Anomalous flow in correlated quantum systems: No-go result and multiple-charge scenario

Junjie Liu; Rui Guan

REVIEW 3 major objections 5 minor 82 references

Anomalous flow in correlated quantum systems: No-go result and multiple-charge scenario

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Energy always flows along the temperature gradient in uncorrelated, energy-conserving systems—but additional conserved charges can drag it against the gradient.

desk verdict Solid no-go theorem, but the advertised drag-induced AEF rests on numerical models where energy and the 'non-energy charge' are the same observable, so the central positive claim is not actually demonstrated. read the letter →

arxiv 2506.05995 v2 pith:WDDHXOKR submitted 2025-06-06 quant-ph cond-mat.stat-mech

Rui Guan , Junjie Liu This is my paper

classification quant-phcond-mat.stat-mech

keywords anomalousenergyflowchargequantumthermodynamicscorrelationscatalystmultipleconservedchargesdrageffectgeneralizedGibbsstate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether energy can flow from a cold subsystem to a hot one when the two subsystems start uncorrelated and total energy is conserved. For energy alone, the answer is no: the authors prove an inequality, $(\beta_A - \beta_B) \Delta E_A(t) \geq \Delta I(t) \geq 0$, that rules out anomalous energy flow at every time, and they show that a quantum catalyst cannot overturn this no-go result. The answer changes when the system conserves additional charges: a normal flow of a non-energy charge can drag energy against its own temperature gradient, without consuming any initial correlation. The same logic extends to every conserved charge, producing a general phenomenon the paper calls anomalous charge flow.

What carries the argument

The central object is the authors' 'global-local thermodynamic description': a pair of exact equalities for the entropy change of each subsystem, one written with quantum relative entropy against the global state (global description) and one with relative entropy against the local Gibbs states (local description). Combining both and using the non-negativity of relative entropy converts the equalities into lower bounds on the flow figure of merit, Eq. (5) in the energy-only case and Eq. (17) with multiple charges. The multiple-charge generalization replaces Gibbs states by generalized Gibbs states $\gamma_j = e^{-\sum_i \lambda_j^i C_j^i}/Z_j$ and uses charge-conserving unitaries, which is what introduces the drag term that can make the bound negative.

What would settle it

Run the same two-mode or two-qubit transport with initial grand-canonical states but choose $H_i$ not proportional to $N_i$, for instance by adding an interaction or different on-site spectra that keep both charges conserved. If $(\beta_A - \beta_B) \Delta E_A(t)$ never becomes negative while particle number flows normally, drag-induced anomalous energy flow would be shown not to survive beyond the paper's proportional-charge examples. A simpler check: scan parameters in the paper's own models to test the predicted threshold $\sum_{i\neq 0}(\lambda_A^i - \lambda_B^i) \Delta C_A^i(t) > \Delta I(t)$ for the onset of negative flow.

Watch

Extended reading notes

Core claim

The paper's central claim is a sharp asymmetry between one-charge and multi-charge quantum thermodynamics. For a bipartite, energy-conserving system initialized in a product of thermal states, the authors derive two exact entropy-balance expressions—one in terms of the global state, one in terms of local states—and show that only their combination yields the constraint $(\beta_A - \beta_B) \Delta E_A(t) \geq \Delta I(t) \geq 0$, so energy always moves along the initial temperature gradient. This no-go result survives the addition of a catalyst that returns to its initial state, since the final lower bound is nonnegative. In systems with multiple conserved charges prepared in generalized Gibbs states, the analogous lower bound contains an extra term, $-\sum_{i\neq 0}(\lambda_A^i - \lambda_B^i) \Delta C_A^i(t)$, which can be negative; numerical models of two bosonic modes and two qubits show that a normal flow of particle number or excitation number makes energy or excitation flow against its own gradient while mutual information rises monotonically. The paper concludes that no initial correlations are needed for anomalous flow, and that energy holds no privileged position among conserved charges.

