🧊 Resource-Theoretical Unification of Mpemba Effects: Classical and Quantum

Notebooks accompanying:

Alessandro Summer, Mattia Moroder, Laetitia P. Bettmann, Xhek Turkeshi, Iman Marvian, and John Goold, “Resource-Theoretical Unification of Mpemba Effects: Classical and Quantum,” Physical Review X 16, 011065 (2026).

TL;DR. We recast anomalous relaxation — the Mpemba effect — as resource dissipation in a quantum resource theory. Thermal and symmetry Mpemba effects become parallel instances of the same phenomenon: a state that is more resourceful initially can lose its resource faster than a less resourceful one, driven by its overlap with the slowest eigenmodes of the dissipative dynamics. This repository provides an executable companion to every worked example in the paper, from classical Markov chains to non-Abelian SU(2) random circuits. Each walkthrough identifies whether it is an exact reproduction, a raw-data-backed calculation, or an independent reconstruction necessitated by unavailable manuscript-era inputs.

Under the same free dynamics, an initially more resourceful state can lose the chosen resource faster when it has less overlap with the slowest decay mode relevant to that resource, producing a crossing of resource monotones. This mode-overlap mechanism provides a common description of thermal, symmetry, coherence, and nonstationarity Mpemba effects in classical and quantum settings.

📰 Media coverage and background

• Science: “Hot things can freeze faster than cool ones. Now, this paradox has gone quantum” — a popular-science feature connecting the classical freezing paradox to recent quantum Mpemba research and this resource-theoretic framework.
• Phys.org: “Framework unifies the classical and quantum Mpemba effects” — an accessible overview of this work and its common slow-mode mechanism.
• Wikipedia: This work can also be found among the references on John Goold (physicist). For a general introduction to the effect, see Mpemba effect, and for historical and biographical context, see Erasto B. Mpemba.

🎯 Main contributions

• 🧩 Resource theory supplies a common definition of different Mpemba effects. Thermal and symmetry Mpemba effects are both reversals of resource ordering under free dynamics, specialized to athermality and asymmetry. Crucially, the two canonical diagnostics used here share the relative entropy of resource construction, $R_{\mathrm{rel}}^{\mathcal F}(\rho)=\min_{\sigma\in\mathcal F}D(\rho\Vert\sigma)$, with only the free-state set changing. For athermality it gives $D(\rho\Vert\pi_\beta)=\beta[F(\rho)-F(\pi_\beta)]$, the excess nonequilibrium free energy in units of $k_{\mathrm B}T$. For asymmetry it gives $D(\rho\Vert\mathcal G[\rho])=S(\mathcal G[\rho])-S(\rho)$, the relative entropy of asymmetry (called entanglement asymmetry in the reduced-subsystem setting). These are distinct monotones, but instances of the same construction: a Mpemba crossing is precisely the initially more resourceful state becoming the less resourceful one. Manuscript: Sec. I.1 · Sec. IV · Walkthrough

• 📉 A common spectral mechanism links thermal and symmetry Mpemba effects. We show that anomalous relaxation is governed by the initial state’s overlap with the resourceful eigenmodes of the dissipative map — a mechanism previously recognized in thermal and nonstationary Mpemba effects by Nava and Fabrizio and Carollo, Lasanta, and Lesanovsky. In the symmetry setting, a strongly symmetry-broken state can relax faster when its overlap with the slowest symmetry-restoring mode is small or vanishes. Manuscript: Sec. II · Sec. III.3 · Thermal walkthrough · Symmetry-mode walkthrough

• ⚖️ The symmetry Mpemba effect is not inherently quantum. It already occurs in classical Markovian systems. Entanglement can accompany particular realizations, but it is not required for the effect. Manuscript: Sec. III.4 · Walkthrough

• 🌐 Open vs. closed is not the fundamental distinction; Markovian vs. non-Markovian is. Dissipating a resource requires an environment. In "closed" Mpemba settings — entanglement-asymmetry circuits, ETH baths, any subsystem of a globally unitary composite — the joint universe conserves total asymmetry (or athermality) exactly; what decreases locally is the reduced-state monotone, because the resource flows into non-local system-environment correlations the partial trace discards. Whether the environment is an external bath or the complementary subsystem, the reduced dynamics is the same G-covariant CPTP map: its memory — Markovian vs. non-Markovian — sets the phenomenology, not the openness of the universe. Appendix F makes this precise: whenever the universe is strongly symmetric — a G-invariant joint unitary acting on a G-invariant environment — the reduced system dynamics is weakly symmetric (a G-covariant CPTP map). Manuscript: Sec. III.1 · Sec. III.7 · Sec. V · Appendix (G-invariance → G-covariance) · Non-Markovian U(1) circuits · ETH isolated bath

• 🎲 Random circuits reveal the role of modes of asymmetry. In random symmetry-preserving circuits — both Abelian U(1) (magnetization-conserving) and non-Abelian SU(2) — sectors carrying higher modes of asymmetry decay faster on average. Restricting the initial state to those sectors suppresses its overlap with the slowest symmetry-restoring mode and produces a spectrum-agnostic strong Mpemba effect: the mechanism operates statistically across random realizations without diagonalizing any specific channel. Manuscript: Sec. III.7 · Markovian U(1) · Non-Markovian U(1) · Non-Abelian SU(2)

📁 Repository contents

The thirteen notebooks in notebooks/ are publication-oriented walkthroughs of the main-text and appendix examples. Each identifies its manuscript location and reproduction status, constructs the model, verifies the relevant structural properties, and only then tests the Mpemba crossing. The SU(2) notebook recomputes the five Fig. 11 curves from 100 raw full-SU(2) circuit realizations and reproduces all ten pairwise crossings. It verifies covariance under $S_x,S_y,S_z$ and uses the exact non-Abelian Haar twirl.

