Quantum Information

In one word, this is a responsible book; the rest is commentary. Around 1992 a few of us were led by Charles Bennett into a Garden of Eden of quantum information, communication, and computation. No sooner had we started exploring our surroundings and naming the birds and the beasts, than Peter Shor put an end to that apparent innocence by showing that factoring could be turned¿by means of quantum hardware¿into a po- nomial task. Fast factoring meant business; everybody seemed to be awfully interested in factoring. Not that anyone had any use for factoring per se, but it seemed that all the world¿s secrets were protected by factor-keyed padlocks. Think of all the power and the glory (and something else) that you might get by acting as a consultant to big businesses and government agencies, helping them pick everyone else¿s locks and at the same time build unpickable ones (well, nearly unpickable) for themselves. And if one can get an exponential advantage in factoring, wouldn¿t an exponential advantage be lying around the corner for practically any other computational task? Quantum infor- tion ¿and all that¿ has indeed blossomed in a few years into a wonderful new chapter of physics, comparable in ?avor and scope to thermodynamics. It has alsoturnedintoaveritable¿industry¿¿producingpapers,conferences,exp- iments, e?ects, devices¿even proposals for quantum computer architectures.

Foreword.- Section 1 Qubits: Quantum state purity.- The representation of qubits.- Stokes parameters.- Single-qubit gates.- The double-slit experiment.- The Mach-Zehnder interferometer.- Multiple qubits.- Section 2 Measurements and quantum operations: The von Neumann classification of processes.- The Pauli classification of measurements.- Maximal measurements and expectation values.- The Lueders rule and non-selective measurements.- Reduced statistical operators.- General operations.- Positive operator valued measures.- Section 3 Quantum non-locality and interferometry: Hidden variables and state completeness.- Von Neumann's 'no-go' theorem.- The Einstein-Podolsky-Rosen argument.- Gleason's theorem.- Bell inequalities.- Interferometric complementarity.- The Franson interferometer.- Two-qubit quantum gates.- Section 4 Classical information and communication: Communication channels.- Shannon entropy.- Renyi entropy.- Coding.- Error correction.- Data compression.- Communication complexity.- Section 5 Quantum information: Quantum entropy.- Quantum relative and conditional entropies.- Quantum mutual information.- Coherent information.- Quantum Renyi and Tsallis entropies.- Section 6 Quantum entanglement: Basic definitions.- The Schmidt decomposition.- Special bases and decompositions.- Stokes parameters and entanglement.- Partial transpose and reduction criteria.- The 'fundamental postulate'.- Entanglement monotones.- Distillation and bound entanglement.- Entanglement and majorization.- Concurrence.- Entanglement witnesses.- Entanglement as a resource.- The thermodynamic analogy.- Information and the foundations of physics.- The geometry of entanglement.- Creating entangled states of light.- Section 7 Entangled multipartite systems.- Stokes and correlation tensors.- N-tangle.- Generalized Schmidt decomposition.- Lorentz-group isometries.- Entanglement classes.- Algebraic invariants of multipartite systems.- Three-qubit states and residual tangle.- Three-qubit quantum logic gates.- States of higher qubit number.- Section 8 Quantum state and process estimation.- Quantum state tomography.- Quantum process tomography.- Direct estimation methods.- Section 9 Quantum communication: Quantum channels.- Channel capacities.- Holevo's theorem.- Discrimination of quantum states.- The no-cloning theorem.- Basic quantum channels.- The GHJW theorem.- Quantum dense coding.- Quantum teleportation.- Entanglement swapping.- Entanglement purification.- Quantum data compression.- Quantum communication complexity.- Section 10 Quantum decoherence and its mitigation: Quantum decoherence.- Decoherence and mixtures.- Decoherence-free subspaces.- Quantum coding, error detection and correction.- The 9-qubit Shor code.- Stabilizer codes.- Concatenation of quantum codes.- Section 11 Quantum broadcasting, copying and deleting: Quantum broadcasting.- Quantum copying.- Quantum deleting.- Landauer's principle.- Section 12 Quantum key distribution: Cryptography.- QKD systems.- The BB84 (four-state) protocol.- The E91 (Ekert) protocol.- The B92 (two-state) protocol.- The 6-state protocol.- Eavesdropping.- Security proofs.- Section 13 Classical and quantum computing: Classical computing.- Deterministic Turing machines.- Probabilistic Turing machines.- Multi-tape Turing machines . 13.5 Quantum Turing machines.- Quantum computational complexity.- Fault-tolerant quantum computing.- The KLM proposal.- Section 14 Quantum algorithms: The Deutsch-Jozsa algorithm.- The Grover search algorithm.- The Shor factoring algorithm.- The Simon algorithm.- Appendix A Mathematical elements: Boolean algebra and Galois fields.- Random variables.- Hilbert space.- The standard quantum formalism.- Dirac notation.- Groups o