Quantum Thermodynamic Uncertainty Relations

• Quantum Thermodynamic Uncertainty Relations are bounds that quantify the trade-off between precision and dissipation in nonequilibrium systems by linking current fluctuations to entropy production.
• Quantum extensions incorporate coherence, entanglement, and non-Markovian dynamics, enabling scenarios where the TUR bound is reduced by up to 40% compared to classical limits.
• Experimental implementations in quantum conductors and thermal engines demonstrate how TURs can guide the optimization of precision, energy efficiency, and device diagnostics in quantum technologies.

Quantum Thermodynamic Uncertainty Relations (TURs) formalize fundamental lower bounds on the relative fluctuations of thermodynamic observables in terms of entropy production, establishing trade-offs between precision and dissipation in both classical and quantum nonequilibrium systems. Quantum extensions of TURs fundamentally diverge from their classical counterparts by incorporating quantum coherence, entanglement, non-Markovian effects, quasiprobabilistic statistics, and the structure of quantum channels, revealing new mechanisms to surpass the classical precision-dissipation limitations.

1. Classical TUR and Its Quantum Generalizations

Classically, for a stochastic current $J$ with mean $\langle J \rangle$, variance $\mathrm{Var}(J)$, and mean entropy production rate $\sigma$, the TUR is expressed as: $$Q \equiv \sigma\,\frac{\mathrm{Var}(J)}{\langle J \rangle^2} \geq 2$$ where the bound $2$ is saturated for classical Markovian or Langevin processes in the linear response regime. The bound arises from detailed balance and fluctuation theorem symmetry, enforcing that higher precision in estimating $J$ is paid for by increased dissipation.

Quantum generalizations replace classical rates and probabilities with steady-state currents and noise susceptibilities computed via quantum master equations, incorporating coherence, entanglement, and non-classical probability structures. Bounds analogous to $Q_{\min} \geq 2$ persist in certain limits, but can be loosened by quantum effects (Singh et al., 2021, Cho et al., 2 May 2025).

2. Coherence-Induced Loosening of TUR Bounds

Quantum coherence fundamentally modulates TUR bounds. In dissipative two-level systems described by a Lindblad master equation, the imaginary component of the off-diagonal density matrix suppresses current fluctuations, reducing the TUR bound below its classical limit (Singh et al., 2021). Explicitly, for a field-driven TLS,

$$Q^{\mathrm{TLS}} = A\,\coth(A/2)\,\left(1-6\rho_I^2\right)$$

with $A$ the affinity and $\rho_I$ the steady-state coherence. At resonance and optimal drive, $Q_{\min}^{\mathrm{TLS}} \approx 1.25$ ($A^* \approx 3.61$), showing up to $\sim 40\%$ reduction relative to the classical case.

In a quantum-coupled two-qubit system (TQS), inter-qubit coherent pathways introduce additional fluctuating channels. At weak coupling ($K \lesssim 1$), the bound matches the TLS scenario, but under strong coupling and driving ($K \gg 1$, $\Omega \gtrsim 3\Omega_0$), joint coherences yield a looser but higher sub-classical bound: $Q_{\min}^{\mathrm{TQS}} \approx 1.36$. These effects are absent in classical analogs (e.g., coupled oscillators) where $Q_i \geq 2$ strictly, even under synchronization (Cho et al., 2 May 2025).

3. Quantum TUR Violations in Transport and Thermoelectric Devices

Quantum TUR violations manifest in quantum thermoelectric junctions, superconducting contacts, and atomic-scale conductors:

• Non-interacting quantum conductors with coherent transport exhibit TUR violations whenever quantum noise contributions exceed the classical counterpart, which arises when the transmission function $\mathcal{T}(E)$ and energy-level structure allow exchange correlations and coherence beyond sequential tunneling (Liu et al., 2019, Agarwalla et al., 2018). In serial double quantum dot systems, the TUR is violated for intermediate bias and resonance widths; however, the violation vanishes at Carnot efficiency.
• In superconducting junctions, the coexistence of quasiparticle and multiple Andreev reflection processes drives drastic TUR breakdowns (Ohnmacht et al., 2 Aug 2024). The ratio $Q$ can fall far below $2$, scaling super-extensively with the order of MAR processes, and is amplified by transparency, low temperature, and non-Markovian dynamics. These regimes enable the construction of quantum thermal engines with low dissipation and high stability.
• Optimization of transmission functions in the Landauer-Büttiker framework reveals that boxcar-like energy-filtered transmission profiles minimize current fluctuations for fixed power and efficiency, producing arbitrarily large TUR violations at large bias or temperature differences, which are unattainable classically (Timpanaro et al., 2021).

