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arxiv: 2509.20347 · v3 · submitted 2025-09-24 · 🪐 quant-ph

Quantum speed limits based on Jensen-Shannon and Jeffreys divergences for general physical processes

Jucelino Ferreira de Sousa , Diego Paiva Pires This is my paper

Pith reviewed 2026-05-18 14:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum speed limitsJensen-Shannon divergenceJeffreys divergenceopen quantum systemsentropic measuresSchatten speedunitary evolutionnonunitary channels
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The pith

Quantum speed limits for any physical process follow from the square root of Jensen-Shannon divergence and the square root of Jeffreys divergence.

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

The paper derives quantum speed limits that apply to finite-dimensional quantum systems evolving under arbitrary physical processes, whether the evolution is unitary or open and noisy. It uses the square root of the Jensen-Shannon divergence, which serves as a faithful distance between states, and the square root of the quantum Jeffreys divergence to bound how fast a state can change. The bounds are written in terms of the Schatten speed of the state and simple functions of its smallest and largest eigenvalues. A sympathetic reader would care because these limits could constrain the pace of quantum evolution in realistic environments with noise, offering tools for thermodynamics and complexity estimates. The claims are checked analytically and numerically on single-qubit states under unitary driving and under depolarizing, phase-damping, and amplitude-damping channels.

Core claim

The authors obtain quantum speed limits for general completely positive trace-preserving maps by upper-bounding the time derivative of the square root of the Jensen-Shannon divergence and the square root of the quantum Jeffreys divergence. The resulting inequalities are expressed using the Schatten speed of the evolved state together with cost functions built from the minimal and maximal eigenvalues of the initial and instantaneous states. For unitary qubit evolution the limits recover Mandelstam-Tamm forms that scale inversely with the variance of the driving Hamiltonian. For the three chosen noisy channels the divergences admit closed-form expressions that permit direct numerical tests of

What carries the argument

The square root of the Jensen-Shannon divergence, treated as a faithful distance on quantum states, and the square root of the quantum Jeffreys divergence; these distances supply the quantities whose rates of change are bounded by the dynamical speed.

Load-bearing premise

The square root of the Jensen-Shannon divergence defines a faithful distance on quantum states, and the same functional form of the speed limit holds for arbitrary completely positive trace-preserving maps without extra restrictions on the generator.

What would settle it

Evolve a mixed qubit under the generalized amplitude-damping channel, compute the actual time derivative of the square root Jensen-Shannon divergence, and check whether it ever exceeds the product of the Schatten speed and the eigenvalue cost function given by the paper.

Figures

Figures reproduced from arXiv: 2509.20347 by Diego Paiva Pires, Jucelino Ferreira de Sousa.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Overview of the role played by quan [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Density plot of the entropic QSLs [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Density plot of the entropic QSLs [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Density plot of the entropic QSL [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Density plot of the normalized relative error [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Density plot of the entropic QSL [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Density plot of the normalized relative error [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We discuss quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing general physical processes. These QSLs were obtained using two families of entropic measures, namely the square root of the Jensen-Shannon divergence, which in turn defines a faithful distance of quantum states, and the square root of the quantum Jeffreys divergence. The results apply to both closed and open quantum systems, and are evaluated in terms of the Schatten speed of the evolved state, as well as cost functions that depend on the smallest and largest eigenvalues of both initial and instantaneous states of the quantum system. To illustrate our findings, we focus on the unitary and nonunitary dynamics of mixed single-qubit states. In the first case, we obtain speed limits $\textit{\`{a} la}$ Mandelstam-Tamm that are inversely proportional to the variance of the Hamiltonian driving the evolution. In the second case, we set the nonunitary dynamics to be described by the noisy operations: depolarizing channel, phase damping channel, and generalized amplitude damping channel. We provide analytical results for the two entropic measures, present numerical simulations to support our results on the speed limits, comment on the tightness of the bounds, and provide a comparison with previous QSLs. Our results may find applications in the study of quantum thermodynamics, entropic uncertainty relations, and also complexity of many-body systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper derives quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing general physical processes, using the square root of the Jensen-Shannon divergence (presented as a faithful distance) and the square root of the quantum Jeffreys divergence. The bounds are expressed in terms of the Schatten speed of the evolved state and cost functions depending on the smallest and largest eigenvalues of the initial and instantaneous states. The results are claimed to apply to both closed and open systems; they are illustrated analytically and numerically for unitary evolution (yielding Mandelstam-Tamm-like bounds inversely proportional to Hamiltonian variance) and for specific non-unitary channels (depolarizing, phase damping, and generalized amplitude damping), with comparisons to prior QSLs.

