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Completely positive master equation for arbitrary driving and small level spacing

Evgeny Mozgunov, Daniel Lidar

arXiv:1908.01095v2quant-phmath-phmath.MP

TL;DR

Markovian master equations trade complete positivity against accuracy guarantees and applicability to small level spacings or fast driving. This paper derives and bounds a completely positive coarse-grained equation that accommodates arbitrary driving. Its applicability avoids the Davies-Lindblad level-spacing requirement, and it supports analyses of fast operations such as dynamical decoupling.

  • Problem

    Existing Markovian equations have competing constraints: Redfield lacks complete positivity, while Davies-Lindblad requires more restrictive conditions, including a sufficiently large level spacing.

    Source: §7 · p. 45

  • Method

    The authors derive the CGME from Redfield by neglecting part of the time-integration domain to restore complete positivity, then generalize it to arbitrary time-dependent driving.

    Source: §7 · p. 44

  • Results

    The CGME is completely positive and, like Redfield, applies when gτB ≪ 1, whereas Davies-Lindblad additionally requires the minimum level spacing to be large relative to g^2τB.

    Source: §7 · p. 45

  • Takeaways & Limitations

    The time-dependent CGME can describe δ-function pulses and provides a basis for rigorous Markovian analyses of fast operations in quantum computing.

    Source: §7 · p. 45

    The Born-approximation error has no rigorous bound in this treatment; the authors estimate its first-order contribution and assume the error converges.

    Source: §3.5 · p. 14

Abstract

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Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.

1 Introduction

Open-system modeling needs approximations, but established Markovian equations trade accuracy guarantees against complete positivity and applicability to small level spacings or fast driving. The paper develops and bounds a completely positive coarse-grained equation that addresses these trade-offs, including for time-dependent Hamiltonians.

1 Introduction

  • Redfield handles fast environments but is not completely positive, whereas Davies preserves complete positivity only within a restricted weak-coupling, finite-level-spacing regime.The introduction identifies the rotating wave approximation as the largest contribution to the Davies error after a relevant relaxation timescale.Source: §1 · pp. 1–2
  • Davies-generator error bounds can become exponentially large with system size, limiting guaranteed applicability, especially when level spacings shrink.The Lindblad equation with Davies generators is often used beyond its formal range as a semi-empirical model, without correctness guarantees.Source: §1 · p. 2
  • The time-dependent CGME is locally generated, completely positive, and has an error bound valid for arbitrary time-dependence of the system Hamiltonian.For local observables its error does not diverge with system size; for nonlocal observables the prefactor is polynomial, O(n^α) with 1/2 ≤ α ≤ 1.Source: §1 · p. 2
  • The paper derives the time-dependent CGME and compares rigorous error bounds for Redfield, Davies, and coarse-grained equations.The paper also considers dynamical decoupling and quantum error-correction settings involving fast operations.Source: §1 · p. 2

2 Three master equations

The paper introduces Redfield, Davies-Lindblad, and coarse-grained master equations, and compares their applicability using bath and system timescales. The CGME is completely positive, extends to time-dependent Hamiltonians, and avoids the level-spacing requirement in the Davies-Lindblad bound.

2.1 Background

  • The system and bath are modeled through a total Hamiltonian with an interaction term, and the bath is assumed stationary.The formalism allows a more general sum of interaction terms; thermal equilibrium is not needed for the proofs.Source: §2.1 · pp. 3–4

2.2 The bath correlation time τB and the fastest system decoherence timescale τSB

  • The bounds use the bath correlation time τB and fastest system decoherence timescale τSB; their ratio is sufficient for CGME applicability, while other equations require additional assumptions.The time-dependent CGME accuracy bound is independent of the norm and time-dependence of Htot.Source: §2.2 · pp. 4–5

2.3 Redfield master equation

  • The Redfield equation uses one generator per system-bath interaction, but is not completely positive and cannot be written in Lindblad form.The paper states that when Redfield preserves positivity, it is more accurate than the Davies-Lindblad equation.Source: §2.3 · p. 5

