Quantum information-cost relations and fluctuations beyond thermal environments: A thermodynamic inference approach

TL;DR

This work develops a thermodynamic inference framework based on the maximum entropy principle to bound information costs in arbitrary nonequilibrium quantum systems, including those coupled to non-thermal or unknown environments. It derives two key results: (i) an information-cost upper bound on mean charge loss $- extstyleig( extstyleigsum_i \lambda_i(0) riangle C_i^S(t)ig) le \\mathcal{U}_M(t)$ that complements a generalized Landauer lower bound, and (ii) an information-fluctuation lower bound on the change in variance $ riangle ext{Var}[ ext{C}_i^S](t)$ when second-order observables are available, via a quadratic reference state and inequality $oxed{ ext{Eq. (fluctuation_inequality)}}$. These relations hold at finite times and reduce to the conventional bounds in appropriate limits, with saturation in quasi-static reversible processes. The authors validate the framework through numerical models of coupled qubits, an information-erasing driven qubit, and a driven double quantum dot, illustrating broad applicability to quantum information processing and non-thermal quantum thermodynamics. The approach provides environment-agnostic constraints on thermodynamic costs using only system observables, enabling practical thermodynamic reasoning for nanoscale quantum technologies.

Abstract

The Landauer's principle, a cornerstone of information thermodynamics, provides a fundamental lower bound on the energetic cost of information erasure in terms of the information content change. However, its traditional formulation is largely confined to systems exchanging solely energy with an ideal thermal bath. In this work, we derive general information-cost trade-off relations that go beyond the scope of Landauer's principle by developing a thermodynamic inference approach based on the maximum entropy principle. These relations require only information about the system and are applicable to complex quantum scenarios involving multiple conserved charges and non-thermal environments. Specifically, we present two key results: (i) In scenarios where only the mean values of observables are accessible, we derive an information-content-informed upper bound on the thermodynamic cost which complements an existing generalized Landauer lower bound. (ii) When second-order fluctuations can also be measured, we obtain an information-content-informed lower bound on the change in variances of observables, thereby extending the Landauer's principle to constrain higher-order fluctuation costs. We numerically validate our information-cost trade-off relations using a coupled-qubit system exchanging energy and excitations, a driven qubit implementing an information erasure process, and a driven double quantum dot system that can operate as an inelastic heat engine. Our results underscore the broad utility of maximum-entropy inference in constraining thermodynamic costs for generic finite-time quantum processes, with direct relevance to quantum information processing and quantum thermodynamic applications.

Figures (4)

• Figure 1: Motivation and schematic of the study. (a) The celebrated Landauer's principle--a cornerstone information-cost relation--establishes a lower bound on heat cost $Q$ dissipated into a thermal bath at temperature $T$ in terms of the change in the system’s information content $\Delta S$ with $S=-\mathrm{Tr}[\rho_S\ln\rho_S]$. (b) However, at the nanoscale, quantum systems often couple to complex environments (E) such as non-thermal ones or those with limited experimental accessibility, where temperature becomes ill-defined. This raises a fundamental question: can an information-cost relation still be formulated under such conditions? (c) Here we address the challenge of limited experimental access to microscopic details by developing a thermodynamic inference framework that leverages available information about a set of system observables $\{\mathcal{O}_i\}$. We introduce a reference state $\rho_r$ [Eq. (\ref{['eq:reference_state']})] which is constructed via the maximum entropy principle. The difference between reference information content $S_r=-\mathrm{Tr}[\rho_r\ln\rho_r]$ and the system's actual information content $S$ is just the quantum relative entropy $D[\rho_S||\rho_r]=\mathrm{Tr}[\rho_S(\ln\rho_S-\ln\rho_r)]$ between the reference state $\rho_r$ and the actual state $\rho_S$. With this relation, we can connect expectation values $\langle \mathcal{O}_i\rangle$ (observations) with the system's actual information content $S$ [Eq. (\ref{['eq:equality']})]. We apply the approach to two scenarios that differ in the available system observations: (i) Quantum systems where only mean values $\langle\mathcal{C}_i^S\rangle$ of multiple conserved charges $\{\mathcal{C}_i^S\}$ (energy, particle number, etc) are accessible, for which we establish an information-cost relation between cost $-\Delta\langle\mathcal{C}_i^S\rangle(t)=\langle\mathcal{C}_i^S\rangle(0)-\langle\mathcal{C}_i^S\rangle(t)$ and system actual information content change $\Delta S(t)$ [Eq. (\ref{['eq:Our_ex']})], (ii) Quantum systems where both the mean values $\langle \mathcal{C}_i^S\rangle$ and the variances $\mathrm{Var}[\mathcal{C}_i^S]\equiv\langle (\mathcal{C}_i^S-\langle \mathcal{C}_i^S\rangle)^2\rangle$ are accessible, for which we establish an information-cost relation between fluctuation cost $\Delta\mathrm{Var}[\mathcal{C}_i^S](t)=\mathrm{Var}[\mathcal{C}_i^S](t)-\mathrm{Var}[\mathcal{C}_i^S](0)$ and system actual information content change $\Delta S(t)$ [Eq. (\ref{['eq:var_eq']})].
• Figure 2: (a) Time-dependent results for the weighted cost $-\sum_{i=0}\lambda_i(0)\Delta C_i^S(t)$ (green solid line) and the inferred upper bound $\mathcal{U}_M(t)$ (orange dashed line) given in Eq. (\ref{['eq:Our_ex']}). (b) Deviation $\mathcal{U}_M(t)+\sum_{i=0}\lambda_i(0)\Delta C_i^S(t)$ as a function of time. Parameters are $\beta_A=0.5$, $\beta_B=2$, $\mu_A=0.5$, $\mu_B=1$, $\varepsilon=2$, $\eta=0.2$.
• Figure 3: (a) Results for the energy fluctuation cost $\Delta \mathrm{Var}[H_S](t)$ (green solid line) and the inferred lower bound $\mathcal{L}_v(t)$ (orange solid line) [cf. Eq. (\ref{['eq:var_eq']})] as functions of time. (b) Deviation of $\Delta \mathrm{Var}[H_S](t)-\mathcal{L}_v(t)$. Parameters are $\varepsilon_0=0.4$, $\varepsilon_{\tau}=4$, $\tau=10$, $\gamma_{1,2}=0.2$, and $T_E=0.25$.
• Figure 4: (a) Results for the energy fluctuation cost $\Delta \mathrm{Var}[H_{\rm{DQD}}](t)$ (green solid line) and the inferred lower bound $\mathcal{L}_v(t)$ (orange solid line) [cf. Eq. (\ref{['eq:var_eq']})] as functions of time. (b) Deviation of $\Delta \mathrm{Var}[H_{\rm{DQD}}](t)-\mathcal{L}_v(t)$. Parameters are $\mu_L=\mu_R=0$, $\Omega=1$, $\varepsilon_{L0}=2$, $\varepsilon_{R0}=2$, $\varepsilon_{L\tau}=1.5$, $\varepsilon_{R\tau}=1.5$, $\phi=\pi/10$, $\Gamma_L=\Gamma_R=\Gamma_{\rm ph}=0.1$, $\delta=0.2$, $T_L=0.2$, $T_R=0.3$, $T_{\rm ph}=0.4$ and the initial system Gibbsian state is fixed by setting $T_0=0.125.$