Thermodynamic Speed Limits in Isolated Quantum Systems

Rikuya Kobashi

REVIEW

Thermodynamic Speed Limits in Isolated Quantum Systems

Rikuya Kobashi

Statistical Mechanics · Submitted 11 Sep 2026 · 60 references

Pith's verdict: Solid thermodynamic speed limits for isolated quantum systems; the core observational entropy bound is rigorous and the examples confirm it, though the Vu–Saito distance's O(1) scaling limits the bound's strength in large systems.

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arXiv ID2609.12520 (version 1)

Pith ID6NSZY44Z, Pith's permanent record of this paper and its review

CitationRikuya Kobashi. "Thermodynamic Speed Limits in Isolated Quantum Systems." arXiv:2609.12520 (2026). https://pith.science/paper/6NSZY44Z

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Editor's note

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This paper does what good theory papers should: it takes a clear question (how does coarse-grained entropy production constrain finite-time unitary dynamics?), formulates it mathematically, and produces verifiable inequalities. The central result—that observational entropy production is bounded below by twice the squared Vu–Saito quantum Wasserstein distance—is derived cleanly from the Lenard–Tasaki inequality, Pinsker's inequality, and the closed-form expression for the VS-QW distance. The Gibbs entropy bound is a bit more involved (concavity argument and Birkhoff–von Neumann) but also checks out. The numerical and analytical examples (Rabi cycle, Landau–Zener with counterdiabatic driving, mixed-field Ising chain) actually confirm the bounds, which is more than many papers in this area do.

The truly new thing here is the connection between the VS-QW distance and entropy increase in isolated systems. Earlier speed-limit work on isolated systems used energy fluctuations or Hilbert-space metrics; this one uses a transport-theoretic distance and ties it to the irreversibility that emerges from coarse-graining. That is a genuine conceptual step, and the derivations are self-contained enough that I trust the results.

The main soft spot is one the authors themselves flag but I want to underscore: the Vu–Saito distance is O(1) in system size. That means the squared lower bound remains O(1) while the entropy production typically scales with system volume. In large systems the bound becomes trivial unless the entropy production rate is also small. This is not a flaw per se—the bound is correct—but it limits practical relevance for macroscopic thermodynamics. The authors mention this as an open problem (need an extensive quantum Wasserstein distance), and I think that is the right framing.

Minor: the Gibbs entropy bound requires a passive initial state. That is plausible for many protocols but not universal. The observational entropy bound only requires the initial state to be macroscopic with respect to the coarse-graining, which is a weaker condition.

Overall, this is a well-constructed paper. It is written clearly, the math is careful, and the limits are honestly discussed. I would bring it to my reading group and would cite it. A serious editor should send it to peer review; it deserves referee time and will likely result in a publication after minor revisions (mostly clarifying the scaling limitations).

Referee report

Significance

The work bridges two important areas—thermodynamic irreversibility in isolated quantum systems and optimal transport theory—by providing a quantitative link between coarse-grained entropy production and the Vu–Saito quantum Wasserstein distance. The bounds are parameter-free and falsifiable, and they clarify how macroscopic state transformations are constrained by entropic cost. The explicit treatment of three physically motivated coarse-grainings (system-bath, local-energy, diagonal entropy) and the numerical verification lend credibility to the approach. An honest discussion of the O(1) scaling of the Vu–Saito distance and the need for extensive versions in macroscopic systems adds to the paper's scientific integrity. If correct, this work establishes a new class of thermodynamic speed limits that are complementary to standard quantum speed limits and are directly relevant to finite-time driving, quantum quenches, and shortcuts to adiabaticity.