Load-bearing premise

The paper's broadest claim—that drag-induced anomalous energy and charge flow works for arbitrary conserved charges—rests on numerical examples in which energy and the non-energy charge are proportional, and on an asserted extension to noncommuting charges that is not backed by numerical or analytic proof.

Editorial extensions

If this is right

In any energy-conserving bipartite process that starts from a product of thermal states, energy flows along the initial temperature gradient at all times, even with catalytic assistance; any attempt to cool the colder side must therefore involve additional conserved charges or initial correlations.
A normal flow of a non-energy charge can create a transient reversal of energy flow, so anomalous energy flow becomes achievable without preparing correlated initial states.
For every conserved charge, there is a corresponding anomalous charge flow: a charge can move against its conjugate affinity when the net normal flows of the other charges overcome the mutual information cost.
The anomalous flows identified here are transient and coexist with monotonically increasing mutual information, so they are not powered by correlation consumption.
In setups without strict charge conservation or with arbitrary initial states, the same drag mechanism persists and can combine with correlation-consumption mechanisms.

Reading between the lines

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

The numerical evidence for drag-induced anomalous flow uses Hamiltonians $H_i = \varepsilon_i N_i$ in which energy and particle number are proportional, so energy flow is a constant multiple of particle flow; extending the mechanism to dynamically independent charges is a nontrivial assumption the paper states but does not demonstrate.
A direct test of that extension would be a model with, say, different dispersion relations for energy and particle number, where $H_i$ is not proportional to $N_i$; if the lower bound in Eq. (17) stays nonnegative there, the drag effect would be limited to proportional-charge settings.
The inequalities suggest a quantitative threshold: anomalous flow should appear only when the net normal-flow term of the other charges exceeds the mutual information growth $\Delta I(t)$; this could be probed experimentally by tuning chemical-potential and temperature gradients independently.
Because the generalized Gibbsian framework is the stated basis for noncommuting charges, a numerical simulation with noncommuting charges (e.g., energy and a spin component) would clarify whether the drag mechanism survives without commutativity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: the no-go theorem in Sec. II is rigorous and worth taking seriously; the drag-induced AEF advertised in the abstract is not supported by the paper's own numerics. The two numerical models use H_j = ε_j N_j up to a constant, so energy and the 'non-energy charge' are the same observable. The initial grand-canonical states are then exactly thermal states for H_j at a shifted inverse temperature β'_j = β_j(1 − μ_j/ε). The Sec. II no-go applies to those effective temperatures, and the negative values of (β_A − β_B)ΔE_A in Fig. 1 are just normal energy flow under the effective gradient. The multi-charge language is redundant in these examples; no drag between dynamically independent charge species is shown. The final remark explicitly defers noncommuting charges, so the central existence claim rests entirely on this proportional-charge limit.

What the paper does well: the global-local derivation of Eq. (5) is clean and rigorous, and the catalyst extension is a genuine complement to Lipka-Bartosik et al. The multiple-charge identity (16) and the necessary condition (17) are correct as far as I can tell. The ACF concept is a legitimate generalization. The paper is well organized and the math is careful.

Soft spots beyond the numerical issue: the claim that ΔI_tot(τ) > 0 in the catalyst proof is not generally true — a trivial identity evolution gives ΔI_tot = 0 — though the no-go still holds because the bound in Eq. (12) is nonnegative. That's minor. The bigger problem is that Eq. (17) is only a lower bound; the paper treats it as if it guarantees existence, but it does not. Without an example with non-proportional charges, the drag mechanism is speculative.

Who this is for: people working on anomalous heat flow and resource-theoretic thermodynamics will want the no-go theorem. They should be cautious about the multi-charge claims. The paper deserves a serious referee — the no-go theorem is citable and the framework is thought-provoking — but the authors should be pushed to either provide a genuinely independent-charge example or substantially soften the universality claims.

