pith. sign in

arxiv: 2606.20790 · v1 · pith:V6IEB73Gnew · submitted 2026-06-18 · ✦ hep-th · cond-mat.stat-mech· quant-ph

Complexity Inequalities for Quantum Subsystems

Pawel Caputa , Giuseppe Di Giulio , Tran Quang Loc This is my paper

Pith reviewed 2026-06-26 15:47 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechquant-ph
keywords subsystem complexitytripartite complexitycomplexity gapholographic complexityFisher-Rao complexityKrylov complexityquantum informationAdS/CFT
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The pith

The complexity gap between a full quantum state and its three subsystems has a definite sign in every tested measure, while tripartite complexity does not.

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

The paper defines two combinations of subsystem complexities for three disjoint regions: a tripartite complexity modeled on tripartite information, and a complexity gap that isolates extra complexity present only in the full state. It evaluates both quantities in three concrete settings: volume complexity in AdS, Fisher-Rao complexity for Gaussian states, and a Krylov-space construction for reduced density matrices. In all three settings the tripartite complexity can be positive or negative. The complexity gap, by contrast, always carries a fixed sign within each setting, although the direction of that sign changes with the chosen complexity definition. The authors therefore propose the gap as a candidate starting point for a future set of complexity inequalities that would constrain allowable combinations of subsystem complexities.

Core claim

Across holographic volume complexity, Fisher-Rao complexity for Gaussian states, and a Krylov-inspired framework for reduced density matrices, the tripartite complexity lacks a definite sign. The complexity gap, however, maintains a definite sign in all examined cases, with the specific sign varying by the complexity definition. This positions the complexity gap as a potential building block for a hierarchy of subsystem complexity inequalities.

What carries the argument

The complexity gap, obtained by subtracting the sum of the three subsystem complexities from the complexity of the full state.

If this is right

  • Tripartite complexity can change sign and therefore does not supply a universal inequality.
  • The complexity gap supplies a signed quantity that may serve as the seed for further inequalities.
  • The sign of the gap is not universal but depends on the operational definition of complexity.
  • The gap isolates complexity that cannot be attributed to any of the three reduced states alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the gap remains signed under additional definitions, it could function as the first member of a complexity cone analogous to the holographic entropy cone.
  • States whose gap violates the observed sign in a given framework might be forbidden from having simple gravitational duals.
  • The dependence of the sign on the complexity measure suggests that different operational notions of complexity may correspond to different physical regimes.

Load-bearing premise

The three chosen notions of subsystem complexity are representative enough that the observed sign pattern of the complexity gap will hold more generally.

What would settle it

An explicit example, in any of the three frameworks or a fourth one, of a tripartite quantum state where the complexity gap takes both positive and negative values.

read the original abstract

Motivated by the role of the holographic entropy cone in constraining the entanglement structure of states with classical gravitational duals, we investigate combinations of subsystem complexities associated with reduced density matrices in multipartite quantum systems. Focusing on subsystems composed of three disjoint regions, we introduce two quantities: a tripartite complexity, inspired by the tripartite information, and a complexity gap, designed to characterize emergent complexity in the full quantum state beyond that of its constituents. We study the sign structure of these quantities in three selected approaches to subsystem complexity. In holography, we employ the complexity=volume proposal in AdS spacetimes; for Gaussian many-body states, we use Fisher-Rao subsystem complexity; and we further develop a Krylov-space inspired, effective framework for reduced density matrices, which we test in few-qubit systems and coherent-state dynamics. Across all three approaches, we find that the tripartite complexity is not sign-definite in general. By contrast, the complexity gap exhibits a definite sign in every example we analyze, although the sign itself depends on the underlying notion of subsystem complexity. Our results suggest that the complexity gap could be a natural candidate building block for a prospective hierarchy of subsystem complexity inequalities.

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

0 major / 1 minor

Summary. The manuscript introduces tripartite complexity (inspired by tripartite information) and complexity gap (to capture emergent complexity beyond constituents) for three disjoint subsystems. It examines the sign structure of these quantities across three frameworks: the complexity=volume proposal in AdS holography, Fisher-Rao subsystem complexity for Gaussian states, and a Krylov-space effective framework tested on few-qubit systems and coherent-state dynamics. The central empirical claim is that tripartite complexity is not sign-definite in general, while the complexity gap exhibits a definite (framework-dependent) sign in every example analyzed. The results position the complexity gap as a prospective building block for subsystem complexity inequalities.

Significance. If the reported sign structure of the complexity gap holds more generally, the work supplies concrete evidence that certain combinations of subsystem complexities obey sign constraints, potentially enabling a hierarchy of inequalities analogous to the holographic entropy cone. The multi-framework approach (holographic CV, information-geometric Fisher-Rao, and Krylov) provides independent checks on the same qualitative observation and strengthens the case that the gap may serve as a useful primitive. The limited scope—explicitly framed as observations in selected examples rather than a general theorem—is appropriately cautious.

minor comments (1)
  1. [Abstract] The abstract and introduction could briefly note the total number of explicit examples computed in each of the three frameworks to allow readers to gauge the breadth of the sign-structure evidence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of its significance, and their recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper reports empirical sign observations for tripartite complexity and complexity gap obtained from explicit calculations across three distinct computational frameworks (holographic CV, Fisher-Rao on Gaussians, Krylov on qubits). These are presented as numerical outcomes of the chosen models rather than as predictions derived from definitions, fitted parameters, or self-citations that would reduce the claimed sign structure to a tautology or input by construction. No equations or uniqueness theorems are invoked that collapse the reported results to the input data or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on two newly introduced quantities whose general properties are asserted from example calculations rather than from independent derivations or external benchmarks.

axioms (1)
  • domain assumption Subsystem complexity measures can be consistently defined for reduced density matrices of disjoint regions.
    Invoked when the paper applies complexity=volume, Fisher-Rao, and Krylov constructions to three-region subsystems.
invented entities (2)
  • tripartite complexity no independent evidence
    purpose: Combination of subsystem complexities inspired by tripartite information
    New quantity defined for three disjoint regions; no independent evidence supplied beyond the examples.
  • complexity gap no independent evidence
    purpose: Measure of emergent complexity in the full state beyond its constituents
    New quantity defined by subtracting summed subsystem complexities; sign property asserted from examples only.

pith-pipeline@v0.9.1-grok · 5742 in / 1469 out tokens · 25517 ms · 2026-06-26T15:47:27.186865+00:00 · methodology

discussion (0)

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Reference graph

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