Quantum Fluctuation Relations

Updated 29 October 2025

Quantum fluctuation relations are exact statistical equalities that constrain energy exchange probabilities in quantum systems far from equilibrium, incorporating quantum coherence and measurement-induced effects.
They extend classical thermodynamic predictions by using two-point measurement schemes, operator formulations, and accounting for non-unitary, open-system dynamics.
Experimental validations using NV centers, quantum conductors, and feedback-controlled systems demonstrate their role in linking equilibrium ensembles with nonequilibrium processes.

Quantum fluctuation relations are exact statistical equalities constraining the probability distributions of thermodynamic quantities such as work, heat, and entropy production in quantum systems driven far from equilibrium. Rooted in the principles of microreversibility and the structure of equilibrium ensembles, quantum fluctuation relations generalize classical results to account for quantum coherence, measurement-induced stochasticity, and non-unitary, open-system dynamics. They provide a unifying framework connecting equilibrium and nonequilibrium thermodynamics, response theory, and resource-theoretic perspectives, and have been experimentally tested across closed and open quantum systems.

1. Mathematical Structure and Formalism

Quantum fluctuation relations take a variety of forms, but their foundations rest on symmetry properties between forward and backward processes rooted in microreversibility (antiunitary time-reversal invariance) and the structure of initial states (often Gibbsian). For a system initially in equilibrium, subjected to an external protocol, the canonical quantum Crooks and Jarzynski equalities are

$\boxed{ P[w; \Lambda] = e^{\beta(w - \Delta F)}\, P[-w; \tilde\Lambda] }$

and

$\boxed{ \langle e^{-\beta W} \rangle_\Lambda = e^{-\beta \Delta F} }$

where $P[w; \Lambda]$ is the probability distribution of work $w$ performed by protocol $\Lambda$ , $\tilde\Lambda$ is its time-reverse, $\beta$ the (initial) inverse temperature, and $\Delta F$ the free energy difference between equilibrium reference points (Facchi et al., 2017, Campisi et al., 2010).

Work in quantum systems is not a Hermitian observable but is operationally defined via a two-point measurement (TPM) scheme. The probability distribution is

$P[w; \Lambda] = \sum_{m,n} \frac{ e^{-\beta E_n^{\Lambda_0}} }{Z(\Lambda_0)} \left| \langle m | U_{\tau,0}[\Lambda] | n\rangle \right|^2 \delta\left( w - [E_m^{\Lambda_\tau} - E_n^{\Lambda_0}] \right).$

These relations extend to open quantum systems governed by Lindblad master equations, where the dynamics is non-unitary due to environmental coupling. For these, operator-based fluctuation relations (e.g., Chetrite-Mallick's formalism (Chetrite et al., 2011, Chetrite et al., 2010)) involve time-ordered exponentials of generalized injected power operators, leading to Jarzynski-type equalities

$\left\langle \overrightarrow{\exp}\left( - \int_0^T W_u du \right) \right\rangle_{\pi_0} = 1,$

with $W_u$ the non-Hermitian quantum "power" operator, and the ensemble average over the Lindblad evolution from initial state $\pi_0$ .

2. Fluctuation Relations in Driven and Open Quantum Systems

Quantum fluctuation relations have been generalized and tested in settings where dynamics combine coherent driving and engineered dissipative (thermalizing) channels. An archetypal model is the single electronic spin of a nitrogen-vacancy (NV) center in diamond subject to cyclic driving and engineered dissipation:

In the regime of an effective infinite-temperature reservoir ( $\beta_R = 0$ ), the fluctuation relation

$\left\langle e^{-\beta \Delta E} \right\rangle = e^{-\beta \Delta F}$

is experimentally verified, with $\Delta E$ comprising both work and heat (Hernández-Gómez et al., 2021).

Under protocols where total work vanishes at stroboscopic times but power delivered is finite, the relation

$\left\langle e^{-(\beta-\beta_R)\Delta E} \right\rangle = e^{-\beta \Delta F}$

holds, and the statistics of $\Delta E$ suffice to confirm QFRs even though distinguishing work and heat is typically experimentally intractable.

