Hybrid Quantum Thermodynamic TUR

• The hybrid quantum thermodynamic uncertainty relation generalizes classical precision bounds by coupling classical intensive parameters with quantum observables.
• It employs measures like entropy production, quantum relative entropy, and Fisher information to bound fluctuations in both equilibrium and nonequilibrium settings.
• Applications span multipartite open quantum systems, superconducting devices, and thermoelectric engines, integrating dissipation, coherence, and information flow.

Searching arXiv for papers on hybrid quantum thermodynamic uncertainty relations and closely related formulations. arxiv_search(query="hybrid quantum thermodynamic uncertainty relation", max_results=10, sort_by="submittedDate") arxiv_search(query="thermodynamic uncertainty relation quantum hybrid superconducting uncertainty", max_results=10, sort_by="submittedDate") The hybrid quantum thermodynamic uncertainty relation denotes a family of bounds that extend classical thermodynamic uncertainty relations into regimes where thermodynamic variables, quantum observables, coherence, information flow, or heterogeneous transport channels appear simultaneously. In the literature, the term “hybrid” is used in several distinct but related senses: for a classical intensive parameter paired with a quantum extensive operator in equilibrium, for transport through mixed Fermi–Bose or normal–superconducting devices, for multipartite settings where dissipation, information flow, and coherence jointly constrain fluctuations, and for formulations that couple quantum-state distinguishability to observable variances (Meng et al., 7 Nov 2025, Vidal et al., 19 Jun 2026, Honma et al., 6 Oct 2025, Salazar, 2023).

1. Terminological scope and recurrent structure

A common structural feature across these formulations is a lower bound on precision in terms of a thermodynamic or information-theoretic cost. In classical exchange-fluctuation settings, the standard scalar form is a lower bound on relative variance by a function of the mean entropy production, with the small-dissipation limit reproducing the familiar $2/\langle \Sigma\rangle$ form. In quantum and hybrid settings, the role of the “cost” can be played by entropy production, symmetric quantum relative entropy, a susceptibility, partial entropy production corrected by information flow, or a measurement-related activity (Timpanaro et al., 2019, Salazar, 2023).

The surveyed literature does not present a single canonical definition of “hybrid.” Instead, the expression labels different mixtures of classical and quantum elements, or different coexistence mechanisms within one device. This suggests that the topic is best understood as a research area organized by a shared dissipation–precision logic rather than by one unique formula.

Hybridization sense	Representative bound	Representative setting
Classical parameter with quantum observable	$\Delta\theta\,\overline{\Delta O} \ge k_B T$	Equilibrium Gibbs states (Meng et al., 7 Nov 2025)
Coexisting transport channels	$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$	Normal–superconductor conductor (Vidal et al., 19 Jun 2026)
Dissipation, information, and coherence combined	$$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$	Interacting multipartite open systems (Honma et al., 6 Oct 2025)

2. Foundational precursors

A major precursor is the exchange-fluctuation-theorem route to thermodynamic uncertainty relations. For charges $\{\mathcal Q_i\}$ obeying

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$

the tightest saturable matrix-valued TUR is

$$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$

Its diagonal sector yields $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$, and for small $\langle \Sigma\rangle$ this expands to the usual $2/\langle \Sigma\rangle$ bound. The result applies to classical or quantum systems, with exact exchange fluctuation theorems under factorized grand-canonical initial states and arbitrary global unitary dynamics (Timpanaro et al., 2019).

A different precursor derives a quantum uncertainty relation from thermodynamic fluctuations. In the canonical ensemble,

$\Delta\theta\,\overline{\Delta O} \ge k_B T$0

and Mandelbrot’s classical fluctuation result gives

$\Delta\theta\,\overline{\Delta O} \ge k_B T$1

Assuming de Broglie’s conjecture $\Delta\theta\,\overline{\Delta O} \ge k_B T$2, equivalently $\Delta\theta\,\overline{\Delta O} \ge k_B T$3, this becomes

$\Delta\theta\,\overline{\Delta O} \ge k_B T$4

and then, with $\Delta\theta\,\overline{\Delta O} \ge k_B T$5,

$\Delta\theta\,\overline{\Delta O} \ge k_B T$6

The derivation is explicitly conditioned on de Broglie’s temperature–time conjecture, on a valid small-fluctuation expansion $\Delta\theta\,\overline{\Delta O} \ge k_B T$7, and on treating the time uncertainty as a clock-tick resolution (Roupas, 2021). It is not itself a hybrid quantum TUR in the modern nonequilibrium sense, but it shows how thermodynamic fluctuation theory can generate a quantum uncertainty bound.

3. Equilibrium hybridization: intensive parameters and quantum observables

An explicit equilibrium formulation of a hybrid thermodynamic uncertainty relation appears for thermodynamically conjugate variables $\Delta\theta\,\overline{\Delta O} \ge k_B T$8 and $\Delta\theta\,\overline{\Delta O} \ge k_B T$9. In a Gibbs state

$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$0

one identifies

$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$1

The uncertainty of the classical parameter $\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$2 is quantified by the quantum Fisher information $\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$3, which admits the exact integral representation

$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$4

with $\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$5 the symmetrized autocorrelation spectrum of $\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$6 (Meng et al., 7 Nov 2025).