🚀 Quick start

Create a virtual environment, install the dependencies, and execute the lightweight single-qubit introduction:

git clone https://github.com/alessum/resource_theory_mpemba.git
cd resource_theory_mpemba
python -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt
python tools/execute_notebooks.py atherm_ex2.ipynb

The final command checks the notebook entirely in memory and does not modify the tracked file. See the notebook guide for the full notebook map and computational scope.

Appendix companions

• 🌀 Mpemba_nonstatioinarity.ipynb — independent reconstruction of the Appendix A nonstationarity example. Both initial states are generated transparently; no pickle or stored trajectory is used.
• ⛓️ Mpemba_ETH.ipynb — manuscript-scale Appendix B calculation for an $N=15$ ETH bath, including the full sparse $2^{16}$-dimensional unitary evolution and Rényi crossing-time inset.
• 📐 Mpemba_QFI_monotone.ipynb — exact Appendix QFI reconstruction showing crossings for SLD and Wigner-Yanase metrics but no crossing for the harmonic-mean metric.
• NOTEBOOKS.md gives the recommended reading order and maps every notebook to the relevant manuscript section and figure.
• tools/notebook_utils.py contains shared numerical routines.
• tools/circuit_data.py implements the reproducible random-circuit simulations.
• data/circuit_examples/ contains raw, per-realization circuit data rather than fitted or illustrative curves.
• tools/ contains notebook builders, data generators, and the top-to-bottom execution check.

▶️ Running the notebooks

Run commands from the repository root. The notebook checker executes in memory by default, so it does not alter tracked outputs:

python -m pip install -r requirements.txt
python tools/execute_notebooks.py

Pass notebook filenames to check only a subset. Add --write only when you intentionally want to refresh stored outputs:

python tools/execute_notebooks.py --write atherm_ex2.ipynb

To rebuild and refresh only the appendix companions:

python tools/build_publishable_notebooks.py \
  Mpemba_nonstatioinarity.ipynb Mpemba_ETH.ipynb Mpemba_QFI_monotone.ipynb
python tools/execute_notebooks.py --write \
  Mpemba_nonstatioinarity.ipynb Mpemba_ETH.ipynb Mpemba_QFI_monotone.ipynb

To rebuild the publishable notebooks, regenerate the standard local circuit datasets, and refresh every stored output:

python tools/build_publishable_notebooks.py
python tools/generate_circuit_data.py
python tools/execute_notebooks.py --write

Paper-scale circuit datasets have explicit commands, runtimes, provenance, and scope in data/circuit_examples/README.md. The exact-size non-Markovian U(1) reference dataset uses archived gate parameters from the companion repository:

python tools/generate_circuit_data.py \
  --only u1-reference \
  --paper-scale \
  --reference-checkout /path/to/mpemba_circuits

The generator verifies the upstream parameter file by SHA-256 before running. The exact occupied-U(1)-sector engine evolves each charge sector once, reuses it for all five tilts, and parallelizes independent circuits. The stored file contains the complete 100-circuit, 1000-layer Fig. 10 calculation.

📊 Circuit-data scope

The stored data are deliberately explicit about their statistical scope:

• The Markovian U(1) evolution (Fig. 9) is a paper-scale independent reproduction using the manuscript system ($N_s{=}4$), environment ($N_e{=}2$), and ensemble size ($100$ deterministic circuit realizations); it is not a replay of an unpublished original random ensemble.
• The non-Markovian U(1) walkthrough is reference-backed: it uses every archived parameter row from the pinned companion repository for the complete 100-circuit $N_s=8,N_e=12$, 1000-layer ensemble. The smaller 24-circuit $N_s=4,N_e=8$ file is retained only for the explicit channel audit of Marvian--Spekkens Eq. (3.10) as a mode-transfer reconstruction identity.
• The SU(2) walkthrough is a paper-scale, figure-matching independent reconstruction at $N_s=8,N_e=12$ with 100 realizations. It evolves isotropic partial-SWAP gates with an invariant singlet bath and computes asymmetry with the exact SU(2) Schur-basis Haar twirl. Its inferred initial-angle and displayed-time conventions are recorded explicitly because they are absent from the printed equations and public original data.
• The U(1) and SU(2) non-Markovian checks audit the same Marvian--Spekkens Eq. (3.10): as an explicit mode-transfer reconstruction identity for U(1), and as a contraction bound on the rank-1 irreducible tensor for SU(2).

📚 Citation

Use GitHub’s Cite this repository control, backed by CITATION.cff, for machine-readable citation metadata. The published article remains the preferred scholarly citation:

@article{Summer2026Mpemba,
  title   = {Resource-Theoretical Unification of Mpemba Effects: Classical and Quantum},
  author  = {Summer, Alessandro and Moroder, Mattia and Bettmann, Laetitia P. and Turkeshi, Xhek and Marvian, Iman and Goold, John},
  journal = {Physical Review X},
  volume  = {16},
  pages   = {011065},
  year    = {2026},
  doi     = {10.1103/rbt4-psfd}
}

⚖️ License

The repository code is available under the MIT License. Third-party materials retain their original terms.