Quantum Thermodynamic Uncertainty Relations

• Quantum thermodynamic uncertainty relations are rigorous inequalities that link the fluctuations of observables with a dissipative cost in quantum systems.
• They incorporate quantum coherence, noncommutativity, and quantum fluctuations using metrics like quantum relative entropy, WYD skew information, and generalized χ²-divergences.
• These relations inform the design of nanoscale thermometers, quantum transport devices, and thermal machines by setting ultimate precision limits.

Quantum thermodynamic uncertainty relations (TURs) constitute a set of rigorous inequalities that quantify trade-offs between the precision of thermodynamic observables and the associated dissipation in quantum systems. These relations generalize classical TURs, accounting for quantum coherence, strong system–environment interactions, noncommutativity, and irreducible quantum fluctuations, and are formulated using advanced concepts such as the quantum relative entropy, the Wigner–Yanase–Dyson skew information, generalized χ²-divergences, and quasiprobabilities. Quantum TURs underpin the design and analysis of nanoscale thermometers, quantum transport devices, and thermodynamic machines, and delineate the ultimate precision limits imposed by quantum mechanics.

1. Formal Definitions and Framework

Quantum TURs relate the fluctuations (variance) of a thermodynamic observable, such as a current, energy, or entropy production, to a dissipative cost (e.g., quantum entropy production, heat, or dynamical activity) through inequalities rooted in quantum information theory and estimation theory.

A canonical quantum TUR for entropy production is given by

$$\Sigma := S(\rho'_{SE} || \rho'_S \otimes \rho_E) = \operatorname{tr}[\rho'_{SE} (\ln \rho'_{SE} - \ln (\rho'_S \otimes \rho_E))]$$

where $\rho'_{SE}$ is the final joint state of system and environment after a unitary evolution, and $\rho'_S \otimes \rho_E$ is the effective backward state. The quantum relative entropy $S(\cdot\|\cdot)$ quantifies irreversibility, and in the absence of coherence reduces to the classical limit (Salazar, 28 Apr 2024, Salazar, 2023).

A general quantum TUR lower-bounds $\Sigma$ by a function of the mean and variance of a quantum observable $\hat{\theta}$ acting on the composite space: $$\Sigma \geq F(\langle \hat{\theta} \rangle_\rho - \langle \hat{\theta} \rangle_\sigma, \langle\langle\hat{\theta}\rangle\rangle_\rho, \langle\langle\hat{\theta}\rangle\rangle_\sigma)$$ with

$$F(x, y, z) = \int_0^1 \frac{\lambda x^2}{(1-\lambda)y + \lambda z + (1-\lambda)\lambda x^2} d\lambda.$$

This result imposes a lower bound on entropy production for any observable, generalizing stochastic thermodynamic TURs (Salazar, 28 Apr 2024).

2. Quantum Fluctuations, Noncommutativity, and Generalizations

Quantum TURs intrinsically depend on operator noncommutativity and irreducible quantum noise inherent to quantum states. Under strong coupling or in the presence of quantum coherence (noncommutativity between the effective energy operator and the system state), additional quantum fluctuations increase the lower bound for estimation errors.

For temperature estimation, the uncertainty in inverse temperature is given by (Miller et al., 2018): $$\Delta\beta \geq \frac{1}{\sqrt{ \Delta \hat{H}_*^2 - Q[\hat{\pi}, \hat{H}_*] } }$$ where $\hat{H}_*$ is the effective energy operator, $\hat{\pi}$ is the system state, and $Q[\hat{\pi}, \hat{H}_*]$ is the mean Wigner–Yanase–Dyson (WYD) skew information: $$Q[\hat{\pi}, \hat{H}_*] = \int_0^1 da\, Q_a[\hat{\pi}, \hat{H}_*], \qquad Q_a[\hat{\pi}, \hat{H}_*] = -\frac{1}{2} \operatorname{tr}\left([\hat{H}_*, \hat{\pi}^a][\hat{H}_*, \hat{\pi}^{1-a}]\right)$$ The total variance splits into a classical (commuting) part and a quantum (skew) part: $$\operatorname{Var}[\rho, A] = K[\rho, A] + Q[\rho, A]$$ As $Q[\rho, A]$ increases due to enhanced coherence or strong coupling, the achievable precision in thermometric estimation is further limited (Miller et al., 2018, Abuali et al., 2021).

