Quantum Relative Entropy TUR

• Quantum Relative Entropy TUR is a framework that quantifies trade-offs between observable fluctuations and entropy production in quantum nonequilibrium processes, uniting thermodynamics and information theory.
• The formalism derives lower bounds on entropy production via relations between mean differences and variances of quantum observables using generalized divergence measures.
• Implications include universal precision limits on quantum currents and design constraints for quantum thermodynamic devices, with applications in nanoscale heat engines and metrological protocols.

Quantum relative entropy Thermodynamic Uncertainty Relations (TURs) constitute a rigorous framework quantifying the fundamental trade-offs between fluctuations and entropy production in quantum nonequilibrium processes. By linking the quantum relative entropy between pairs of quantum states to the statistics of observables, these relations generalize their classical stochastic analogs and reveal deep connections between quantum thermodynamics, information theory, and estimation theory. The formalism captures universal limits on the precision of quantum currents, the irreversibility cost of quantum operations, and the ultimate constraints governing precision quantum thermodynamic devices.

1. Foundations: Quantum Relative Entropy and Its Thermodynamic Role

Quantum relative entropy for density operators $\rho$ and $\sigma$ on a Hilbert space $\mathcal{H}$ is defined as

$$S(\rho\|\sigma) = \operatorname{Tr}[\rho\ln\rho - \rho\ln\sigma].$$

This non-negative quantity vanishes if and only if $\rho = \sigma$, and reduces to the classical Kullback-Leibler divergence for diagonal states in a common basis. In the context of quantum thermodynamics, quantum relative entropy plays the role of average entropy production, denoted $\Pi = S(\rho\|\sigma) \geq 0$ (Salazar, 2023, Salazar, 28 Apr 2024).

For nonequilibrium steady states (NESS), the McLennan–Zubarev ensemble provides an exact description: $$\rho_{\mathrm{NESS}} = Z_\mathrm{NESS}^{-1}\exp\big[-\bar\beta(H-\bar\mu N) + \Sigma\big],$$ where $\Sigma$ is the entropy-production operator, $\bar\beta$ and $\bar\mu$ are effective thermodynamic parameters, and the steady-state entropy production is given by $$\Sigma = D(\rho_{\mathrm{NESS}}\Vert \rho_{\mathrm{eq}})$$ (1901.10428).

2. Derivation and Structure of Quantum Relative Entropy TURs

Quantum relative entropy TURs bound entropy production from below in terms of mean and variance differences of quantum observables. Let $A$ be a Hermitian operator; define

$$\Delta = \langle A \rangle_\rho - \langle A \rangle_\sigma,\quad \mathrm{Var}_\rho(A) = \langle A^2 \rangle_\rho - \langle A \rangle_\rho^2,$$

and similarly for $\sigma$.

The general TUR takes the form

$$S(\rho\|\sigma) \geq \int_0^1 \frac{\lambda\,\Delta^2}{ (1-\lambda)\mathrm{Var}_\rho(A) + \lambda\mathrm{Var}_\sigma(A) + (1-\lambda)\lambda \Delta^2} d\lambda,$$

where $0\leq\lambda\leq 1$ (Salazar, 28 Apr 2024). This is derived via quantum generalizations of the $\chi^2$-divergence and classical variation-representation techniques, embedding quantum states into corresponding Nussbaum–Szkoła joint distributions.

For the variance-to-square-difference ratio

$$U(A;\rho,\sigma) = \frac{\mathrm{Var}_\rho(A)+\mathrm{Var}_\sigma(A)}{\frac{1}{2}\,(\langle A\rangle_\rho-\langle A\rangle_\sigma)^2},$$

a bound follows in terms of a function of the symmetric quantum relative entropy: $$U(A;\rho,\sigma) \geq f(\widetilde{S}(\rho,\sigma)),$$ where $$\widetilde{S}(\rho,\sigma) = \frac{1}{2}\big[S(\rho\|\sigma) + S(\sigma\|\rho)\big]$$ and $f$ encodes geometric information (Salazar, 2023).

3. Relations to Classical TURs and the Quantum Cramér–Rao Bound

In the classical limit—when $\rho$, $\sigma$, and $A$ are simultaneously diagonal—the quantum TUR reduces to established forms, specifically the stochastic TUR for Markovian processes: $$D(p\|q) \geq F(\Delta_{pq},\,\mathrm{Var}_p(A),\,\mathrm{Var}_q(A)),$$ where $F$ is the classical counterpart of the quantum bound (Salazar, 28 Apr 2024).

