Information-Thermodynamic Speed Limits

• Information-thermodynamic speed limits are rigorous, geometric bounds that constrain the rate of change in probability distributions by linking them to entropy production and dissipation costs.
• The framework employs the Fisher information metric and statistical manifolds to quantify state distinguishability and trade-offs between protocol duration and thermodynamic cost.
• This unified approach integrates insights from stochastic thermodynamics and quantum dynamics, offering practical implications for optimizing non-equilibrium processes.

Information‐thermodynamic speed limits formalize fundamental trade‐offs between the rate of change of probability distributions over states (or thermodynamic observables) and the associated entropy production or dissipation cost. These bounds are geometric and information-theoretic, operating in both classical and quantum nonequilibrium systems, and are tightly connected to Fisher information, statistical manifolds, and irreversibility. Recent research establishes unified frameworks and sharp inequalities, some of which extend to trajectory level, multipartite networks, relativistic information transfer, and quantum autonomous systems.

1. Information-Geometric Formulation and Fisher Metric

The foundational approach constructs the statistical manifold of probability distributions using the Fisher information metric (Ito, 2017). The infinitesimal line element is

$ds^2 = \sum_{x=0}^n \frac{(dp_x)^2}{p_x}$

Over a protocol of duration $\tau$, the statistical length quantifies the path traversed in probability space:

$$\mathcal{L} = \int_0^\tau \sqrt{\frac{ds^2}{dt^2}}\, dt$$

The geodesic distance (minimal information-geometric distance) between initial and final distributions $r(0)=\sqrt{p(0)}$, $r(\tau)=\sqrt{p(\tau)}$ is

$$\mathcal{D} = 2\,\arccos\bigl(r(0)\cdot r(\tau)\bigr)$$

The Fisher information metric directly controls the speed limit and quantifies the distinguishability rate of probability distributions in time (Nishiyama et al., 7 Apr 2025).

2. Speed–Cost Inequality and Physical Trade-Offs

Central to the topic is the inequality connecting the transition time, statistical length, and thermodynamic cost (Ito, 2017):

$$\tau \ge \frac{\mathcal{L}^2}{2\,\mathcal{C}} \quad\Longrightarrow\quad \tau \ge \frac{\mathcal{D}^2}{2\,\mathcal{C}}$$

where

$$\mathcal{C} = \frac{1}{2} \int_0^\tau \frac{ds^2}{dt^2} dt$$

The cost $\mathcal{C}$ is the integrated rate of entropy production (or dissipation). These bounds imply that decreasing protocol time $\tau$ at fixed statistical (informational) displacement $\mathcal{D}$ requires increasing entropy production. Equality $\tau=\mathcal{L}^2/(2 \mathcal{C})$ is achieved for constant-speed (time-independent Fisher metric rate) protocols. For near equilibrium, $\mathcal{C}$ is approximately the time-integrated bath entropy, recovering classical analogs of the Mandelstam–Tamm quantum speed limit.

3. Universal Information–Thermodynamic Bound and Extensions

Tighter universal bounds are obtained via Kullback–Leibler divergences and trajectory-level statistics (Vo et al., 2020). For a process with forward distribution $P^F(\omega)$ and backward $P^B(\omega^\dagger)$:

$$\Sigma_F = D[P^F(\omega)\|P^B(\omega^\dagger)] \ge D[P^F(\phi)\|P^B(\phi)]$$

Specializing to Markov jump processes, the classical speed limit (CSL) derived is

$$\tau \ge \frac{L^2}{2\,\Sigma^{HS}\langle A \rangle_\tau}$$

with $L$ the total-variation distance between endpoints, $\Sigma^{HS}$ the Hatano–Sasa entropy production, and $A$ the dynamical activity. These bounds are tighter than previous results and relate information-theoretic distinguishability (distance between forward and backward process statistics) to thermodynamic irreversibility.

4. Temporal Fisher Information and Unified Bounds

Temporal Fisher information $\mathcal{I}_t(t)$ quantifies the instantaneous rate of change of probability distributions (Nishiyama et al., 7 Apr 2025):

$$\mathcal{I}_t(t) = \sum_i p_i(t) \bigl(d_t\ln p_i(t)\bigr)^2$$

It admits upper bounds in terms of entropy production:

$\mathcal{I}_t(t) \le \frac{\Sigma(t)}{2 t^2}$

and geometric lower bounds via Bhattacharyya angle:

$$\frac{1}{2}\int_0^\tau \sqrt{\mathcal{I}_t(t)} \,dt \ge \mathcal{L}_P(P(0), P(\tau))$$

Combining, one arrives at trade-off relations:

$$\tau \ge \frac{2\,\mathcal{L}_P(P(0), P(\tau))}{\overline{\sqrt{\Lambda(t)}}}$$

These unify classical and quantum speed limits, making the temporal Fisher information a central resource for both speed and precision bounds.

