Quantum Thermodynamic Speed Limits

• Quantum thermodynamic speed limits are rigorous bounds that determine the minimum time for a quantum system to evolve based on energy fluctuations and thermodynamic metrics.
• They generalize classic Mandelstam–Tamm and Margolus–Levitin limits to incorporate dissipation, decoherence, and feedback in open quantum dynamics.
• These speed limits offer practical insights for optimizing energy conversion, quantum control, and error correction by linking state geometry to dynamic thermodynamic costs.

Quantum thermodynamic speed limits (QTSLs) formalize the ultimate lower bound on the time required for a quantum system to undergo a specified transformation, with explicit dependence on thermodynamic quantities such as energy fluctuations, entropy production, and dynamical activity. QTSLs subsume the classical Mandelstam–Tamm and Margolus–Levitin bounds but rigorously generalize them to incorporate dissipation, decoherence, feedback, and non-Hermitian system-bath interactions. These inequalities are foundational in the paper of open quantum dynamics, energy conversion, metrology, and quantum information processing, relating the geometric structure of state space to thermodynamic resources.

1. Foundational Quantum Speed Limits and Their Thermodynamic Extensions

The archetypal quantum speed limits for isolated systems are the Mandelstam–Tamm (MT) and Margolus–Levitin (ML) bounds. For pure-state, closed-unitary dynamics, the MT bound is

$$\tau \ge \frac{\mathcal{L}_D(\psi_0,\psi_\tau)}{\Delta E}$$

where $\mathcal{L}_D$ is the Bures or Fubini–Study angle, and $\Delta E$ is the energy variance. The ML bound reads

$$\tau \ge \frac{\hbar}{\langle E \rangle - E_\text{gs}} \sin^2 \mathcal{L}_D$$

where $E_\text{gs}$ is the ground-state energy.

Quantum thermodynamic speed limits generalize these bounds to open (dissipative) quantum systems, described by non-Hermitian or Lindblad-type generators, by incorporating additional terms for entropy production and dynamical activity. For Lindblad evolution (Funo et al., 2018), the main thermodynamic speed limit is

$$\tau \geq \frac{T(\rho(0),\rho(\tau))}{\hbar^{-1}\Delta E_\tau + \hbar^{-1}\Delta E_{D,\tau} + \tau^{-1}\int_0^\tau \sqrt{\dot{\sigma}(t)A(t)}dt}$$

with $T(\rho(0),\rho(\tau))$ a trace or Bures distance, $\Delta E_\tau$ and $\Delta E_{D,\tau}$ the time-averaged variances of the Hamiltonian and bath-induced counterdiabatic Hamiltonian, $\dot{\sigma}(t)$ the entropy production rate, and $A(t)$ the dynamical activity. This formalism rigorously encodes the minimum time for a finite system to traverse a specified path in state space given its thermodynamic budget.

2. Non-Hermitian Dynamics and Continuous Measurement Formalism

Speed limits for dynamics governed by non-Hermitian Hamiltonians, which describe open systems and quantum jump processes, retain the MT and ML form but operate on normalized purifications of the system (Nishiyama et al., 25 Apr 2024). The instantaneous speed operator is

$$\Delta\mathcal H(t) = \sqrt{\langle\mathcal H^\dagger(t)\mathcal H(t)\rangle_t - |\langle\mathcal H(t)\rangle_t|^2}$$

yielding the MT-type bound

$$\int_{\tau_1}^{\tau_2}dt\,\Delta\mathcal H(t) \geq \mathcal L_D(\widetilde\rho(\tau_1),\widetilde\rho(\tau_2))$$

and the ML-type bound (for time-independent $\mathcal{H}$)

$$2\,\sin^2\left(\frac{\mathcal L_D}{2}\right) \leq \tau(\langle H\rangle-E_g+\langle\Gamma\rangle)$$

with $\Gamma$ the decay (anti-Hermitian) part of $\mathcal{H}$.

Continuous matrix product state (cMPS) methods generalize these bounds to quantum trajectory ensembles: $$\tau \geq \frac{\mathcal L_D(\rho(0),\rho(\tau))}{\overline{\mathcal V}}$$ where the average “velocity” $\overline{\mathcal V}$ is determined by the quantum Fisher information and dynamical activity of the joint system–environment trajectory (Hasegawa, 2022, Yunoki et al., 13 Feb 2025).

3. Thermodynamic Uncertainty Relations and Speed-Activity Trade-offs

Quantum thermodynamic speed limits tightly interconnect with thermodynamic uncertainty relations (TURs). For any observable current $\mathcal{C}$, the MT-type TUR for non-Hermitian evolution is

$$\left[\tan\Theta_{\rm MT}\right]^2 \geq \left(\frac{\langle\mathcal C\rangle(\tau_2)-\langle\mathcal C\rangle(\tau_1)}{\Delta\mathcal C(\tau_2)+\Delta\mathcal C(\tau_1)}\right)^2$$

and analogously for the ML-type (Nishiyama et al., 25 Apr 2024): $$\frac{1}{\left[e^{-\langle\Gamma\rangle\tau}-\tau(\langle H\rangle-E_g)\right]^2}-1 \geq \left(\frac{\langle\mathcal C\rangle(\tau)-\langle\mathcal C\rangle(0)}{\Delta\mathcal C(\tau)+\Delta\mathcal C(0)}\right)^2$$ which matches the structure of the TUR for classical Markov processes. In open quantum systems, these bounds quantify the fluctuation–dissipation trade-off and link achievable precision with cost (dynamical activity and entropy production).

