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Open quantum complexity is a sub-Finsler geometry: dissipation makes geodesics irreversible and restricts allowed directions.

2026-07-10 07:50 UTC pith:Q2OMYZDX

load-bearing objection Clean, usable extension of Nielsen geometry to Lindblad dynamics that correctly yields sub-Finsler structure and explicit flag-curvature dependence on penalties.

arxiv 2607.08411 v1 pith:Q2OMYZDX submitted 2026-07-09 quant-ph hep-th

The Geometry of Quantum Complexity in Open Systems

Ezra Acalapati , Kausik Ghosh , Giuseppe Policastro This is my paper
classification quant-ph hep-th
keywords quantum complexityopen systemsLindblad equationFinsler geometrysub-Finslerian geometryflag curvatureoptimal controlNielsen geometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Nielsen's geometric complexity assigns a cost to paths of unitary gates and measures difficulty by the shortest path length. This paper carries the same idea to open systems whose continuous evolution is a Lindblad equation. Complexity is redefined as the minimal cost of a controlled trajectory in the space of mixed states, where both unitary rotations and non-unitary noise channels can be used as controls and each is given a penalty weight. The resulting geometry is typically sub-Finslerian rather than Riemannian: the allowed velocities form only a cone or half-plane inside the full tangent space, geodesics cannot be run backwards because of dissipation, and the fundamental length function depends on both position and direction. Explicit calculations for a depolarizing qubit, an amplitude-damped qubit, and a damped oscillator show that the same penalty factors that produce negative sectional curvature in the closed case now change the sign of the flag curvature, controlling how neighbouring optimal trajectories diverge. The construction therefore supplies a concrete geometric language for complexity once decoherence and environmental coupling are admitted.

Core claim

When Nielsen's cost-functional minimization is extended from unitary generators to Lindbladian generators acting on mixed states, the natural geometry is in general sub-Finslerian: admissible velocities are restricted by the physically allowed controls, geodesics become non-reversible under dissipation, and the flag curvature of this geometry is controlled by the penalty factors that weight unitary versus non-unitary controls.

What carries the argument

The Finslerian function induced by Pontryagin's Maximum Principle on the free-final-time optimal-control problem. For a quadratic control cost plus terminal cost, this function is obtained by rescaling admissible velocities onto the indicatrix (the unit-cost surface); its explicit form is given for both driftless and drifted Lindbladians (Eqs. 6.9, 6.12, 6.15) and becomes the length functional whose geodesics are the complexity-minimizing trajectories.

Load-bearing premise

The instantaneous cost is taken to be a quadratic function of the control amplitudes plus a constant terminal cost, with free final time, so that optimal trajectories sit on the zero level set of the Pontryagin Hamiltonian; a different homogeneous cost or fixed-time problem would produce a different geometry.

What would settle it

Compute the flag curvature for a concrete single-qubit depolarizing or amplitude-damping Lindbladian with chosen penalty factors; if the curvature fails to change sign (or to vanish after the polar-coordinate change used for the driftless depolarizing case) as the penalties are varied across the regimes reported in the examples, the claimed geometric dependence is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper extends Nielsen’s geometric circuit complexity from unitary evolution to open systems governed by Lindblad dynamics. Complexity is defined as the minimal cost of a controlled trajectory on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. Using the autonomous optimal-control formulation and Pontryagin’s Maximum Principle, the authors show that the induced geometry is typically sub-Finslerian rather than Riemannian: dissipation renders geodesics non-reversible and the admissible velocities are restricted to a cone (or affine half-plane when drift is present). Explicit Finsler functions are derived for under-actuated single-qubit dynamics (Eqs. 6.9, 6.12, 6.15). Five worked examples—depolarizing and amplitude-damping channels with and without unitary/non-unitary drift, plus the Gaussian sector of a damped harmonic oscillator—illustrate indicatrices, flag curvature, and numerical optimal trajectories, and demonstrate that the flag curvature depends on the penalty factors in the cost functional.

Significance. If the construction is accepted, it supplies a concrete geometric language for complexity in dissipative quantum systems that is directly tied to experimentally relevant channels (depolarizing, amplitude damping, damped oscillator). The derivation from PMP to the Finsler function is standard and carefully executed; the explicit reconstruction of the Finslerian function for under-actuated qubit dynamics and the numerical geodesics for the drifted depolarizing case are reproducible and falsifiable. The observation that flag curvature can change sign with the penalty factors parallels the known unitary story and opens a route to studying directional sensitivity of optimal open-system trajectories. The framework is therefore a useful addition to the complexity literature and has clear potential links to NISQ control and holographic models of evaporating black holes.