Quantum generalizations of the classical χ²-divergence also underpin tight bounds: $$\chi^2_\lambda[\rho, \sigma] \geq \frac{\lambda^2 (\langle \hat{\theta} \rangle_\rho - \langle \hat{\theta} \rangle_\sigma)^2}{\lambda \langle\langle \hat{\theta} \rangle\rangle_\rho + (1-\lambda)\langle\langle \hat{\theta} \rangle\rangle_\sigma + \lambda(1-\lambda)(\langle \hat{\theta} \rangle_\rho - \langle \hat{\theta} \rangle_\sigma)^2}$$ Integration over $\lambda$ in the above yields a lower bound on the quantum relative entropy, and hence on entropy production, entirely in terms of observable statistics (Salazar, 28 Apr 2024).

3. Operator-Based TURs and Generalized Uncertainty Relations

Quantum thermodynamic currents—such as work, heat, and internal energy rates—can be promoted to well-defined Hermitian operators, yielding a full quantum operator formalism for TURs (Sathe et al., 17 Jun 2024). For a Hamiltonian $H_S(t)$ and interaction $V_{SE}$, power, heat rate, and internal energy rate operators are: $$\dot{W}(t) = \frac{d H_S(t)}{dt}, \quad \dot{Q}(t) = -\frac{i}{\hbar}[H_S(t), V_{SE}], \quad \dot{U}(t) = \dot{W}(t) + \dot{Q}(t)$$ Their noncommutativity yields strictly quantum constraints on fluctuations via the Robertson–Schrödinger uncertainty relation: $$\sigma_A^2 \sigma_B^2 \geq \frac{1}{4}|\langle [A, B] \rangle|^2 + |\operatorname{cov}(A, B)|^2$$ For example, in a quantum battery protocol, the energy-power uncertainty is: $$\sigma_{E_B}^2 \sigma_{P_B^c}^2 \geq \frac{1}{4\hbar^2}|\langle [H_0, [H_0,V_S]] \rangle|^2 + |\operatorname{cov}(H_0, -\frac{i}{\hbar}[H_0, V_S])|^2$$ These relations enforce trade-offs between energy storage precision and charging rate, imposing constraints that have no classical analog (Sathe et al., 17 Jun 2024, Dong et al., 2022).

4. Quantum TUR Violation: Physical Mechanisms and Model Systems

Quantum systems can violate the classical TUR lower bound (commonly $\mathcal{Q}_{\text{min}}=2$), enabled by coherence, strong coupling, and non-Markovianity.

In field-driven two-level systems, the minimal achievable TUR product is reduced due to quantum coherence: $$\mathcal{Q}^{\rm(TLS)} = \mathcal{A}\coth(\mathcal{A}/2)\left(1 + 2\rho_R^2 - 6\rho_I^2\right)$$ with a quantum limit $\mathcal{Q}_{\rm min}^{\rm(TLS)} \approx 1.25$ (Cho et al., 2 May 2025). For two quantum-coupled qubits, further reductions are possible, with a minimal bound reaching $\mathcal{Q}_{\rm min}^{\rm(TQS)} \approx 1.36$ under strong coupling; the imaginary part of quantum coherences directly controls current generation and fluctuation suppression.

Violations also occur in quantum collisional models, where discrete system–environment collisions induce non-Markovianity. Classical TURs are violated once excess auxiliary collisions generate strong memory effects or when auxiliaries interact, but quantum-corrected TURs (depending on quantum dynamical activity and coherence) remain respected (Maity et al., 31 Dec 2024). Empirically, experimental studies on two-qubit NMR systems demonstrate that generalized TURs based on fluctuation symmetry remain valid even as specialized TURs break down under strong nonequilibrium or non-Gaussian conditions (Pal et al., 2019).

