Quantum Mpemba Speedups in the Thermodynamics of Landauer Erasure

Abstract: We investigate how nonequilibrium quantum initial states can reduce the finite-time thermodynamic cost of Landauer erasure. Considering a general finite-dimensional quantum memory coupled to a thermal reservoir via a Davies generator, we show that the dissipated heat at a fixed operational erasure fidelity is largely controlled by the overlap of the initial state with the slowest Liouvillian relaxation mode. We derive a modified finite-time Landauer bound in which the excess dissipation above the quasistatic limit scales quadratically with this slow mode projection, and we prove a sufficient Mpemba Landauer condition under which a hotter state can erase faster and dissipate less heat than a colder preparation, without violating Landauer's principle. A minimal qutrit model illustrates these quantum Mpemba speedups and reveals broad parameter regimes where coherence and Hamiltonian-induced modes conspire to suppress finite-time entropy production. We further identify practical control knobs, including temperature tuning, coherence engineering, and Hamiltonian shaping of Liouvillian spectra, which enable Mpemba-enhanced erasure to be directly tested on platforms such as superconducting circuits, trapped ions, semiconductor quantum dots, and solid-state spins.

• The paper establishes that a hotter quantum-memory preparation can erase faster than a colder one when its overlap with the slowest Liouvillian mode is sufficiently smaller.
• The analysis combines Davies-generator spectral theory with Spohn’s entropy balance, showing that excess dissipation scales quadratically with slow-mode overlap while heat reduction also requires an independent entropy-change condition.
• A qutrit model demonstrates broad parameter regimes with simultaneous speed and heat advantages, while the framework identifies experimental routes using superconducting circuits, trapped ions, quantum dots, and NV centers.

Overview

This paper by Chattopadhyay (2608.16254) connects two bodies of work in nonequilibrium quantum statistical mechanics: the finite-time Landauer principle and the Liouvillian spectral theory of the quantum Mpemba effect (QME). The central claim is that a hotter initial preparation of a quantum memory can, under well-defined spectral conditions, reach a prescribed erasure fidelity faster and dissipate less heat than a colder one — without violating the quasistatic Landauer bound of $k_B T \ln 2$. The mechanism is not a new microscopic effect; rather, it exploits the established fact that relaxation under a Davies generator is governed by the projection of the initial state onto the slowest Liouvillian mode. The paper's contribution is to show that this same modal overlap controls both the operational erasure time and the leading-order excess entropy production above equilibrium.

Setup: erasure as autonomous Davies relaxation

The memory is a $d$-dimensional system with logical zero encoded in a subspace $\mathcal H_0$ (projector $\Pi_0$). The protocol has three stages. First, the memory is prepared in a thermal state $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$ of an input Hamiltonian $H_i$ at a tunable temperature $T_{\mathrm{prep}}$. Second, an instantaneous quench replaces $H_i$ with an erasure Hamiltonian $H_f$, energetically biasing the logical-zero manifold; this quench injects work $W_{\mathrm{quench}} = \Tr[\rho(0)(H_f - H_i)]$. Third, for all $d$0 the system evolves autonomously under a primitive, thermodynamically consistent Davies generator at bath temperature $d$1, with unique fixed point $d$2.

Erasure is declared successful when the logical population $d$3 crosses a fidelity threshold $d$4 (e.g., 0.95), defining the operational time $d$5. Because $d$6 is held fixed during relaxation, no work is performed after the quench, and the Landauer cost is identified with the heat discharged into the bath, $d$7. This identification makes the cost exactly equal to the free-energy decrease of the system during autonomous relaxation.

Spectral condition for dynamical speedup

Expanding the deviation from equilibrium in biorthogonal right/left eigenmodes of the (generally non-normal) Liouvillian gives

$d$8

with coefficients $d$9. For the erasure observable, each mode contributes through the product $\mathcal H_0$0 where $\mathcal H_0$1 — a point the paper emphasizes, since a large coefficient $\mathcal H_0$2 is irrelevant to erasure if the corresponding right mode has vanishing support on $\mathcal H_0$3.

The key controlled approximation requires that the higher-mode remainder satisfy $\mathcal H_0$4 over the interval containing the target crossing, with $\mathcal H_0$5. Within this regime, the erasure time obeys the explicit bounds

$\mathcal H_0$6

and the sufficient Mpemba condition becomes $\mathcal H_0$7: a hotter preparation erases faster whenever its slow-mode amplitude is smaller. When this bound fails, the paper provides a two-mode transcendental correction to be solved numerically. Notably, the appendix shows that pure coherence modes typically have $\mathcal H_0$8 for nondegenerate Hamiltonians with diagonal projectors, so coherence-assisted erasure requires degenerate Bohr frequencies, engineered dissipative couplings, or a non-diagonal logical projector — a structural constraint on how broadly the mechanism applies.