Recommendation: send to peer review, but expect significant revision on the numerical demonstrations and the interpretation of the drag effect.

Referee Report

3 major / 5 minor

Summary. The paper develops a 'global-local' thermodynamic description for charge exchange between correlated quantum systems. In Section II it proves a no-go result: for a bipartite system initialized in an uncorrelated thermal product state and evolving under an energy-conserving unitary, anomalous energy flow (AEF) is impossible, even when a quantum catalyst is used. In Section III the authors generalize the framework to multiple conserved charges and derive inequality (17), arguing that a net normal flow of non-energy charges can drag energy against the temperature gradient, and introduce the concept of anomalous charge flow (ACF). Two numerical models, a two-mode bosonic system and a two-spin system, are presented as evidence. Section IV extends the formalism to charge non-conservation and arbitrary initial states, and Section V concludes. The no-go derivations are exact, while the positive drag mechanism rests on the numerical examples.

Significance. The no-go result for initially uncorrelated energy-conserving systems is a clean and rigorous contribution: Eqs. (1)-(5) follow from non-negativity of relative entropy and mutual information, with no fitted parameters. The multiple-charge generalization is conceptually attractive and the inequality (17) is correct. If a genuine example of drag-induced AEF/ACF with dynamically independent conserved charges were provided, the paper would be a significant advance. However, the current numerical evidence does not establish the central existence claim, and the final remark in Sec. V explicitly leaves the noncommuting-charge case open. The theoretical framework itself is well structured and may be useful for future work.

major comments (3)

[Sec. III.A.2 and Sec. III.B.2, Eqs. (18), (20), (23), (25)] The numerical examples do not demonstrate drag between independent charge species because the energy and the non-energy charge are proportional. In the bosonic model H_j = ε N_j, so the initial grand-canonical state γ_j = exp[-β_j(H_j - μ_j N_j)]/Z_j equals a thermal state for H_j at the effective inverse temperature β'_j = β_j(1 - μ_j/ε). With the parameters β_A = 20, β_B = 18, μ_A = 0.8, μ_B = 0.4, ε = 1.5, one finds β'_A ≈ 9.33 < β'_B ≈ 13.2, so the negative value of (β_A - β_B)ΔE_A(t) in Fig. 1 corresponds to normal energy flow from the effectively hotter system A to the effectively colder system B. The same reduction applies to the spin model of Sec. III.B.2: H_i = ε N_i - (ε/2)I, so the initial state is thermal for H_i at β'_i = β_i(1 - μ_i/ε), and the observed 'ACF' is the same normal energy flow. The no-go theorem of Sec. II therefore applies to the effective temperatures, and the examples reduce to a single-charge problem; the particle/energy identity makes the multi-charge description redundant.
[Sec. III.A.1, Eq. (17)] Equation (17) is only a necessary condition for AEF: the lower bound L_l^E(t) being negative does not by itself guarantee that (β_A - β_B)ΔE_A(t) is negative. The existence claim requires a constructive example or a proof that the lower bound can be saturated or exceeded by a valid dynamics. Since the numerical examples in Sec. III.A.2 and III.B.2 fail to demonstrate drag between independent charges (as noted above), and the final remark in Sec. V explicitly defers the noncommuting-charge case, the paper does not currently establish the central positive claim that normal flows of non-energy charges can induce AEF without initial correlations.
[Sec. II.B, after Eq. (12)] The assertion that ΔI_tot(τ) > 0 for any nontrivial catalytic process is not correct in general. A unitary that swaps the states of subsystems A and B, with an inert catalyst returning to its initial state, is energy-conserving, leaves the final state in product form for A, B, and C, and gives ΔI_tot(τ) = 0 while still transferring energy between A and B. The no-go conclusion itself remains valid because the right-hand side of Eq. (12) is nonnegative (positive when S_C(0) > 0), but the proof as written needs to be corrected; the claim that residual A-B correlations necessarily make ΔI_tot(τ) positive is false.

