dorsal/arxiv
View SchemaEntanglement in phase space
| Authors | A. M. Ozorio de Almeida |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0612029 |
| URL | https://arxiv.org/abs/quant-ph/0612029 |
Abstract
Classical surfaces in phase space correspond to quantum states in Hilbert space. Subsystems specify factor spaces of the Hilbert space. An entangled state corresponds semiclassically to a surface that cannot be decomposed into a product of lower dimensional surfaces. Such a classical factorization never exists for ergodic eigenstates of a chaotic Hamiltonian. The space of quantum operators corresponds to a double phase space. The various representations of the density operator then result from alternative choices of allowed coordinate planes. In the case of the Wigner function and its Fourier transform, the chord function, or the quantum characteristic function, this is a phase space on its own. The reduced Wigner function, representing the partial trace of a density operator over a subsystem is the projection of the original Wigner function; the reduced chord function is obtained as a section. The purity of the reduced density operator, the square of its trace, is a measure of entanglement, obtained by integrating either the square of the reduced Wigner function, or the square-modulus of the reduced chord function. Bell inequalities for general parity measurements can be violated even for classical looking states with positive Wigner functions that have evolved classically from product states. These include the original EPR state. Entanglement with an unknown environment results in decoherence. An example is that of the centre of mass of a large number of independent particles, entangled with internal variables. In this case, the Central Limit Theorem for Wigner functions leads to some aspects of Markovian evolution for the reduced system.
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"abstract": "Classical surfaces in phase space correspond to quantum states in Hilbert\nspace. Subsystems specify factor spaces of the Hilbert space. An entangled\nstate corresponds semiclassically to a surface that cannot be decomposed into a\nproduct of lower dimensional surfaces. Such a classical factorization never\nexists for ergodic eigenstates of a chaotic Hamiltonian. The space of quantum\noperators corresponds to a double phase space. The various representations of\nthe density operator then result from alternative choices of allowed coordinate\nplanes. In the case of the Wigner function and its Fourier transform, the chord\nfunction, or the quantum characteristic function, this is a phase space on its\nown. The reduced Wigner function, representing the partial trace of a density\noperator over a subsystem is the projection of the original Wigner function;\nthe reduced chord function is obtained as a section. The purity of the reduced\ndensity operator, the square of its trace, is a measure of entanglement,\nobtained by integrating either the square of the reduced Wigner function, or\nthe square-modulus of the reduced chord function. Bell inequalities for general\nparity measurements can be violated even for classical looking states with\npositive Wigner functions that have evolved classically from product states.\nThese include the original EPR state. Entanglement with an unknown environment\nresults in decoherence. An example is that of the centre of mass of a large\nnumber of independent particles, entangled with internal variables. In this\ncase, the Central Limit Theorem for Wigner functions leads to some aspects of\nMarkovian evolution for the reduced system.",
"arxiv_id": "quant-ph/0612029",
"authors": [
"A. M. Ozorio de Almeida"
],
"categories": [
"quant-ph"
],
"title": "Entanglement in phase space",
"url": "https://arxiv.org/abs/quant-ph/0612029"
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