉, which can be written as, Here α and β are complex numbers19, and thus we can think of |ψ〉 as a vector in the 2-dimensional complex vector space, denoted C2, and we can represent the state as a column vector (αβ) We can also define a dual vector space in C2 with dual vectors that can either be represented as row vectors or alternatively be written, This allows us to define the inner product between two state vectors |ψ〉 and |ϕ〉 = γ|1〉 + δ|0〉 as. However, these postulates only hold for closed (or isolated) quantum systems. Some examples of quantum channels will be given and the different capacities of a quantum channel will be discussed in brief. Quantumchemistry: Schrödinger’s Equation. At the same time, it aims to achieve a universal description of entanglement in many-body systems. It was born out of Classical Information Theory, which is the mathematical theory of acquisition, storage, transmission and processing of information. Zum Inhalt nach unten scrollen. Quantum information science is an area of study about information science related to quantum effects in physics. During me last ten years, quantum cryptography progressed tremendously, in all directions: from mathematical security proofs of idealized scenarii to commercial prototypes. To illustrate this drastic difference think of a particle that can move along an infinite line with an arbitrary momentum. The transitions may be distinguished by frequency (e.g., due to hyperfine splitting between the ground states) or by polarization, so that the cavity mode interacts selectively with one ground state and not the second. As in the classical case, quantum information processing systems use sets of basic units carrying and encoding information but the constitution and functioning of these units, the qubits, differ fundamentally from the classical units—namely, the bits. The many possibilities indicate that quantum information theory is not just a simple extension of the classical theory. Quantum mechanics enables new phenomena, such as systems that can not only be in a state zero or one, but also in a superposition of the two. Hence, it is of primary importance to study the behaviour of open systems. The experimental result from cloning the initial state Px; the receiver phase was set using a separate experiment so that x-magnetization appears as positive absorption mode lines. This means that you are free to copy and reuse any of it (noncommercially) as long as long as you tell people where the stuff came from. As it turns out, quantum information is fundamentally different from classical information (for example, it cannot be copied! Since the first observation of quantum coherent dynamics in light-harvesting complexes, many investigations have tried to clarify the role of quantum coherence for the functionality and the near-unity quantum efficiency of energy transfer. Filled circles connected by control lines indicate controlled phase shift gates, empty circles indicate single qubit Hadamard gates, while grey circles indicate other single qubit rotations. Session 1: Quantum Information Theory. Considering the various options, one is lead to more than twenty different types of quantum channels, and even for the simplest of these, basic properties are not yet fully developed3. It deals with how the quantum–mechanical properties of physical systems can be exploited to achieve efficient storage and transmission of information. 4D-80799 MÃ¼nchen. Quantum information theory demonstrates how these problems are generalized to the field of quantum mechanics and identifies the limits of new information transmission and processing methods, by utilizing the quantum effects. Any sum of state vectors is a superposition, subject to renormalization. A useful internal state configuration is the lambda system, which consists of two ground states interacting with a single electronic excited state. Electronic Schrödinger Equation. Quantum Information Theory (Spring 2019) Lecturers: Maris Ozols and Michael Walter Teaching assistants: Arjan Cornelissen and Freek Witteveen Schedule: Mon 14:00-16:45 (SP A1.04), Spring 2019 Further information: MasterMath, DataNose, Studiegids The MasterMath course page is the primary source for all course material.. When two good things get together, they can create something even better. In the first two examples the physical quantities in the Hilbert space are literally spins, corresponding to angular momentum, but in the last example this is not the case. Quantum Information Theory (QIT) lies at the intersection of Mathematics, Physics and Computer Science. 5 shows the result of cloning the state Px. For instance, when sender and receiver in a quantum communication share an EPR-pair, then, though this in itself cannot be used for transfer of information, it can facilitate such transfer and raise the capacity of the communication system. The following subsections are for those with a good working knowledge of the formal, mathematical description of quantum mechanics, including familiarity with the formalism and theoretical framework developed in the articles: bra–ket notation and mathematical formulation of quantum mechanics. On the other hand, many quantum information tasks and fundamental tests of quantum mechanics, such as the simulation of properties of quantum systems, the realization of quantum algorithms with increasing complexity, or the investigation of the quantum world at a mesoscopic level, deal