5. Trajectory-Level and Hierarchical Speed Limits

Single-trajectory stochastic Fisher information (SFI) quantifies the entropic “velocity” at the realization level (Melo et al., 29 Apr 2025):

$\iota_x(t) = \left(\partial_t\ln p_x(t)\right)^2$

The corresponding speed limit at trajectory level is

$\Delta t \ge \frac{\ell^2[x(t)]}{\,j[x(t)]}$

where $\ell[x(t)]$ is the stochastic length and $j[x(t)]$ the action along trajectory $x(t)$. Averaging over ensembles establishes a hierarchy of bounds:

$$\Delta t \ge \frac{\mathcal{L}^2}{\mathcal{J}} \ge \frac{\langle\ell^2\rangle}{\mathcal{J}} \ge \frac{\langle\ell\rangle^2}{\mathcal{J}}$$

This structure shows not all trajectories saturate the tightest average bound, which is always the conventional FI-based speed limit.

6. Generalized Activity and Infinite Family of Speed Limits

By parameterizing “activity” through generalized means of forward and backward fluxes, Nagayama et al. derive an infinite family of thermodynamic speed limits (Nagayama et al., 30 Dec 2024). For homogeneous symmetric mean $m(a, b)$, total activity $\mu_m$, and current–force $\Psi_m$:

$$\left\langle \sigma \right\rangle_\tau \ge \left\langle v_1 \right\rangle_\tau \Psi_m^{-1}\left(\frac{\left\langle v_1 \right\rangle_\tau}{\left\langle \mu_m \right\rangle_\tau}\right)$$

Choosing different means yields distinct lower bounds on entropy production, each with different metrics and tightness properties. The framework subsumes previously known speed limits and exhibits systematic achievability, with clear geometric relations to optimal transport and Fisher–Rao structures.

7. Quantum Thermodynamic Speed Limits

Quantum systems admit speed limits bounding the evolution of system/memory density operators, incorporating Hilbert-space dimensions and Schatten $p$-norms (Tang et al., 12 Nov 2025):

$T^\star_p = \frac{d_s^{1-1/p}\ln d_s\,\Lambda_s^{(p)}T_s^{(p)} + d_m^{1-1/p}\ln d_m\,\Lambda_m^{(p)}T_m^{(p)} {d_s^{1-1/p}\ln d_s\,\Lambda_s^{(p)} + d_m^{1-1/p}\ln d_m\,\Lambda_m^{(p)}}$

These QTSLs are derived from autonomous Hamiltonian frameworks with catalytic constraints. The associated dynamical Landauer’s bound relates entropy changes, heat flow, and the QTSL time, providing operational interpretations in quantum hypothesis testing.

8. Extensions: Relativistic, Multipartite, and Highly Irreversible Regimes

Speed limits are extended to relativistic contexts where critical behaviors emerge (Tsuruyama, 3 Jul 2025). Under Lorentz transformations, Kullback–Leibler divergence and Fisher information diverge as the sender’s velocity approaches $c$, imposing a critical velocity beyond which decoding becomes thermodynamically impossible.

Multipartite networked systems exhibit sharper bounds than monolithic analogs (Tasnim et al., 2021). Refined inequalities account for subsystem constraint structure, quantifying the minimal entropy production required for distributed information processing.

For systems with unidirectional (resetting) transitions, the speed limit captures contributions from resetting entropy production and admits optimization refinements (Gupta et al., 2020). Tight finite-time Landauer bounds reveal sharply increased dissipation during highly irreversible operations.

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In summary, information-thermodynamic speed limits express rigorous, geometric inequalities that bound the rate and precision of state transformations in terms of entropy production, Fisher information, and activity metrics. These bounds unify notions from stochastic thermodynamics, information geometry, and quantum open systems, with extensions to trajectory statistics, networked dynamics, relativistic constraints, and adaptive protocols. The interplay between statistical distance, information fluctuation, and dissipation cost is now established as a core principle governing non-equilibrium system evolution.