Second law of thermodynamics under control restrictions

The second law of thermodynamics, formulated as an ultimate bound on the maximum extractable work, has been rigorously derived in multiple scenarios. However, the unavoidable limitations that emerge due to the lack of control on small systems are often disregarded when deriving such bounds, which is specifically important in the context of quantum thermodynamics. Here we study the maximum extractable work with limited control over the working system and its interaction with the heat bath. We derive a general second law when the set of accessible Hamiltonians of the working system is arbitrarily restricted. We then apply our bound to particular scenarios that are important in realistic implementations: limitations on the maximum energy gap and local control over many-body systems. We hence demonstrate in what precise way the lack of control affects the second law. In particular, contrary to the unrestricted case, we show that the optimal work extraction is not achieved by simple thermal contacts. Our results not only generalize the second law to scenarios of practical relevance, but also take first steps in the direction of local thermodynamics.

Figure 1

An example exhibiting an energy constraint. $p$ is the excitation probability of a classical bit, and $\mathrm{\Delta }$ is the energy in the excited state. The dark orange region is forbidden. (a) Example of a general protocol using WTC. The total work extracted work (solid blue region) is negative. (b) The black arrow denotes an initial thermalizing map in ${\mathcal{M}}_{\text{TO}}$. Then the transition $(p^{*},{\mathrm{\Delta }}_{\text{max}})\to (p^{*},{\mathrm{\Delta }}_1)$ takes place, followed by an isothermal transition back to ${\mathrm{\Delta }}_{\text{max}}$. The total work extracted (red solid area) is positive.

Figure 2

The curves $g$ [in blue; see Eq. (55)] and $f$ [in orange; see Eq. (56)] illustrating the thermo-majorization condition for the two-qubit example. Since the orange curve (corresponding to the state ${\omega }_{{\sigma }_z\otimes {\sigma }_z+t\mathbb{1}\otimes {\sigma }_z}$ for $t<t_c$) lies below the blue curve (corresponding to the maximally mixed state) the transition from the maximally mixed state to ${\omega }_{{\sigma }_z\otimes {\sigma }_z+t\mathbb{1}\otimes {\sigma }_z}$ is possible using a thermal operation. The green curve (identity function) corresponds to the thermal state ${\omega }_{{\sigma }_z\otimes {\sigma }_z}$.