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arXiv:quant-ph/0412024 · AIP conference proceedings

Decoherence Properties of Arbitrarily Long Histories

Artur Scherer

Published Jan 1, 2004• 2 citations
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Abstract

Within the decoherent histories formulation of quantum mechanics, we consider arbitrarily long histories constructed from a fixed projective partition of a finite‐dimensional Hilbert space. We review some of the decoherence properties of such histories including simple necessary decoherence conditions and the dependence of decoherence on the initial state. Here we make a first step towards generalization of our earlier results [Scherer and Soklakov, e‐print: quant‐ph/0405080, (2004) and Scherer et al., Phys. Lett. A 326, 307, (2004)] to the case of approximate decoherence.

DOI
10.1063/1.1834466

Related papers

Initial states and decoherence of histories

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We study decoherence properties of arbitrarily long histories constructed from a fixed projective partition of a finite dimensional Hilbert space. We show that decoherence of such histories for all initial states that are naturally induced by the projective partition implies decoherence for arbitrary initial states. In addition we generalize the simple necessary decoherence condition [Scherer et al., Phys. Lett. A 326, 307 (2004)] for such histories to the case of arbitrary coarse graining.

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Initial states and decoherence of histories

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We study decoherence properties of arbitrarily long histories constructed from a fixed projective partition of a finite dimensional Hilbert space. We show that decoherence of such histories for all initial states that are naturally induced by the projective partition implies decoherence for arbitrary initial states. In addition we generalize the simple necessary decoherence condition [Scherer et al., Phys. Lett. A (2004)] for such histories to the case of arbitrary coarse-graining.

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Decoherent Histories and Quantum Maps

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Discrete maps play an important role in the investigation \nof dynamical features of complex classical systems, especially \nwithin the theory of chaos. Similarly, quantum maps have proven \nto be a very useful mathematical tool within the study of complex \nquantum dynamical systems. Using the decoherent histories formulation \nof quantum mechanics we consider a particular framework for studying \nquantum maps which is motivated by the method of classical symbolic \ndynamics. Symbolic dynamics is known to be a very powerful method \nspecifically invented for the purpose of representing classical dynamical \nsystems by a discrete model that is suitable for information theoretic studies. \nOur framework uses the decoherent histories formalism which, \nsimilarly to classical symbolic dynamics, allows one to introduce information \ntheoretic quantities with respect to system dynamics. \nOur research within this framework can be viewed as a contribution \ntowards the development of a general theory of "quantum \nsymbolic dynamics". \n\nWe start by considering a special but very important example \nfor a quantum map: the quantum baker's map, invented for \nthe theoretical investigation of quantum chaos. \nHere we use the decoherent histories formalism to examine \nthe coarse-grained evolution of this map \nwith regard to the question of how classical predictability of \nthe evolution depends on the character of coarse-graining. \n%is affected by the choice of coarse-graining. \nWe demonstrate that hierarchical coarse-grainings display \ninteresting features with respect to this question. \n\nWe proceed with the investigation of decoherence properties \nof arbitrary unitary quantum maps. A number of interesting results \nis obtained within the framework of arbitrarily long histories \nconstructed from a fixed projective partition of a finite \ndimensional Hilbert space. In particular, we derive simple \nnecessary decoherence conditions, which employ only a single \niteration of a given unitary quantum map. Furthermore, \na surprising result is obtained with regard to the \nfundamental question of how the choice of the \ninitial state affects decoherence of histories. \nWithin the considered framework we show that if decoherence \nis established for arbitrary history lengths and all initial \nstates from the smallest natural set of states that \ncan be associated with the framework, then we get decoherence \nof such histories for arbitrary initial states. \nFinally, we make first steps towards proving analogous\nresults for approximate decoherence and suggest \nvarious questions for future research.\n

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Completeness of decoherence functionals

Stephan Schreckenberg

The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.

arXiv:gr-qc/9502036•Journal of Mathematical Physics•Sep 1, 1995•9 citations
Within the decoherent histories formulation of quantum mechanics, we consider arbitrarily long histories constructed from a fixed projective partition of a finite‐dimensional Hilbert space. We review some of the decoherence properties of such histories including simple necessary decoherence conditions and the dependence of decoherence on the initial state. Here we make a first step towards generalization of our earlier results [Scherer and Soklakov, e‐print: quant‐ph/0405080, (2004) and Scherer et al., Phys. Lett. A 326, 307, (2004)] to the case of approximate decoherence.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
Discrete maps play an important role in the investigation \nof dynamical features of complex classical systems, especially \nwithin the theory of chaos. Similarly, quantum maps have proven \nto be a very useful mathematical tool within the study of complex \nquantum dynamical systems. Using the decoherent histories formulation \nof quantum mechanics we consider a particular framework for studying \nquantum maps which is motivated by the method of classical symbolic \ndynamics. Symbolic dynamics is known to be a very powerful method \nspecifically invented for the purpose of representing classical dynamical \nsystems by a discrete model that is suitable for information theoretic studies. \nOur framework uses the decoherent histories formalism which, \nsimilarly to classical symbolic dynamics, allows one to introduce information \ntheoretic quantities with respect to system dynamics. \nOur research within this framework can be viewed as a contribution \ntowards the development of a general theory of "quantum \nsymbolic dynamics". \n\nWe start by considering a special but very important example \nfor a quantum map: the quantum baker's map, invented for \nthe theoretical investigation of quantum chaos. \nHere we use the decoherent histories formalism to examine \nthe coarse-grained evolution of this map \nwith regard to the question of how classical predictability of \nthe evolution depends on the character of coarse-graining. \n%is affected by the choice of coarse-graining. \nWe demonstrate that hierarchical coarse-grainings display \ninteresting features with respect to this question. \n\nWe proceed with the investigation of decoherence properties \nof arbitrary unitary quantum maps. A number of interesting results \nis obtained within the framework of arbitrarily long histories \nconstructed from a fixed projective partition of a finite \ndimensional Hilbert space. In particular, we derive simple \nnecessary decoherence conditions, which employ only a single \niteration of a given unitary quantum map. Furthermore, \na surprising result is obtained with regard to the \nfundamental question of how the choice of the \ninitial state affects decoherence of histories. \nWithin the considered framework we show that if decoherence \nis established for arbitrary history lengths and all initial \nstates from the smallest natural set of states that \ncan be associated with the framework, then we get decoherence \nof such histories for arbitrary initial states. \nFinally, we make first steps towards proving analogous\nresults for approximate decoherence and suggest \nvarious questions for future research.\n
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.
The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.