Thermodynamic length for far-from-equilibrium quantum systems

We consider a closed quantum system initially at thermal equilibrium and driven by arbitrary external parameters. We derive a lower bound on the entropy production which we express in terms of the Bures angle between the nonequilibrium and the corresponding equilibrium state of the system. The Bures angle is an angle between mixed quantum states and defines a thermodynamic length valid arbitrarily far from equilibrium. As an illustration, we treat the case of a time-dependent harmonic oscillator for which we obtain analytic expressions for generic driving protocols.

Figure 1

Irreversible entropy production $\langle \Sigma \rangle$ (25) (blue, solid line) together with the lower bound (8) corresponding to the lowest-order term of the expansion (7) (red, dashed line), or the exact function $s\left(x\right)$ (6) (green, dotted line), as a function of the (numerical) adiabaticy parameter $Q^{*}$ (13). Parameters are $\hslash =1$, $\beta =1.2$, ${\omega }_0=0.9$, ${\omega }_1=0.5$.

Figure 2

Irreversible entropy production $\langle \Sigma \rangle$ (25) (blue, solid line) together with the lower bound (8) evaluated for the Bures angle $\mathcal{L}$ (1) (green, dotted line) and the trace distance $\mathcal{T}$ (26) (purple, dot-dashed line), as a function of the (numerical) adiabaticy parameter (13) for the sudden switch $Q^{*}\left({\omega }_1\right)=({\omega }_0^2+{\omega }_1^2)/\left(2{\omega }_0{\omega }_1\right)$. Parameters are $\hslash =1$, $\beta =4.8$, ${\omega }_0=0.9$.

Figure 3

Lower bounds for the nonequilibrium entropy production based on the Bures angle, $(8/{\pi }^2){\mathcal{L}}^2$ (8) (blue solid), and on the Bures distance, ${\mathcal{D}}^2$ (A7) (red dashed), as a function of the fidelity $F$.