5. Alternative and Complementary Quantum TUR Formulations

Quantum TUR formulations have been developed using information-theoretic metrics and quantum estimation theory, expanding their generality:

• Quantum Relative Entropy TUR: The uncertainty of any observable $\hat{\theta}$ is fundamentally lower-bounded by the symmetric quantum relative entropy between two states,

$$U(\hat{\theta}; \rho, \sigma) \geq f(\tilde{S}(\rho,\sigma)), \quad \tilde{S}(\rho, \sigma) = [S(\rho\|\sigma) + S(\sigma\|\rho)]/2$$

enabling TURs for arbitrary processes, including those far from equilibrium and with non-thermal environments (Salazar, 2023).

• Quasiprobability-Based TUR: The Terletsky–Margenau–Hill (TMH) quasiprobability captures quantum dynamical fluctuations (including negativity). The main result,

$$\dot{\epsilon}(\rho(t)) \geq \frac{2|J_X^{(d)}(t)|^2}{m_X(t)}$$

shows that anomalously enhanced fluctuations, possible due to negative quasiprobabilities, are necessary to suppress entropy production below classical bounds; mere quantum coherence is not sufficient (Yoshimura et al., 20 Aug 2025).

• Quantum First Passage and Feedback TURs: TURs have been specialized for first-passage processes in quantum Markov chains using the Loschmidt echo and quantum Fisher information, with violation or tightening governed by the degree of quantum irreversibility (Hasegawa, 2021). For systems under continuous measurement and feedback control, quantum Fisher information–based TURs retain universal validity and reveal how feedback can enhance precision by increasing quantum dynamical activity (Hasegawa, 2023, Hasegawa, 2019).

6. General Mapping Frameworks and Experimental Realizations

The coherent-incoherent correspondence (CIC) framework transfers TURs derived for incoherent (classical-like) Lindblad systems directly to coherent quantum settings under continuous measurement, sidestepping the need for explicit quantum correction terms if certain commutation relations (secular approximation) are satisfied. The mapping ensures that statistical moments of jump observables are identical in both systems, yielding tight, classical-form TURs directly applicable to quantum trajectories (Nishiyama et al., 15 May 2025). This approach simplifies both conceptual understanding and experimental inference of entropy production in open quantum systems.

Recent quantum-computer-based experiments have validated general quantum TURs by simulating open quantum dynamics on real qubit hardware. These platforms confirm the lower bounds—even under realistic noise—and demonstrate bound saturation for optimal observables, underscoring the applicability of quantum TURs to noisy intermediate-scale quantum (NISQ) devices (Ishida et al., 29 Feb 2024).

7. Classical Limits, Modified TURs, and Open Questions

Quantum TURs uniformly recover classical stochastic thermodynamic TURs in the limit of commuting operators and vanishing quantum coherence (Salazar, 28 Apr 2024, Salazar, 2023, Abuali et al., 2021). When states and observables are diagonal in the same basis, or strong system–environment coupling or memory effects are absent, TURs reduce to their classical forms.

In driven, finite-time quantum thermal machines (such as the Otto cycle), both classical and quantum TURs can be violated in regimes of small entropy production, as non-Gaussian fluctuations and resonance effects dominate. In these cases, uncertainty products can drop below classical lower bounds, making a modified, dynamics-sensitive TUR necessary (Lee et al., 2020). Whether a general tightness criterion exists for all types of quantum TUR violations, and how quantum resources (entanglement, negativity in quasiprobabilities, etc.) can be harnessed for optimal precision-cost trade-offs, constitutes an open line of inquiry.

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This comprehensive theoretical framework elucidates the structure and implications of quantum thermodynamic uncertainty relations, revealing both the unavoidable quantum-imposed constraints on measurement precision and the rich array of physical mechanisms that allow, or prohibit, violation of classical bounds. The interplay between quantum coherence, operator noncommutativity, irreversibility, and their impact on precision-dissipation trade-offs is central to quantum thermodynamics, nanoscale metrology, and emerging quantum technologies.