Joint measurability in non-equilibrium quantum thermodynamics.

Quantum work and fluctuation theorems are mostly discussed in the framework of projective two-point measurement (TPM) schemes. According to a well known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. In this Letter we consider a generalized version of the scheme and analyze it from the viewpoint of quantum measurement theory. We extend the no-go theorem to the case of generalized energy measurements that are incompatible for at least some intermediate unitary evolution. On the other hand, we can show that the conditions (i) and (ii) above can be satisfied simultaneously if the observables are jointly measurable. We explicitly construct an example with noisy energy measurements and derive bounds for the noise. In such a noisy scenario a single work measurement apparatus can be used to determine the correct average work and to obtain free energy differences with the help of a Jarzynski equality.