2.4 Davies-Lindblad master equation

  • The Davies-Lindblad equation is derived from the total system-bath Hamiltonian, but its generators are exponentially numerous in the number of qubits.The passage identifies the number of Davies generators as unfortunately exponential in n.Source: §2.4 · p. 6

2.5 Coarse-grained master equation (CGME) for time-independent or time-dependent system Hamiltonians

  • The paper generalizes the coarse-graining approach to time-dependent system Hamiltonians while retaining complete positivity.The earlier approach in Ref. considered only the time-independent case.Source: §2.5 · p. 6

2.5.1 The time-independent case

  • For time-independent Hamiltonians, the CGME uses a continuous family of Lindblad terms and is manifestly completely positive.Its applicability range is only slightly smaller than Redfield’s, while discretizing the continuous integral can keep generator count constant in system size per interaction term.Source: §2.5.1 · p. 6

2.5.2 The time-dependent case

  • For time-dependent Hamiltonians, the CGME has the same form and range of applicability, with its operators made time-dependent.The coarse-graining time Ta is a free parameter; the paper says a nearly optimal choice minimizes the derived Redfield-to-CGME solution-distance bound.Source: §2.5.2 · pp. 6–7

2.6 Error bounds and range of applicability

  • The CGME bound does not involve the energy gap, unlike the Davies-Lindblad bound, so it can apply to systems whose gap shrinks with size.The stated CGME bound depends on τB/τSB and time; its constants are universal, but the bound is not uniform in time.Source: §2.6 · p. 7

3 (Re-)Derivation of the coarse-grained master equation

This section re-derives the Redfield, Davies–Lindblad, and coarse-grained master equations, constructs a completely positive Lindblad-form CGME without the rotating-wave approximation, and bounds its error. It also develops a discretized, locally generated form and examines how applicability and locality scale with system size, noting limits of the Born-error bound and the leading-order effect of higher-order corrections.

3.1 Setting up

  • The derivation assumes a rapidly decaying bath correlation function, τB ≪ τSB, and a stationary bath state.The stationary-bath assumption makes the two-time correlation function invariant under simultaneous shifts of its arguments.Source: §3.1 · pp. 8–9

3.2 Redfield master equation

  • The Markov approximation yields the Redfield equation, which is not completely positive, with a Markov-approximation error of the same order as the Born-approximation error.The section also derives the Redfield equation’s Schrödinger-picture form.Source: §3 · p. 8, §3.1 · p. 9

3.3 Davies-Lindblad master equation

  • The Davies–Lindblad derivation applies the rotating-wave approximation, discarding rapidly oscillating terms; the resulting right-hand-side error is large but rapidly oscillating, so the solutions remain close.Earlier time-dependent versions of the Davies–Lindblad equation require adiabatic driving.Source: §3.3 · p. 11

3.4 Coarse-grained master equation

  • The CGME derivation uses state time-averaging in lieu of the RWA, reducing to the RWA in the limit of large time-averaging scale, and removes a small integration-domain part to restore complete positivity.Source: §3.4 · p. 11

3.4.1 Time-averaging of the state

  • Time-averaging over Ta, with τB ≪ Ta ≪ τSB, gives solutions within O(Ta/τSB) of the unaveraged equation for t ≤ cτSB.Although the derivative error is O(1/τSB), the bound concerns the difference between solutions.Source: §3.4.1 · pp. 11–12

3.4.3 Complete positivity

  • Expressing the CGME in Lindblad form establishes its complete positivity, and the Davies–Lindblad equation arises in the Ta → ∞ limit.The derivation also shows that the Davies–Lindblad equation’s error can be controlled for sufficiently large but finite coarse-graining time.Source: §3.4.3 · p. 14