minor comments (5)

[Sec. III.B.2] The text uses 'ACT' in two places ('numerical evidence for the ACT' and 'drag-induced ACT'); this should be 'ACF'.
[Sec. II.A, before Eq. (11)] There is a missing space in 'justItot(t)'; the sentence should read 'Since we just have Itot(t) ≤ 2 min{S(t), S_C(t)}...'.
[Sec. I, Introduction] The sentence '...and entanglement enhancement [46] among others; For more developments...' uses a semicolon where a period should be: '...among others. For more developments...'.
[Sec. III.A.2] 'a minimum bosonic system' should be 'a minimal bosonic system'.
[Sec. II.B, Eq. (12)] The definition of the multipartite mutual information I_tot(t) is introduced only after Eq. (12); introducing it before Eq. (8) would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Numerical demonstrations of drag-induced AEF/ACF use H_j = ε_j N_j, so the initial GGE is thermal at shifted inverse temperatures; the predicted anomalous flow reduces by construction to normal flow under the effective gradient.

renaming known result [Sec. III.A.2, Eqs. (18)-(21), Fig. 1; Sec. III.B.2, Eqs. (23)-(25), Fig. 2]
"we employ a minimum bosonic system where the energy and particle number serve as charges [64] ... H0 = ∑_{j=A,B} ε_j a†_j a_j. (18) ... each subsystem is prepared in a grand-canonical state, γ_j = e^{−β_j(H_j−μ_j N_j)}/Z_j. (20) ... Fig. 1 (a) provides clear evidence of AEF, demonstrated by the negative value of the figure of merit (βA−βB)∆EA(t) < 0."

With H_j = ε_j N_j, the initial state Eq. (20) equals e^{−β'_j H_j}/Z_j with β'_j = β_j(1 − μ_j/ε_j), i.e., a thermal state at shifted inverse temperature. Hence the central bound Eq. (21), (βA−βB)∆EA ≥ ∆I + (βAμA−βBμB)∆NA, is identically the no-go bound (β'_A−β'_B)∆EA ≥ ∆I ≥ 0. The chosen parameters (βA=20, βB=18, μA=0.8, μB=0.4, ε=1.5) give β'_A=9.33 < β'_B=13.2, so A is effectively hotter and the negative (βA−βB)∆EA is normal heat flow under the physically relevant gradient, not a drag effect. The μ-term is not an independent non-energy charge flow; it is the same observable as ∆EA up to the constant ε. The ACF example, Eqs. (23)-(25), repeats the same construction with H_i = ε_i N_i up to a c-number.

full rationale

The no-go theorem of Sec. II is derived self-containedly from quantum relative entropy identities and the non-negativity of relative entropy and mutual information; it involves no fitted parameters and no load-bearing self-citations. The catalytic extension likewise follows from the stated self-consistency condition and entropy inequalities. The multiple-charge inequalities, Eqs. (17) and (22), are exact consequences of the global-local framework and are not circular. However, the positive existence claim of drag-induced AEF and ACF is supported only by numerical models in which energy and the non-energy charge are proportional (H_j = ε_j N_j). In those models the initial generalized Gibbs state is exactly thermal for H_j at the shifted inverse temperature β'_j = β_j(1 − μ_j/ε_j), so the displayed 'anomalous' flow is the normal energy flow predicted by the paper's own no-go result applied to β'_A and β'_B. The drag term is therefore a relabeling of the energy flow rather than an independent charge species producing a novel effect. Because the central affirmative claim is confirmed only through this reduction, the paper exhibits partial circularity, even though the core derivations are independent and mathematically sound.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The core inequalities are derived without fitting; the only hand-chosen numbers are the simulation parameters used to exhibit the effects. The proof relies on standard properties of relative entropy, unitarity, and generalized Gibbs states. The noncommuting-charge generalization is asserted on the basis of Refs. [37,38] but not numerically demonstrated, and the ΔI_tot(τ)>0 step is not generally true.

