Quantum Macroscopic Fluctuation Theory

• The paper introduces a quantum extension of MFT that incorporates operator-valued fields to reconcile large-deviation principles with quantum stochastic dynamics.
• It employs advanced techniques such as Keldysh path integrals and free probability to derive novel fluctuation relations and quantify quantum corrections in transport.
• The work unifies ballistic and diffusive transport frameworks, demonstrating the role of quantum noise, entanglement statistics, and hydrodynamic scaling in many-body systems.

A quantum extension of Macroscopic Fluctuation Theory (MFT) generalizes the classical statistical framework for non-equilibrium transport phenomena to account for quantum coherence, entanglement, and operator-valued noise in many-body systems. Classical MFT provides action functionals and large-deviation principles for diffusive and ballistic systems through hydrodynamic and stochastic equations governing density and current fields, together with their fluctuations. Quantum extensions incorporate operator-valued fields, quantum noise kernels, and non-commutative statistics, yielding new large-deviation functionals, fluctuation relations, and entanglement statistics tailored to quantum many-body transport. Key models include the quantum symmetric simple exclusion process (QSSEP), noisy quantum diffusive chains, integrable systems under ballistic dynamics, and quantum hierarchical systems analyzed via coarse-graining and stochastic operator equations. This synthesis involves the unification of quantum stochastic (Lindblad) dynamics, path-integral/Keldysh actions, and free probability techniques to describe fluctuations and correlations at hydrodynamic scales.

1. Foundational Principles: From Classical to Quantum MFT

Classical MFT considers stochastic lattice gases (e.g., SSEP/ASEP), where the density field $n(x,t)$ and current $j(x,t)$ obey continuity equations and local constitutive laws subject to Gaussian noise, resulting in path-integral weights: $$P[n,j] \sim \exp\left\{-\frac{1}{\epsilon} \int dt\, dx\, \left[ \frac{(j + D \partial_x n)^2}{2 \sigma(n)} \right] \right\}$$ with $\epsilon = a_0/L \rightarrow 0$, $D(n)$ diffusion, and $\sigma(n)$ mobility (Bernard, 2021).

A quantum extension first requires promoting these fields to operator-valued densities and currents, e.g., $\hat\rho(x)$ and $\hat j(x)$ defined in terms of fermionic creation-annihilation operators (Hruza, 2024). Quantum models introduce operator algebra non-commutativity, quantized statistics (Bose/Fermi), and decoherence phenomena.

In the quantum regime, both closed and open system settings are considered. Evolution may follow quantum stochastic differential equations (QSDEs) or Lindblad master equations (GKSL). For example, in QSSEP one evolves the density matrix under both coherent hopping and noise: $$d\rho_t = -i[dH_t, \rho_t] - \frac{1}{2} [dH_t, [dH_t, \rho_t]]$$ with $dH_t$ a sum of hopping terms weighted by classical Brownian trajectories (Albert et al., 23 Jan 2026, Bernard, 2021).

Quantum large-deviation functionals can be constructed via Keldysh path integrals over operator-valued histories, yielding actions of the form: $$S_K[\rho_c, \rho_q] = \int dt\, dx\, [ \rho_q (i \hbar \partial_t - L[\rho_c]) \rho_c + \tfrac{i}{2} \rho_q \mathcal{N}[\rho_c] \rho_q ]$$ where the noise kernel $\mathcal{N}$ embodies the quantum fluctuation-dissipation relation and operator-valued correlations (Hu, 2020).

2. Quantum Fluctuations, Large Deviations, and Hydrodynamics

Quantum MFT manifests large-deviation rate functions for mesoscopic observables (densities, currents, coherence profiles) that generalize classical results. For free fermion models in disordered media, the moment-generating function for the integrated current

$$\Pi(\alpha) = \lim_{L\to\infty}\frac{1}{|\Lambda_L|} \ln \left\langle e^{\alpha |\Lambda_L| \mathcal{I}_{\Lambda_L}} \right\rangle_{\beta,\omega}$$

yields a deterministic, convex, $C^\infty$ rate function via Legendre transform, with asymptotic exponential suppression of quantum uncertainty around the macroscopic current (Bru et al., 2020): $I(j) = \sup_\alpha [ j \alpha - \Pi(\alpha) ]$ and for $j \approx j_*$, $I(j) \sim \frac{(j - j_*)^2}{2 \Pi''(0)}$, where $\Pi''(0)$ encodes quantum current fluctuations.

In noisy quantum diffusive models (QSSEP/QSSIP), the finite-size corrections to current statistics—encapsulating genuine quantum effects unattainable in classical analogs—become explicit in the deviation from the classical large-deviation rate function,

$F_{qu,w}(u) = F_{cl}(u) + O(1/N)$

where $F_{cl}$ matches the SSEP/SSIP results and subleading terms arise from loop-correlation functions of quantum coherences (Albert et al., 23 Jan 2026).

In diffusive transport settings, the continuity and constitutive equations for quantum models (QSSEP, open quantum spin chains) maintain classical hydrodynamic forms at leading order, while sub-leading terms encode quantum entanglement and coherence: $$\partial_t n(x, t) + \partial_x j(x, t) = 0, \quad j(x, t) = -D(n) \partial_x n(x, t) + \sqrt{\sigma(n)} \xi(x, t)$$ where $\xi(x, t)$ now represents operator-valued quantum noise, e.g., with correlators parameterized by non-local statistical structures (Shpielberg, 2019, Hruza, 2024).