This demonstrates that rigorous fluctuation relations remain valid in the presence of complex, open, and driven quantum dynamics, provided the protocol and measurement paradigm are matched to system observables.

3. Role of Measurements, Decoherence, and Feedback

The statistics of energy changes under repeated quantum measurements, including those at random intervals, also satisfy robust fluctuation theorems. In generic measurement protocols applying $M$ projective measurements between initial and final energy observables, the characteristic function of the exchanged heat $Q$

$G(u) = \overline{ \left\langle \mathrm{Tr}[ e^{iuH} \mathcal{V}_{\mathbf{k}, \boldsymbol{\tau}}e^{-iuH} \rho_0 \mathcal{V}_{\mathbf{k}, \boldsymbol{\tau}}^\dagger ] \right\rangle }$

encodes all moments of $Q$ . The Jarzynski equality

$\langle e^{-\beta Q} \rangle = 1$

remains valid for a thermal initial state and arbitrary measurement sequence, including stochastic waiting times (Gherardini et al., 2022). In the asymptotic limit $M \to \infty$ , generic systems thermalize to a maximally mixed (infinite-temperature) state unless symmetries (shared invariant subspaces) lead to partial thermalization.

For feedback-controlled quantum dynamics, detailed fluctuation relations incorporate both work and mutual information densities, e.g.,

$\frac{P_F(k, l; W)}{P_B(l, k; -W)} = \exp\left[\beta (W - \Delta F^{(k)}) + I^{(k, l)}\right],$

with $I^{(k, l)}$ quantifying the information gained and used in feedback, broadening the QFR scope to include Maxwell-demon-like protocols (Camati et al., 2018).

4. Symmetry-Induced Fluctuation Relations and Trajectory Ensembles

Beyond standard TPM schemes, quantum fluctuation relations extend to dynamical observables defined on quantum jump trajectories, especially relevant for Markovian open systems. Large-deviation and tilted-ensemble formalisms analyze the statistics of trajectory observables (e.g., quantum jump counts), where the scaled cumulant generating function (SCGF) obeys symmetry-induced fluctuation relations of the general form

$\theta_s(\lambda) = \theta_s\left[ (U^{-1})^T (\lambda + s) - s \right].$

Here, $U$ is the matrix representing symmetry of the observable under a group action, and $\theta_s(\lambda)$ is obtained from the spectrum of the Doob-transformed Lindbladian (Marcantoni et al., 2021). Experimentally accessible examples include photon counting in quantum optics and current statistics in mesoscopic conductors.

Time-reversal symmetry in non-equilibrium steady states leads to fluctuation relations for large deviation functions $F(\lambda)$ of transferred quantities (e.g., energy, charge): $F(\lambda, \sigma) = F(i \gamma - \lambda, -\omega - \sigma),$ with extended relations possible under pure transmission conditions, whereby all cumulants can be generated from equilibrium mean currents evaluated at shifted thermodynamic parameters (Bernard et al., 2013).

5. Generalizations: Conserved Quantities, Coherence, and Athermality

Standard quantum fluctuation relations assume canonical (Gibbs) equilibrium as the initial state. In systems with extra (non-commuting) conserved quantities, the steady state is a Generalized Gibbs Ensemble (GGE), and fluctuation relations must be correspondingly generalized: $\frac{P_\mathrm{FW}(\mathcal{W})}{P_\mathrm{BW}(-\mathcal{W})} e^{-\mathcal{W}} = e^{-\Delta\mathcal{F}},$ with

$\mathcal{W}_\mathrm{FW} = \beta' E' + \sum_l \beta_l' M_l' - [\beta E + \sum_k \beta_k M_k],$

and $\Delta\mathcal{F}$ the generalized free energy difference (Mur-Petit et al., 2017). Systematic violation of such generalized QFRs signals missing integrals of motion in integrable systems.

