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arXiv:quant-ph/0509174 · Physical Review A

Predictability sieve, pointer states, and the classicality of quantum trajectories

Diego A. R. Dalvit, Jacek Dziarmaga, Wojciech Hubert Zurek

Published Dec 1, 2005• 33 citations
arXivHTMLPDF

Abstract

We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states---the most predictable states of an open quantum system selected by the predictability sieve---are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).

DOI
10.1103/physreva.72.062101

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Einselection and Quantum Darwinism: Pointer States via Predictability Sieve — E8 Intelligence Research

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FINDING: Zurek's einselection and quantum Darwinism formalize the emergence of classicality via environment-induced superselection, with pointer states selected by a predictability sieve that minimizes entropy production. | MATH: The core is the von Neumann entropy \( S(\rho) = -\mathrm{Tr}(\rho \ln \rho) \). Einselection uses the environment–system Hamiltonian \( H_{SE} \) to define pointer states as eigenstates of the system's reduced density matrix \( \rho_S = \mathrm{Tr}_E(\rho_{SE}) \) that remain stable under decoherence. The predictability sieve maximizes \( S(\rho_S(t)) \) over initial states to find those with minimal entropy increase — a variational principle: \( \delta S(\rho_S(t))/\delta \rho_S(0) = 0 \) for pointer states. Quantum Darwinism quantifies redundancy \( R_\delta \) via mutual information \( I(S:F) = S(S) + S(F) - S(S,F) \) between system and environment fragments, where \( R_\delta \sim \delta^{-1} \) for small \( \delta \). | CONNECTION: The predictability sie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Joint momenta-coordinates states as pointer states in quantum decoherence

Nomenjanahary Tanjonirina Manampisoa, Ravo Tokiniaina Ranaivoson, R. Raboanary +3 more

Quantum decoherence provides a framework to study the emergence of classicality from quantum systems by showing how interactions with the environment suppress interferences and select robust states known as pointer states. Earlier studies have linked Gaussian coherent states to pointer states. More recently, it was conjectured that more general quantum states called joint momenta-coordinates states may serve as more suitable candidates to be pointer states. These states are associated to the concept of quantum phase space and saturate, by definition, generalized uncertainty relations. In this work, we rigorously prove this conjecture. Building on the Lindblad framework for the damped harmonic oscillator, and applying Zurek's predictability-sieve criterion, we analyze both underdamped and overdamped regimes. We show that only in the underdamped case do joint momenta-coordinates states remain pure and robust for all times, establishing them as the true pointer states. This extends Isar's earlier underdamped treatment, generalizes the concept beyond Gaussian approximations, and embeds classical robustness in the quantum phase space formalism, with potential applications in error-resistant quantum information.

arXiv:2509.18206•Sep 20, 2025•0 citations
We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states---the most predictable states of an open quantum system selected by the predictability sieve---are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).
We study various measures of classicality of the states of open quantum systems subject to decoherence and continuous quantum measurement. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection (einselection) are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states, the most predictable states of an open quantum system selected by the predictability sieve, are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, predictability sieve does not select well defined pointer states.)
Selection of the preferred classical set of states in the process of decoherence is discussed with an emphasis on the role of correlations, information loss and entropy. Persistence of correlations between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records and utilising information) predictability is a measure of such persistence. \texitPredictability sieve – a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states, and to locate the islands of predictability including the preferred pointer basis, is proposed. The relation between the consistent histories approach and the preferred sets of states is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent. The correspondence between the perceived reality and the evolving state of the quantum Universe based on a prolonged existence of states of individual systems and correlations between them is proposed. This emphasis on predictable existence of states of classical systems and records of classical observers suggests existential interpretation of quantum theory. Existential interpretation addresses the long-standing problem of the “collapse of the wave function” in a manner which goes beyond Everett's relative state and Bohr's Copenhagen interpretations, by identifying the collapse with the decay of non-classical superpositions of states (and of the observers records) into mixtures of preferred states chosen by the environment-induced superselection.
Selection of the preferred classical set of states in the process of decoherence -- so important for cosmological considerations -- is discussed with an emphasis on the role of information loss and entropy. {\it Persistence of correlations} between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records) {\it predictability} is a measure of such persistence. {\it Predictability sieve} -- a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states and to locate the islands of predictability including the preferred {\it pointer basis} is proposed. Predictably evolving states of decohering systems along with the time-ordered sequences of records of their evolution define the effectively classical branches of the universal wavefunction in the context of the ``Many Worlds Interpretation". The relation between the consistent histories approach and the preferred basis is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent.
The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ``Schr\"odinger-cat states.'' The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ``collapse of the wave packet.'' The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ``classical reality'' as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ``found out'' without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ``fitness'' in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.
FINDING: Zurek's einselection and quantum Darwinism formalize the emergence of classicality via environment-induced superselection, with pointer states selected by a predictability sieve that minimizes entropy production. | MATH: The core is the von Neumann entropy S(ρ)=−Tr(ρln⁡ρ) S(\rho) = -\mathrm{Tr}(\rho \ln \rho) S(ρ)=−Tr(ρlnρ). Einselection uses the environment–system Hamiltonian HSE H_{SE} HSE​ to define pointer states as eigenstates of the system's reduced density matrix ρS=TrE(ρSE) \rho_S = \mathrm{Tr}_E(\rho_{SE}) ρS​=TrE​(ρSE​) that remain stable under decoherence. The predictability sieve maximizes S(ρS(t)) S(\rho_S(t)) S(ρS​(t)) over initial states to find those with minimal entropy increase — a variational principle: δS(ρS(t))/δρS(0)=0 \delta S(\rho_S(t))/\delta \rho_S(0) = 0 δS(ρS​(t))/δρS​(0)=0 for pointer states. Quantum Darwinism quantifies redundancy Rδ R_\delta Rδ​ via mutual information I(S:F)=S(S)+S(F)−S(S,F) I(S:F) = S(S) + S(F) - S(S,F) I(S:F)=S(S)+S(F)−S(S,F) between system and environment fragments, where Rδ∼δ−1 R_\delta \sim \delta^{-1} Rδ​∼δ−1 for small δ \delta δ. | CONNECTION: The predictability sie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Quantum decoherence provides a framework to study the emergence of classicality from quantum systems by showing how interactions with the environment suppress interferences and select robust states known as pointer states. Earlier studies have linked Gaussian coherent states to pointer states. More recently, it was conjectured that more general quantum states called joint momenta-coordinates states may serve as more suitable candidates to be pointer states. These states are associated to the concept of quantum phase space and saturate, by definition, generalized uncertainty relations. In this work, we rigorously prove this conjecture. Building on the Lindblad framework for the damped harmonic oscillator, and applying Zurek's predictability-sieve criterion, we analyze both underdamped and overdamped regimes. We show that only in the underdamped case do joint momenta-coordinates states remain pure and robust for all times, establishing them as the true pointer states. This extends Isar's earlier underdamped treatment, generalizes the concept beyond Gaussian approximations, and embeds classical robustness in the quantum phase space formalism, with potential applications in error-resistant quantum information.