[Paper Review] Decoherence in closed and open systems

Mario Castagnino, Roberto Laura+1 more|ArXiv.org|Jan 19, 2006

Quantum Information and Cryptography8 references3 citations

TL;DR

This paper proposes a unified formal framework for decoherence in both closed and open quantum systems, showing that self-induced decoherence (SID) in closed systems and environment-induced decoherence (EID) in open systems are complementary rather than contradictory. It defines distinct decoherence times—t_DU for closed systems and t_DS for open systems—and demonstrates that for macroscopic systems, t_DU ≫ t_DS, resolving long-standing conceptual issues in decoherence theory by unifying the two approaches under a single theoretical perspective grounded in weak limits and coarse-graining.

ABSTRACT

A generalized formal framework for decoherence, that can be used both in open and closed quantum systems, is sketched. In this context, the relationship between the decoherence of a closed system and the decoherence of its subsystems is studied, and the corresponding decoherence times are defined: for macroscopic systems, the decoherence time of the closed system is much greater than the decoherence time of its subsystems. Finally, it is shown that the application of the new formal framework to a well-known model leads to physically adequate results.

Motivation & Objective

• To resolve the conceptual divide between decoherence in closed systems (self-induced decoherence, SID) and open systems (environment-induced decoherence, EID), which are often treated as mutually exclusive.
• To establish a generalized theoretical framework that accommodates both SID and EID under a single formalism, enabling consistent comparison and unification.
• To define and compare decoherence times t_DU (for closed systems) and t_DS (for open systems), showing that t_DU ≫ t_DS for macroscopic systems.
• To clarify the role of coarse-graining and weak limits in defining pointer bases and equilibrium states in both types of systems.
• To demonstrate the compatibility and consistency of SID and EID results through a detailed analysis of a well-known model.

Proposed method

• Introduces a generalized formalism based on weak limits and coarse-graining, where the evolution of expectation values of relevant observables is studied under unitary dynamics.
• Defines a coarse-grained state ρ_G(t) as the projection of the full state ρ(t) onto the space of relevant observables, using a projector π that selects macroscopic variables.
• Applies the weak limit W-lim_{t→∞} ρ_G(t) = ρ_G* to define the final equilibrium state, ensuring convergence in the limit of infinite time.
• Uses the structure of the Hamiltonian to guide the choice of partitions and relevant observables, enabling the study of subsystems within a closed system.
• Applies the formalism to a well-known model, showing that previously misinterpreted results can be correctly understood in this unified framework.
• Demonstrates that the pointer basis in both closed and open systems emerges naturally from the weak limit of the coarse-grained state, providing a general criterion for pointer basis selection.

Experimental results

Research questions

• RQ1 How can self-induced decoherence in closed quantum systems and environment-induced decoherence in open systems be reconciled within a single theoretical framework?
• RQ2 What is the relationship between the decoherence times t_DU (closed system) and t_DS (open system), and how do they scale for macroscopic systems?
• RQ3 Can a unified definition of the pointer basis be derived for both closed and open systems using weak limits and coarse-graining?
• RQ4 Why do previous interpretations of certain models fail to correctly describe decoherence, and how does the new framework resolve these inconsistencies?
• RQ5 How does the choice of relevant observables and partitions affect the emergence of classical behavior in both closed and open systems?

Key findings

• For macroscopic systems, the decoherence time of a closed system (t_DU) is significantly longer than that of an open system (t_DS), i.e., t_DU ≫ t_DS, indicating that decoherence occurs much more slowly in isolated systems.
• The final state ρ_G* of the coarse-grained system is obtained as the weak limit of ρ_G(t) as t → ∞, ensuring convergence in the space of relevant observables.
• The pointer basis in both closed and open systems is uniquely defined by the weak limit of the coarse-grained state, providing a general and consistent criterion for pointer basis selection.
• The formalism successfully reproduces physically adequate results when applied to a well-known model, correcting misinterpretations from previous studies.
• The framework demonstrates that SID and EID are not contradictory but complementary, with both approaches yielding compatible results under the same theoretical structure.
• In finite-dimensional subspaces of relevant observables, the weak limit of ρ_G(t) converges strongly to ρ_G*, confirming the robustness of the coarse-graining procedure.