Dynamical Landauer Bound

• Dynamical Landauer’s bound is a framework that quantifies the maximum speed of quantum state evolution in open systems by linking entropy production and thermodynamic resources.
• It decomposes the state change rate into classical contributions from entropy production and quantum enhancements measured by the quantum Fisher information.
• This bound informs practical applications in quantum thermodynamics, control protocols, and quantum computing by defining operational limits on speed and efficiency.

The dynamical version of Landauer's bound constitutes a set of rigorous inequalities restricting the speed at which open quantum systems can evolve, based on intrinsic thermodynamic quantities and quantum resource measures. Rather than expressing Landauer's principle as a static lower bound on entropy production for erasure or state transformation, modern theory has developed “dynamical” Landauer-type bounds that constrain the instantaneous velocity of quantum state change in open-system settings. The central advance is a decomposition of the rate of state change into quantum and classical contributions, manifesting as quantifiable speed limits that can be saturated or enhanced by exploiting quantum coherence resources, such as those measured by the quantum Fisher information.

1. Classical Landauer's Bound and Its Dynamical Analogue

The classic form of Landauer's principle states that erasing one bit of information in a system at inverse temperature $\beta$ incurs a minimal entropy production of $\Delta S \geq k_B \ln 2$, which operationally corresponds to a minimal heat dissipated $\Delta Q \geq k_B T \ln 2$ [Landauer 1961]. In dynamical quantum evolution, this is generalized: instead of bounding total entropy for a process, one bounds the rate of entropy production together with the rate of change of the system state, specifically characterized by contractive metrics such as the trace distance or Bures angle.

For a quantum open system governed by a GKSL master equation,

$$\dot{\rho} = -\frac{i}{\hbar}[H, \rho] + \mathcal{D}[\rho],$$

the instantaneous entropy production rate is given by

$$\dot{\sigma} = \dot{S} - \beta \dot{Q},\quad \dot{S} = -\operatorname{Tr}[\dot{\rho} \ln \rho],\quad \dot{Q} = \operatorname{Tr}[H \dot{\rho}].$$

Speed limits can then be formulated as bounds on $\|\dot{\rho}\|_\text{Tr}$ in terms of $\dot{\sigma}$ and system-specific activity and mobility parameters.

2. Entropy-Production-Based Speed Limits

Entropy production, together with activity (the sum over transition rates times populations), has provided the foundation for classical dynamical Landauer bounds. The Funo–Shiraishi–Saito (FSS) and Vu–Saito (VS) inequalities,

$$\|\dot{\rho}\|_\text{Tr} \leq \frac{1}{\hbar}\left[\sqrt{V_\rho(H)} + \sqrt{V_\rho(H_\mathcal{D})}\right] + \sqrt{\frac{1}{2}\dot{\sigma} M},$$

where $V_\rho(H)$ and $M$ are, respectively, the variance of the Hamiltonian in state $\rho$ and the mobility constructed from transition rates, represent the standard entropy-based speed limits [New J. Phys. 21, 013006 (2019); Phys. Rev. X 13, 011013 (2023)]. These bounds translate the irreversibility inherent in open-system evolution into a direct constraint on how rapidly a quantum state can move away from its initial configuration.

3. Quantum Coherence, Resource Theory of Asymmetry, and Fisher Information

Recent advances, culminating in Sekiguchi et al. (Sekiguchi et al., 15 Oct 2024), show that standard entropy-based bounds are generically loose, especially when quantum coherence (asymmetry with respect to energy) contributes substantially. By combining entropy–production theory with the resource theory of asymmetry (RTA), one decomposes the evolution rate into classical and quantum components. The purely quantum contribution is governed by the quantum Fisher information,

$$I_F(H, \rho) = 2 \sum_{j,k} \frac{(p_j - p_k)^2}{p_j + p_k} |\langle j|H|k\rangle|^2,$$

which quantifies quantum fluctuations of energy, i.e., the extent to which $\rho$ exhibits coherence between different energy eigenstates.

