  When studying thermalization of quantum systems, it is typical to ask whether
a system interacting with an environment will evolve towards a local thermal
state. Here, we show that a more general and relevant question is "when does a
system thermalize relative to a particular reference?" By relative
thermalization we mean that, as well as being in a local thermal state, the
system is uncorrelated with the reference. We argue that this is necessary in
order to apply standard statistical mechanics to the study of the interaction
between a thermalized system and a reference. We then derive a condition for
relative thermalization of quantum systems interacting with an arbitrary
environment. This condition has two components: the first is state-independent,
reflecting the structure of invariant subspaces, like energy shells, and the
relative sizes of system and environment; the second depends on the initial
correlations between reference, system and environment, measured in terms of
conditional entropies. Intuitively, a small system interacting with a large
environment is likely to thermalize relative to a reference, but only if,
initially, the reference was not highly correlated with the system and
environment. Our statement makes this intuition precise, and we show that in
many natural settings this thermalization condition is approximately tight.
Established results on thermalization, which usually ignore the reference,
follow as special cases of our statements.