Significance. If the central derivations hold, the work supplies new entropic QSLs applicable to open quantum systems, with potential utility in quantum thermodynamics, entropic uncertainty relations, and many-body complexity. Strengths include the use of faithful distances, parameter-free expressions for the illustrated cases, analytical results for concrete channels, and numerical support with comparisons. The approach is internally consistent with standard quantum-information techniques for the specific continuous evolutions considered.

major comments (1)
  1. [Abstract and general claims] Abstract and general-processes claim: the manuscript states that the QSL functional form holds for arbitrary CPTP maps without additional restrictions on the generator. However, the Schatten speed and instantaneous-state construction presuppose a continuous differentiable family ρ(t) with a well-defined derivative; a single arbitrary CPTP map Φ applied to ρ_0 supplies no natural time parametrization or generator. The concrete illustrations are restricted to unitary evolution and specific continuous channels, so the unrestricted generality claim is not supported by the provided derivations and requires either explicit restriction to differentiable paths or additional justification.
minor comments (2)
  1. [Notation and definitions] Clarify in the text whether the eigenvalue-based cost functions are evaluated only at the endpoints or integrated along the path, and ensure consistent notation for the two entropic measures across sections.
  2. [Numerical results] The numerical simulations for the non-unitary channels would benefit from explicit discussion of how the time parametrization is chosen and whether the bounds remain tight for other generators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the single major comment below and agree that a clarification is warranted to align the stated scope with the technical assumptions of the derivations.

read point-by-point responses

  1. Referee: [Abstract and general claims] Abstract and general-processes claim: the manuscript states that the QSL functional form holds for arbitrary CPTP maps without additional restrictions on the generator. However, the Schatten speed and instantaneous-state construction presuppose a continuous differentiable family ρ(t) with a well-defined derivative; a single arbitrary CPTP map Φ applied to ρ_0 supplies no natural time parametrization or generator. The concrete illustrations are restricted to unitary evolution and specific continuous channels, so the unrestricted generality claim is not supported by the provided derivations and requires either explicit restriction to differentiable paths or additional justification.

    Authors: We thank the referee for this observation. Our derivations define the Schatten speed via the time derivative of a continuous family ρ(t) and employ instantaneous eigenvalues of both ρ(0) and ρ(t); these constructions presuppose a differentiable parametrization. An isolated arbitrary CPTP map indeed lacks a natural time scale or generator. We therefore agree that the abstract and general-processes statements should be restricted to continuous, differentiable evolutions generated by CPTP maps. In the revised manuscript we will explicitly qualify the claims to this setting, while noting that all concrete illustrations (unitary dynamics and the listed continuous channels) already satisfy the differentiability requirement. This change will remove any ambiguity between the stated generality and the supporting mathematics. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from divergence definitions and standard bounding techniques

full rationale

The paper constructs QSLs by taking the square root of the Jensen-Shannon divergence (or Jeffreys) as a distance measure on states, then bounding its time derivative via the Schatten norm of the generator and eigenvalue cost functions. This follows directly from the triangle inequality or integral form of the distance along a continuous path ρ(t), without any fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the central result to its inputs. The application to general CPTP maps is framed via the instantaneous evolved state and its derivative, which is consistent with the continuous-time setting used in the derivations and examples; no step equates a claimed prediction to a tautological renaming or construction from the same quantity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the property that the square root of Jensen-Shannon divergence is a faithful distance and on the assumption that the same functional bound form holds for arbitrary CPTP maps. No free parameters or new entities are introduced.

axioms (2)
  • domain assumption The square root of the Jensen-Shannon divergence defines a faithful distance on the set of quantum states.
    Explicitly invoked in the abstract to justify the use of this quantity as the basis for the speed limit.
  • domain assumption The derived speed-limit functional form remains valid for any completely positive trace-preserving evolution without further restrictions on the generator.
    Required for the claim that the results apply to general physical processes including the listed noise channels.

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    upper bound DJ,JS(ρ0,ρτ) ≤ sqrt(∫ dt fJ,JS(ρ0,ρt) ||dρt/dt||1) with fJ involving |ln(κmin(ρ0)κmin(ρt))| + κmax(ρ0)/κmin(ρt) and fJS involving |ln(κmin(ρt)κmin(ϖ))|

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