Universal Landauer-like inequality from the first law of thermodynamics

We demonstrate that the equality-based first law of thermodynamics inherently implies a universal Landauer-like inequality, connecting variations in system entropy and energy. This Landauer-like inequality is shown to depend solely on system information and is highly applicable in scenarios where the implementation of the conventional Landauer principle becomes challenging. Moreover, we unveil that this Landauer-like inequality complements the Landauer principle by establishing an alternative upper bound on heat dissipation. To underscore its practicality, we illustrate the utility of the Landauer-like inequality in contexts such as dissipative quantum state preparation and quantum information erasure applications. Our findings offer insights into identifying thermodynamic constraints, with particular relevance to the domains of quantum thermodynamics and the energetics of quantum information processing. Additionally, this approach paves the way for investigating systems coupled to nonthermal baths or those characterized by limited access to bath information.

Figure 1

Results for the time-dependent dissipated heat $Q\left(t\right)$ (blue solid curve), the upper bound ${\mathcal{Q}}_u\left(t\right)$ given by Eq. (6) (red dashed curve), the quantum coherence contribution $T_R\mathrm{\Delta }\mathrm{Coh}\left(t\right)$ (green dash-dotted curve), and the diagonal entropy contribution $-T_R\mathrm{\Delta }{S}^{\prime }\left(t\right)$ (magenta dotted curve). Main panel: ${\rho }_S\left(0\right)={\rho }_{\mathrm{th}}$ with ${\beta }_R=30$. Inset: ${\rho }_S\left(0\right)$ has the same diagonal elements with ${\rho }_{\mathrm{th}}$ but sorted in an increasing order with respect to an ordered energy basis with increasing eigenenergies. We set $\mathrm{\Omega }=2\pi$ MHz as the unit and adopt experimental values $({\mathrm{\Omega }}_2,\omega ,\gamma )=2\pi \times (0.02,0.01,0.03)$ MHz [56].

Figure 2

Validating Eq. (10) with an initial thermal state ${\rho }_S\left(0\right)=e^{-\beta H_S\left(0\right)}/\mathrm{Tr}\left[e^{-\beta H_S\left(0\right)}\right]$. Upper panel: The dissipated heat $Q\left(t\right)$ (blue solid curve), the derived upper bound ${\tilde{\mathcal{Q}}}_u\left(t\right)+W\left(t\right)$ (red dashed curve), the LP bound $-T\mathrm{\Delta }S\left(t\right)$ (green dash-dotted curve), and the quantum coherence contribution $T_R\left(0\right)\mathrm{\Delta }\mathrm{Coh}\left(t\right)$ (right axis, magenta dotted curve). Lower panel: The time-dependent inverse reference temperature ${\beta }_R\left(t\right)$ (left axis), and the work $W\left(t\right)$ and $-{\tilde{\mathcal{Q}}}_u\left(t\right)$ (right axis). Parameters are $\gamma =0.2$, $\beta =T^{-1}=1$, ${\varepsilon }_0=0.4$, ${\varepsilon }_{\tau }=10$, and $\tau =10$.