von Neumann entropy and quantum version of thermodynamic entropy

• URL: http://arxiv.org/abs/2412.15316v2
• Date: Sat, 18 Jan 2025 11:31:40 GMT
• Title: von Neumann entropy and quantum version of thermodynamic entropy
• Authors: Smitarani Mishra, Shaon Sahoo,
• Abstract summary: The debate whether the von Neumann (VN) entropy is a suitable quantum version of the thermodynamic (TH) entropy has a long history. We analyze in detail whether and when the VN entropy is the same or equivalent to the TH entropy.
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• Abstract: The debate whether the von Neumann (VN) entropy is a suitable quantum version of the thermodynamic (TH) entropy has a long history. In this regard, we briefly review some of the main reservations about the VN entropy and explain that the objections about its time-invariance and subadditivity properties can be addressed convincingly. In a broader context, we here analyze in detail whether and when the VN entropy is the same or equivalent to the TH entropy. We find that the VN entropy of a large isolated system is equivalent to its TH entropy if the system is in a mixed state described by a microcanonical (MC) ensemble. This equivalence also holds for open system and subsystem if the full system (i.e. system-bath or subsystem-bath combination) is in a mixed state described by an MC ensemble or in a {\it typical} pure state (or if the {\it Eigenstate Thermalization Hypothesis}, {\it ETH}, holds for the full system). Unlike isolated and open system, demonstrating the equivalence for a subsystem is not straightforward. We take a one-dimensional spin-1/2 model to numerically show that, if the full system is in a pure state, then the VN entropy of a subsystem is equivalent to its TH entropy if the {\it ETH} holds for the full system. We also demonstrate that this equivalence always works (irrespective of whether the {\it ETH} holds) if the full system is in a mixed state describe by an MC ensemble.

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