Major objections (3)

[III B, Eq. (14)] The definition of the Gibbs entropy for nonequilibrium states, Eq. (14), involves a double sum over i (energy eigenvalues) and k (a label within each degenerate eigenspace) and uses indices n_i and d_i. The text should explicitly explain how this expression reduces to the equilibrium Gibbs entropy (Eq. (7)) when ρ is microcanonical, and how the ordering of states within degenerate eigenspaces is fixed. This is important because the subsequent passivity assumption (Eq. (16)) and the derivation of the bounds rely on this ordering; a brief justification would improve clarity.
[VI, derivations of Eqs. (32) and (35)] The concavity argument using the functions V_1 and V_2 (Eqs. (126)–(127)) is mathematically sound but presented in a very compact form. The derivation of Eqs. (130)–(131) and the positivity argument that follows would benefit from more explicit steps, especially to convince the reader that the terms involving permutations indeed yield the claimed lower bounds. Given the centrality of this result, the level of detail should be increased, or a reference to a standard majorization argument should be added.
[IV B 1, Theorem 2 (Eq. (40))] The theorem assumes that the initial state is macroscopic with respect to the initial coarse-graining C_i, i.e., ρ(0) = ρ_{C_i}(0). While this is part of the protocol (the initial measurement is applied), the authors should remark that this assumption is not restrictive because the measurement step can always be performed before the unitary evolution. The bound is then a statement about the combined measurement-and-evolution process. A brief note clarifying this point would prevent potential misinterpretation.

Minor objections (6)

[III B, Eq. (14) and surrounding text] The symbols n_i and d_i are defined in Eq. (15) but the recurrence n_i = n_{i-1} + d_i is not explained in words; it would help to state that n_i is the cumulative dimension up to the i-th energy eigenspace.
[V A, Fig. 3 caption] Both axes are labeled with '× 10^3', but the figure panels use different vertical scales (one shows 0–1.2, the other 0–0.5). The caption should specify explicitly that the numerical values are multiplied by 10^3 for readability, and indicate the actual physical units (Jτ on the horizontal axis).
[V A, Eq. (68) and surrounding text] The local Hamiltonian H_loc is defined as -h_x σ^x - h_z σ^z, but in the mixed-field Ising model (Eq. (67)) the transverse field is h_x and longitudinal field is h_z. It would be clearer to note that the local temperature profile (Eq. (69)) breaks translation invariance and that no interaction term is included in the local Gibbs factor, even though the full evolution uses the interacting Hamiltonian.
[VI, Eq. (112)–(117)] The derivation of the doubly stochastic matrix B uses the fact that the initial measurement projects onto the PVM C_0. If the initial coarse-graining is not energy-diagonal but arbitrary, the matrix B is still doubly stochastic. The text could note this generality more explicitly.
[Appendix A 1, Eq. (A13)] The definition of ᴌᴴB in terms of the mesoscopic projector Π_B^j involves a dimensionless inverse temperature β and a shifted energy E_j. The relation between ᴌᴴB and the physical bath Hamiltonian H_B should be clarified; in particular, the justification that π_B is uniform within each bath-energy window relies on this redefinition. A sentence explaining that this is a standard coarse-graining of the bath would be helpful.
[Appendix D, Eq. (D13)–(D14)] The entanglement entropy example is interesting but the final bound involves the Wasserstein distance between ρ_{C_ent(Ψ(τ))}(τ) and the initial product state. Since the initial state is pure and product, its coarse-grained state is just itself. The bound then reads ΔS_ent ≥ 2 W_q(ρ_{C_ent(Ψ(τ))}(τ), |Ψ(0)⟩⟨Ψ(0)|)^2. The text should note that the coarse-graining depends on the final state's Schmidt decomposition, which is not known a priori; this limits the predictive power of the bound. A brief caveat would be appropriate.

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The paper

The paper claims

Quantum Wasserstein distance sets a thermodynamic speed limit for isolated systems

The reading

This paper derives lower bounds on entropy production for isolated quantum systems undergoing finite-time unitary driving. The bounds use the Vu–Saito quantum Wasserstein distance between the initial and final coarse-grained states. For the Gibbs entropy and for observational entropy (with suitable coarse-graining), the entropy production is at least 2 (W_q)^2, and for the Gibbs entropy also a linear bound in W_q. Dividing by the average entropy-production rate yields a lower bound on the operation time. This gives a thermodynamic characterization of speed limits, complementing the usual energy-based quantum speed limits.

What carries the argument

The Vu–Saito quantum Wasserstein distance W_q(ρ,σ) = (1/2) min_{V^†V=I} ||VρV^† - σ||_1, which admits a closed form as half the L1-distance between the sorted eigenvalue lists of ρ and σ. This distance captures the optimal unitary rotation needed to align two spectra, and the paper uses it to quantify how far the coarse-grained state has moved. The other key ingredient is the observational entropy S_obs^C(ρ) = -Σ_x p_x ln(p_x/V_x), which, under the assumption that the initial state is macroscopic with respect to the coarse-graining C, obeys a majorization relation under unitary evolution via a doubly stochastic map.

