Quantum TUR Violations in Transport

• Quantum TUR violations in transport are defined as the breakdown of classical bounds on noise-to-current ratios due to quantum coherence and multi-charge transfer processes.
• Full-counting statistics reveal that mechanisms like Andreev reflections, non-Markovian dynamics, and coherent multi-electron events suppress fluctuations and enhance precision-dissipation trade-offs.
• These insights inform the design of quantum thermal machines and sensors, enabling operational regimes that outperform classical limitations in efficiency and accuracy.

Quantum TUR violations in transport refer to the breakdown of the classical thermodynamic uncertainty relation (TUR) in quantum coherent conductors. The TUR sets a lower bound on the product of the relative fluctuations of particle or energy currents and the thermodynamic cost (entropy production) in steady-state nonequilibrium systems. While classically this bound is strictly enforced by Markovian stochastic dynamics, quantum transport systems can violate the TUR, especially in the presence of quantum coherence, non-Markovian dynamics, or multi-particle charge transfer processes such as Andreev and multiple Andreev reflections. These violations have profound implications for understanding the limits of precision and efficiency in quantum thermal machines and quantum electronic devices.

1. Formulation of the Thermodynamic Uncertainty Relation in Quantum Transport

The TUR is conventionally stated for a steady-state current $I$ with variance (zero-frequency noise) $S$, and entropy production rate $\Sigma$: $\frac{S}{I^2}\,\Sigma \ge 2k_B,$ with $k_B$ Boltzmann's constant. For an isothermal voltage-biased setup at temperature $T$ and bias $V$, $\Sigma = IV/T$ and the TUR becomes: $\frac{S}{I} \frac{V}{T} \ge 2k_B.$ In classical Markovian processes, this bound is universally satisfied. However, the quantum regime admits steady-state transport via coherent processes and multi-charge events not captured by the classical stochastic picture. Evaluation of $I$ and $S$0 in this context employs the full-counting statistics (FCS) framework, encoding the charge transfer probability $S$1 for $S$2-electron processes at energy $S$3, leading to: $S$4 Classical TUR violation occurs whenever this FCS-generated TUR ratio $S$5 drops below the classical bound $S$6 (Ohnmacht et al., 2024, Mayo et al., 3 Jun 2025).

2. Mechanisms and Physical Origins of TUR Violation

The breakdown of the TUR in quantum transport arises predominantly from the coexistence of multi-particle transport channels and quantum coherences:

• Multi-charge transfer processes (e.g., Andreev Reflection): In normal-metal–superconductor (NS) and superconductor–superconductor (SS) junctions, Andreev reflection (AR) and multiple Andreev reflections (MAR) allow transfer of $S$7-electron groups (with $S$8), effectively enhancing dissipation relative to shot noise due to quartic weighting of $S$9 in the variance formula. The cross-terms in the FCS further suppress noise, driving the TUR ratio below the classical threshold (Ohnmacht et al., 2024, Ohnmacht et al., 23 Dec 2025).
• Quantum coherence: Macroscopic superconducting coherence results in pair or non-local Cooper-pair mixing (quantum dot-coupled coherent dots), allowing channels that alter current–noise relations in ways inaccessible to classical or single-particle quantum scatterers (Prech et al., 2022, Mayo et al., 3 Jun 2025).
• Non-Markovianity and Dynamical Coherence: Markovian stochastic dynamics enforce TUR compliance, but non-Markovian behavior (e.g., memory effects, retardation, hybrid system-reservoir couplings, non-Abelian conserved quantities) enables regimes where backflow of information and dynamical coherences facilitate violation of classical bounds (Maity et al., 2024, Scandi et al., 21 Aug 2025). Tur violations in such settings can be further enhanced by specifically tuning the degree of non-Markovianity or environmental correlations.

3. Quantitative Conditions and Types of Violating Systems

Violations of the TUR are present in a broad but system-dependent range of quantum transport regimes:

• NS Junctions: TUR violation is achieved only for very high transparency ($\Sigma$0), strongly near gap bias ($\Sigma$1). The effect amplifies as temperature decreases: in the limit $\Sigma$2, exactly at $\Sigma$3, $\Sigma$4 (Ohnmacht et al., 2024).
• SS Junctions: Large violations persist at moderate transparencies ($\Sigma$5–$\Sigma$6), maximized near MAR onset voltages $\Sigma$7, and diverging at perfect transmission and low bias, regularized only by subgap mechanisms. The deepest violation (most negative $\Sigma$8) is achieved at low temperatures, high transparency, and small voltage, where multiple MAR channels are simultaneously active (Ohnmacht et al., 2024).
• Hybrid NSN or CPS Devices: Quantified both for classical and quantum TURs, violation depth correlates with the induced pair amplitude across the dot or nanowire. In the Cooper-pair splitter geometry, nonlocal crossed Andreev reflection (CAR) significantly amplifies the TUR violation compared to local AR alone (Mayo et al., 3 Jun 2025).
• Double Quantum Dot (DQD) and Maser Models: TUR violations are strictly bounded to the regime where coherent tunnel coupling is comparable to or exceeds reservoir-induced decoherence. Entanglement and dynamical coherence (two-time correlation functions) are necessary and sufficient for TUR breaking; in the absence of these, classical bounds are restored (Prech et al., 2022, Kalaee et al., 2021).
• Engineered Transmission Functions: In noninteracting coherent thermoelectric devices, step-like ("boxcar") energy filtering enables the vanishing of noise at finite current and dissipation, causing $\Sigma$9 for large bias—an extreme limit of TUR breakdown. Any realistic broadening, asymmetry, or imperfection restores TUR compliance at sufficiently high affinity (Gerry et al., 2022, Timpanaro et al., 2021).
• Non-Abelian and Noncommuting Charges: Quantum collisional models with non-Abelian conserved quantities exhibit an even further relaxation of the classical TUR, with the allowed current fluctuations bounded below by a denominator including the non-Abelian correction (e.g., Eq. (20) in (Scandi et al., 21 Aug 2025)), whose positive value strictly lowers the relative fluctuation floor.

