This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. More information about the AMS Short Course can be found at http://www.csee.umbc.edu/~lomonaco/ams/Announce.html
This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. More information about the AMS Short Course can be found at http://www.csee.umbc.edu/~lomonaco/ams/Announce.html
Space searches with a quantum robot by P. Benioff Perturbation theory and numerical modeling of quantum logic operations with a large number of qubits by G. P. Berman, G. D. Doolen, D. I. Kamenev, G. V. Lopez, and V. I. Tsifrinovich Inconclusive rate with a positive operator valued measure by H. E. Brandt Quantum amplitude amplification and estimation by G. Brassard, P. Hoyer, M. Mosca, and A. Tapp Manipulating the entanglement of one copy of a two-particle pure entangled state by L. Hardy Geometric algebra in quantum information processing by T. F. Havel and C. J. L. Doran Quantum computing and the Jones polynomial by L. H. Kauffman Quantum hidden subgroup algorithms: A mathematical perspective by S. J. Lomonaco, Jr. and L. H. Kauffman Improved two-party and multi-party purification protocols by E. N. Maneva and J. A. Smolin Quantum games and quantum algorithms by D. A. Meyer A proof that measured data and equations of quantum mechanics can be linked only by guesswork by J. M. Myers and F. H. Madjid Quantum computation by geometrical means by J. Pachos Pauli exchange and quantum error correction by M. B. Ruskai Relative entropy in quantum information theory by B. Schumacher and M. D. Westmoreland An unentangled Gleason's theorem by N. R. Wallach Entangled chains by W. K. Wootters.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
An invitation to quantum computation: A Rosetta stone for quantum mechanics with an introduction to quantum computation by S. J. Lomonaco, Jr. Qubit devices by H. E. Brandt Quantum algorithms and quantum complexity theory: Introduction to quantum algorithms by P. W. Shor Shor's quantum factoring algorithm by S. J. Lomonaco, Jr. Grover's quantum search algorithm by S. J. Lomonaco, Jr. A survey of quantum complexity theory by U. V. Vazirani Quantum error correcting codes and quantum cryptography: An introduction to quantum error correction by D. Gottesman A talk on quantum cryptography or how Alice outwits Eve by S. J. Lomonaco, Jr. More mathematical connections: Topological quantum codes and anyons by A. Kitaev Quantum topology and quantum computing by L. H. Kauffman An entangled tale of quantum entanglement by S. J. Lomonaco, Jr. Index.
This volume is based on lectures delivered at the 2009 AMS Short Course on Quantum Computation and Quantum Information, held January 3-4, 2009, in Washington, D.C. Part I of this volume consists of two papers giving introductory surveys of many of the important topics in the newly emerging field of quantum computation and quantum information, i.e., quantum information science (QIS). The first paper discusses many of the fundamental concepts in QIS and ends with the curious and counter-intuitive phenomenon of entanglement concentration. The second gives an introductory survey of quantum error correction and fault tolerance, QIS's first line of defence against quantum decoherence. Part II consists of four papers illustrating how QIS research is currently contributing to the development of new research directions in mathematics. The first paper illustrates how differential geometry can be a fundamental research tool for the development of compilers for quantum computers. The second paper gives a survey of many of the connections between quantum topology and quantum computation. The last two papers give an overview of the new and emerging field of quantum knot theory, an interdisciplinary research field connecting quantum computation and knot theory. These two papers illustrate surprising connections with a number of other fields of mathematics. In the appendix, an introductory survey article is also provided for those readers unfamiliar with quantum mechanics.
The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation and quantum information theory. This paper is a written version of the first of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000, and will appear in the AMS PSAPM volume entitled "Quantum Computation." Part 1 of the paper is an introduction the to the concept of the qubit. Part 2 gives an introduction to quantum mechanics covering such topics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schrodinger's equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schrodinger picture, quantum entanglement, EPR paradox, quantum entropy. Part 3 gives a brief introduction to quantum computation, covering such topics as elementary quantum computing devices, wiring diagrams, the no-cloning theorem, quantum teleportation, Shor's algorithm, Grover's algorithm. Many examples are given. A table of contents as well as an index are provided for readers who wish to "pick and choose." Since this paper is intended for a diverse audience, it is written in an informal style at varying levels of difficulty and sophistication from the very elementary to the more advanced.