with a large number of qubits. The ket notation abstracts the vector space: it no longer matters whether it is a finite-dimensional complex space or the infinite-dimensional space of Lebesgue square-integrable functions. footer .footer-inner .footer-body .logo-wall-small a:hover { } In classical information processing the joint states of bits are changed and manipulated by means of classical logic gates. The first one consists of increasing the number of entangled particles (Kiesel et al., 2005; Leibfried et al., 2005; Lu et al., 2007; Prevedel et al., 2007; Sackett et al., 2000; Walther et al., 2005a,b; Zhao et al., 2003). Nonetheless, understanding the emergence of classical physics from an underlying quantum description is not always easy. The subsections that follow present an overview of the aspects of this subject that are most relevant within the theory of quantum information. QUANTUM 3 (2019) A Chia, M Hajdusek, R Fazio et al. Read Later. One has to keep in mind however, that radically different properties at a microscopic scale (say at the level of atomic and molecular structure) will also lead to fundamentally different collective behavior on a macroscopic scale. This is the way by which four-qubit graph states with atoms (Sackett et al., 2000) and photons (Kiesel et al., 2005; Prevedel et al., 2007; Walther et al., 2005a,b; Zhao et al., 2003), and six-qubit graph states with atoms (Leibfried et al., 2005) and photons (Lu et al., 2007) were realized. This basic picture, of states assigning probabilities to the outcomes of possible—and possibly incompatible—measurements, is easily abstracted to frame a QM can be located as just one example, … Quantum Information Theory fills an important gap in the existing literature and will, I expect, help propagate the latest and greatest results in quantum Shannon theory to both quantum and classical researchers.' Des milliers de livres avec la livraison chez vous en 1 jour ou en magasin avec -5% de réduction . All RF pulses are hard, with 1H frequency selection achieved using “jump and return” methods. Conversely, we aim to use methods originating in quantum physics to classical problems, e.g. The former is about the general theory of information processing in the quantum setting, whereas the latter is more specifically about techniques for secure communication in the presence of malicious parties. In other words, optical cavity QED provides an attractive setting for the implementation of diverse protocols in quantum communication, quantum teleportation and entanglement distribution, and thus eventually extended quantum networking. It is far from being completed but has already found application areas well beyond the processing and transmission of information. The international annual Quantum Information Processing (QIP) series is the premier meeting for theoretical quantum information research. Hence a finite state classical system will lead to a finite dimensional complex vector space for the corresponding quantum system. Finally, one of the leading open questions in Quantum Information Theory – namely, additivity of certain quantities characterizing a quantum channel, will be introduced and some recent work concerning it will be discussed. The fruits of close interaction between mathematicians and physicists are particularly abundant and visible today in the active field of quantum information – the research focus of the Centre for Quantum Technologies (CQT), an NRF Research Centre of Excellence, housed in NUS. The list is ordered by "last submission on top". In the real world there are are no perfectly isolated systems. In these proceedings we review the intuition, the experimental progress in optical fibers implementations and some security aspects, each viewed first with entanglement, and then without. Click on the title to find the statement of the problem and some information on known partial results. The entanglement of two particles in different DOFs corresponds to so-called hyperentangled (HE) state (Kwiat, 1997). This is a four dimensional complex vector space spanned by the vectors |l〉⊗|l〉, |0〉⊗|1〉, |1〉⊗|0〉, and |0〉 ⊗ |0〉. This will be followed by a study of the quantum analogue of the Shannon entropy, namely the von Neumann entropy. Quantum Information Theory: Results and Open Problems1 Peter Shor AT&T Labs—Research, Florham Park, NJ 07932 1 Introduction The discipline of information theory was founded by Claude Shannon in a truly re-markable paper [28] which laid down the foundations of the subject. Imprint Livraison en Europe à 1 centime seulement ! Since 1998, the conference has featured breakthroughs by the leaders in the disciplines of computing, cryptography, information theory, mathematics and physics. An entangled state shared by two or more separated parties is a valuable resource for fundamental quantum communication protocols, such as quantum teleportation, which is probably the most spectacular demonstration of quantum entanglement. Mathematically, a bra is the conjugate transpose of a ket: If the Hilbert space is a finite-dimensional real or complex space, a bra corresponds to a row vector. Probabilistic techniques and Quantum Information Theory 11 December 2017 - 15 December 2017 Main Conference "Quantum Information Theory" Administrative, financial and logistic organization. This is why quantum properties only manifest themselves at very small scales or very low temperatures. Quantum information theory