  Passive states, i.e., those states from which no work can be extracted via
unitary operations, play an important role in the foundations and applications
of quantum thermodynamics. They generalize the familiar Gibbs thermal states,
which are the sole passive states being stable under tensor product. Here, we
introduce a partial order on the set of passive states that captures the idea
of a passive state being virtually cooler than another one. This partial order,
which we build by defining the notion of relative passivity, offers a
fine-grained comparison between passive states based on virtual temperatures
(just like thermal states are compared based on their temperatures). We then
characterize the quantum operations that are closed on the set of virtually
cooler states with respect to some fixed input and output passive states.
Viewing the activity, i.e., non-passivity, of a state as a resource, our main
result is then a necessary and sufficient condition on the transformation of a
class of pure active states under these relative passivity-preserving
operations. This condition gives a quantum thermodynamical meaning to the
majorization relation on the set of non-increasing vectors due to Hoffman. The
maximum extractable work under relative passivity-preserving operations is then
shown to be equal to the ergotropy of these pure active states. Finally, we are
able to fully characterize passivity-preserving operations in the simpler case
of qubit systems, and hence to derive a state interconversion condition under
passivity-preserving qubit operations. The prospect of this work is a general
resource-theoretical framework for the extractable work via quantum operations
going beyond thermal operations.
