Decomposing information into copying versus transformation

In many real-world systems, information can be transmitted in two qualitatively different ways: by copying or by transformation. Copying occurs when messages are transmitted without modification, e.g., when an offspring receives an unaltered copy of a gene from its parent. Transformation occurs when messages are modified systematically during transmission, e.g., when non-random mutations occur during biological reproduction. Standard information-theoretic measures do not distinguish these two modes of information transfer, although they may reflect different mechanisms and have different functional consequences. Starting from a few simple axioms, we derive a decomposition of mutual information into the information transmitted by copying and by transformation. Our decomposition applies whenever the source and destination of the channel have the same set of outcomes, so that a notion of message identity exists, although generalizations to other kinds of channels and similarity notions are explored. Furthermore, copy information can be interpreted as the minimal work needed by a physical copying process, relevant to better understand the physics of replication. We use the proposed decomposition to explore a model of amino acid substitution rates. Our results apply to any system in which the fidelity of copying, rather than simple predictability, is of critical relevance.