We derive a thermodynamic quantum speed limit for irreversible processes: (kB ∆Σ)·τ ≥ ■/(2T), where ∆Σ is the entropy produced in units of kB , τ is the process duration, T is the temperature, and ■ is thereduced Planck constant. We call this the Parker Relation. It is derived by two routes for thermalLindblad channels in the weak-coupling limit (γ/ω ■ 1): (1) the Deffner–Lutz quantum speed limit foropen systems applied to the amplitude damping channel, with a Hilbert–Schmidt norm bound on theLindblad generator rigorous to leading order in γ/ω; and (2) the open-system Margolus–Levitin bound,where the effective driving energy T × kB × ∆Σ follows from the free energy budget of a Markovianthermal channel with no external work. We establish explicitly when the relation is novel versus when itreduces to existing bounds. For thermal relaxation (∆Σ = ■ω/kB T) the Parker bound reproduces thestandard Mandelstam–Tamm result τ ≥ 1/(2ω) — a consistency check. For Landauer logical erasure (∆Σ= ln 2) the bound is independent of existing quantum speed limits. The classical bridge is established viathe incoherent limit of the quantum amplitude damping channel, which reduces exactly to the classicaltwo-state master equation governing bistable erasure devices. ■ appears in the classical bound becauseevery thermal bath has a quantum mechanical substrate; the classical Fisher–Rao speed limit combinedwith the quantum-limited attempt frequency ν0~ kBT/■ gives τmin ~ ■/(2kB T) to within a numericalprefactor of order unity (see Appendix A). The primary novel prediction is the minimum time forirreversible bit erasure at 300 K: τ min = ■/(2kB T ln 2) = 18.4 femtoseconds — a quantum-derivedfoundational floor that real CMOS devices exceed by 8 orders of magnitude. For a 5 GHz transmon at20 mK the bound gives τ min ≈ 16 ns — 60 000× below current best T2 values, a consistency check not anovel prediction. Four open questions are identified: tightening the generator norm bound numericallyvia QuTiP; completing the Route 2 formal connection to del Campo–Funo; extending beyond thermalLindblad channels; and completing the classical bridge via Nicholson et al. or Dechant. The relation isconjectured to hold generally for all irreversible processes at temperature T.
We derive a thermodynamic quantum speed limit for irreversible processes: (kB ∆Σ)·τ ≥ ■/(2T), where ∆Σ is the entropy produced in units of kB , τ is the process duration, T is the temperature, and ■ is thereduced Planck constant. We call this the Parker Relation. It is derived by two routes for thermalLindblad channels in the weak-coupling limit (γ/ω ■ 1): (1) the Deffner–Lutz quantum speed limit foropen systems applied to the amplitude damping channel, with a Hilbert–Schmidt norm bound on theLindblad generator rigorous to leading order in γ/ω; and (2) the open-system Margolus–Levitin bound,where the effective driving energy T × kB × ∆Σ follows from the free energy budget of a Markovianthermal channel with no external work. We establish explicitly when the relation is novel versus when itreduces to existing bounds. For thermal relaxation (∆Σ = ■ω/kB T) the Parker bound reproduces thestandard Mandelstam–Tamm result τ ≥ 1/(2ω) — a consistency check. For Landauer logical erasure (∆Σ= ln 2) the bound is independent of existing quantum speed limits. The classical bridge is established viathe incoherent limit of the quantum amplitude damping channel, which reduces exactly to the classicaltwo-state master equation governing bistable erasure devices. ■ appears in the classical bound becauseevery thermal bath has a quantum mechanical substrate; the classical Fisher–Rao speed limit combinedwith the quantum-limited attempt frequency ν0~ kBT/■ gives τmin ~ ■/(2kB T) to within a numericalprefactor of order unity (see Appendix A). The primary novel prediction is the minimum time forirreversible bit erasure at 300 K: τ min = ■/(2kB T ln 2) = 18.4 femtoseconds — a quantum-derivedfoundational floor that real CMOS devices exceed by 8 orders of magnitude. For a 5 GHz transmon at20 mK the bound gives τ min ≈ 16 ns — 60 000× below current best T2 values, a consistency check not anovel prediction. Four open questions are identified: tightening the generator norm bound numericallyvia QuTiP; completing the Route 2 formal connection to del Campo–Funo; extending beyond thermalLindblad channels; and completing the classical bridge via Nicholson et al. or Dechant. The relation isconjectured to hold generally for all irreversible processes at temperature T.
