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arXiv:2411.00944 · Physical Review Letters

Minimizing Dissipation via Interacting Environments: Quadratic Convergence to Landauer Bound

Patryk Lipka-Bartosik, Martí Perarnau-Llobet

Published Sep 8, 2025• 2 citations
arXivHTMLPDF

Abstract

We explore the fundamental limits on thermodynamic irreversibility when cooling a quantum system in the presence of a finite-size reservoir. First, we prove that, for any noninteracting $n$-particle reservoir, the entropy production $\mathrm{\ensuremath{\Sigma}}$ decays at most linearly with $n$. Instead, we derive a cooling protocol in which $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}1/{n}^{2}$, which is, in fact, the best possible scaling. This becomes possible due to the presence of interactions in the finite-size reservoir, which must be prepared at the verge of a phase transition. We further show numerically that an intermediate scaling of $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}1/{n}^{\ensuremath{\delta}}$ with $\ensuremath{\delta}\ensuremath{\in}(1,2)$ can be achieved using a star-network configuration for the reservoir. Our results open the possibility of cooling with a higher energetic efficiency via interacting reservoirs.

DOI
10.1103/5w1r-1nzs

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We explore the fundamental limits on thermodynamic irreversibility when cooling a quantum system in the presence of a finite-size reservoir. First, we prove that, for any noninteracting nnn-particle reservoir, the entropy production \ensuremathΣ\mathrm{\ensuremath{\Sigma}}\ensuremathΣ decays at most linearly with nnn. Instead, we derive a cooling protocol in which \ensuremathΣ\ensuremath∝1/n2\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}1/{n}^{2}\ensuremathΣ\ensuremath∝1/n2, which is, in fact, the best possible scaling. This becomes possible due to the presence of interactions in the finite-size reservoir, which must be prepared at the verge of a phase transition. We further show numerically that an intermediate scaling of \ensuremathΣ\ensuremath∝1/n\ensuremathδ\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}1/{n}^{\ensuremath{\delta}}\ensuremathΣ\ensuremath∝1/n\ensuremathδ with \ensuremathδ\ensuremath∈(1,2)\ensuremath{\delta}\ensuremath{\in}(1,2)\ensuremathδ\ensuremath∈(1,2) can be achieved using a star-network configuration for the reservoir. Our results open the possibility of cooling with a higher energetic efficiency via interacting reservoirs.
The erasure of quantum information is governed by two fundamental thermodynamic bounds: Landauer’s principle, which establishes the absolute minimum energetic cost of resetting a state, and Nernst’s unattainability principle, which dictates that perfect cooling to absolute zero entropy requires infinite resources. Under finite constraints, conventional cooling protocols utilizing thermal baths composed of n non-interacting particles are bottlenecked by a linear dissipation barrier (O(1/n)). In this thesis, we establish the ultimate thermodynamic limits of finite-resource multiqubit erasure and introduce the most thermodynamically efficient cooling protocol currently known. Building upon the max-cooling framework—a global unitary operation that systematically sorts the eigenvalues of the joint system-bath state to pack the highest probability mass into the target’s ground state—we engineer a bath Hamiltonian with specific, target-dependent degeneracies for arbitrary m-qubit systems. This collective protocol achieves an optimal asymptotic entropy production scaling of Σ = O(mˆ2/nˆ2), successfully saturating the scaling of the fundamental theoretical lower bound. This degeneracy-optimized approach strictly outperforms both standard non-interacting reservoirs and the best-known protocol for the independent cooling of single qubits using a reservoir of interacting particles (O(mˆ3/nˆ2)). Furthermore, we establish that surpassing conventional dissipation barriers requires an operational threshold of just n ≥ 2mˆ2 bath particles to achieve this optimal scaling.
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Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state | φ 1 ⟩ be no smaller than p φ 1 max − δ . Here p φ 1 max is the maximum probability of information erasure that is permissible by the physical context, and δ ⩾ 0 the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework.
We investigate and compare two particle number-conserving protocols for the preparation of a topologically nontrivial state. The first is derived from thermally coupling the system to a cold bath, while the second is based on engineered dissipation. We numerically study the time required to reach the target state as well as its robustness against physically important perturbations. Crucially, in both protocols, the cooling capability is limited by dissipatively induced diffusion processes. The resulting quadratic scaling of the cooling time with system size is also corroborated analytically using mean-field approximations and a purely classical random-walk model. Furthermore, we find that the engineered protocol admits a unique and stable dark state, which contributes to an ongoing discussion regarding the applicability of dissipative state preparation to many-body systems.