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We have mass-energy equivalence.

Do we also have information-energy equivalence? Can we use the Landauer bounds to prove by transitivity that information and energy are equivalent?

Straying a bit off topic, but I think one of the more sensible approaches to information-energy equivalence is a thermodynamic engine with an information reservoir (in addition to the heat and work reservoirs normally considered). https://arxiv.org/pdf/1408.1224.pdf https://journals.aps.org/prx/pdf/10.1103/PhysRevX.3.041003

Maxwell's demon?

more of information-entropy equivalence, but close enough