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The British Journal for the Philosophy of Science

The Von Neumann Entropy: A Reply to Shenker

Leah Henderson

Published Jun 1, 2003• 21 citations
Publisher

Abstract

Shenker has claimed that Von Neumann's argument for identifying the quantum mechanical entropy with the Von Neumann entropy, S(ρ) = − ktr(ρ log ρ), is invalid. Her claim rests on a misunderstanding of the idea of a quantum mechanical pure state. I demonstrate this, and provide a further explanation of Von Neumann's argument.

DOI
10.1093/bjps/54.2.291

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Shenker has claimed that Von Neumann's argument for identifying the quantum mechanical entropy with the Von Neumann entropy, S(ρ) = − ktr(ρ log ρ), is invalid. Her claim rests on a misunderstanding of the idea of a quantum mechanical pure state. I demonstrate this, and provide a further explanation of Von Neumann's argument.
In quantum mechanics, the expression for entropy is usually taken to be -kTr(ρlnρ), where ρ is the density matrix. The convention first appears in Von Neumann's Mathematical Foundations of Quantum Mechanics. The argument given there to justify this convention is the only one hitherto offered. All the arguments in the field refer to it at one point or another. Here this argument is shown to be invalid. Moreover, it is shown that, if entropy is -kTr(ρlnρ), then perpetual motion machines are possible. This and other considerations support the conclusion that this expression is not the quantum-mechanical correlate of thermodynamic entropy. Its usefulness in quantum-statistical mechanics can be explained by its being a convenient quantification of information, but information and entropy are not synonymous. As the present paper shows, one can change while the other is conserved.
Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Meir Hemmo and Orly Shenker have recently argued against this view by attacking von Neumann’s argument. I argue that Hemmo and Shenker’s arguments fail because of several misunderstandings about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite-particles case, and the infinite-particles case.
In 1932, John von Neumann argued for the equivalence of the thermodynamic entropy and −Trρlnρ, since known as the von Neumann entropy. Meir Hemmo and Orly R. Shenker recently challenged this argument by pointing out an alleged discrepancy between the two entropies in the single-particle case, concluding that they must be distinct. In this article, their argument is shown to be problematic as it (a) allows for a violation of the second law of thermodynamics and (b) is based on an incorrect calculation of the von Neumann entropy.
On the Equivalence of −Trρln⁡ρ-\text{Tr}\rho\ln\rho−Trρlnρ and Thermodynamic Entropy.
In 1939, von Neumann argued for the equivalence of the thermodynamic entropy and −Trρln⁡ρ-\text{Tr}\rho\ln\rho−Trρlnρ, since known as the von Neumann entropy. Hemmo and Shenker (2006) recently challenged this argument by pointing out an alleged discrepancy between the two entropies in the single particle case, concluding that they must be distinct. In this article, their argument is shown to be problematic as it a) allows for a violation of the second law of thermodynamics and b) is based on an incorrect calculation of the von Neumann entropy.
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.
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$ (SVNS_{VN}SVN​), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy (STDS_{TD}STD​). 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 SVNS_{VN}SVN​ and STDS_{TD}STD​, as is assumed throughout the current debate, but rather to establish a correspondence between SVNS_{VN}SVN​ and the Gibbs statistical mechanical entropy SGS_{G}SG​. 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.