We derive a fundamental uncertainty principle relating entropy change and time: for any quantum system coupled to a thermal environment, the entropy change ΔS and the characteristic timescale Δt satisfy ΔS·Δt ≥ kᴪ/2. We call this the Parker Relation. The derivation proceeds from the quantum Thermodynamic Uncertainty Relation (quantum TUR) of Hasegawa and Van Vu [Phys. Rev. E 99, 062126 (2019)], which holds for all quantum channels without restriction to quasi-static or equilibrium processes. We provide a second independent derivation from the canonical commutation relations of the entropy field operator Ś = ∫Φ√(-g) d³x, which yields the same bound through the Robertson inequality. Four immediate consequences are derived with explicit numerical predictions: (1) a minimum information erasure time of 18.4 femtoseconds at 300 K, establishing a fundamental floor on computational switching speed beyond any engineering constraint; (2) a minimum decoherence time of 53 nanoseconds for a 5 GHz superconducting transmon qubit at 20 mK; (3) an exponential suppression of high-frequency vacuum modes that removes 119 orders of magnitude from the cosmological constant ultraviolet divergence; and (4) a characteristic thermodynamic timescale of ~15 ms for living cells at physiological metabolic rates, coinciding with observed neural and cellular signaling timescales. The Parker Relation is interpretation-independent: it holds in Copenhagen, many-worlds, and objective collapse frameworks equally. It is the temporal complement of Landauer’s principle and the thermodynamic analog of the Heisenberg energy-time uncertainty relation.
As one of the fundamental questions, quantum speed limit (QSL) is of vital role in the community of quantum physics, which sets the ultimate, maximal speed with which a quantum system can evolve. QSL originated from the understanding of the Heisenberg uncertainty relation. In 1945, Mandelstam and Tamm found the time Δt in the celebrated relation ΔEΔt≥ℏ could be thought as the minimum time that a quantum system needs to evolve from an initial state to the orthogonal state for unitary evolution. QSL sets the lower bound of the evolution. They established the relationship between the minimum evolution time ( τQSL ) and the energy uncertainty of the system (ΔE), namely τQSL=πℏ/(2ΔE) , the Mandelstam-Tamm (MT) bound. Later, Margolus and Levitin derived another different bound for a closed system which connected the τQSL with the average energy of the system ( 〈E〉 ), i.e., τQSL=πℏ/(2〈E〉) , the Margolus-Levitin (ML) bound. Combining the MT and ML bounds together, we can have a unified bound for a unitary system and orthogonal states. In the case of open system dynamics, because there is energy and/or coherence exchange between the system and the environment, the orthogonal final state of the initial state can hardly be reached any longer. An investigation of the QSL on non-unitary quantum evolution needs to be considered. The generalized QSLs are established by employing the distance between the distinguishable states. Different metrics define different distances, leading to different QSL bounds. We show a few typical generalized QSL bounds based on different metrics. And the unified QSL bound for non-unitary evolution is obtained by combining the generalized MT and ML bounds under the same metric. The generations and developments of QSL have successfully generalized the MT bounds. Even the ML bounds have been attempted to generalize, the current studies of ML bound are thought as the significantly harder task, and keep as the open questions. The influences of external environment and quantum system dynamics which include the role of entanglement in QSL for open dynamics and many-body systems, the relation between the maximum interaction speed and QSL in quantum spin systems, the non-Markovianity effect of the environment on accelerating the speed of evolution have been investigated. Furthermore, by employing the gauge invariant distance in the Riemannian metric, we establish a distinct QSL bound, which relates the quantum speed limit time to the geometry phase, and presents the geometric and gauge invariant properties of it. The bound shows the impacts of the transmission modes of the state vectors on the evolution path in the manifold. Different from the established MT and ML bounds, our bound establishes the relation between the QSL and the Pancharatnam connection 1-form, and our results show that the QSL can be controlled via manipulating the generalized Pahcharatnam connection. Also, our results of QSL are holonomy via the generalized Pahcharatnam connection, which show us a possible technique to speed up the holonomic quantum computation. The bound is also interesting to the metrology and non-equilibrium thermodynamics, etc. QSL has also been studied in the quantum representation of Wigner function. In the future research of quantum speed limit, there are many issues that need to be considered, such as the generalized ML bound, the unified QSL bound under different metrics, the application of maximum rate of quantum information derived from the QSL, and the relation between the quantum speed limit and the quantum heat engine, etc.
The emergence of irreversibility in physical processes, despite the fundamentally reversible nature of quantum mechanics, remains an open question in physics. This thesis explores the intricate relationship between quantum mechanics and thermodynamics, with a particular focus on minimizing entropy production in finite-time processes. By employing tools from quantum information theory and geometric thermodynamics, we tackle the challenge of deriving irreversible thermodynamic behavior from the reversible microscopic framework of quantum mechanics. We begin with a comprehensive review of the laws of thermodynamics, setting the stage for the subsequent analyses. We introduce novel developments in quantum thermodynamics through a generalized framework for geometric thermodynamics, which enables the derivation of finite-time corrections beyond the Markovian regime. Building on this foundation, we extend Landauer's principle by incorporating a finite-time correction that highlights the necessity of strong coupling for optimal information erasure processes. This result underscores the emergence of Planckian time as a fundamental speed limit to thermalization. Additionally, we explore how collective effects can be harnessed to reduce energy dissipation in thermodynamic operations, revealing that classical correlations between systems can significantly mitigate dissipation, though this may pose new questions regarding the third law of thermodynamics. Finally, we optimize thermodynamic processes in mesoscopic systems, including quantum dot engines and information engines. These findings not only enhance our understanding of the fundamental limits of irreversibility but also open new avenues for research. Future works will focus on fully exploiting collective effects, aligning these with the third law of thermodynamics, and understanding the thermodynamic consistency of master equations.