4. Decomposition into Quantum and Classical Cost Components

Recent advances have split the speed-limit cost into quantum (coherent) and classical (dissipative) contributions. The quantum term is governed by the state’s quantum Fisher information (QFI), measuring U(1) asymmetry: $$I_F(H,\rho) = 2\sum_{j,k} \frac{(p_j-p_k)^2}{p_j+p_k} |\langle j|H|k\rangle|^2$$ This term sets the ultimate speed for coherent transformations, while the classical term is determined by the entropy production rate $\dot{\sigma}$ and “mobility” $M'$ associated with population reshuffling (Sekiguchi et al., 15 Oct 2024): $$\|\dot\rho\|_{Tr} \leq \frac{1}{2\hbar} \sqrt{I_F(H+\mathscr{H}_D,\rho)} + \sqrt{\frac{1}{2}\dot{\sigma} M'}$$ The minimal time for a target state change satisfies

$$\tau_{min} \geq D_T(\rho_0,\rho_\tau) \big/ \left[\frac{1}{2\hbar}\langle \sqrt{I_F} \rangle + \langle \sqrt{\dot{\sigma} M'} \rangle\right]$$

This decomposition clarifies the quantum enhancement in speed relative to classical Markovian processes and provides tighter bounds via resource theory of asymmetry.

5. Non-Markovian, Feedback, and Many-Body Generalizations

Quantum thermodynamic speed limits have been extended to non-Markovian evolution, where memory effects modify transition rates and entropy production (Das et al., 2021). General bounds incorporate renormalized quantum dynamical activity and effective entropy production rates: $$\tau \geq \frac{\|\rho(\tau)-\rho(0)\|_{Tr}}{\langle \Delta E \rangle_{\tau} + \langle \Delta E_{\mathcal{D}} \rangle_\tau + \sqrt{\frac{1}{2}\langle \dot{S}_{tot}^M\rangle_\tau \langle A\rangle_\tau}}$$ Feedback control introduces further enhancements in quantum dynamical activity, shortening the minimal time for error correction and improving precision at fixed thermodynamic cost (Yunoki et al., 13 Feb 2025). All bounds remain valid under physically relevant scenarios including quantum error correction, continuous measurement, and strongly coupled system–bath interactions.

In many-body thermal systems, local time–energy uncertainty relations yield robust speed limits that scale with effective local heat capacity: $$\frac{2\,\sqrt{k_B T^2 C_{v,i}}}{\hbar} \geq \frac{|\langle \dot{Q}_i^H \rangle|}{\sigma_{Q_i}}$$ and bound rates of change of observables, transport coefficients, and collapse times irrespective of interaction range or dimensionality (Nussinov et al., 2021).

6. Practical Applications and Numerical Tightness

QTSLs have been numerically demonstrated to be tight and universal, for example in driven two-level (qubit) systems (Yamauchi et al., 27 Feb 2025). For work extraction far beyond the adiabatic regime, the minimal cost to accelerate non-adiabatic processes is rigorously separated into quantum-coherent and classical-dissipative terms: $$|\dot{W} - \dot{W}_{ad}| \leq \sqrt{4g_{tt}^{QF}V_{\rho_t}([\dot{H}_t])} + \sqrt{2A(\rho_t)\dot{\sigma}}$$ Separate bounds apply to classical and quantum work components, and the overall minimal time remains below explicit numeric bounds in all protocols tested.

Quantum absorption refrigerators exemplify the impact of QTSLs on device performance: the bounding second order cooling rate (BSOCR) encapsulates the trade-off between steady cooling rate and approach time; initial coherence and strong interaction enhance BSOCR but cannot violate the fundamental speed constraint (Mukhopadhyay et al., 2017).

7. Unification, Outlook, and Open Directions

Quantum thermodynamic speed limits provide a unifying framework for quantum control, nonequilibrium statistical mechanics, quantum thermodynamics, and resource theory. All major forms are rigorously derived from operator semi-norms, geometric state-distance, and contractive metrics, and incorporate both unitary (quantum) and dissipative (classical) contributions.

Outstanding directions include the extension of QTSLs to time-nonlocal generators, systems with long-range interactions, cyclic or driven protocols, and resource-theoretic quantification in quantum information tasks. Current and future bounds will inform the design of faster quantum technologies, elucidate limits on energy conversion, and clarify the operational role of nonclassical resources in thermodynamic tasks.