Von Neumann's Entropy Does Not Correspond to Thermodynamic Entropy

Abstract

Von Neumann argued by means of a thought experiment involving measurements of spin observables that the quantum mechanical quantity$S_{VN}=-k\mathrm{Tr}\,(\rho \mathrm{log}\,\rho) $ is conceptually equivalent to thermodynamic entropy. We analyze Von Neumann's thought experiment and show that his argument fails.

Entropy and Computation: The Landauer-Bennett Thesis Reexamined

“…The reason is that we are not sure what is the quantum mechanical counterpart of thermodynamic entropy. We know it is not the von Neumann entropy; see our [21].) An influential argument to the effect that a Maxwellian Demon is incompatible with classical mechanics was given by Bennett [1], in which the LB thesis is put to work.…”

On the Equivalence of von Neumann and Thermodynamic Entropy

“…This article considers von Neumann's introduction of −Trρ ln ρ as the quantum mechanical generalisation of thermodynamic entropy. In particular, it shows that an argument raised by Shenker (1999) and Hemmo and Shenker (2006) against the equivalence of von Neumann and thermodynamic entropy is problematic because a) their reasoning allows for a violation of the second law of thermodynamics and b) the alleged disparate behaviour in von Neumann and thermodynamic entropy during the Step 4 location measurement is in fact due to a wrong calculation of the von Neumann entropy. It is in fact the system's conditional entropy that decreases during the step, leading to the seemingly disparate behaviour of the two entropies.…”

“…The following argument, including any conceptual ambiguities, has been taken unamended from (Hemmo and Shenker, 2006), with the exception that we represent the position of the particle by the two orthogonal states |L and |R , where L and R stand for 'Left' and 'Right', as opposed to Hemmo and Shenker's mixed state ρ(L) representation. For the remainder of the article I furthermore assume, just like H&S, that it is in fact possible to treat a single quantum particle as a genuine thermodynamic system.…”

“…Figure 2: Illustration of the Gedankenexperiment following Hemmo and Shenker (2006). (1) the particle is prepared in a spin-x up eigenstate.…”

“…In this article I demonstrate that their argument against the equivalence of thermodynamic and von Neumann entropy is 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. The article is structured as follows: after a summary of von Neumann's original argument, I will quickly revisit the debate that has been lead to date by Shenker (1999) and Henderson (2003) before moving on to an analysis of Hemmo and Shenker's (2006) most recent contribution, which will be shown to be problematic.…”