Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.

arXiv:2412.15316Dec 19, 20240 citations

A Man Misunderstood: Von Neumann did not claim that his entropy corresponds to the phenomenological thermodynamic entropy

Erin Sheridan

Recently, attention has returned to the now-famous 1932 thought experiment in which John von Neumann establishes the form of the quantum mechanical von Neumann entropy -Tr $ ρ\ln ρ$ ( $S_{VN}$ ), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy ( $S_{TD}$ ). Hemmo and Shenker (2006) reconstruct von Neumann's thought experiment and argue that it fails to establish this desired correspondence. Prunkl (2019) and Chua (2019) challenge Hemmo and Shenker's result in turn. This paper aims to provide a new foundation for the current debate by revisiting the original text (von Neumann (1996, 2018)). A thorough exegesis of von Neumann's cyclical gas transformation is put forth, along with a reconstruction of two additional thought experiments from the text. This closer look reveals that von Neumann's goal is not to establish a link between $S_{VN}$ and $S_{TD}$ , as is assumed throughout the current debate, but rather to establish a correspondence between $S_{VN}$ and the Gibbs statistical mechanical entropy $S_{G}$ . On these grounds I argue that the existing literature misunderstands and misrepresents his goals. A revised understanding is required before the success of von Neumann's reversible gas transformation can be definitively granted or denied.Recently, attention has returned to the now-famous 1932 thought experiment in which John von Neumann establishes the form of the quantum mechanical von Neumann entropy -Tr $ ρ\ln ρ$ ($S_{VN}$), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy ($S_{TD}$). Hemmo and Shenker (2006) reconstruct von Neumann's thought experiment and argue that it fails to establish this desired correspondence. Prunkl (2019) and Chua (2019) challenge Hemmo and Shenker's result in turn. This paper aims to provide a new foundation for the current debate by revisiting the original text (von Neumann (1996, 2018)). A thorough exegesis of von Neumann's cyclical gas transformation is put forth, along with a reconstruction of two additional thought experiments from the text. This closer look reveals that von Neumann's goal is not to establish a link between $S_{VN}$ and $S_{TD}$, as is assumed throughout the current debate, but rather to establish a correspondence between $S_{VN}$ and the Gibbs statistical mechanical entropy $S_{G}$. On these grounds I argue that the existing literature misunderstands and misrepresents his goals. A revised understanding is required before the success of von Neumann's reversible gas transformation can be definitively granted or denied.

arXiv:2007.06673Jul 13, 20200 citations

Von Neumann's Two Quantum Entropies

Biao Wu

Von Neumann proposed two quantum entropies, one in 1927 and the other in 1929. He intended to define thermodynamic entropy for quantum states in both cases. The 1927 entropy is the well-known von Neumann entropy, which is zero for all pure states. The 1929 entropy, which is nonzero for almost all pure states, has largely been ignored. After reviewing the history and properties of both entropies, we argue that the 1927 entropy is not thermodynamic entropy and should be used simply as a measure of entanglement, whereas the 1929 entropy is the true thermodynamic entropy for quantum states.Von Neumann proposed two quantum entropies, one in 1927 and the other in 1929. He intended to define thermodynamic entropy for quantum states in both cases. The 1927 entropy is the well-known von Neumann entropy, which is zero for all pure states. The 1929 entropy, which is nonzero for almost all pure states, has largely been ignored. After reviewing the history and properties of both entropies, we argue that the 1927 entropy is not thermodynamic entropy and should be used simply as a measure of entanglement, whereas the 1929 entropy is the true thermodynamic entropy for quantum states.

cond-mat.stat-mechCross-listedarXiv:2609.30284Sep 4, 20260 citations

On the characterization of the von Neumann entropy via the entropies of measurements

P. Staszewski

Reports on Mathematical PhysicsFeb 1, 19783 citations

Recently, attention has returned to the now-famous 1932 thought experiment in which John von Neumann establishes the form of the quantum mechanical von Neumann entropy -Tr $ ρ\ln ρ$ (

S_{VN}

), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy (

S_{TD}

). Hemmo and Shenker (2006) reconstruct von Neumann's thought experiment and argue that it fails to establish this desired correspondence. Prunkl (2019) and Chua (2019) challenge Hemmo and Shenker's result in turn. This paper aims to provide a new foundation for the current debate by revisiting the original text (von Neumann (1996, 2018)). A thorough exegesis of von Neumann's cyclical gas transformation is put forth, along with a reconstruction of two additional thought experiments from the text. This closer look reveals that von Neumann's goal is not to establish a link between

S_{VN}

and

S_{TD}

, as is assumed throughout the current debate, but rather to establish a correspondence between

S_{VN}

and the Gibbs statistical mechanical entropy

S_{G}

. On these grounds I argue that the existing literature misunderstands and misrepresents his goals. A revised understanding is required before the success of von Neumann's reversible gas transformation can be definitively granted or denied.