3.5 Ranges of applicability of the three master equations

  • The Born approximation error lacks a rigorous bound here, though its first-order contribution is estimated and is subleading if the error converges.The authors note that a rigorous bound can be obtained.Source: §3.5 · p. 14
  • For 0<t<cτSB and sufficiently small τB/τSB, the CGME error is bounded by C(c,δ)τB/τSB.The analysis does not give C(c,δ) explicitly because it does not track the Born-error contribution; higher-order corrections improve only subleading terms.Source: §3.5 · p. 15
  • For Redfield, CGME, and Davies-Lindblad, the error bound has a nonzero transient as t→0, while its slope is O(1/τSB).The error itself starts at zero; the bounds are not tight.Source: §3.5 · p. 15

3.6 Discretization, spatial locality, and the Lieb-Robinson bound

  • A discrete CGME approximation has system-size-independent parameters when [H,A]=O(1), but the Lamb shift is not discretized in this work.The authors leave a discrete approximation for the Lamb shift to future work.Source: §3.6 · p. 16
  • The Lieb-Robinson bound yields local CGME generators with radius x=vTa/2+δr, retaining simultaneous locality and complete positivity.The operators have exponentially decaying tails, and the local-plus-small-nonlocal decomposition uses the Lieb-Robinson bound.Source: §3.6 · pp. 16–17
  • In one dimension, the stochastic-trajectory evolution has an MPS structure, but fixing bond dimension by truncation is generally uncontrolled.With fixed bond dimension, the described method uses O(n,t) computation time and O(n) memory; averaging over randomness adds cost.Source: §3.6 · p. 17

3.7 Scaling with size

  • For n qubits, the applicability bound scales as Λ∼n²/τSB for correlated noise and Λ∼n/τSB for uncorrelated noise.Under locality, the error in local density matrices can obey a stronger bound consistent with the isolated-qubit result.Source: §3.7 · p. 18
  • At times ∼τSB, the required coupling range is between √n gτB≪1 and n gτB≪1 depending on noise correlations, while local observables are expected to retain gτB≪1.The CGME and Redfield ranges switch the single-qubit τB/τSB requirement from exponential in system size for Davies-Lindblad to polynomial.Source: §3.7 · p. 18

4 The CGME with a time-dependent system Hamiltonian

The authors derive a time-dependent coarse-grained master equation in manifestly completely positive form, valid for arbitrary time-dependent system Hamiltonians. The derivation adapts the Redfield-based coarse-graining steps, with a time average and an integral-bound change to recover complete positivity.

4 The CGME with a time-dependent system Hamiltonian

  • The derivation assumes a stationary bath state for simplicity, but the authors state that it also goes through without that assumption.It begins with a time-dependent system Hamiltonian and uses the Markov approximation ρB,I(τ) → ρB,I(t).Source: §4 · p. 19
  • The time average is accurate when Ta is small compared with the fastest timescale over which ρI(t) changes.Under this condition, the associated error is small.Source: §4 · pp. 19–20
  • Changing an integral's lower limit to recover complete positivity introduces an error of order O(τB/Ta).The transformation is otherwise exact.Source: §4 · p. 20
  • The time-dependent CGME is manifestly completely positive and valid for arbitrary time-dependent system Hamiltonians.It has the same form as the time-independent CGME, except that its operators depend on time.Source: §4 · p. 21

5 Applications and Examples

This section presents applications and examples of the time-dependent CGME, including dynamical decoupling, CGME accuracy for a slow bath, and simplified Lambless equations. It also examines the Lamb shift’s effects on density-matrix elements and decay, including the error from dropping HLS.

