Thermodynamics of quantum systems with multiple conserved quantities

We consider a generalisation of thermodynamics that deals with multiple conserved quantities at the level of individual quantum systems. Each conserved quantity, which, importantly, need not commute with the rest, can be extracted and stored in its own battery. Unlike in standard thermodynamics, where the second law places a constraint on how much of the conserved quantity (energy) that can be extracted, here, on the contrary, there is no limit on how much of any individual conserved quantity that can be extracted. However, other conserved quantities must be supplied, and the second law constrains the combination of extractable quantities and the trade-offs between them which are allowed. We present explicit protocols which allow us to perform arbitrarily good trade-offs and extract arbitrarily good combinations of conserved quantities from individual quantum systems.