Extended reading notes

Core claim

The central claim is that, for a class of coarse-grained entropies (Gibbs entropy and observational entropy), the entropy production ΔS over a finite-time unitary protocol is bounded from below by the square of the Vu–Saito quantum Wasserstein distance between the initial and final coarse-grained states: ΔS ≥ 2 $W_q^{2}$, and for the Gibbs entropy additionally ΔS ≥ ln(d/(d-1)) W_q. These inequalities imply that, at a fixed average entropy-production rate, the operation time τ must satisfy τ ≥ (2 $W_q^{2}$)/⟨σ⟩, so a finite change of the coarse-grained state cannot be achieved arbitrarily fast.

Load-bearing premise

For the observational entropy bound, the initial state must be macroscopic with respect to the coarse-graining (i.e., of the form ρ = Σ_x p_x (P_x / V_x)), which ensures that the observational entropy equals the von Neumann entropy at the start.

Editorial extensions

If this is right

For driven isolated systems, the bounds provide a quantitative link between irreversibility and the rate of change of the macroscopic state, complementing the Mandelstam–Tamm and Margolus–Levitin quantum speed limits.
The inequalities yield lower bounds on operation times for shortcuts to adiabaticity, showing that even when endpoint entropy production vanishes, intermediate times must involve positive entropy production if the Wasserstein distance is positive.
For many-body systems with local-energy coarse-graining, the speed limit gives a finite-size bound on hydrodynamic relaxation times.
The results offer a unified framework that recovers known open-system thermodynamic speed limits in the system–bath coarse-graining limit.
The bounds are complementary to standard Fannes–Audenaert continuity bounds for entropy, which give upper limits on entropy production in terms of trace distance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the Vu–Saito distance is O(1) in system size while entropy production is extensive, the lower bounds become weak for macroscopic systems; an extensivized Wasserstein distance would be needed for strong bounds.
The work suggests that optimal transport theory, already powerful in stochastic thermodynamics, can also constrain unitary dynamics through coarse-graining, opening a route to quantum thermodynamic geometry.
Experimental platforms such as trapped ions or ultracold atoms could test the bounds by reconstructing coarse-grained entropies and spectrally estimating the Wasserstein distance from tomographic data.

Original abstract

We derive thermodynamic speed limits for isolated quantum systems undergoing finite-time unitary driving. The key ingredient is a set of generalized second-law inequalities: for the Gibbs entropy and for several thermodynamically motivated observational entropies, the entropy production is bounded from below by a function of the Vu--Saito quantum Wasserstein distance between the initial and final coarse-grained states. These inequalities yield lower bounds on the operation time in terms of the average entropy-production rate. We apply the framework to system--bath coarse-graining, local-energy coarse-graining, and diagonal entropy, and illustrate the resulting bounds with numerical and analytically solvable examples. Our results provide a thermodynamic characterization of finite-time state transformations in isolated quantum systems and clarify how coarse-grained entropy production constrains macroscopic reachability.

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We derive thermodynamic speed limits for isolated quantum systems undergoing finite-time unitary driving. The key ingredient is a set of generalized second-law inequalities: for the Gibbs entropy and for several thermodynamically motivated observational entropies, the entropy production is bounded from below by a function of the Vu--Saito quantum Wasserstein distance between the initial and final coarse-grained states. These inequalities yield lower bounds on the operation time in terms of the average entropy-production rate. We apply the framework to system--bath coarse-graining, local-energy coarse-graining, and diagonal entropy, and illustrate the resulting bounds with numerical and analytically solvable examples. Our results provide a thermodynamic characterization of finite-time state transformations in isolated quantum systems and clarify how coarse-grained entropy production constrains macroscopic reachability.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We derive thermodynamic speed limits for isolated quantum systems undergoing finite-time unitary driving.	contribution 0.99	0.97	0.15	asserted 1.00
s1	The key ingredient is a set of generalized second-law inequalities: for the Gibbs entropy and for several thermodynamically motivated observational entropies, the entropy production is bounded from below by a function of the Vu--Saito quantum Wasserstein distance between the initial and final coarse-grained states.	contribution 0.39	0.84	0.26	asserted 1.00
s2	These inequalities yield lower bounds on the operation time in terms of the average entropy-production rate.	result 0.97	0.90	0.25	asserted 1.00
s3	We apply the framework to system--bath coarse-graining, local-energy coarse-graining, and diagonal entropy, and illustrate the resulting bounds with numerical and analytically solvable examples.	contribution 0.59	0.96	0.07	asserted 1.00
s4	Our results provide a thermodynamic characterization of finite-time state transformations in isolated quantum systems and clarify how coarse-grained entropy production constrains macroscopic reachability.	interpretation 1.00	0.93	0.84	asserted 1.00