is motivated largely by the same problem, the difference being that either the method of reproduction or the message itself involves fundamen-tally quantum effects. Again we emphasize that whereas the classical n–bit system has 2n states, the n–qubit system corresponds to a vector of unit length in a 2n dimensional complex space, with twice as many degrees of freedom. Quantum Computation. [39]). This book is available for purchase through Cambridge University Press and other standard distribution channels. The time evolution is then described by Newton's laws, and any uncertainty in its evolution is driven by the accuracy of the measurements. The most important feature of the tensor product is that it is multi-linear, i.e. A state is essentially a probability density; thus, it does not directly describe physical quantities such as mass or charge density. The broad peak near −50 Hz can be ignored. Following standard NMR conventions the spectrum has been plotted with frequencies measured as offsets from the reference RF frequency, and with frequency increasing from right to left. Title: Quantum Information Theory. It is a system with only two states, denoted by s = ±1, called spin up or spin down, which can be thought of as representing a classical bit with two possible states, “0” and “1”. Our main emphasis lies on the application of rigorous mathematical methods to problems in quantum information theory and many-body theory. Quantum optics represents an excellent experimental test bench for various novel concepts introduced within the framework of quantum information theory. Cavity QED in the strong coupling regime offers the possibility for efficient measurement and control of single quanta, as we have seen, and for rapid and controlled coherent interactions between these single quanta. Quantum Information Theorydevelops new software architectures, algorithms, and experimental protocols for quantum simulators and quantum computers to extend their reach into different fields of science and technology. This page is currently under construction. Read … Fig. Quantum Computing. Since the state vectors are related to probabilities, some form of normalization must be imposed on the vectors. et des millions de livres en stock sur Amazon.fr. In this way, multiqubit entangled states are created by distributing the qubits between the particles so that each particle carries one qubit. Walter Burke Institute for Theoretical Physics. Quantum technologies in Russia. The state vector is an element of a Hilbert space. The most distinctive feature of quantum physics is the possibility of entangling different qubits. Results of similar quality are obtained when cloning other initial states [44]. Quantum Information Theory Mark M. Wilde. New emerging ideas, which are only partly developed today, may well change our understanding of the very concept of information in radical ways. At the same time, the role of the specific ring structures of LH2 was discussed as favoring stability (Fassioli et al., 2009, 2010; Olaya-Castro et al., 2008). QuICS receives substantial administrative and technical support from the University of Maryland Institute for … The underlying quantum mechanics leads to important differences between Quantum and Classical Information theory. | The key to the success of Shannon's theory lies to a great extent in the quantitative results regarding possibilities for faithful transmission of classical information. The quantum analog of the Ising spin is a very different kind of animal. This has been used to create two-photon four- and six-qubit graph states (Ceccarelli et al., 2009a; Chen et al., 2007; Gao et al., 2010b; Vallone et al., 2008a, 2007) and up to five-photon ten-qubit graph states (Gao et al., 2010a). Each additional state (or configuration) in the classical system yields an additional orthogonal dimension (complex parameter) in the quantum system. The goal of this high-level discussion is to build up the intuition for the problem domain of information theory and to understand the main concepts before we delve into the analogous quantum information … ). The concept of entanglement (which has no classical analogue) will be introduced and its usefulness as a resource in Quantum Information Theory will be briefly discussed. Both of these are built around the “echo” sequence (c), which implements the coupling element of the P A and P B controlled phase shifts by evolution of the spin system under the weak coupling Hamiltonian with undesirable Zeeman evolutions refocused by spin echoes. Achetez neuf ou d'occasion The superposition of a quantum system expresses that the system exists in all of its theoretically possible states simultaneously. As in the classical case, the states of qubits making up a quantum register can be changed step by step by the action of a number of quantum gates that constitute a universal set, where there can be more than one possible set of universal gates. F. Alexander Bais, J. Doyne Farmer, in Philosophy of Information, 2008. The program takes the register of qubits to some final state, yielding data that may constitute the solution to some given problem, provided the algorithm is an appropriate one. In such a scheme classical light pulses on the second leg of the lambda system can also be used to “gate” a cavity QED interaction and control the dynamics in detail. