  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
mutational biases occur during genetic replication. 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 versus the information transmitted by transformation. We begin with a
decomposition that applies when the source and destination of the channel have
the same set of messages and a notion of message identity exists. We then
generalize our decomposition to other kinds of channels, which can involve
different source and destination sets and broader notions of similarity. In
addition, we show that copy information can be interpreted as the minimal work
needed by a physical copying process, which is relevant for understanding 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.