5.1 Application: dynamical decoupling

  • The authors establish that dynamical decoupling works in the strictly Markovian Redfield and CGME frameworks, extending the result beyond the Davies-Lindblad equation.The underlying bath may have a nonzero correlation time, which lets the Markovian master equation describe pulse-sequence effects.Source: §5.1 · p. 21

5.1.1 Example 1: time-dependent Redfield master equation with a rectangle bath correlation function

  • For a rectangle bath correlation function, the decoherence rate is reduced by at least ∆t/τc under pulses when (gτc)^2 ≪1, but this bath is not physically permissible.Its Fourier transform γ(ω) is negative for some ω, so the resulting equation may fail to describe a completely positive map.Source: §5.1.1 · p. 22

5.1.2 Example 2: time-dependent CGME with an Ohmic spectral density

  • The time-dependent CGME predicts dephasing suppression under dynamical decoupling for a KMS-compatible Ohmic bath when the sufficient cutoff condition is satisfied.The paper notes that suppression is predicted consistently when Eq. (127) holds; Fig. 3 plots the suppression factor across pulse intervals, cutoffs, and temperatures.Source: §5.1.2 · p. 23, Figure 3 · p. 24

5.3 Toy example of a slow bath at the boundary of the range of applicability, with explicit correlation function and spectral density

  • The numerically optimal averaging time is Ta,num. opt ≈2.87 ≈3Ta,theory opt, and an adjusted Ta outperforms Davies at moderate τB/τSB.The slow-bath toy example also finds that the Original Redfield solution becomes negative at t ≈2.57τSB, limiting comparisons to its positive interval.Source: §5.3 · pp. 26–28, Figure 6 · p. 27

5.4 Role of the Lamb shift in the CGME

  • In the CGME, the Lamb shift affects the phases of off-diagonal density-matrix elements in the energy eigenbasis, with its effect on the norm difference eventually disappearing as those elements decay.The reported comparison uses Ta,adj = 1.15Ta,opt.Source: §5.4 · p. 28
  • Figure 7 compares the error from dropping the Lamb shift HLS for Ta,adj with differences in matrix-element magnitudes and phases and the decay of off-diagonal elements.The plotted quantities include ∥ρC(t) − ρCL(t)∥1, maxij δ|ρC,ij|, and the maximum phase difference.Source: Figure 7 · p. 28

6 Error bounds and estimates

This section develops and combines error bounds for the Born and Markov approximations, time averaging, enforcing complete positivity, and changes to integration limits. It optimizes the coarse-graining time to bound CGME error and compares finite-time CGME with the Davies–Lindblad limit, including bounds on Davies–Lindblad and Redfield errors.

6.1 Bound on the difference of solutions of differential equations differing by a bounded operator

  • The solution-comparison lemma bounds trace-norm differences for equations with bounded generators and extends to non-Markovian kernels.The usual choice is Λ = 4/τSB; the resulting exponential bound can be tightened to linear in time under additional assumptions, but the authors do not recast all results that way.Source: §6.1 · pp. 29–30

6.2 Born approximation error

  • For Gaussian baths, integrability of |C(t)| and t|C(t)| controls the Born-error estimate and supports the estimate in Eq. (33).Exponential correlation decay is sufficient, but the authors introduce this weaker nonexponential-bath condition; extending the requirement to non-Gaussian baths involves higher-order correlations.Source: §6.2 · pp. 30, 36
  • The all-orders Born-error argument assumes a finite convergence radius and that the perturbative series converges to the true solution; proving convergence with the required integral condition is left for future work.The authors note that a proof would require controlling the factorial number of Wick-theorem terms, adapting Davies’s proof.Source: §6.2 · p. 34
  • The Born-approximation analysis uses Eq. (197) to bound cB, an upper bound on ∥ρB,I∥1.The following section combines the Born and Markov errors into a total expression.Source: §6.2 · p. 36, §6.3 · p. 36

6.3 Markov approximation error

  • The Markov approximation error is estimated after replacing ρB,I(τ) with ρB,I(t), and cB is O(1).The section gives a total expression for the Born and Markov errors and then states a looser bound.Source: §6.3 · pp. 36–37

6.4.1 General analysis of time-averaging

  • For a first-order linear differential equation, the time-averaged solution’s deviation is bounded using b2, cρ, Λ, Ta, and time.The bound depends on the generator norm and solution norm bounds stated alongside Lemma 2.Source: §6.4.1 · p. 37