For quantum NESS, the derivation connects to the quantum Cramér–Rao bound. By parameterizing the ensemble with a counting field $\theta$, the quantum Fisher information $I(\theta)$ quantifies precision under parameter shifts. The TUR emerges as

$$\frac{\mathrm{Var}(J)}{\langle J\rangle^2} \geq \frac{1}{2\Sigma}$$

($J$: steady-state current, $\Sigma$: entropy production), which is universally valid even when purely quantum transport violates the classical factor of $2$ TUR. The loosened (quantum) bound reflects diminished constraints compared to the classical regime, while still precluding unbounded improvements to precision without increased entropy production (1901.10428).

4. Generality, Assumptions, and Extensions

Quantum relative entropy TURs hold under minimal structural conditions:

• $\rho$ and $\sigma$ are arbitrary density operators; no commutativity or Markovianity is required.
• Observables $A$ are any Hermitian operator on the relevant Hilbert space.
• No restrictions are imposed on environmental structure, system–bath coupling strength, or proximity to equilibrium (Salazar, 28 Apr 2024, Salazar, 2023).

These properties ensure applicability to strongly coupled open quantum systems, arbitrary quantum channels (CPTP maps), and general many-body systems, aligning with the geometric structure of the relevant quantum state manifolds.

Several extensions have been formulated, including the use of symmetric quantum divergences, alternative information measures (Rényi, Tsallis), and resource-theoretic quantities (coherence, asymmetry) as prospective alternatives to the Umegaki (relative) entropy (Salazar, 2023).

5. Physical Implications and Applications

Quantum relative entropy TURs set universal lower bounds on entropy production for observable changes, constraining both classical and quantum fluctuation regimes:

• Nanoscale Heat Engines: The bounds fix a minimum entropy cost for a given output current precision, forbidding indefinite suppression of both fluctuations and dissipation. Quantum engines may exceed classical classical “Pietzonka–Seifert” performance bounds by a factor of two, subject to the quantum Fisher information constraint (1901.10428).
• Quantum Thermodynamic Protocols: For composite system–environment processes, the bound

$$\Sigma = S(\rho\|\sigma) \geq F(\Phi, \mathrm{Var}_\rho(\ln\rho_E), \mathrm{Var}_\sigma(\ln\rho_E))$$

(with $\Phi$: entropy flux through the environment) yields nontrivial efficiency and irreversibility constraints (Salazar, 28 Apr 2024, Salazar, 2023).

• Quantum Metrology and Measurement: These inequalities quantify the minimal uncertainty in current or work measurements consistent with entropy production, and underpin quantum Cramér–Rao-type tradeoffs in unitary parameter estimation tasks (Salazar, 2023, Salazar, 28 Apr 2024).

6. Special Cases, Tightness, and Saturation

Quantum relative entropy TURs are exact and nonperturbative. Their saturation is achieved in minimal scenarios (e.g., two-level systems where observables are block-diagonal with distinct eigenvalues for $\rho$ and $\sigma$). In the symmetric ($\lambda=1/2$) case, the quantum triangular discrimination appears, and for infinitesimal dynamics generated by a unitary, the TUR reduces to the quantum Cramér–Rao bound in the short-time limit. When quantum coherence between $\rho$ and $\sigma$ is nonzero, the TURs become strictly tighter than the classical bounds, but still always hold as $S(\rho\|\sigma) \geq 0$ (Salazar, 28 Apr 2024, Salazar, 2023).

7. Broader Connections: Entropic and Information-Theoretic Uncertainty Relations

Quantum relative entropy TURs complement relative entropic uncertainty relations (REURs), as in (Floerchinger et al., 2020), which bound the sum of relative entropies between observable statistics and maximum-entropy reference models. Unlike TURs, which furnish lower bounds on entropy production via observable statistics, REURs provide upper bounds on overall distinguishability from maximum-entropy models, incorporating both measurement incompatibility and state mixedness.

These interrelated frameworks unify classical and quantum uncertainty principles and can accommodate constraints from side-information, opening avenues for enhanced quantification of uncertainty in non-Gaussian and strongly nonclassical regimes (Floerchinger et al., 2020). This suggests a broad scope for future generalizations involving alternative divergence measures and operational tasks in quantum thermodynamics.