free parameters (1)

Numerical parameters (β_A, β_B, μ_A, μ_B, ε, J) = Fig. 1: βA=20, βB=18, μA=0.8, μB=0.4, ε=1.5, J=0.2; Fig. 2: βA=1, βB=2, μA=0.4, μB=0.8, ε=2, J=0.2
Chosen by hand to satisfy the necessary conditions for AEF (Fig. 1) and ACF (Fig. 2); not fitted to external data, but required to exhibit the effects in simulation.

assumptions (6)

standard math Quantum relative entropy is non-negative: D[ρ||σ] ≥ 0.
Used throughout to convert exact equalities (3), (4), and (16) into lower bounds such as Eqs. (5) and (17).
standard math Von Neumann entropy is invariant under unitary evolution.
Used in deriving Eqs. (1), (A6), (A7), and the catalytic generalizations.
domain assumption Initial state is a product of local (generalized) Gibbs states with well-defined affinities.
This defines the uncorrelated scenario for the no-go theorem and the multiple-charge transport setup; the paper partially relaxes it in Sec. IV B using reference Gibbs states.
domain assumption The unitary U conserves total energy (no-go) or all charges: [U, Σ_j C_i^j] = 0 for all i.
Defines the charge-conserving transport scenarios; relaxed in Sec. IV A.
domain assumption A valid quantum catalyst returns exactly to its initial state at the end of the process, so ΔE_C(τ) = 0 and ΔS_C(τ) = 0.
Standard catalyst condition from Eq. (7) and Ref. [50]; used in deriving Eq. (12).
ad hoc to paper For any nontrivial catalytic process, ΔI_tot(τ) > 0.
Asserted after Eq. (12) to make the lower bound strictly positive; not generally true, since swap-like energy-conserving unitaries can leave a product state with zero mutual information. The no-go conclusion still follows from the nonnegative bound.

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Cite this review

Pith. "Pith review of Anomalous flow in correlated quantum systems: No-go result and multiple-charge scenario." pith.science (2026). https://pith.science/paper/WDDHXOKR

@misc{pith2026250605995,
  author       = {Pith},
  title        = {Pith review of: Anomalous flow in correlated quantum systems: No-go result and multiple-charge scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDDHXOKR}},
  note         = {Machine review of arXiv:2506.05995}
}

read the original abstract

Correlated quantum systems can exhibit thermodynamic behaviors that defy classical expectations, with anomalous energy flow (AEF) against temperature gradients serving as a paradigmatic example. While AEF has been shown to arise from the consumption of initial quantum correlations, little is known about whether AEF can occur without correlation depletion, or if analogous anomalous transport exists for conserved quantities--dubbed charges--other than energy. Here, we develop a general global-local thermodynamic approach to describe charge exchange between arbitrary correlated quantum systems. For energy-conserving systems, we analytically rule out AEF in initially uncorrelated states, even with the involvement of quantum catalysts, thereby complementing existing studies. In contrast, in systems with multiple conserved charges, we uncover a mechanism for AEF that requires no initial correlations but is instead induced by a drag effect from normal flows of non-energy charges. Furthermore, by treating all conserved charges on equal footing, we generalize AEF to a broader concept of anomalous charge flow, applicable to any conserved charge. We confirm theoretical expectations with numerical examples. These findings deepen our understanding of nonequilibrium quantum thermodynamics and open new avenues for controlling transport phenomena in correlated quantum systems.

Figures

Figures reproduced from arXiv: 2506.05995 by the authors.

