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Communications in Theoretical Physics

Collapse of Entangled State and Entropy Increase in an Isolated System

Qiren Zhang

Published Apr 1, 2008• 0 citations
Publisher

Abstract

We show that the collapse of the entangled quantum state makes the entropy increase in an isolated system. The second law of thermodynamics is thus proven in its most general form.
DOI
10.1088/0253-6102/49/4/12

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The irreversible entropy increase described by the second law of thermodynamics is fundamentally tied to thermalization and the emergence of equilibrium. In the first part of our work (Ref: arXiv.2503.04152), we constructed an isolated gas system model and numerically demonstrated irreversible growth of entanglement entropy caused by erasure of spread non-equilibrium state information. Here, we mathematically prove that for a typical macroscopic system in a non-equilibrium state ∣ϕ0⟩|ϕ_0\rangle∣ϕ0​⟩, the quantum state ∣ϕ0′⟩=O^(t)∣ϕ0⟩|ϕ'_0\rangle = \hat{O}(t)|ϕ_0\rangle∣ϕ0′​⟩=O^(t)∣ϕ0​⟩ will inevitably evolve toward equilibrium. Our work demonstrates that the second law of thermodynamics, and consequently the ergodic hypothesis in statistical physics, can be understood and proven from a quantum information perspective. From this perspective, the second law can be stated as: In typical macroscopic physical systems, the spreading and erasure of non-equilibrium information is inevitable.
The notion of entropy has been at the core of thermodynamics and statistical physics since the 19th century, originally mentioned in the statement of the second law of thermodynamics as ``entropy is non-decreasing in an isolated system.'' In the context of both classical and quantum mechanics, there are cases where the second law is violated. In other words, entropy of the system may fluctuate downward. The question of how low or high the entropy of a quantum system can get is of interest in the context of small systems that can maintain quantum coherence for short times, and in the context of our universe, assuming it is an isolated quantum system. We explore the rare and extreme fluctuations of two well-developed notions of entropy that are relevant and interesting in isolated thermodynamic quantum systems, namely the observational and entanglement entropy. We find several fundamental differences in the behavior of the two when fluctuating to their extremes. We also provide an analytical proof of a tight upper bound on entanglement entropy for systems constrained to conservation laws.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
The law of increase of entropy is one of the most important laws in science but no exact explanation has been given to the origin of it. Statistical thermodynamics only explains it by using Boltzmann’s statistical interpretation together with not sufficiently verified assumptions. This paper shows that the increase of entropy in an isolated macroscopic system can be reasonably explained with no assumption based on quantum mechanics if we take into account that microscopic particles such as electrons and atomic nuclei not only move in space but also have the basic quality of spontaneously spreading into every possible spatial position. The present work allows us to give a plausible definition to the entropy of an isolated system by using the Hamiltonian operator and a wave function of it and provides a firm basis for thermodynamics. It also gives a new physical meaning of entropy effective for further studies of irreversible processes in nature.
Academically, among students, an apparent paradox may arise when one tries to interpret the second law of thermodynamics within the context of the quantum mechanical wave function collapse. This is so because a quantum mechanical system suddenly seems to undergo, from a less restrictive state constructed from a superposition of eigenstates of a given operator, to a more restrictive state: the collapsed state. This paper is intended to show how this picture turns out to be a misconception and, albeit briefly, furtherly discuss the scope of Max Born’s probabilistic interpretation within the second law of thermodynamics.