Download Quantum Measurement Theory and its Applications by Kurt Jacobs PDF

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By Kurt Jacobs

Contemporary experimental advances within the keep an eye on of quantum superconducting circuits, nano-mechanical resonators and photonic crystals has intended that quantum size thought is now an necessary a part of the modelling and layout of experimental applied sciences. This publication, geared toward graduate scholars and researchers in physics, supplies a radical creation to the fundamental thought of quantum dimension and plenty of of its vital glossy purposes. size and keep watch over is explicitly handled in superconducting circuits and optical and opto-mechanical platforms, and strategies for deriving the Hamiltonians of superconducting circuits are brought intimately. extra purposes lined comprise suggestions keep an eye on, metrology, open platforms and thermal environments, Maxwell's demon, and the quantum-to-classical transition.

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Quantum Measurement Theory and its Applications

Contemporary experimental advances within the regulate of quantum superconducting circuits, nano-mechanical resonators and photonic crystals has intended that quantum size thought is now an fundamental a part of the modelling and layout of experimental applied sciences. This ebook, aimed toward graduate scholars and researchers in physics, supplies an intensive creation to the fundamental concept of quantum dimension and lots of of its very important sleek functions.

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To apply the polar decomposition theorem to a quantum measurement described by a set of operators {An }, we write each of these operators as An = Un Pn . 76) and the final state can be written as ρ˜ n = Un Pn ρPn pn Un† . 77) The requirement upon the operators An = Un Pn for them to represent a measurement becomes A†n An = n (Pn )2 = I. 78) n This representation of a quantum measurement has a number of important ramifications. First note that since the unitary operators Un do not change the eigenvalues of the density matrix, they play no role in describing the information extraction by the measurement.

For emphasis we state this result as a theorem. Theorem 2 The fundamental theorem of quantum measurement: Every set of operators {An }, n = 1, . . , N, that satisfies n A†n An = I, describes a possible measurement on a quantum system, where the measurement has n possible outcomes labeled by n. If ρ is the state of the system before the measurement, ρ˜ n is the state of the system upon obtaining measurement result n, and pn is the probability of obtaining result n, then ρ˜ n = An ρA†n , pn pn = Tr[A†n An ρ].

For outcome n the final state is ρ˜ n = Un An ρA†n Un† /pn , where ρ is the initial state and pn = Tr[A†n An ρ]. Thus the probability that the system is in the desired state is Qn = ψ|ρ˜ n |ψ . To obtain the overall probability that the feedback procedure will put the system in the desired state we must average over all the possible outcomes. Thus the overall probability of success is P= pn ψ|ρ˜ n |ψ = n ψ|Un An ρA†n Un† |ψ . 3. In this case the joint state of the two systems following the interaction is w = Uρ ⊗ σ U † .

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