For quantum systems initially in athermal or coherently prepared states, quantum fluctuation relations acquire correction factors encoding the informational and quantum-coherence content. For a harmonic oscillator in a photon-added/subtracted thermal state, the Crooks-like equality is (Holmes et al., 2020): $\frac{(E_f | \gamma_i^\pm, E_i)}{(E_i | \gamma_f^\pm, E_f)} = \mathcal{N}_\pm(W) \, \exp\left[\beta \left( W - 2\Delta F \mp \Delta E_{\mathrm{vac}} \right)\right],$ with $\mathcal{N}_\pm(W)$ representing additional suppression or enhancement of irreversibility beyond classical predictions, and $\Delta E_{\mathrm{vac}}$ the vacuum energy shift.

The presence of quantum coherence and athermality in the initial system or energy supply ("battery") modifies fluctuation relations via terms involving quantum distortion factors, effective potentials, or temperature-dependent mean coherence, leading to a hierarchy of corrections to the Crooks relation.

6. Nonequilibrium Fluctuation-Dissipation Relations

The fluctuation-dissipation theorem (FDT) is generalized in the quantum regime to include fundamentally nonequilibrium corrections arising from the breakdown of detailed balance. In quantum Markovian dynamics,

$\operatorname{Im} R^{(1)}(\omega) = \text{(equilibrium fluctuation)} + \text{(curl flux contribution)},$

where the curl quantum flux

$c_{mn} = L_{nn,mm}\rho_{mm} - \min\left( L_{nn,mm}\rho_{mm}, L_{mm,nn}\rho_{nn} \right)$

explicitly quantifies cyclic probability currents driving irreversible dynamics (Zhang et al., 2021). The conventional FDT is recovered only when the curl flux vanishes (equilibrium).

7. Experimental Realizations and Applications

Quantum fluctuation relations have been experimentally verified using diverse platforms:

Quantum coherent conductors (Aharonov-Bohm rings) exhibit higher-order current-noise correlations obeying fluctuation relations even as Onsager-Casimir symmetries are broken by magnetic fields (0911.3470).
Single NV centers in diamond, with controlled dissipation, demonstrate fluctuation relations in regimes where distinguishing work and heat is infeasible (Hernández-Gómez et al., 2021).
Protocols involving repeated quantum measurements reproduce robust Jarzynski equalities (Gherardini et al., 2022).
Feedback-controlled systems, such as manipulated qubits, enable verification of information-inclusive detailed fluctuation relations (Camati et al., 2018).

Fluctuation relations are applied to probe many-body correlations in interacting quantum pumps (measured deviation from stationary fluctuation-dissipation), identify hidden conserved charges in quantum simulators, and paper the interplay of coherence and thermodynamic irreversibility.

Table: Examples of Quantum Fluctuation Relations

System/Scenario	Fluctuation Relation	Special Feature
Driven, open NV center (Hernández-Gómez et al., 2021)	$\langle e^{-\beta\Delta E} \rangle = e^{-\beta\Delta F}$	Energy change includes work and heat
Repeated measurements (Gherardini et al., 2022)	$\langle e^{-\beta Q} \rangle = 1$	Robust to measurement randomness
Quantum transport (0911.3470)	$S_1 = 2k_B T G_2$ , $S_1^A = 6k_B T G_2^A$	Valid even when symmetries break
GGE initial state (Mur-Petit et al., 2017)	Generalized Crooks/Jarzynski QFRs	Includes arbitrary conserved charges
Energy-coherent battery (Holmes et al., 2020)	Quantum Crooks-like equality with distortion factor $q(\chi)$	Explicit quantum-coherence correction

Quantum fluctuation relations thus serve as central, experimentally accessible constraints on quantum dynamics far from equilibrium, unifying principles from microreversibility, measurement theory, information thermodynamics, and open-system quantum dynamics. Their rigorous formulation and verification underpin foundational advances in quantum thermodynamics, nanoscale devices, and quantum information technologies.