Quantum control theory and applications: A survey <s> Projective measurement <s> An inexpensive faucet aerator is provided by three molded parts and a thin metal cup that holds two of the molded parts in assembled relationship. The first molded part is an elongated annulus provided with upstream and downstream recesses separated by an inner ring that helps to break up the liquid flow and serves as an abutment to support a second molded, jet-forming, part in the upstream recess and is arranged to be engaged by the third, molded, part located in the downstream recess and aiding in defining air intake means to the aerator. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Projective measurement <s> PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics <s> BIB002
To better control a quantum system, it is often desirable to extract information from the controlled quantum system by means of measurements. Measurement theory in quantum mechanics is essentially different from that in classical mechanics since a measurement on a quantum system unavoidably affects the measured system (a detailed discussion of this issue can be found in BIB002 ). In quantum mechanics, an observed physical quantity (called an observable) is expressed as a Hermitian operator on the Hilbert space H and one cannot simultaneously and precisely measure two noncommutative observables according to the Heisenberg uncertainty principle. A widely used measurement model is projective measurement (or von Neumann measurement). For an observable M , one can select a complete set of orthogonal projectors {P m : m P m = I, P m = P † m , PmP m = δm ,m P m } to describe a projective measurement, where P m is the projector onto the eigenspace of M with eigenvalue m, M = m mP m and δm ,m is the Kronecker delta BIB001 . For a quantum system in the state |ψ , the measurement outcome will correspond to one of the eigenvalues m of the observable M . The probability of occurrence of outcome m is p(m) = ψ|P m |ψ . Once the outcome m has occurred, the state of the measured system changes to (this process is known as the collapse postulate). For a mixed state ρ, the probability of outcome m is p(m) = tr{P m ρ} and the state is changed into PmρPm p(m) if we obtain the outcome m.