minor comments (5)
  1. The quadratic cost plus free-final-time terminal cost Ct (Eqs. 6.2, 7.1 and §5.4) is a modeling choice that is stated clearly but could be flagged more prominently in the abstract or introduction as the source of the particular Finsler structure obtained.
  2. In §7.1 the claim that the reduced metric is flat is verified by a coordinate change; a short remark that the same change fails for amplitude damping (§7.2) would make the contrast sharper.
  3. Figures 1–3 and 7 show indicatrices at a single base point; adding one sentence on how the shape varies across the Bloch ball would help the reader assess global features.
  4. A few typographical inconsistencies appear (e.g., “Werecallheresomebasicfacts”, missing spaces after periods in the introduction and §2). A light copy-edit would improve readability.
  5. The discussion of non-Markovian extensions and feedback (§8) is interesting but brief; a pointer to existing non-Markovian optimal-control literature would strengthen the outlook.

Circularity Check

0 steps flagged

No significant circularity: Finsler/sub-Finsler structure and flag-curvature dependence are derived from the optimal-control problem and explicit reconstruction, not forced by definition or self-citation.

full rationale

The paper defines open-system complexity as the minimal cost of a Lindbladian trajectory (Nielsen-style optimal control on mixed states). Section 5 applies the standard Pontryagin Maximum Principle to an autonomous control system with quadratic cost plus free final time/terminal cost Ct, inducing a (sub-)Finsler function F on the cone of admissible velocities exactly as in the external reference [16]. Section 6 then solves the resulting algebraic equations for a single qubit (two controls, one constraint) and obtains the explicit formulae (6.9) without drift, (6.12) with drift, and (6.15) on the singular locus; these are direct rearrangements of the cost and the left-inverse of the control matrix, not tautologies. The five examples of §7 simply specialise the same formulae, plot the resulting indicatrices, and evaluate the flag curvature of the induced metric; the penalty factors pγ, pω, pγω remain free parameters that the user chooses, and the observed sign changes of K are computed outputs, not fitted inputs re-labelled as predictions. No uniqueness theorem is imported from the authors’ prior work, no ansatz is smuggled via self-citation, and no equation reduces a claimed prediction to a quantity already fixed by construction. The derivation is therefore self-contained against its own modelling choices.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard optimal-control and Finsler geometry plus the modeling choice that continuous open-system dynamics is Markovian Lindblad and that the cost is quadratic in the controls. No new physical entities are postulated; the free parameters are the user-chosen penalty weights and terminal cost that define the geometry.

free parameters (2)
  • penalty factors p_γ, p_ω, p_γω and terminal cost C_t
    These numbers are chosen by hand to define the cost functional (Eq. 7.1); different choices produce different flag curvatures and geodesics. They are not fitted to data but remain free inputs of the geometry.
  • drift strength Δ
    Appears as an uncontrolled unitary or non-unitary term in the examples; its value is set by the modeler and affects the shape of the indicatrix.
axioms (4)
  • domain assumption Continuous open-system evolution is generated by a time-homogeneous Lindblad (GKSL) master equation
    Stated in §2; non-Markovian or non-CP maps are excluded by construction.
  • standard math Pontryagin's Maximum Principle supplies the necessary conditions for the optimal controls (normal case p0 = −1)
    Invoked throughout §5; abnormal extremals are set aside.
  • ad hoc to paper The instantaneous cost is quadratic in the control amplitudes plus a constant terminal cost
    Eqs. 6.2 and 7.1; chosen for convenience so that the Finsler function can be written in closed form.
  • standard math Definitions of Finsler structure, spray, and flag curvature
    Reviewed in §4 from standard references (Bao–Chern–Shen).

pith-pipeline@v1.1.0-grok45 · 32814 in / 2523 out tokens · 27739 ms · 2026-07-10T07:50:43.176892+00:00 · methodology

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read the original abstract

We extend Nielsen's geometric approach for quantum complexity from closed to open quantum systems, whose dynamics is governed by Lindbladian evolution. In this framework, complexity is defined through an optimal-control problem on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. We show that the resulting geometric structure differs fundamentally from the Riemannian geometry that emerges in the case of unitary evolution. In the open-system setting, the natural geometry is typically sub-Finslerian. Dissipation makes the geodesics non-reversible, while the admissible tangent directions are restricted by the physically allowed controls. We analyze several physically motivated examples, including a single qubit subject to depolarizing and amplitude-damping channels, as well as the damped harmonic oscillator. We show that, similarly to the unitary case, varying the penalty factors in the cost functional modifies the geometric properties through changes in the flag curvature, the Finslerian analog of sectional curvature. Our results provide a geometric framework for quantifying the abstract notion of complexity in dissipative quantum systems, with potential connections to experimentally realizable setups.

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