Geometry of Quantum States: An Introduction to Quantum EntanglementCambridge University Press, 06/12/2007 Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied. |
المحتوى
| 1 | |
Outline of quantum mechanics | 135 |
Coherent states and group actions | 156 |
The stellar representation | 182 |
The space of density matrices | 209 |
Purification of mixed quantum states | 233 |
Quantum operations | 251 |
maps versus states | 281 |
Distinguishability measures | 323 |
Monotone metrics and measures | 339 |
Quantum entanglement | 363 |
1 | ix |
References | 437 |
24 | 440 |
62 | 446 |
81 | 452 |
Density matrices and entropies | 297 |
عبارات ومصطلحات مألوفة
3-sphere affine algebra arbitrary bistochastic Bloch ball bundle Bures classical coherent completely positive compute cone convex set coordinates corresponding CP map curve defined definition denote density matrices diagonal difficult dimension dimensional dynamical matrix eigenstates eigenvalues equal equation fibre field Figure find finite first fixed flat geodesic geometry given Hence Hermitian matrices Hilbert space Hilbert–Schmidt Horodecki Husimi function inequality infinity invariant K¨ahler linear maximally mixed measure Monge distance monotone Neumann entropy observe obtain octant operator monotone orbit orthogonal parameter partial trace phase space plane points positive maps positive operators POVM probability distribution Problem projective space pure quantum mechanics qubit R´enyi random real numbers relative entropy reshuffling rotation Schmidt decomposition Section Shannon entropy sphere stochastic submanifold subspace symmetric tangent vector tensor theorem transformations unitary unitary matrix vector space von Neumann entropy Wehrl entropy Wigner function zero Zyczkowski
مقاطع مشهورة
الصفحة 28 - Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalized, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary. They are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of...
الصفحة 35 - You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage.
الصفحة 150 - Now, using the facts that the absolute value of a product is the product of the absolute values...
الصفحة 364 - Another way of expressing the peculiar situation is : the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated and therefore virtually capable of being "best possibly known ", ie of possessing, each of them, a representative of its own.
الصفحة 5 - G coA can be expressed as a convex combination of at most n + 1 points from A.
الصفحة 35 - My greatest concern was what to call it. I thought of calling it 'information', but the word was overly used, so I decided to call it 'uncertainty'.
الصفحة 139 - H, it follows from (16-56) that the ratio of the minor to the major axis of the polarization ellipse...
الصفحة 47 - It is assumed that the interchange of the order of integration with respect to x and differentiation with respect to 9 is allowed.
الصفحة 33 - We will discuss this gap in the framework of mixing "similar" polytopes described in the next section. II. MIXTURB OF SIMILAR POLYTOPBS A convex polytope is a set which can be expressed as the intersection of a finite number of closed half spaces. In our discussion, all polytopes are non-empty convex polytopes in Rn.
الصفحة 102 - In the house of mathematics there are many mansions and of these the most elegant is projective geometry.