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arXiv:2103.02337 · Physical Review A

Impossibility of achieving Landauer's bound for almost every quantum state

Paul M. Riechers, Mile Gu

Published Jul 19, 2021• 17 citations
arXivHTMLPDF

Abstract

The thermodynamic cost of resetting an arbitrary initial state to a particular desired state is lower bounded by Landauer's bound. However, here we demonstrate that this lower bound is necessarily unachievable for every initial state (except possibly the single minimally dissipative input) for any reliable reset mechanism. Since local heating threatens rapid decoherence, this issue is of substantial importance beyond mere energy efficiency. For the case of qubit reset, we find the minimally dissipative state analytically for any reliable reset protocol, in terms of the entropy-flow vector introduced here. This allows us to verify a recent theorem about initial-state dependence of entropy production for any finite-time transformation, as it pertains to quantum state preparation.

DOI
10.1103/physreva.104.012214

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The thermodynamic cost of resetting an arbitrary initial state to a particular desired state is lower bounded by Landauer's bound. However, here we demonstrate that this lower bound is necessarily unachievable for every initial state (except possibly the single minimally dissipative input) for any reliable reset mechanism. Since local heating threatens rapid decoherence, this issue is of substantial importance beyond mere energy efficiency. For the case of qubit reset, we find the minimally dissipative state analytically for any reliable reset protocol, in terms of the entropy-flow vector introduced here. This allows us to verify a recent theorem about initial-state dependence of entropy production for any finite-time transformation, as it pertains to quantum state preparation.
In practice, qubit reset must be operated in an extremely short time, which incurs a thermodynamic cost within multiple orders of magnitude above the Landauer bound. We present a general framework to determine the minimal thermodynamic cost and the corresponding optimal protocol for memory erasure under arbitrary erasure speeds. Our study reveals the divergent behavior of minimal entropy production in the short-time limit depends on the convergence and divergence of the jump operator. There is an inherent trade-off between the minimal required time and the set error probability for the convergent class. Moreover, we find the optimal protocol exhibits general features in the fast-driving regime. To illustrate these findings, we employ fermionic and bosonic baths as examples. Our results suggest that the super-Ohmic bosonic heat bath is suitable for qubit reset.
The Landauer principle sets a lower bound on the thermodynamic cost of qubit reset, which is only attainable for the quasistatic process. In this Letter, we explore the collective advantage of qubit reset of coherent qubits in three aspects. First, for the quasistatic process, the thermodynamic cost of collective reset is remarkably lower than parallel reset because of the reduced Hilbert space dimension due to entanglement effects. Second, for the finite-time qubit reset, we prove that the error probability fades away and per-qubit heat production tends the Landauer bound for initially continuous protocols in the thermodynamic limit. Third, we show that qubit reset performance enhances with the increase in the number of qubits. Our results, illustrated by different protocols, provide a blueprint for future quantum device fabrication.
Quantum information reset is an intrinsically irreversible process constrained by the second law of thermodynamics. In contemporary superconducting processors, dissipation during qubit reset is frequently dominated not only by quantum limits but by a classical energy floor on the order of 10^{-21} J. In contrast, the entropic bound T \Delta S at dilution refrigerator temperatures (\sim 15 mK) corresponds to a lower energy scale of approximately 10^{-23} J, establishing entropy production as the organizing physical principle that limits both the efficiency and the speed of reset operations. We introduce an entropy-limited efficiency estimator, \eta_S = 1 - (T \Delta S / W_{\rm diss}), which quantifies the thermodynamic distance from the entropic (Landauer) limit. By construction, \eta_S = 0 corresponds to the saturation of the entropy production limit, while \eta_S \to 1 indicates dissipation dominated by non-entropic (classical) contributions. Maximizing \eta_S is equivalent to a constrained variational problem in which dissipated work is minimized subject to the second law of thermodynamics. For typical superconducting qubits (\omega_q/2\pi \sim 5 GHz, T_1 \sim 50–100 \mus, T \sim 15 mK), the entropic bound T \Delta S \sim 10^{-23} J lies two orders of magnitude below the experimentally observed classical dissipation floor W_{\rm floor} \sim 10^{-21} J. Figures 1 and 2 were generated using the provided scripts_generate_figures.py script, using parameters: \omega_q/2\pi = 5 GHz, T_1 = 80 \mus, and T = 15 mK.
Landauer's principle provides a perspective on the physical meaning of information as well as on the minimum working cost of information processing. Whereas most studies have related the decrease in entropy during a computationally irreversible process to a lower bound of dissipated heat, recent efforts have also provided another lower bound associated with the thermodynamic fluctuation of heat. The coexistence of the two conceptually independent bounds has stimulated comparative studies of their close relationship or tightness; however, these studies were concerned with finite quantum systems that allowed the revival of erased information because of a finite recurrence time. We broaden these comparative studies further to open quantum systems with infinite recurrence times. By examining their dependence on the initial state, we find the independence of the thermodynamic bound from the initial coherence, whereas the entropic bound depends on both the initial coherence and the population. A crucial role is indicated by the purity of the initial state: The entropic bound is tighter when the initial condition is sufficiently mixed, whereas the thermodynamic bound is tighter when the initial state is close to a pure state. These trends are consistent with previous results obtained for finite systems.
The Landauer principle states that any logically irreversible information processing must be accompanied by dissipation into the environment. In this Letter, we investigate the heat dissipation associated with finite-time information erasure and the effect of quantum coherence in such processes. By considering a scenario wherein information is encoded in an open quantum system whose dynamics are described by the Markovian Lindblad equation, we show that the dissipated heat is lower bounded by the conventional Landauer cost, as well as a correction term inversely proportional to the operational time. To clarify the relation between quantum coherence and dissipation, we derive a lower bound for heat dissipation in terms of quantum coherence. This bound quantitatively implies that the creation of quantum coherence in the energy eigenbasis during the erasure process inevitably leads to additional heat costs. The obtained bounds hold for arbitrary operational time and control protocol. By following an optimal control theory, we numerically present an optimal protocol and illustrate our findings by using a single-qubit system.
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.