4. TURs from Fluctuation Theorems and Quantum Relative Entropy

Exchange fluctuation theorems (EFTs) in quantum systems yield matrix-valued and saturable scalar TURs that generalize classical bounds: $$\frac{\mathrm{Var}(Q)}{\langle Q \rangle^2} \geq f(\langle \Sigma \rangle)$$ with $f(x) = \mathrm{csch}^2(g(x/2))$, where $g$ is the inverse function of $x = u \tanh u$. This relation is tight, saturable, and applies far from equilibrium, encompassing non-Markovian and non-stationary processes (Timpanaro et al., 2019, Timpanaro, 15 Jul 2024, Moustos et al., 12 Nov 2025). Covariances and cross-fluctuations obey corresponding matrix inequalities.

The quantum relative entropy uncertainty relation further generalizes TURs using quantum divergences: $$\frac{\mathrm{Var}_\rho(A) + \mathrm{Var}_\sigma(A)}{(1/2)(\langle A \rangle_\rho - \langle A \rangle_\sigma)^2} \geq f\Big(\frac{1}{2} [S(\rho\|\sigma) + S(\sigma\|\rho)]\Big)$$ where $A$ is a Hermitian observable, tracing the bound to the symmetry and distinguishability of quantum states. Limits recover the classical TUR and quantum Cramér–Rao inequalities for parameter estimation (Salazar, 2023, Salazar, 28 Apr 2024).

5. Non-Markovianity, Synchronization, and Quasiprobabilistic Quantum TURs

Quantum TUR violations are amplified by non-Markovianity, bath correlations, and synchronization protocols. In quantum collisional models, strong memory effects and auxiliary correlations produce both smooth (collision-duration driven) and sharp (bath-correlated) TUR violations below the classical bound, reaching minima $\mathcal{Q}_{\min} \approx 1.6$ (Maity et al., 31 Dec 2024). Nevertheless, the QTUR bound based on dynamical activity and coherence remains respected.

Periodic synchronization of coupled quantum oscillators driven by modulated non-Markovian baths produces strong (up to order-of-magnitude) TUR violations for local currents, enabled by collective mode formation and anti-phase correlations, significantly reducing the dissipation required for finite output power (Razzoli et al., 25 Apr 2024).

The Terletsky–Margenau–Hill quasiprobability approach establishes a quantum TUR for arbitrary observables, revealing that negativity in quasiprobabilities—not coherence alone—is necessary to beat classical dissipation limits, as dissipationless currents at large degeneracy emerge only when TMH fluxes attain strong negativity (Yoshimura et al., 20 Aug 2025).

6. Experimental Implementations and Applications

Quantum TURs have been validated in NMR SWAP engines (Shende et al., 21 Oct 2024, Pal et al., 2019), quantum Ising model drives (Motta et al., 5 Mar 2025), atomic-scale quantum conductors (Friedman et al., 2020), and relativistic quantum thermal machines (Moustos et al., 12 Nov 2025). These experiments confirm both generalized and tighter TURs, reveal violation regimes, and measure the impact of quantum resources, such as coherence and quasiprobability negativity, on the precision–dissipation landscape.

Applications include:

• Metrological bounds on quantum sensors and clocks (Kwon et al., 6 Dec 2024)
• Design of quantum heat engines and refrigerators with minimized fluctuations
• Guidance for efficiency–power–precision optimization in quantum nano-technology (Timpanaro et al., 2021, Ohnmacht et al., 2 Aug 2024)
• Quantum device diagnostics via TUR ratio analysis in transport experiments (Friedman et al., 2020)

7. Outlook and Theoretical Extensions

Quantum TUR research continues to expand with developments in:

• Universal matrix-valued TURs incorporating cross-correlations
• TURs for open, strongly correlated, or feedback-controlled quantum systems
• TURs incorporating higher moments and nonstationary protocols (Timpanaro, 15 Jul 2024)
• Interconnections with quantum speed limits, contextuality, and resource theories
• Hybrid quantum–classical computational approaches to large-scale simulations (Motta et al., 5 Mar 2025)

Quantum TURs robustly encode the interplay between fluctuation, dissipation, and irreversibility, placing rigorous constraints on the design, performance, and control of small-scale quantum thermodynamic systems.