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Physica A Statistical Mechanics and its Applications

Systems with separable many-particle interactions. II

L.W.J. den Ouden, Hans W. Capel, Jacques H. H. Perk

Published Jan 1, 1976• 41 citations
Publisher
DOI
10.1016/0378-4371(76)90019-4

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Usual arguments on EPR state assume that, after particles I and II interact somewhere, they are spatially separated, which is why EPR correlation was considered to be a paradox. That is, after the interaction, particle I exists only in a spatial region A, while particle II exists only in a spatial region B located at a distance from A. However, though the separation imposes certain restriction on possible quantum state, its implication has not been properly considered. That is, though quantum state of A is a tensor product of quantum state of particle I and II in A, as quantum state of particle II in A is ground state only, the product is identical to quantum state of particle I in A. Similarly, quantum state of B is quantum state of particle II in B. Then, as entire quantum state is Cartesian, not tensor, product of quantum state of A and B, the entire quantum state is Cartesian product of quantum state of particle I and II, which means there is no tensor product term to represent quantum entanglement. For example, using binary state, (|0>, |0>) + (|1>, |1>) = (|0>, |1>) + (|1>, |0>) = (|0> + |1>, |0> + |1>). Separation process resolves entanglement. If the entire quantum state is calculated differently, by first taking Cartesian product and, then, tensor product, a tensor product term appears. However, as the term represents action at a distance and, thus, unphysical, its coefficient must always be 0. That is, quantum entanglement of EPR correlation assuming the spatial separation is a superficial mathematical artifact representing action at a distance.

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Usual arguments on EPR state assume that, after particles I and II interact somewhere, they are spatially separated, which is why EPR correlation was considered to be a paradox. That is, after the interaction, particle I exists only in a spatial region A, while particle II exists only in a spatial region B located at a distance from A. However, though the separation imposes certain restriction on possible quantum state, its implication has not been properly considered. That is, though quantum state of A is a tensor product of quantum state of particle I and II in A, as quantum state of particle II in A is ground state only, the product is identical to quantum state of particle I in A. Similarly, quantum state of B is quantum state of particle II in B. Then, as entire quantum state is Cartesian, not tensor, product of quantum state of A and B, the entire quantum state is Cartesian product of quantum state of particle I and II, which means there is no tensor product term to represent quantum entanglement. For example, using binary state, (|0>, |0>) + (|1>, |1>) = (|0>, |1>) + (|1>, |0>) = (|0> + |1>, |0> + |1>). Separation process resolves entanglement. If the entire quantum state is calculated differently, by first taking Cartesian product and, then, tensor product, a tensor product term appears. However, as the term represents action at a distance and, thus, unphysical, its coefficient must always be 0. That is, quantum entanglement of EPR correlation assuming the spatial separation is a superficial mathematical artifact representing action at a distance.
We investigate the conditions which can lead to the occurrence of "decoherence" (i.e., of "environment-induced superselection"), in a quantum system interacting with another, infinite quantum system (an apparatus, environment, etc.), in the context of the nonrelativistic quantum mechanics. We are particularly concerned with: a kind of interaction, its relative strength, initial state of the apparatus (environment), and compatibility of the self-Hamiltonian and the interaction Hamiltonian of the composite system, "system plus apparatus", including time dependent interactions. As necessary conditions concerning occurrence of decoherence appear the two requirements: (i) That the interaction Hamiltonian is of the separable kind , and (ii) That the complete Hamiltonian, Ĥ u satisfies commutator relation [Ĥ u ( t ), Ĥ u ( t ')] = 0 ( t and t ' arbitrary instants). Since non-separable interaction cannot produce the same, we conclude that occurrence of decoherence is neither a rule (as it is sometimes assumed in the context of the "environment theory"), nor an exceptional consequence of the dynamics of a system interacting with its environment (as it is sometimes claimed in the criticism of the environment theory). Exhaustiveness and generality of our analysis extend applicability of related results (which are independent on a particular model of (infinite) environment), at least as a qualitative test and/or quantitative estimation of the results of the calculations obtained by the use of the different methods (semiclassical methods, path-integration technique, etc.) in quantum many-particle physics.