**3. Weak Invariant of Spin in Time-Dependent Magnetic Field and Lindbladian Operators**

In recent years, quantum thermodynamics of a two-level system has been discussed in the literature [15,16]. Here, we consider a single spin in a time-dependent magnetic field **B**(*t*). The Hamiltonian reads

$$H(t) = \mathbf{B}(t) \cdot \boldsymbol{\sigma} \tag{7}$$

with σ being the Pauli-matrix vector. The constant involving the gyromagnetic ratio is set equal to unity. In the thermodynamic context, variation of the magnetic field should be slow. That is, the time scale of the variation is much longer than those of relaxation and quantum dynamics.

From the linearity of Equation (6) with respect to the Hamiltonian and the su(2) Lie algebra of the spin, it is natural to examine the following Lindbladian operators:

$$L\_{\ i} = \sigma\_{i}\ (i = 1, \ 2, \ 3). \tag{8}$$

These are Hermitian, and accordingly the subdynamics is unital [6]. Substituting Equation (8) into Equation (6) and using linear independence of the Pauli matrices, we have

$$
\dot{\mathbf{B}}(t) = -2\operatorname{A}\left.\mathbf{B}(t)\right| + 2c\left.\mathbf{B}(t)\right|,\tag{9}
$$

where the overdot denotes differentiation with respect to time, and *A* and *c* are given by

$$A = \begin{pmatrix} a\_1 & 0 & 0 \\ 0 & a\_2 & 0 \\ 0 & 0 & a\_3 \end{pmatrix} \tag{10}$$

$$\csc A = \text{tr}\,A = a\_1 + a\_2 + a\_3. \tag{11}$$

respectively. From Equation (9), we see that

$$\mathbf{B}(t) \cdot \dot{\mathbf{B}}(t) = 2(a\_2 + a\_3) \, B\_1^{\, 2}(t) + 2(a\_3 + a\_1) \, B\_2^{\, 2}(t) + 2(a\_1 + a\_2) \, B\_3^{\, 2}(t),\tag{12}$$

which is positive since *a <sup>i</sup>*'s are nonnegative and not all of them can be zero. Therefore, the magnitude *B*(*t*) = **<sup>B</sup>**(*t*) has to monotonically increase in time. In addition, from Equation (9), we find the coefficients to be given by

$$a\_{1} = \frac{1}{4} \Big( -\frac{\dot{B}\_{1}}{B\_{1}} + \frac{\dot{B}\_{2}}{B\_{2}} + \frac{\dot{B}\_{3}}{B\_{3}} \Big), \\ a\_{2} = \frac{1}{4} \Big( \frac{\dot{B}\_{1}}{B\_{1}} - \frac{\dot{B}\_{2}}{B\_{2}} + \frac{\dot{B}\_{3}}{B\_{3}} \Big), \\ a\_{3} = \frac{1}{4} \Big( \frac{\dot{B}\_{1}}{B\_{1}} + \frac{\dot{B}\_{2}}{B\_{2}} - \frac{\dot{B}\_{3}}{B\_{3}} \Big). \tag{13}$$

Thus, without detailed information about the interaction between the subsystem and the environment, the explicit form of the Lindblad equation is now fully determined as follows:

$$\text{sif } \frac{\partial}{\partial t} \frac{\rho}{t} = [H(t), \ \rho] - \frac{i}{2} \sum\_{i=1}^{3} a\_i [\sigma\_{i\prime}, [\sigma\_{i\prime}, \rho]]. \tag{14}$$

We note that the second term on the right-hand side, termed the dissipator, is small and is of the order . *B <sup>i</sup>*(*t*)'s. Equation (14) belongs to a class of finite-level systems in contact with a singular reservoir consisting of particles with the vanishing correlation time [17]. On the other hand, what is peculiar here is highlighted in Equation (13) that purely comes from isoenergeticity, without recourse to the detailed property of the energy bath.

Hermiticity of the Lindbladian operators in Equation (8), or equivalently the fact that the subdynamics is unital, can have a significant implication for the isoenergetic process. As shown in Reference [18] (see, also Reference [11] and a discussion generalized to the Rényi entropy in Reference [19]), time evolution of the von Neumann entropy

$$S\left[\rho\right] = -\text{tr}\left(\rho \ln \rho\right) \tag{15}$$

under the Lindblad equation in its general form in Equation (1) with Equation (4) satisfies

$$\frac{d}{dt}\frac{S}{t} = \sum\_{i} a\_{i\cdot} \Gamma\_{i\prime} \tag{16}$$

$$\Gamma\_i \ge \left\langle \left| L\_i^\dagger \right|, L\_i \right\rangle \right\rangle \tag{17}$$

where the Boltzmann constant is set equal to unity. Since the Lindbladian operators in the present case are Hermitian, the subdynamics tends to produce the entropy in time,

$$\frac{d\ S}{dt} \ge 0,\tag{18}$$

as desired.