4. Mathematical Characterization, FCS Structure, and Restoration Mechanisms

The fate of the TUR in quantum systems is fully captured by the FCS description. The key insight is that higher-order cumulants and mixing terms (e.g., $\frac{S}{I^2}\,\Sigma \ge 2k_B,$0 in variance and cross-terms) favor multi-charge transfer for TUR violation. The violation arises in those regions of parameter space where the equilibrium contributions of multi-charge or coherent transport cross the critical threshold ($\frac{S}{I^2}\,\Sigma \ge 2k_B,$1):

• For any process supporting channels up to maximum charge $\frac{S}{I^2}\,\Sigma \ge 2k_B,$2, the physically tightest quantum TUR reads:

$\frac{S}{I^2}\,\Sigma \ge 2k_B,$3

with standard TUR corresponding to $\frac{S}{I^2}\,\Sigma \ge 2k_B,$4; any process with $\frac{S}{I^2}\,\Sigma \ge 2k_B,$5 can violate the single-charge bound (Ohnmacht et al., 23 Dec 2025).

• Quantum coherence is a necessary, though not always sufficient, condition: For the violation of the "quantum TUR" (Landauer–Büttiker, non-interacting phase-coherent bound), macroscopic pairing correlations are essential. Single-particle fermionic coherence alone is insufficient (Mayo et al., 3 Jun 2025).
• Any real-world imperfection (finite lifetime, soft transmission cutoffs, environmental decoherence, inelastic scattering, partial transparency) prevents perfect TUR violation at sufficiently large affinity, restoring the cost-precision tradeoff in the far-from-equilibrium regime (Gerry et al., 2022, Timpanaro et al., 2021).

5. Implications for Quantum Thermal Machines and Quantum Technologies

Quantum TUR violations allow construction of devices with precision-dissipation tradeoff fundamentally superior to classical limitations:

• Superconducting devices: Exploit AR/MAR-induced TUR violations to realize stable quantum heat engines or refrigerators with low entropy production per unit current and vanishingly small current noise. Junctions with high transparency driven near gap edge voltages and at low temperature present the optimal configuration (Ohnmacht et al., 2024).
• Quantum-limited sensors and pumps: Implementing boxcar or step-filtered transmissions supports ultra-precise transport with minimal energetic cost, foundational for next-generation metrological standards and quantum-limited thermoelectric designs (Timpanaro et al., 2021).
• Quantum information and coherence engineering: Devices set at or near the threshold for maximal coherence or with well-controlled nonlocal CAR can act as on-demand sources of nonclassical current fluctuations, offering new operational regimes for quantum control schemes (Mayo et al., 3 Jun 2025, Scandi et al., 21 Aug 2025).
• Experimental verification: The quantitative FCS-derived TUR ratios, Fano factor suppression, and phase diagrams present robust signatures and experimental targets for platform verification in superconducting nanocontacts, quantum dots, masers, and collisional models (Ohnmacht et al., 2024, Prech et al., 2022, Kalaee et al., 2021, Maity et al., 2024).

6. Generalizations, Boundaries, and Theoretical Refinements

All existing quantum TUR violations trace to at least one of:

• The presence of multi-particle processes (raising the minimum attainable noise-to-current ratio);
• Noncommutativity or non-Abelian charges, relaxing the minimum bound via quantum corrections in fluctuation theorems;
• Quantum coherence or memory effects, invalidating local-detailed-balance-imposed trade-offs;
• Hard-filtered or boxcar-like transmission functions, generating sharp deviation from equilibrium FCS structure.

The tightest general bound is always governed by the maximal charge per process and the commutation properties of the conserved quantities. Charge-dependent quantum TURs classify the optimal achievable trade-off for any physically allowed mechanism, with the minimal TUR ratio set by $\frac{S}{I^2}\,\Sigma \ge 2k_B,$6, the largest elementary transport event (Ohnmacht et al., 23 Dec 2025). For all processes contained within $\frac{S}{I^2}\,\Sigma \ge 2k_B,$7 and described by scattering theory, the corresponding modified TUR is provably unbreakable.

A plausible implication is that future extensions of TUR-based precision-dissipation theory in quantum thermodynamics must recognize channel-dependent and operator algebraic structure (such as degrees of commutativity and maximum charge transfer), not merely the presence of coherence or non-Markovianity in isolation.