6.4.2 Time averaging of the Redfield ME

  • For the Redfield master equation, the time-averaging error is bounded using Λ = 4/τSB, with a tighter bound also given by Lemma 2.The section provides both a compact bound and the repeated Lemma 2 bound.Source: §6.4.2 · p. 38

6.4.3 Bound on the error due to enforcing complete positivity

  • The complete-positivity step drops part of the time-averaged integral, and its error bound uses an evolution time with a Ta/2 collar.For an Ohmic bath, the integral defining τB has a logarithmic divergence in T.Source: §6.4.3 · pp. 38–39
  • The tightest stated bound on ∥ρBM,I(t) − ρC,I(t)∥1 combines the time-averaging and complete-positivity errors.The bound is given in Eq. (240a), with its right-hand side expressed using b2, cBM, Λ, Ta, and t.Source: §6.4.3 · p. 40

6.4.4 Optimization of Ta

  • For sufficiently small τB/τSB, the total error can be small at any t/τSB, within the stated region τSB ≤ C.The passage says this also proves validity in the weak-coupling limit, while the inequality is more general.Source: §6.4.4 · p. 42

6.5 The Davies-Lindblad error

  • The Davies–Lindblad equation is the Ta → ∞ limit of the CGME; the comparison’s dimension factor is 2n, and 1/δE may depend exponentially on n.The passage says the accumulated solution error is bounded, but the bound itself is not included here.Source: §6.5 · p. 43
  • For t < cτSB, the Davies-Lindblad error scales as O(τB/δE) under the stated constant-system-size and level-spacing conditions.The bound is presented for n = O(1); when 4nδE ≫ τB, a second term can be neglected.Source: §6.5 · p. 43

6.6 Redfield limits of integration error

  • Accounting for changing integration limits adds a logarithmic correction to the Redfield O(τB/τSB) error scaling.Because the bound on the error is time-dependent, the analysis repeats the differential-equation argument; the resulting estimate is summarized in Eq. (26).Source: §6.6 · pp. 43–44

7 Summary and Conclusions

The paper derives a completely positive coarse-grained master equation (CGME), including for arbitrary time-dependent driving, and bounds its validity alongside Redfield and Davies-Lindblad equations. The bounds show that CGME avoids the minimum-level-spacing cost of Davies-Lindblad while retaining complete positivity.

7 Summary and Conclusions

  • The authors derive the CGME from Redfield by omitting part of the time-integration domain, restoring complete positivity, and generalize it to arbitrary time-dependent driving.The time-dependent equation remains completely positive and has essentially the same form as its time-independent counterpart.Source: §7 · p. 44
  • The optimal coarse-graining time Ta scales as the geometric mean of the bath correlation time τB and fastest system decoherence timescale τSB.The bounds for the three equations are expressed using τB/τSB and, for Davies-Lindblad, also 1/√τSBδE, where δE is the minimum level spacing.Source: §7 · pp. 44–45
  • When [H, A] = O(1), the three-parameter approximation has no system-size dependence in ∆ϵ or in k*, the number of terms replacing the continuous integral.Alongside the one-parameter CGME family, the authors derive an approximation parameterized by (Ta, ∆ϵ, k*).Source: §7 · p. 45
  • Redfield and CG master equations apply when gτB ≪1, while Davies-Lindblad additionally requires minimum level spacing large relative to g²τB; CGME retains complete positivity without that cost.These are sufficient conditions; the Davies-Lindblad bound can become severely restrictive as level spacing shrinks with system size.Source: §7 · p. 45
  • The time-dependent CGME accommodates nonadiabatic and δ-function-pulse drives, supporting analyses of dynamical decoupling and fast gates in quantum error correction.The authors state that this permits a rigorous analysis of fault tolerance in a first-principles Markovian setting.Source: §7 · p. 45
  • The CGME provides local generators for finite-temperature dynamics of non-commuting models and extends many-body localization studies to open-system quantum annealing.The paper identifies topological subsystem-code lifetimes and finite-temperature open-system quantum annealing as potential applications.Source: §7 · p. 45

A Proof that γ(ω) > 0

The proof establishes non-negativity of the bath spectral-density matrix using positivity arguments, including a Bochner-theorem proof that avoids bath eigenstates and Dirac delta functions. The derivations assume a stationary bath, and more complicated system-environment interactions require a different proof.

