[Paper Review] Decoherence according to Environment and Self Induced Decoherences

Mario Castagnino, Roberto Laura|ArXiv.org|Dec 16, 2005

Quantum Information and Cryptography5 references3 citations

TL;DR

This paper proposes a unified formalism that reconciles Environment-Induced Decoherence (EID) and Self-Induced Decoherence (SID) by showing they are complementary aspects of a single framework. It demonstrates that decoherence in open systems (EID) and closed systems (SID) can coexist and are mathematically equivalent under a common weak limit formalism, resolving long-standing disputes about their compatibility and showing that decoherence times and pointer bases are consistently defined across both approaches.

ABSTRACT

A generalized decoherence formalism that can be used both in open (using Environment Induced Decoherence-EID) and closed (using Self Induced Decoherence-SID) quantum systems is sketched

Motivation & Objective

• To resolve the long-standing perception that EID and SID are antagonistic approaches to decoherence.
• To establish a generalized formalism that applies uniformly to both open (EID) and closed (SID) quantum systems.
• To demonstrate that decoherence in open subsystems implies decoherence in the full closed system, and vice versa.
• To clarify the role of the environment and system partitioning in decoherence, especially in the absence of energy dissipation.
• To provide a consistent framework for defining decoherence times and pointer bases across different system-environment decompositions.

Proposed method

• Formalizes decoherence as a three-step process: defining relevant observables, computing their expectation values, and proving weak limit convergence to a final equilibrium state.
• Applies the weak limit $ W\lim_{t\rightarrow\infty}\rho(t) = \rho_{*} $ to both open and closed systems to define decoherence in a unified way.
• Uses a model Hamiltonian with a central spin $ S_0 $ coupled to $ N $ environmental spins $ S_j $, where $ S_0 $ is coupled to all $ S_j $, but $ S_j $ are not coupled to each other.
• Analyzes decoherence via the predictability sieve criterion and compares EID and SID decoherence times $ t_{DS} $ and $ t_{DU} $, showing $ t_{DS_0} \ll t_{DU} = \infty $ in the trivial environment case.
• Demonstrates that when the environment Hamiltonian is trivial (no internal dynamics), $ t_{DU} \to \infty $, so the full system does not decohere, even if a subsystem does.
• Establishes that the decoherence of all open subsystems $ S_i $ implies the decoherence of the full closed system $ U $, and vice versa, under the weak limit formalism.

Experimental results

Research questions

• RQ1 Can Environment-Induced Decoherence (EID) and Self-Induced Decoherence (SID) be unified under a single theoretical framework?
• RQ2 Under what conditions does a closed system decohere via SID, and how does this relate to EID in its subsystems?
• RQ3 Why do criticisms of SID in prior literature (e.g., [7]) fail when viewed through the lens of a complete system analysis?
• RQ4 How do decoherence times $ t_{DS} $ (EID) and $ t_{DU} $ (SID) compare, and what determines their finiteness or infinity?
• RQ5 Can a consistent, dynamical pointer basis be defined for both EID and SID using the same formalism?

Key findings

• The decoherence of all open subsystems $ S_i $ interacting with their respective environments $ E_i $ implies the decoherence of the full closed system $ U $, as shown by the weak limit $ W\lim_{t\to\infty}\rho(t) = \rho_* $.
• The decoherence of the full closed system $ U $ implies the decoherence of all its open subsystems $ S_i $, demonstrating the mutual consistency of EID and SID.
• In the model with a trivial environment (no internal dynamics), the decoherence time $ t_{DU} \to \infty $, so the full system does not decohere, even though $ S_0 $ does with finite $ t_{DS_0} $.
• The result that $ t_{DS_0} \ll t_{DU} = \infty $ resolves the apparent contradiction in prior work (e.g., [7]), showing that the failure of the full system to decohere is not a flaw in SID but a consequence of trivial environment dynamics.
• The paper shows that criticisms of SID based on the non-decoherence of the full system are invalid when the system is fully analyzed, as the subsystem $ S_0 $ does decohere while the environment remains non-decohering.
• The unified formalism supports both EID and SID, with EID having extensive experimental confirmation and SID providing a complete description of the classical limit, and both yielding decoherence times of similar order of magnitude in non-trivial cases.