The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS3
Abstract
In spread complexity, the growth rate of the average position of a time-evolved state along its Krylov chain recovers the proper radial momentum of an infalling particle in AdS. It is, however, a measure of the first moment only, irrespective of the spread of the wavepacket away from its classical trajectory. The rate of a normalized Krylov-Wigner negativity can be proposed as a diagnostic of the second moment of the boundary state that captures this spreading. Starting with the seed-normalized Krylov-Wigner distribution, that is, the Wigner transform of the descendant cloud with the decaying return amplitude divided out, we obtain an analytic Bessel form in the macroscopic limit and compute its total negativity explicitly. Retaining the Bessel variable all the way through, we find that the negativity goes as sinh4Δ(πt/β). But the raw normalized-state negativity saturates to an O(1) constant, well below the O(√D) bound. Using the exact negative binomial statistics of the Krylov chain, the normalized negativity is at late times a fixed power of the second moment of the Krylov wavepacket, 𝒩(t)∝[Var(N̂Krylov)]Δ, for every Δ>1/2. The relation linearizes precisely at Δ=1. In this regime via the momentum dictionary of Caputa et al. (arXiv:2410.23334), the rate becomes the product of the proper radial position and momentum, ℛ∝𝒞 Pρ, which we interpret as the rate of the tidal stretch of nearby geodesics falling into the horizon. We also comment briefly on a speculative direction for future research, in particular the interpretation of the transverse string size operator in terms of the Krylov number operator through the common SU(1,1) discrete series.
Versions
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BibTeX
@misc{basu2026normalised,
title = {{The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS$_3$}},
author = {Ritam Basu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04065},
eprint = {2607.04065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04065},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS_3
AU - Ritam Basu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.04065
UR - https://arxiv.org/abs/2607.04065
AB - In spread complexity, the growth rate of the average position of a time-evolved state along its Krylov chain recovers the proper radial momentum of an infalling particle in AdS. It is, however, a measure of the first moment only, irrespective of the spread of the wavepacket away from its classical trajectory. The rate of a normalized Krylov-Wigner negativity can be proposed as a diagnostic of the second moment of the boundary state that captures this spreading. Starting with the seed-normalized Krylov-Wigner distribution, that is, the Wigner transform of the descendant cloud with the decaying return amplitude divided out, we obtain an analytic Bessel form in the macroscopic limit and compute its total negativity explicitly. Retaining the Bessel variable all the way through, we find that the negativity goes as sinh^4Δ(πt/β). But the raw normalized-state negativity saturates to an O(1) constant, well below the O(√D) bound. Using the exact negative binomial statistics of the Krylov chain, the normalized negativity is at late times a fixed power of the second moment of the Krylov wavepacket, 𝒩(t)∝[Var(N̂_Krylov)]^Δ, for every Δ>1/2. The relation linearizes precisely at Δ=1. In this regime via the momentum dictionary of Caputa et al. (arXiv:2410.23334), the rate becomes the product of the proper radial position and momentum, ℛ∝𝒞 P_ρ, which we interpret as the rate of the tidal stretch of nearby geodesics falling into the horizon. We also comment briefly on a speculative direction for future research, in particular the interpretation of the transverse string size operator in terms of the Krylov number operator through the common SU(1,1) discrete series.
ER -