  In a thermodynamic process with measurement and feedback, the second law of
thermodynamics is no longer valid. In its place, various second-law-like
inequalities have been advanced that each incorporate a distinct additional
term accounting for the information gathered through measurement. We
quantitatively compare a number of these information measures using an
analytically tractable model for the feedback cooling of a Brownian particle.
We find that the information measures form a hierarchy that reveals a web of
interconnections. To untangle their relationships, we address the origins of
the information, arguing that each information measure represents the minimum
thermodynamic cost to acquire that information through a separate, distinct
measurement protocol.
