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Thermodynamic equilibrium is a unique state characterized by time-reversalsymmetry, which enforces zero fluxes and prohibits work extraction from asingle thermal bath. By virtue of being microscopically out of equilibrium,active matter challenges these defining characteristics of thermodynamicequilibrium. Although time irreversibility, fluxes, and extractable work havebeen observed separately in various non-equilibrium systems, a comprehensiveunderstanding of these quantities and their interrelationship in the context ofliving matter remains elusive. Here, by combining experiments, simulations, andtheory, we study the correlation between these three quantities in a singlesystem consisting of swimming Escherichia coli navigating through funnel-shapedobstacles. We show that the interplay between geometric constraints andbacterial swimming breaks time-reversal symmetry, leading to the emergence oflocal mass fluxes. Using an harmonically trapped colloid coupled weakly tobacterial motion, we demonstrate that the amount of extractable work depends onthe deviation from equilibrium as quantified by fluxes and entropy production.We propose a minimal mechanical model and a generalized mass transfer relationfor bacterial rectification that quantitatively explains experimentalobservations. Our study provides a microscopic understanding of bacterialrectification and uncovers the intrinsic relation between time irreversibility,fluxes, and extractable work in living systems far from equilibrium.

Bacteria Through Obstacles: Unifying Fluxes, Entropy Production, and Extractable Work in Living Active Matter