**Figure 2.** Verification of ACF: (a) Results of (βBµB − βAµA)∆NA(t) (green solid line) and ∆I(t) − (βA − βB)∆EA(t) (blue dashed line) as a function of time, (b) Evolution of charges of subsystem A: energy EA(t) = Tr[HAρ(t)] (red dashed line) and excitation NA(t) = Tr[NAρ(t)] (blue solid line), (c) The monotonic increase of the quantum mutual information I(t) between the two spins. Parameters are βA = 1, βB = 2, µA = 0.4, µB = … view at source ↗

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Here we extend the thermodynamic framework by in- corporating multiple conserved charges besides energy

Theory Previously, we have demonstrated that AEF cannot occur in initially uncorrelated systems where only energy is con- served. Here we extend the thermodynamic framework by in- corporating multiple conserved charges besides energy. This generalization allows us to investigate the potential influence of additional charges on the direction and magnitude ...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Here we extend the thermodynamic framework by in- corporating multiple conserved charges besides energy

Theory Previously, we have demonstrated that AEF cannot occur in initially uncorrelated systems where only energy is con- served. Here we extend the thermodynamic framework by in- corporating multiple conserved charges besides energy. This generalization allows us to investigate the potential influence of additional charges on the direction and magnitude ...

[2] [2]

The bosonic model we consider consists of two bosonic modes with annihilation operators aj (j = A, B) and on-site energyεj, with its Hamiltonian reading H0 = ∑ j=A,B εj a† jaj

Numerical example To illustrate the occurrence of AEF due to the drag effect of a net normal flow of non-energy charges, we employ a minimum bosonic system where the energy and particle num- ber serve as charges [64]. The bosonic model we consider consists of two bosonic modes with annihilation operators aj (j = A, B) and on-site energyεj, with its Hamilt...

work page

[3] [3]

A special case: ρj(0)= γj = γR j We observe that correlated initial states with local marginals given by exact generalized Gibbsian states [cf. Eq. (13)] constitute a special case of our general framework de- fined by Eqs. (31)-(38). Here, the subsystems’ initial states coincide with their reference states ρj(0) = γj = γR j , yield- ing well-defined initi...

work page

[4] [4]

Theory Rearranging Eq. (16) and its local equivalence by isolat- ing an arbitrary quantity (λA k − λB k )∆C A k (t) for kth non- energy charge (k ≠ 0) as the focus of demonstrating ACF and invoking the non-negativity of the quantum relative en- tropy, we can obtain two lower bounds on(λA k −λB k )∆C A k (t), namely, LC g (t) = −∆I(t) − ∑i≠k(λA i − λB i )∆...

work page

[5] [5]

(23) Here, εi marks the energy gap of spins and σz i is the Pauli- z operator at site i

Numerical example To numerically demonstrate ACF, we consider a two-qubit model that can be related to the previous bosonic system [65] H0 = − ∑ i=A,B εi 2 σz i . (23) Here, εi marks the energy gap of spins and σz i is the Pauli- z operator at site i. We consider εA = εB = ε. Similar to the bosonic system considered above, the charge-conserving unitary tr...

work page

[6] [6]

Quantum catalyst-free systems We start from considering quantum catalyst-free bipartite systems. In this scenario, the change in von Neumann entropy of subsystem A can be expressed as (see details in Appendix A) ∆SA(t) = D[ρ(t)∣∣ρA(t)⊗ γR B]− D[ρB(0)∣∣γR B] − ∑ i=0 ˜λB i ∆C B i (t)− I(0). (31) Here, I(0) denotes the initial quantum correlations between tw...

work page

[7] [7]

Quantum catalyst-involved systems We then turn to setups with quantum catalyst. We assume that the bipartite system is initially uncorrelated with the quan- tum catalyst such that the total initial state takes a product form, ρtot(0) = ρ(0)⊗ ωC (Recall that we allow for corre- lated states ρ(0)). The quantum catalyst’s state ωC should be determined in a s...

work page

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Perarnau-Llobet, K

M. Perarnau-Llobet, K. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Ac ´ın, Extractable work from correlations, Phys. Rev. X 5, 041011 (2015)

work page 2015