3. Ballistic Macroscopic Fluctuation Theory: Quantum Integrable Systems

For strictly ballistic transport (generic integrable models), the Quantum Ballistic Macroscopic Fluctuation Theory (BMFT) provides a complete construction via mapping to point-particle ensembles (Kethepalli et al., 23 May 2025). Each quasiparticle, parametrized by rapidity $\theta$, propagates with state-dependent $v^{\rm eff}[\rho](x,\theta)$, and all fluctuations propagate deterministically via the Euler-scale GHD equations: $$\partial_t \rho_t(x,\theta) + \partial_x (v^{\rm eff}_{[\rho_t]}(x,\theta) \rho_t(x,\theta)) = 0$$ with $v^{\rm eff}_{[\rho]}(x,\theta)$ satisfying an integral equation defined by the two-body scattering shift $\mathfrak{a}(\theta - \theta')$. The BMFT action functional leverages a change to free ("bare") coordinates and incorporates quantum statistics through the large-deviation cost: $$\tilde{\mathcal{F}}[r_0] = \int dX\, d\theta\, G(r_0, \bar{r}_0),\qquad G(r, \bar r) = r \ln \frac{r}{\bar r} + (1 - \eta r) \ln \frac{1 - \eta r}{1 - \eta \bar r}$$ with $\eta = \pm 1$ for Fermi/Bose statistics. Full-counting statistics (FCS) and long-range correlation functions emerge from saddle-point evaluation of the action, confirming that all late-time fluctuations are determined by initial noise convected by non-linear Euler equations (Kethepalli et al., 23 May 2025, Doyon et al., 2022).

BMFT unifies treatment of full-counting charge statistics and dynamical two-point kernels, producing closed-form expressions valid for both classical and quantum cases. The theory applies rigorously in the limit of large system size and time $T \to \infty$, $x, T \sim O(L)$, and initial states sampled from a generalized Gibbs ensemble (Kethepalli et al., 23 May 2025).

4. Coherence, Entanglement, and Quantum Hydrodynamics

Quantum MFT uniquely accommodates fluctuation statistics of coherences (two-point Green’s functions) and entanglement entropy at diffusive hydrodynamic scales. In QSSEP and similar noisy quantum chains, off-diagonal correlations and their cumulants scale as $N^{1-n}$ and satisfy closed equations that extend the classical MFT structure (Hruza, 2024). The large-deviation principle for coherence profiles $G(x, y, t)$ and their associated currents $J(x, y, t)$ follows: $$\mathbb{P}[\{G, J\}] \sim \exp\left\{ -N \mathcal{S}[G, J] \right\},\qquad \mathcal{S}[G, J] = \int dt\, dx\, dy\, \frac{[J + (\partial_x^2 + \partial_y^2) G]^2}{2\, \Sigma(G)}$$ Moreover, exact NESS results and free-probability techniques characterize loop-cumulants and spectra of subblocks, with non-crossing partition combinatorics underlying the structure of stationary correlations.

Quantum entanglement emerges macroscopically and can be quantified by coarse-grained Rényi-2 entropy $S_2(\ell, t) \propto \ell$ in non-equilibrium steady states of open quantum chains (Shpielberg, 2019). These models demonstrate the impact of quantum coherence and entanglement on fluctuation statistics, often yielding non-local corrections, cross-noise terms, and modified mobility matrices that alter rare-event probabilities and dynamical phase structure.

5. Unified Frameworks, Free Probability, and Experimental Implications

Quantum extensions of MFT rely on structured coarse-graining, operator-valued noise modeling, and path-integral/Keldysh approaches. Operator-valued correlation noise derives from truncation and slaving in quantum BBGKY hierarchies, resulting in quantum Boltzmann-Langevin equations with fluctuation-dissipation relations (Hu, 2020). In ballistic and diffusive settings, the quantum noise kernel may be non-Markovian, colored in time, and non-commuting.

Free probability provides combinatorial tools for describing spectra and moments of coherence matrices in mesoscopic quantum diffusive models, generalizing classical additivity and facilitating large-$N$ asymptotic analysis (Hruza, 2024).

Quantum large-deviation corrections manifest at $O(1/N)$ and encode quantum fluctuations absent in classical MFT, as seen in current cumulant corrections in QSSEP/QSSIP. Such corrections are experimentally accessible in protocols allowing control of noise realizations and measurement of subleading cumulants (Albert et al., 23 Jan 2026).

Ballistic quantum MFT, through explicit mapping and saddle-point principles, produces closed formulas for FCS and time-dependent correlation functions in integrable models, making direct connections to generalized hydrodynamics and hydrodynamic fluctuation theory (Kethepalli et al., 23 May 2025, Doyon et al., 2022).

6. Open Problems and Future Directions

Significant open problems remain in developing universal quantum MFT action functionals, additivity principles, and variational frameworks for arbitrary quantum transport models (Bernard, 2021). Generalizing fluctuation theorems (e.g., Gallavotti–Cohen, Jarzynski) to quantum stochastic differential equation settings and characterizing rate functions for entanglement growth and multi-replica sectors are active research frontiers.

Understanding the precise role of entanglement, combinatorial structures (e.g., associahedra in cyclic cumulants), and universal scaling limits in quantum large-deviation theory will govern future advancements. Direct connections between quantum hydrodynamics, stochastic fluctuation theory, and experimental protocols remain a central focus.

In summary, quantum extensions of Macroscopic Fluctuation Theory, in both ballistic and diffusive regimes, provide a rigorous framework for statistical analysis of quantum transport, coherence, and entanglement in many-body systems, encompassing explicit large-deviation principles, operator-valued stochastic equations, and applications ranging from integrable quantum models to experimentally relevant mesoscopic systems (Kethepalli et al., 23 May 2025, Bernard, 2021, Hruza, 2024, Albert et al., 23 Jan 2026, Bru et al., 2020, Doyon et al., 2022, Shpielberg, 2019, Hu, 2020).