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Figures

**Figure 1.** FIG. 1. Schematic of the two-point coarse-graining protocol for an

**Figure 2.** FIG. 2. Thermodynamic coarse-grainings used for observational entropy. The system–bath coarse-graining resolves the bath energy shell

**Figure 3.** FIG. 3. In panel (a), time evolution of the FOE observational

**Figure 4.** FIG. 4. A plot of the LZ counter-diabatic (CD) protocol. We sim

**Figure 6.** FIG. 6. Entropy production and Wasserstein-distance lower bounds

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.12520 by the authors.

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System–bath coarse-graining Consider a small systemSinteracting with a large bathB. The total Hamiltonian is H(λ) =H S(λ)⊗I B +IS⊗HB +HI,(A1) whereHI denotes the interaction Hamiltonian. The projector onto thej-th bath-energy mesoscopic window and its volume are defined by ΠB j :=1 j(HB),Ω B j := trB[ΠB j ].(A2) For a total stateρ, let ρS := trB[ρ] = X i ...

View on Pith
[59]

Local-energy coarse-graining We next consider an isolated many-body system divided intomspatial regions, H= mO ℓ=1 Hℓ.(A27) The Hamiltonian is decomposed as H= mX ℓ=1 Hℓ +H int,(A28) whereHℓ is the local Hamiltonian of regionℓ, andH int describes interactions between different regions. Let Hℓ = X Eℓ Eℓ ˆP (ℓ) Eℓ (A29) be the spectral decomposition ofHℓ. T...

View on Pith
[60]

Global-energy coarse-graining and diagonal entropy Finally, consider the spectral decomposition of the global Hamiltonian, H(λ) = X i Ei(λ) ˆP (E) i (λ).(A39) The global-energy coarse-graining is CH(λ) := n ˆP (E) i (λ) o i .(A40) The corresponding observational entropy is SD(ρ;H(λ)) :=S CH(λ) obs (ρ) =− X i pi ln pi Vi(λ),(A41) where pi := tr h ˆP (E) i ...

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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General bound a. Theorem 2: Observational entropy.LetC i ={ ˆPx}x andC f ={ ˆQy}y be the initial and final coarse-grainings. Assume that the initial state is macroscopic with respect toCi, ρ(0) =ρCi(0) := X x px(0) ˆPx Vx .(37) After the unitary evolutionρ(τ) =U(τ)ρ(0)U †(τ), the final coarse-grained state is ρCf (τ) := X y qy(τ) ˆQy Vy .(38) The observat...

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Thermodynamic specializations a.C ρ,HB coarse-grainingWe first apply the general in- equality to the system–bath coarse-grainingC ρ,HB. As the initial state, we assume ρ(0) =ρ s(0)⊗π B, π B = e−β ˜HB ˜ZB .(43) We define the corresponding observational entropy produc- tion, the conventional open-system entropy production, and the Vu–Saito quantum Wasserste...

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View on Pith

[59] [59]

Local-energy coarse-graining We next consider an isolated many-body system divided intomspatial regions, H= mO ℓ=1 Hℓ.(A27) The Hamiltonian is decomposed as H= mX ℓ=1 Hℓ +H int,(A28) whereHℓ is the local Hamiltonian of regionℓ, andH int describes interactions between different regions. Let Hℓ = X Eℓ Eℓ ˆP (ℓ) Eℓ (A29) be the spectral decomposition ofHℓ. T...

View on Pith

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View on Pith