Jonathan Barrett

University of Oxford
Matthew Leifer

Thermodynamic vs information theoretic entropies in probabilistic theories

Time has a direction, in the sense that many processes happen one way but not in the reverse. A coffee mug may fall and smash into pieces, but a collection of pieces does not spontaneously leap from the floor and assemble themselves into a mug. This is a consequence of the Second Law of Thermodynamics, which states that if a system is not interfered with, entropy increases, i.e., the system becomes more disordered.
The concept of entropy also appears in information theory, where it quantifies the extent to which data can be compressed. These might look like completely different concepts: what has the compression of data got to do with smashed mugs? But it turns out that they are closely linked. For example, Landauer's Principle states that if information stored in a system is erased, then the environment surrounding the system must become more disordered.
The aim of the project is to investigate how entropy should be defined in a completely arbitrary probabilistic theory. The project will thereby determine whether different notions of entropy are necessarily the same, or whether their relationship is a special feature of classical and quantum theories.

Historically, the concept of entropy emerged from two distinct avenues. In phenomenological thermodynamics, entropy is defined in terms of heat transfer and temperature, whilst in information theory, the entropy of a source is defined in terms of the compressibility of a message. It is generally appreciated that there are close connections between these notions. For example, the statistical mechanical entropy of Gibbs is given by a mathematical expression identical with that for the Shannon entropy. Landauer's principle states that logically irreversible manipulation of information carries an inevitable thermodynamic cost. But there is much debate over how exactly the connections between thermodynamic and information theoretic entropies should be understood.
The aim of the project is to investigate the relationship between thermodynamic and information theoretic entropies. This will be done in the context of a comparatively recent and very active area of research, wherein operational formalisms for probabilistic theories more general than the classical and quantum theories are developed. The project will determine how information theoretic and thermodynamic entropies should be defined in arbitrary probabilistic theories, whether there is a necessary connection between the two, and whether quantum theory can be derived from postulates involving entropy and the Second Law.

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