We explore timekeeping through the lens of Landauer's principle of bit erasure, where each erasure event produces exactly ln(2) bits of entropy. We reformulate the entropy-time connection as dt = dS/k, where k = R ln(2) is the entropy production rate determined by the erasure rate R. We introduce a dimensionless coupling constant alpha = k t_P, where t_P is the Planck time. At the Planck frequency limit, this yields alpha = ln(2) approximately 0.693. We note a numerical coincidence with loop quantum gravity: the Immirzi parameter, recently derived from Landauer's principle, is gamma = ln(2)/(pi sqrt(3)). Both expressions contain ln(2), though whether this indicates a fundamental connection remains an open question. We present new experimental results from IBM Quantum hardware (ibm_torino), testing a proxy for our temperature-independence prediction. Using the reset gate as a Landauer erasure operation, we measure reset fidelity across different initial states parametrized by angle theta. We find a coefficient of variation CV = 0.31%, indicating that reset performance is largely independent of initial state. Additionally, Ramsey interferometry experiments show that Landauer erasure destroys quantum coherence (visibility drops from 98.5% to 4.7%), consistent with irreversible entropy production. Two-qubit experiments demonstrate entropy transfer from system to reservoir. While these results are consistent with our framework, we emphasize that they do not constitute direct verification of the temperature-independence prediction, which requires controlled temperature variation in trapped-ion or similar systems.
Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. Most known quantum speed limits, including the celebrated Mandelstam-Tamm and Margolus-Levitin ones, are general bounds applicable to arbitrary initial states. However, when applied to mixed states of many-body systems, they, as a rule, dramatically overestimate the speed of quantum evolution and fail to provide meaningful bounds in the thermodynamic limit. Here we derive a quantum speed limit for a closed system initially prepared in a thermal state and evolving under a time-dependent Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. In a broad class of many-body setups it proves to be drastically stronger than general quantum speed limits.
The recent mathematical derivation by Abiuso et al. establishes a universal lower bound on quantum thermalization times, τth ≥ τPl /2 = ℏ/(2kB T ), utilizing quantum Fisher information geometry, Bures metric angles, and Hamiltonian estimation limits. While their operational result is mathematically exact, their underlying method relies on an adversarial epistemic formulation—treating thermalization as an information-acquisition game in which an external device or bath must “learn” or “estimate” the system Hamiltonian across a continuous family of operators. Decades of empirical and theoretical physics preceding this proof, from Zaanen’s identification of universal Planckian scattering in cuprate superconductors to Bruin and Mackenzie’s transport scaling across strange metals, alongside the Maldacena-Shenker-Stanford chaos bound and hydrodynamic fluctuation limits by Lucas and Delacrétaz, demonstrate that the Planckian scale is an objective physical feature of interacting systems rather than an artifact of measurement. In this work, we demonstrate that this bound does not arise because an open system requires time to resolve operator distinguishability. Rather, it is the direct mechanical consequence of physical metric work executed against positive manifold impedance in a ten-dimensional self-dual product manifold M3 × (T5 × S5 ). Within this geometry, physical time is a derived parameter (Virtual Time, T ) tracking the accumulation of metric work across an anisotropic 5-Space basis (α, x, y, z, Ω) mediated across the O(5, 5) T-duality boundary (R ↔ 1/R) by non-refundable Planck-floor transfer operations (∆E = ℏωP ). Under this strict zero-debt ledger, instantaneous collisional state swaps are topologically barred. At high temperatures (β∆ ≤ 1), the maximum rate of metric phase-slip across conjugate boundary cycles cannot exceed the energetic throughput permitted by the positive manifold impedance Ω ≥ 0, directly yielding τth ≥ ℏ/(2kB T ). In the low-temperature limit (β∆ ≫ 1), topological strain equilibrium (∆F → 0) halts Virtual Time (∂/∂T → 0), recovering the adiabatic quantum floor τ ≥ ℏ/∆ and guaranteeing vanishing decay widths (Γ ≡ 0). We salute the mathematical achievement of Abiuso et al. while demonstrating that the Planckian bound is an immutable elastodynamic property of spacetime rather than an epistemic limit of quantum estimation theory.