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(c) The quantum Hamiltonian H is essentially self-adjoint on the span of nitely many number eigenstates As a matter of fact, Hamiltonians that are semibounded, symmetric (Hermitian) polynomials of Q f and P f are admissible One can conclude that a continuous-time, Brownian motion regularization of the phase space path integral can be rigorously established It applies to a wide class of Hamiltonians It can also be proved that the formulation is fully covariant under general canonical coordinate transformations Recently, dos Santos and Aguiar [31] constructed a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator Their coherent state propagator provides an explicit connection between the Wigner and the Husimi representations of the evolution operator The dos Santos Aguiar representation is different from the usual path integral forms suggested by Klauder and Skagerstam presented in this section These different representations, although equivalent quantum mechanically, lead to different semiclassical limits

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4 Coherent States in Quantum Information: an Example of Experimental Manipulation

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41 Quantum States for Information

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Quantum information processing is about exploiting quantum mechanical features in all facets of information processing (data communication, computing) Excellent textbooks, monographs, and reviews exist which give all the material necessary to understand this fast-developping eld [32 34] The states act as information carriers, while the communication is processed through a sequence of quantum operations constituting the channel The sender encodes information by preparing the channel into a well-de ned quantum state belonging to an alphabet A = { 0 , 1 , , M } The receiver, following any relevant signal propagation, performs a measurement on the transmission channel to ascertain which state was transmitted by the sender Quantum information theory is mainly based on superposition-basis and entanglement measurements This requires high- delity implementation to be effective in the laboratory Unfortunately, quantum measurements are invasive in the sense that little or no re nement is achieved by further observation of an already measured system Some of the dif culties in implementing communication in quantum information stems from the fragility of Schr dinger-cat-like superpositions Even with transmission of orthogonal codewords, decoherence, energy dissipation, and other imperfections deteriorate orthogonality If the states in the sender s alphabet are not orthogonal, no measurement can distinguish between overlapping quantum states without some ambiguity [21, 35 38] Then errors seem unavoidable: there exists a nonzero probability that the receiver will misinterpret the transmitted codeword However, this impossibility of discriminating between nonorthogonal quantum states presents an advantage for quantum key distribution [39] Indeed, nonorthogonality prevents an eavesdropper from acquiring information without disturbing the state Also, in some cases it has been shown by Fuchs [36] that the classical information capacity of a noisy channel is actually maximized by a nonorthogonal alphabet Mathematically, the question of distinguishing between nonorthogonal states [36, 37] is addressed by optimizing a state-determining measurement over all posCoherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5

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4 Coherent States in Quantum Information: an Example of Experimental Manipulation

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itive-operator-valued measures (POVM) [38] But arbitrary POVMs are not easy to manipulate! In a recent work, Cook, Martin, and Geremia [40] demonstrated that real-time quantum feedback can be used in place of a quantum superposition of the type Schr dinger cat state to implement an optimal quantum measurement for discriminating between optical coherent states This work gives us an excellent opportunity of presenting standard coherent states as they are produced and used in realistic conditions As a preliminary to the description of the experiment, we will also give an account of the theoretical background needed

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42 Optical Coherent States in Quantum Information

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The optical eld produced by a laser provides a convenient quantum system for carrying information Since optical coherent states | are not orthogonal, one would 2 attempt to minimize the overlapping | = e i I ( ) e | | /2 by using large-amplitude regimes However, one faces power limitations and the appearance of nonlinear effects So one is more inclined to develop optimization methods for communication processes based on small-amplitude optical coherent states and photodetection When one tries to distinguish between two nonorthogonal states through some receiver device, there exists a quantum error probability The latter is bounded below by some minimum, named the quantum limit or Helstrom bound [38] in this context Three types of receivers were described by Geremia [42] Kennedy [44] proposed in 1972 a receiver based on simple photon counting to distinguish between two different coherent states However, the Kennedy receiver error probability lies above the quantum mechanics minimum, that is, the Helstrom bound Then, Dolinar [41] proposed in 1973 a measurement scheme capable of achieving the quantum limit Dolinar s receiver, while still based on photon counting, approximates an optimal POVM by superposing a local feedback signal on the channel A serious experimental drawback was that real-time adjustment of the local signal following each photon was considered as quite impracticable As a result, Sasaki and Hirota [45] later proposed an alternative receiver that applies an open-loop unitary transformation to the incoming coherent state signals to render them more distinguishable by simple photon counting Geremia [42] compared, theoretically and numerically, the relative performance of the Kennedy, Dolinar, and Sasaki Hirota receivers under realistic experimental conditions, insisting on the following aspects: (i) subunity quantum ef ciency, where it is possible for the detector to miscount incoming photons, (ii) nonzero dark counts, where the detector can register photons even in the absence of a signal,

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