A Proof that γ(ω) > 0

  • The derivations assume a stationary bath; the proof covers the stated interaction case, while more complicated environment interactions require a different proof.Stationarity is expressed as [ρB, HB] = 0.Source: A Proof that γ(ω) > 0 · p. 47
  • A second positivity proof avoids bath eigenstates and Dirac delta functions, accommodating the smooth spectral densities used for continuous-spectrum baths.The proof uses Bochner’s theorem.Source: A Proof that γ(ω) > 0 · p. 47
  • Bochner’s theorem establishes nonnegative bath spectral-density eigenvalues by showing the associated function is of positive type over arbitrary time partitions.The proof establishes positivity of the function’s matrix and then applies Bochner’s theorem to its Fourier transform.Source: A Proof that γ(ω) > 0 · pp. 47–48

B Lamb shift simplification

The appendix simplifies the Lamb-shift expression and develops the coarse-graining-time limit in which the Davies-Lindblad equation emerges from the CGME.

B Lamb shift simplification

  • A change of variables and the identity C(θ)=C*(−θ) simplify the Lamb-shift integral and yield the main-text expression, whose stated form ensures Hermiticity.The ω+ = 0 expression is understood as a limit.Source: B Lamb shift simplification · p. 49

C The Davies-Lindblad equation is the infinite coarse-graining time limit of the CGME

  • The Davies-Lindblad rotating-wave approximation is the infinite coarse-graining-time limit of the CGME, as shown by Ref..The appendix motivates the derivation by noting that simply dropping terms with different Bohr frequencies lacks rigorous mathematical justification.Source: C The Davies-Lindblad equation is the… · pp. 49–50

C.1 The ω = ω′ case

  • For ω = ω′, the proof uses integration by parts and the symmetry C(v)=C*(−v) to handle the correlation-function integrals.The passages introduce this case as part of establishing the coarse-graining-time limit.Source: §C.1 · p. 50

C.2 The ω̸ = ω′ case

For unequal Bohr frequencies, the off-diagonal CGME terms vanish as the coarse-graining time grows; the Lamb-shift term has the corresponding diagonal limit.

C.2 The ω̸ = ω′ case

  • For ω ≠ ω′, the term vanishes because its boundary contribution does not cancel the overall T_a^-1 factor.The derivation uses integration by parts and identifies the Lamb-shift amplitude S(ω) in the boundary expression.Source: §C.2 · pp. 50–51
  • Together, the vanishing unequal-frequency terms and the Lamb-shift limit establish the Davies-Lindblad equation as the T_a → ∞ limit of the CGME.The conclusion is stated for the Davies-Lindblad (RWA) equation.Source: §C.2 · p. 51
  • The imaginary part of the Lamb-shift coefficient likewise approaches its diagonal form, Im(x_ωω′) → Im(x_ωω)δ_ωω′.The text describes this as a similar calculation for the Lamb-shift term.Source: §C.2 · p. 51

D Derivation of the discrete approximation Eq. (78)

The appendix derives a discrete approximation by discretizing the CGME integral and bounding its errors, including the truncation and discretization contributions.