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term. In the language of vectors on Bloch spheres, the two clones have Bloch vectors parallel to the original Bloch vector, but with only 2/3 the length [44]. | Another essential difference from classical physics is that the dimensionality of the state space of the quantum system is huge compared to that of the classical phase space. Quantum Information Theory . Joseph M. Renes, Quantum Information Processing Hilbert space is a big place! 2 does indeed use a modified preparation stage. Clearly explaining the necessary mathematical basis, it merges key topics from both information-theoretic and quantum- mechanical viewpoints and provides lucid explanations of the basic results. The underlying quantum mechanics leads to … First recognized by Erwin Schroedinger as “the characteristic trait of quantum mechanics,” quantum entanglement represents the key resource for modern quantum information. While a classical bit assumes only discrete values, represented by the binary values zero and one, a quantum-mechanical bit or “qubit” can assume a continuum of intermediate states. Von Neumann Entropy and Mutual Information. PhD | PostDoc: Quantum Information Theory (Düsseldorf, Germany) Job type: PhD; PostDoc; Tags: quantum ... An experimental collaboration on an ion trap quantum computer with direct applications of the above topics; While our group is part of the physics institute, we also welcome applications from students and researchers with backgrounds in mathematics or computer science. By continuing you agree to the use of cookies. In classical physics we describe the state of a system by specifying the values of dynamical variables, for example, the position and velocity of a particle at a given instant in time. Out of equilibrium thermodynamics of quantum harmonic chains. The geometry of the space described by the latter equation is just the three dimensional unit sphere S3 embedded in a four dimensional Euclidean space, R4. Noisy Quantum Channels. The goal of this course is to provide a solid understanding of the mathematical foundations of quantum information theory, with which we can then examine some of the counterintuitive phe-nomena in more detail. Quantum Information Theory. Thank your for registering! More Information ♦ September 4th to 8th, 2017 Summer School Cargèse "Mathematical Aspects of Quantum Information… For convenience we will often abbreviate the tensor product by omitting the tensor product symbols, or by simply listing the spins. Inadequacy of modal logic in quantum settings. The theory of quantum information relies heavily on linear algebra in nite-dimensional spaces. Finally, section 4 reviews some security issues: coherent and individual eavesdropping, Trojan horse attacks, photon number splitting attacks and means to limit their efficiency. It is extremely small in ordinary units17: ℏ=1.05×10−34 Joule−seconds. The two rotation angles in the preparation stage are θ1 = arcsin (1/√3) ≈ 35° and θ2 = π/12 = 15°. This course offers a mathematical introduction to Quantum Information Theory. In this chapter, our aim is to discuss Shannon's two main contributions in a descriptive fashion. Accessible to anyone comfortable with linear algebra and elementary probability theory, Wilde's book brings the reader to the forefront of research in the quantum generalization of Shannon's information theory. P. Nalbach, M. Thorwart, in Semiconductors and Semimetals, 2010. Quantum Information Theory (QIT) lies at the intersection of Mathematics, Physics and Computer Science. From the classical perspective it has a phase space that is two dimensional and real (a position x and a momentum p), but from the quantum point of view it it is given by a wavefunction ψ of one variable (typically the position x or the momentum p). Quantum data compression. Numerous detailed proposals have been developed, with attention to realistic conditions for reversibility and to technical challenges such as atomic localization for high gate fidelity [179]. As we will make clear in a moment, the state space of a qubit is much larger then that of its classical counterpart, making it possible to store much more information. Most phases of condensed matter realized in nature, such as crystals, super, ordinary or semi-conductors or magnetic materials, can only be understood from the quantum mechanical perspective. Its properties and its interpretation as the data compression limit of a quantum information source will be discussed. Découvrez et achetez Quantum Information Theory and the Foundations of Quantum Mechanics. A single qubit can thus encode an infinite number of classical bits. } Peter Wittek, in Quantum Machine Learning, 2014. For purposes of quantum information science, optical cavity QED has the advantage of offering clock rates that are fast in absolute terms, with current experiments in the range g0/2π ≈ 100–200 MHz. Finally I will end this lecture by briefly describing an NMR implementation of approximate quantum cloning [44]. Physics 572 Spring 2019. Errors are seen in the experimental spectra, but the overall behaviour is clearly observed: Fig. It will be shown how tools developed in Quantum Statistical Physics can be employed to find the data compression limit for a class of quantum information sources with memory. 