Finite-time Landauer cost and the main theorem

Using Spohn's entropy balance, the exact heat identity is $\mathcal H_0$9, where $\Pi_0$0 is the integrated entropy production and $\Pi_0$1 the von Neumann entropy change. Expanding the matrix logarithm around the full-rank Gibbs state via its Fréchet derivative yields a quadratic form in the Liouvillian amplitudes:

$\Pi_0$2

with $\Pi_0$3 strictly positive for any nontrivial dissipative mode of a primitive generator. The excess dissipation therefore scales quadratically in the slow-mode overlap $\Pi_0$4 — the same quantity that controls the erasure time.

The main theorem then separates into two logically distinct statements. Under the controlled slow-mode condition, the hotter preparation reaches target fidelity sooner. However, reduced dissipation requires an additional sufficient condition,

$\Pi_0$5

which combines the exact Spohn balance with bounded remainders. The paper is explicit that suppression of the slow Liouvillian mode alone does not imply lower heat dissipation; only if both protocols terminate at equal entropy changes does smaller integrated entropy production translate directly into smaller $\Pi_0$6. This caveat is important because it prevents over-interpretation of the quadratic scaling result.

Qutrit illustration

A minimal three-level model demonstrates the mechanism concretely. The erasure Hamiltonian places logical zero at energy $\Pi_0$7 and couples a hybridized excited doublet ($\Pi_0$8, $\Pi_0$9, coupling $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$0), so that population flow and coherence decay are intertwined and the Liouvillian spectrum acquires nontrivial structure. With parameters $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$1, $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$2, $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$3, and $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$4, parameter sweeps reveal broad regimes where $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$5, yielding simultaneously $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$6 and $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$7. Both quantities depend strongly on $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$8, with many hot–cold pairs exhibiting the Mpemba advantage relative to the dotted quasistatic bound $\rho_{\mathrm{th}}(T_{\mathrm{prep}})$9. The authors stress that the qutrit is illustrative only; the framework itself is formulated purely at the level of the Liouvillian spectral decomposition.

Experimental feasibility

The protocol requires only tunable state preparation, engineered Markovian dissipation, and high-fidelity readout under a common generator — capabilities available on four platforms discussed in detail. Superconducting circuits offer calorimetric measurement of dissipated heat during finite-time erasure and dispersive readout of populations [saira2020nonequilibrium]. Trapped ions have already demonstrated QME inversions under engineered dissipation [PhysRevLett.133.010402] and support nonequilibrium thermodynamic protocols. Semiconductor quantum dots exhibit Mpemba-type anomalies from Liouvillian spectral properties [PhysRevLett.131.080402] and allow continuous charge monitoring plus calorimetry. NV-center systems have recently observed the QME directly without bath engineering [chatterjee2025direct, schnepper2025experimental] and permit full process tomography, enabling reconstruction of the Liouvillian spectrum and the initial-state overlaps entering the theory's conditions.

Limitations and open questions

Several restrictions qualify the results. The single-mode approximation is controlled only after faster modes have decayed relative to the slowest one; near $H_i$0 it fails outright, and if the operational erasure time falls within the higher-mode decay window, the full multimode expansion must be used. The heat-reduction theorem is sufficient rather than necessary, depends on bounded remainders $H_i$1 (estimable numerically or analytically only when a spectral gap exists), and requires the entropy-change inequality — so the dynamical and thermodynamic claims are genuinely separate. The analysis assumes finite-dimensional Hilbert spaces, weak coupling, Markovian secular Davies dynamics, and diagonalizable generators (Jordan blocks would introduce polynomial prefactors requiring generalized eigenspaces). The paper leaves open whether the mechanism survives for unbounded spectra, non-Markovian reservoirs, strong system–bath coupling, or driven environments, and how collective Mpemba mechanisms scale in many-body registers with correlated baths. Connecting the mode picture to thermodynamic length, quantum speed limits, and optimal control to obtain provably optimal protocols also remains unresolved.

Conclusion

The paper establishes that the spectral origin of the quantum Mpemba effect can be repurposed as a thermodynamic resource for information erasure: within a controlled slow-mode regime, reducing the initial-state overlap with the slowest Liouvillian mode shortens the operational erasure time, and — subject to an independent entropy-balance condition — reduces the finite-time heat dissipated above the quasistatic Landauer bound. The quadratic scaling of excess entropy production with the slow-mode projection provides a quantitative design principle, and the identified control knobs (preparation-temperature tuning, coherence engineering, Hamiltonian shaping of the Liouvillian spectrum) map directly onto existing experimental platforms. The framework's restriction to finite-dimensional Markovian dynamics, and the separation between the dynamical and thermodynamic conditions, define the precise scope within which these conclusions hold.