Markus Müller: Exact operational interpretation of the free energy without the thermodynamic limit

Marcus Ruler is introduced as a speaker who will discuss the operational interpretation of entropy and free energy without resorting to thermodynamic limits. He expresses his pleasure at being back at Western and engaging with the philosophy crowd. The talk will cover the standard view on entropy, thermodynamics as a resource theory, and introduce a new interpretation of free energy and entropy for single-shot scenarios. The outline includes a discussion on the thermodynamic limit, the role of fluctuations in small systems, and the potential implications for quantum information theory.

This paragraph delves into the limitations of standard thermodynamics, which are only meaningful in the thermodynamic limit with large systems. It raises the question of what happens in the case of small quantum systems or strongly correlated systems where fluctuations cannot be ignored. The concept of Landauer erasure is introduced as an example where standard free energy does not fully capture the energetics of the process. Bennett's puzzle is also discussed, which challenges the standard understanding of state transitions based on free energy differences.

The speaker introduces the concept of thermodynamics as a resource theory, which provides a framework for understanding state transitions in small and strongly correlated systems. This approach allows for the identification of free operations and the characterization of possible transitions between states. The resource theory approach is contrasted with the standard thermodynamic limit, highlighting the importance of considering additional constraints for small systems.

The paragraph explores the role of catalysts in thermodynamic processes, particularly in the context of small-scale systems where correlations can build up between the catalyst and the system. It discusses how these correlations can be a resource that enables transitions that would otherwise be impossible according to the standard second law of thermodynamics. The concept of work extraction and the cost of work are examined within this framework, revealing the fundamental irreversibility associated with single quantum state operations.

This section presents a one-shot interpretation of free energy and entropy, discussing how these quantities can be understood for single particles or single cycles of an engine, as opposed to the traditional thermodynamic limit. It suggests that the standard Helmholtz free energy can provide operational meaning for single-shot processes when correlations are allowed to build up. The paragraph also presents a conjecture about the role of correlations and coherence in quantum thermodynamics.

The speaker discusses how correlations can be used as a resource in quantum thermodynamics, allowing for more efficient energy transformations in certain situations. An example is provided where a catalyst helps in transforming a quantum state with less energy than what would be naively expected based on the standard free energy difference. The potential for correlations to be beneficial in thermodynamic processes is highlighted, along with the idea that this might have broader implications for quantum information theory.

The paragraph presents a conjecture about the relationship between entropy and unitary transformations in quantum mechanics. It suggests that the von Neumann entropy could have a one-shot operational interpretation, where the entropy of a state determines the possibility of transforming it into another state using a unitary operation and a catalyst. The conjecture implies a deep connection between entropy, unitary operations, and the concept of catalysts in quantum mechanics.

The speaker concludes with speculative thoughts on the broader implications of the findings for quantum information theory. It suggests that the principles discussed in the context of thermodynamics might also apply to quantum information tasks, such as decoupling quantum information. The potential for a single-shot understanding of entropy in quantum information theory is highlighted, along with the possibility of new interpretations of entropy in quantum field theories and string theory.

In this final paragraph, the speaker engages with the audience, addressing their intuitions and questions about the presented concepts. The discussion revolves around the resolution of paradoxes, such as Bennett's puzzle, and the conditions under which the standard thermodynamic intuitions might fail. The importance of considering correlations and stochastic independence in thermodynamic processes is emphasized, along with the potential for these concepts to challenge and extend our understanding of thermodynamics and quantum mechanics.