D Derivation of the discrete approximation Eq. (78)

  • The discrete approximation starts from the time-independent CGME and writes its error as a sum of contributions, including integral truncation and discretization.The discretization error is bounded over intervals of width Δϵ.Source: D Derivation of the discrete… · pp. 51–52
  • When the cutoff ϵ is far from all system frequencies, the truncated-integral contribution becomes small as 1/ϵ.The derivation establishes this by integration by parts.Source: D Derivation of the discrete… · p. 52
  • For a local n-qubit Hamiltonian with O(1) local terms and a local unit-norm operator A, [H,A]=O(1), so the chosen Δϵ and k* have no system-size dependence.The bounds are presented as demonstrating O(1) scaling for these discretization choices.Source: D Derivation of the discrete… · pp. 52–53
  • The authors caution that Eq. (330) is not recommended for numerical calculations because tighter bounds allow larger Δϵ and smaller k*.The stated purpose of Eq. (330) is to show that the parameters can be O(1).Source: D Derivation of the discrete… · p. 53

E An error bound that is linear in t

This section presents a norm-based lemma for bounding the difference between two evolution equations, with error growing linearly in time under a bounded-evolution condition.

E An error bound that is linear in t

  • Lemma 5 bounds the separation between solutions of a pair of linear evolution equations by a term proportional to elapsed time when the evolution remains norm-bounded.The stated result applies for finite t when solutions from bounded initial conditions remain bounded by c_y.Source: E An error bound that is linear in t · pp. 53–54
  • The result also holds for the operator norm and trace norm.The text identifies these as p = ∞ and p = 1, respectively.Source: E An error bound that is linear in t · p. 53
  • In this work, Lemma 5 is applied only to the Markov error; other derivation equations may not be invertible, so the authors instead use Lemma 1 or prove results independently.For the Markov error, the text uses a loose bound to obtain a more compact expression.Source: E An error bound that is linear in t · p. 54

F Proof of Lemma 2

The proof bounds the error introduced by time averaging through transient boundary contributions and the change in the state over nearby times. It then constructs a piecewise bound that applies across the full evolution.

F Proof of Lemma 2

  • Changing variables in the time average creates transient contributions at the beginning and end of the evolution, interpreted as instantaneous kicks.For t ≤ Ta/2, parts of the second-term integrals cancel over an area (Ta/2 − t)^2.Source: F Proof of Lemma 2 · pp. 54–55
  • The state difference at nearby times is bounded by cρΛ|τ′|, using the evolution equation ˙ρ = Lρ.Source: F Proof of Lemma 2 · p. 55
  • The auxiliary bound b2(t) is solved in two time regimes, with the later-time solution initialized by its value at Ta/2.The derivation notes that ˙b2 > 0.Source: F Proof of Lemma 2 · p. 56
  • The all-time error bound combines the bound at min(t, Ta/2) with an exponential factor for t > Ta/2.The resulting expression bounds ∥δρ(t)∥1 for all t.Source: F Proof of Lemma 2 · p. 56

G Derivation of the bound (245)

This derivation establishes a bound for the error between the two evolutions and examines how its usefulness depends on time and the ratio τB/τSB.

G Derivation of the bound (245)

H Optimal time

The section defines a relevance time based on when the error bound ceases to improve on a trivial bound, then chooses the coarse-graining time to maximize that relevance time. The resulting formulas have a restricted regime of use.

H Optimal time

  • The bound relevance time t∗ is the largest time for which the error bound beats the trivial bound 1 + cBM.Source: H Optimal time · p. 58
  • The derivation optimizes Ta at t = t∗ to maximize t∗, yielding an equation for t∗ and then an expression for the error.Source: H Optimal time · p. 58
  • The optimized expressions should be used only when t∗ > Ta/2.For τSB ≪ τB, Ta instead scales as τSB; for τSB ≫ τB, the formulas are expected to give only a minor improvement on the main-text bounds.Source: H Optimal time · p. 58

I Proof of the bound (269b) for the Redfield limits of integration error

The proof bounds the error from the Redfield integration limits by extending an integration limit to infinity and treating the resulting expression in separate regimes. It relates the cases through an upper bound on a tighter bound.

I Proof of the bound (269b) for the Redfield limits of integration error

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