Decomposing information into copying versus transformation

In many real-world systems, information can be transmitted in two qualitatively different ways: by {\em copying} or by {\em transformation}. {\em Copying} occurs when messages are transmitted without modification, for example when an offspring receives an unaltered copy of a gene from its parent. {\em Transformation} occurs when messages are modified in a systematic way during transmission, e.g., when non-random mutations occur during biological reproduction. Standard information-theoretic measures of information transmission, such as mutual information, do not distinguish these two modes of information transfer, even though they may reflect different mechanisms and have different functional consequences. We propose a decomposition of mutual information which separately quantifies the information transmitted by copying versus the information transmitted by transformation. Our measures of copy and transformation information are derived from a few simple axioms, and have natural operationalizations in terms of hypothesis testing and thermodynamics. In this later case, we show that our measure of copy information corresponds to the minimal amount of work needed by a physical copying process, having special relevance for the physics of replication of biological information. We demonstrate our measures on a real world dataset of amino acid substitution rates. Our decomposition into copy and transformation information is general and applies to any system in which the fidelity of copying, rather than simple predictability, is of critical relevance.