General fine-grained uncertainty relation and the second law of thermodynamics

We investigate the fine-grained uncertainty relations for qubit systems by measurements corresponding respectively to two and three spin operators. Then we derive the general bound for a combination of two probabilities of projective measurements in mutually unbiased bases in $d$-dimensional Hilbert space. All of those uncertainty inequalities can be applied to construct different thermodynamic cycles such that the violation of those inequalities will lead to the violation of the second law of thermodynamics. This reveals the relationship between fine-grained uncertainty and the second law of thermodynamics.

Figure 1

A thermodynamic cycle. There are two paths from the initial state to the completely mixed state. The first path $\left(\text{a}\right)\to \left(\text{b}\right)$ denotes the first part of the cycle. This process is irreversible. The second path $\left(\text{a}\right)\to \left(\text{d}\right)\to \left(\text{c}\right)\to \left(\text{b}\right)$ is a reversible process and its reversed process denotes the second part of the cycle.

Figure 2

The thermodynamic cycle for d dimensions. The states are initially separated in (a). Then the membranes are inserted in stage (b). The states are completely mixed reaching equilibrium in (c). From (c) to (a), the process forms a circle.