From this representation one obtains the chain

$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$7

Combining $\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$8 with the quantum Cramér–Rao inequality gives

$\sigma \ge (k_B |I|/e)\,\mathrm{arsinh}(2e|I|/S)$9

This is “hybrid” because $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$0 is a classical intensive parameter while $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$1 is a quantum extensive operator (Meng et al., 7 Nov 2025).

The equality conditions are restrictive. They require both saturation of the Cramér–Rao bound and tightness of the kernel estimate, which occurs in the high-temperature regime $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$2 or when the fluctuation spectrum is sharply peaked near $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$3. In the one-dimensional transverse-field Ising chain, the susceptibility bound is especially tight near and above the quantum critical point, while all bounds collapse onto $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$4 at high temperature and in low-frequency-dominated ferromagnetic regimes (Meng et al., 7 Nov 2025).

4. Quantum entropy production and state-pair formulations

A second major branch formulates hybrid quantum TURs directly in terms of pairs of quantum states. For two density operators $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$5 and any Hermitian observable $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$6, the symmetric uncertainty

$$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$7

satisfies

$$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$8

where $$\mathrm{Var}[J_1]/\langle J_1\rangle^2 \ge 2(1+\delta_{J_1})^2/(\Sigma_1-I_1)$$9 and $\{\mathcal Q_i\}$0 is the inverse of $\{\mathcal Q_i\}$1 (Salazar, 2023).

In thermodynamic notation, with

$\{\mathcal Q_i\}$2

the bound becomes

$\{\mathcal Q_i\}$3

The derivation uses a reduction to classical probability tables $\{\mathcal Q_i\}$4, a generalized Cramér–Rao/Pinsker-type lemma, and the identity $\{\mathcal Q_i\}$5. The result covers arbitrary state pairs, general CPTP maps, strong coupling, and non-thermal environments. In the commuting limit it reduces to the classical exchange-TUR. The bound is tight only in a minimal two-level toy model with commuting $\{\mathcal Q_i\}$6 (Salazar, 2023).

A related result lower-bounds the quantum relative entropy itself. For two quantum states $\{\mathcal Q_i\}$7 and Hermitian $\{\mathcal Q_i\}$8,

$\{\mathcal Q_i\}$9

where $$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$0 and

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$1

In open quantum thermodynamics, taking $$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$2 and $$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$3 yields

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$4

No commutation assumption is needed, and in the incoherent limit the formula reduces to the classical TUR for stochastic entropy production. In the unitary small-$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$5 limit it reproduces the quantum Cramér–Rao inequality (Salazar, 2024).

5. Nonequilibrium transport in hybrid quantum devices

In transient bipartite energy transport, the hybrid coupled oscillator–qubit system provides a concrete mixed-statistics example. The exchange-fluctuation theorem

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$6

implies the generalized bound

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$7

For the Jaynes–Cummings model, however, the specialized TUR ratio becomes

$$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$8

or $$\frac{P(\mathcal Q_1,\dots,\mathcal Q_n)}{P(-\mathcal Q_1,\dots,-\mathcal Q_n)}=\exp(\Sigma),\qquad \Sigma=\sum_i A_i \mathcal Q_i,$$9 with $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$0. By contrast, the bosonic oscillator–oscillator case has $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$1. The generalized version from the universal fluctuation symmetry is always satisfied, while the tighter specialized bound can be violated in the hybrid and qubit–qubit cases for certain parameter regimes. A Keldysh–NEGF proof shows that the tighter bound is recovered in the weak-coupling regime for generic bipartite systems (Saryal et al., 2020).

A more recent transport realization is the coherent normal–superconductor junction. There the average current splits as

$$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$2

with Andreev and quasiparticle contributions, while the zero-frequency noise decomposes as

$$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$3

The interference term $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$4 can change sign and prevents a simple additive bound at the level of individual noise components, but the nonequilibrium excess noise admits a manifestly positive representation, so that $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$5. This leads to the hybrid quantum thermodynamic uncertainty relation

$$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$6

valid for arbitrary real superconducting gap $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$7. In the pure-Andreev limit $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$8 one recovers the previously known $$\mathcal C-f(\langle \Sigma\rangle)\,\bm q\,\bm q^T \succeq 0,\qquad f(x)=\mathrm{csch}^2\!\bigl(g(\tfrac{x}{2})\bigr),\qquad g^{-1}(y)=y\tanh y.$$9-charge QTUR, and for $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$0 one reduces to the normal-lead QTUR with charge $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$1 (Vidal et al., 19 Jun 2026).

Linear-response hybrid superconducting systems with broken time-reversal symmetry provide a related but distinct scenario. In two-terminal transport the charge and heat TUR bounds become

$$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$2

with $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$3. For full time-reversal symmetry, $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$4 and both reduce to $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$5. In the Andreev interferometer example, $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$6 in the heat-engine regime and $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$7 in the refrigerator regime, while purely normal coherent two-terminal conductors still give the standard value $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$8 (Taddei et al., 2023).