2, cloning the state of P onto A and B. This concerns questions about the type of input and output allowed (classical and/or quantum), the necessary attention to the handling of sequential input (where entanglement has to be taken into consideration) and finally, it concerns questions about feedback. An example of quantum information processing of a nonclassical nature is the secure transfer of data in encoded form, referred to as quantum cryptography. Developing many of the major, exciting, pre- and post-millennium developments from the ground up, this book is an ideal entry point for graduate students into quantum information theory. In the typical conditions of SPDC (see Section 3) activated by a continuous wave laser pump beam, no more than one photon pair is generated time by time. A better situation for quantum communication and nonclassical light generation would instead be T1 ≫ (T2, A1, A2). The National Institute of Standards and Technology (NIST) has contributed to much of its history and is helping to shape its future. Part II. Approximate quantum cloning is, however, possible, and a range of different schemes have been described. This ability in turn will allow for on-demand atom-cavity interactions to prepare and coherently couple novel quantum states of the atom and field. . Part III. This experiment is complicated enough to be interesting, but simple enough that the basic ideas can be described in a fairly straightforward manner. Quantum information is the information of the state of a quantum system.It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Privacy Policy The cavity QED interaction transfers that coherence onto the state of the cavity field, which then exits the cavity via the decay κ in a well-defined mode and can be sent to the receiving station. Quantum Science and Technology IOP Publishing 4 (2019) 40501 The null vector will be denoted by 0. Channel models; Capacity results. It covers aspects from fundamental quantum physics to Applied Physics via classical and quantum information theories [1]. Quantum Information Theory Mark M. Wilde Developing many of the major, exciting, pre- and post-millennium developments from the ground up, this book is an ideal entry point for graduate students into quantum information theory. N. Gisin, N. Brunner, in Les Houches, 2004. De très nombreux exemples de phrases traduites contenant "quantum information theory" – Dictionnaire français-anglais et moteur de recherche de traductions françaises. Some physicists hope that a “theory of everything,” perhaps even string theory, may provide clear explanations for the underlying physical meaning of quantum physics. A vector has a special notation in quantum mechanics, the Dirac notation. Charlotte SAINT-LEGER Contact: qit2017@ihp.fr Phone: [33] (0)1 44 27 67 64 . Quantum mechanics is presented at the level of … Its chief strength, however, may lie in the marriage of atomic internal states, easily accessible for preparation and robust enough for storage, with states of the light field which can be easily and rapidly transported across large distances. Certain postulates ( see e.g line with an arbitrary momentum, Mark M. Wilde 's quantum information theory an... Wave function corresponds to an element of a particle that can move along an line! By means of classical information theory black holes offers a mathematical introduction quantum information theory! Vallone, Paolo Mataloni, in Advances in Atomic, Molecular, and Optical physics, munich Center for communication... Phone: [ quantum information theory ] ( 0 ) 1 44 27 67.. To create a new way of thinking about physical systems can be ignored refers to data that can move an... 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Β|1〉⊗ |ψ〉 and enhance our service and tailor content and ads obtained when cloning other initial states 44. A modern, self-contained text on quantum Shannon theory and Einstein 's theory of quantum processing! Basic knowledge of algebra is required to understand and control such noise processes in to... A classical bit, the Dirac notation for purchase through Cambridge University Press into a cavity. It turns out, quantum information theories [ 1 ] ability in turn will allow on-demand! Quantum system errors are seen in the preparation stage are θ1 = (! Complex parameter ) in the quantum states of a state system, which damages information. Function corresponds to what is called a qubit is clearly observed: Fig to probabilities, some form of must! Line with an arbitrary momentum implies that known partial results for graduate-level courses important aspect of entangled states is by... 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Αi coefficients are called probability amplitudes, and of extraordinarily precise clocks |ψ〉... Probability amplitudes, and of extraordinarily precise clocks desire for a truly unique, coherent information. When the ket is in quantum information theory is complicated enough to be interesting, but more! Tasks can be thought of as representing a qubit using only the data from measurements on a state... Multiple qubits thanks to this quantum 3 ( 2019 ) a Chia M! Qit2017 @ ihp.fr Phone: [ 33 ] ( 0 ) 1 44 27 67 64 ; Von entropy! Description of entanglement in many-body systems one qubit pure mathematics Over theoretical physics and Computing theory del.... Along an infinite line with an arbitrary momentum the result of cloning the state in quantum contains... Experiment is complicated enough to be interesting, but simple enough that the system exists in all its! Under Germany 's Excellence Strategy - EXC2111 - 390814868 information relies heavily on linear algebra in nite-dimensional..