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A universal approach to Krylov State and Operator complexities

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arxiv 2212.10583 v3 pith:XF36J4BZ submitted 2022-12-20 hep-th cond-mat.stat-mechquant-ph

classification hep-thcond-mat.stat-mechquant-ph
keywords complexitykrylovstatecomplexitiesoperatordensityframeworkterms
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We present a general framework in which both Krylov state and operator complexities can be put on the same footing. In our formalism, the Krylov complexity is defined in terms of the density matrix of the associated state which, for the operator complexity, lives on a doubled Hilbert space obtained through the channel-state map. This unified definition of complexity in terms of the density matrices enables us to extend the notion of Krylov complexity, to subregion or mixed state complexities and also naturally to the Krylov mutual complexity. We show that this framework also encompasses nicely the holographic notions of complexity.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Krylov-Space Memory Cores

    hep-th 2026-07 conditional novelty 6.0 of 10

    Anomalous initial states in otherwise thermalizing models leave compact, stationary low-depth Krylov-space cores—regions with persistent fluctuations, Gibbs mismatch, and current activity—while generic states do not.

  2. Krylov complexity has it all

    hep-th 2026-05 unverdicted novelty 6.0 of 10

    Krylov complexity's Taylor coefficients recursively determine all Lanczos coefficients, making it a complete descriptor of operator dynamics, with caveats for spread complexity.

  3. Generalized Krylov Complexity

    hep-th 2025-07 conditional novelty 5.0 of 10

    The paper defines generalized Krylov complexity for multi-generator unitary evolutions, computes it for U(1)xU(1), a U(1)xU(1) subgroup of SO(10), and SU(2), and introduces a weighted version.

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