Quantum thermoelectric junctions sharpen the distinction between classical and quantum noise contributions. The charge noise can be decomposed as $$\mathrm{Var}(\mathcal Q_i)/\langle \mathcal Q_i\rangle^2 \ge f(\langle \Sigma\rangle)$$9 in the paper’s notation, with the rigorous statement that only the “classical” component follows the TUR, while the remaining “quantum” component is responsible for potential violation. In thermoelectric engines, these violations are weak and disappear as the Carnot efficiency is approached, where the relative uncertainty diverges (Liu et al., 2019).

6. Information flow, coherence, feedback, and measurement

In interacting multipartite open quantum systems, the hybrid character becomes explicitly tripartite: local dissipation, information flow, and coherence appear in the same bound. For subsystem $\langle \Sigma\rangle$0, the thermodynamic form reads

$\langle \Sigma\rangle$1

where $\langle \Sigma\rangle$2 is the local dissipation, $\langle \Sigma\rangle$3 the information flow into $\langle \Sigma\rangle$4, and $\langle \Sigma\rangle$5 a quantum-coherence correction. A stronger activity-based form also exists in terms of the partial dynamical activity $\langle \Sigma\rangle$6. The local second law is written as $\langle \Sigma\rangle$7. In the autonomous quantum Maxwell’s demon and autonomous quantum clock, the bound with the coherence correction is always satisfied, whereas the corresponding classical form without $\langle \Sigma\rangle$8 can fail (Honma et al., 6 Oct 2025).

For continuously monitored open quantum systems with Markovian feedback, the finite-time current bound takes the parallel form

$\langle \Sigma\rangle$9

Here $2/\langle \Sigma\rangle$0 is a quantum mutual-information contribution extracted by feedback, and a tighter activity-based version involves the total jump rate $2/\langle \Sigma\rangle$1 and the inverse $2/\langle \Sigma\rangle$2 of $2/\langle \Sigma\rangle$3. The framework shows that feedback can suppress fluctuations, but only together with thermodynamic cost and mutual information (Honma et al., 26 Feb 2026).

A coherence-driven loosening of the classical bound also occurs in a driven dissipative two-qubit system. The TUR product

$2/\langle \Sigma\rangle$4

has $2/\langle \Sigma\rangle$5 in the single-qubit limit and $2/\langle \Sigma\rangle$6 for strongly coupled qubits under strong fields. The negative-sign coherence terms in the variance, especially the imaginary coherence and the interqubit coherence $2/\langle \Sigma\rangle$7, are identified as the mechanism suppressing relative fluctuations below the classical limit $2/\langle \Sigma\rangle$8 (Cho et al., 2 May 2025).

A measurement-theoretic extension links thermodynamic activity to indirect measurement noise. For an unbiased indirect measurement of $2/\langle \Sigma\rangle$9, with noise operator $\Delta\theta\,\overline{\Delta O} \ge k_B T$00 and survival activity $\Delta\theta\,\overline{\Delta O} \ge k_B T$01, one obtains

$\Delta\theta\,\overline{\Delta O} \ge k_B T$02

Combined with Ozawa’s universally valid noise–disturbance relation, this yields a joint constraint between measurement noise, disturbance, and thermodynamic activity (Mihashi et al., 2022).

7. Scope, limitations, and outstanding issues

Several recurring caveats organize the field. First, generalized bounds derived from exchange fluctuation symmetry are often more robust than specialized, tighter bounds. In the transient energy-transport models, the generalized TUR is always satisfied, whereas the specialized TUR can be violated for mixed Fermi–Bose statistics or fermionic transport under certain parameter regimes (Saryal et al., 2020).

Second, equality or near-equality is typically confined to special limits. The thermodynamically conjugate-variable relation $\Delta\theta\,\overline{\Delta O} \ge k_B T$03 becomes tight only when both the Cramér–Rao bound and the kernel bound are saturated, notably at high temperature or for low-frequency-dominated fluctuations (Meng et al., 7 Nov 2025). The symmetric-relative-entropy state-pair bound is tight only in a minimal two-level commuting model (Salazar, 2023).

Third, “quantum advantage” in TUR language is not uniform across regimes. In two-qubit and broken-time-reversal hybrid superconducting settings, the conventional classical lower bound $\Delta\theta\,\overline{\Delta O} \ge k_B T$04 can be loosened (Cho et al., 2 May 2025, Taddei et al., 2023). By contrast, in noninteracting thermoelectric engines the apparent violation disappears near the thermodynamic efficiency limit, where fluctuations diverge as Carnot efficiency is approached (Liu et al., 2019).

Finally, the literature identifies several unresolved directions. For interacting multipartite systems, explicit open questions include extension to strong system–bath coupling, non-Markovian memory effects, explicit feedback-control protocols, and quantification of the cost of information processing in autonomous quantum Maxwell demons (Honma et al., 6 Oct 2025). A plausible implication is that future work will continue to treat “hybrid” not as a single mechanism, but as a unifying label for TURs in which thermodynamic irreversibility, quantum coherence, information processing, and heterogeneous transport channels are inseparable parts of the same precision bound.