This leads to the improved dynamical Landauer bound: $$\|\dot{\rho}\|_\text{Tr} \leq \frac{1}{2\hbar} \sqrt{I_F(H+H_\mathcal{D}, \rho)} + \sqrt{\frac{1}{2}\dot{\sigma} M'},$$ where $M'$ is a mobility-like term restricted to off-diagonal transitions. This inequality is strictly tighter than previous forms, isolating quantum enhancement of speed as a function of available coherence.

4. Integrated Bounds, Minimal Time, and State Distances

The dynamical version of Landauer's bound naturally extends to finite-time transformations. One considers a contractive distance $D(\rho_0, \rho_\tau)$—typically trace distance or Bures angle—and integrates the instantaneous speed limit over time: $$D(\rho_0, \rho_\tau) \leq \int_0^\tau \|\dot{\rho}_t\|_\text{Tr}\,dt \leq \int_0^\tau \left[\frac{1}{2\hbar}\sqrt{I_F(H+H_\mathcal{D}, \rho_t)} + \sqrt{\frac{1}{2} \dot{\sigma}(t) M'(t)}\right]dt,$$ yielding a minimal-time bound

$$\tau \geq \frac{D(\rho_0, \rho_\tau)}{(1/2\hbar)\langle \sqrt{I_F(H+H_\mathcal{D})} \rangle + \langle \sqrt{\frac{1}{2}\dot{\sigma} M'} \rangle}.$$

This form rigorously restricts the speed of open system evolution, interpolating smoothly between the closed-system Mandelstam–Tamm or Margolus–Levitin bounds (when classical dissipation vanishes) and entropy-driven classical limits (when quantum coherence is absent).

5. Physical Interpretation and the Quantum-Classical Transition

In the dynamical Landauer bound, the speed of quantum evolution is split into two additive factors:

• Quantum Fisher information term: scales with the system's off-diagonal coherence in the energy basis. When $[\rho, H]=0$, quantum Fisher information vanishes and the bound reduces to entropy-activity-mobility terms—recovering classical stochastic speed limits.
• Entropy-production term: captures the classical irreversibility imposed by the environment; its magnitude is set by bath coupling strength and the mobility of population transitions.

Thus, as quantum coherence (asymmetry) increases, the speed bound is quantum-enhanced, enabling evolution rates that outperform any classical protocol subject only to entropy and activity constraints.

6. Observable Currents and Generalizations

A direct corollary of the dynamical version is a bound on the time-derivative of expectation values for arbitrary observables,

$$\left| \frac{d}{dt} \langle X \rangle \right| \leq \frac{1}{\hbar}\sqrt{I_F(H+H_\mathcal{D}, \rho) V_\rho(X)} + \sqrt{\frac{1}{2}\dot{\sigma} M_X},$$

where $V_\rho(X)$ is the variance of $X$ in $\rho$, and $M_X$ is the mobility for $X$. In the absence of dissipation, this reproduces the Mandelstam–Tamm bound for isolated systems.

7. Implications for Quantum Thermodynamics, Control, and Computation

The dynamical Landauer-type bounds offer a unified perspective for the design and analysis of quantum thermal machines, control protocols, quantum information processing, and error correction. They clarify the resources necessary to attain minimal operation times for state conversion, work extraction, or dissipation-limited transformations. Only genuine quantum coherence—quantified by the quantum Fisher information—is a faithful resource for “breaking” the classical Landauer limit, allowing protocols that outpace any classical (purely entropic) irreversibility. These results are essential both for understanding the thermodynamic cost of quantum information processing and for bounding the efficiency-speed trade-offs in practical quantum technologies.

In contemporary quantum thermodynamics, the dynamical version of Landauer’s bound thus represents a stringent, physically interpretable constraint—expressed as a time-differentiated inequality—on the rate of state change, with precise classical and quantum origins, generalizing the foundational principle to arbitrary driven, open quantum evolutions (Sekiguchi et al., 15 Oct 2024).