The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS 3

arXiv:2607.04065 · hep-th, cond-mat.stat-mech, quant-ph · Submitted 2026-07-05 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS 3".

Kai: In spread complexity, the average position of an operator along its Krylov chain recovers the right radial momentum of an infalling particle in AdS,

Mira: First, who's behind it and why it matters.

Title and authors: Kai: Looking at the title, "The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS three" it immediately signals that this paper is about using a specific quantum measure to tell something about how matter falls into a black hole geometry in three dimensions <ref:2607.04065#pg1>.

Mira: The authors, Ritam Basu and his team, are clearly operating at the intersection of quantum field theory on curved backgrounds and the study of complex operator dynamics, which is where things get really interesting for condensed matter theorists like us.

Lev: What I find intriguing about their approach is that they aren't just looking at static properties; they are analyzing a dynamic process—the infalling particle—and trying to quantify the non-classical aspects of its motion.

Kai: The core idea, as described in the abstract, is taking the rate of this normalized negativity and using it as a proxy for the second moment of the boundary state, which captures how much that state is spreading away from where it started.

Mira: That spreading aspect is what sets this work apart from simpler complexity measures; it’s about quantifying that dispersion beyond just knowing the average position along the Krylov chain.

Lev: If we think about error correction, this implies that the structure of entanglement itself, quantified by this rate, dictates how much information is being generated or lost during a collapse or an evolution process.

Kai: So, if we can measure this rate experimentally on some system—even a simplified one—we could potentially gain insight into the gravitational dynamics dual to that system.

Mira: The implication here is that we are finding a dictionary between the algebraic structure of quantum operators and the geometric properties of AdS spacetime, specifically relating spreading complexity to radial momentum.

Lev: That mapping suggests that certain quantum information measures aren't just abstract mathematical constructs; they have a physical meaning tied to how objects move in curved spacetimes.

Kai: It’s about moving from knowing where things are on average to understanding the full extent of their spatial distribution during a dynamical event.

Mira: The paper sets up a very specific framework, starting with the seed-normalized Krylov-Wigner distribution and then deriving an analytic Bessel form for its negativity, which is quite technical.

Lev: I wonder if that derivation itself introduces any assumptions about the nature of the initial state or the Hamiltonian that we need to be careful about when trying to translate this to real physical systems.

The paper's summary: Kai: To summarize the main thrust, the paper argues that while the average position along the Krylov chain recovers something related to radial momentum, it fundamentally ignores how much that wavepacket is spreading out across space.

Mira: The key finding they present is using the rate of a normalized Krylov-Wigner negativity as a diagnostic tool precisely to capture this second moment of the boundary state's spread.

Lev: In simpler terms, they propose that this negativity rate tracks the variance of the Krylov wavepacket, N(t) proportional to Var(Krylov), which is a powerful way to see how fast things are diverging.

Kai: They show that this relationship becomes very clean at late times, and critically, it linearizes precisely at a specific critical dimension, = one where the rate directly corresponds to the proper radial position multiplied by momentum <ref:2607.04065#pg0>.

Mira: That linearization at = one is where they connect everything: it links the negativity rate to R proportional to C P rho, which describes the rate of tidal stretching of neighboring infalling geodesics <ref:2607.04065#pg0>.

Lev: If this connection to tidal stretching is robust, it suggests that we might be able to measure spacetime curvature effects using observable quantities in a quantum system, even if that system is an effective model for AdS space.

Kai: The method involves starting with the seed-normalized distribution and computing the total negativity explicitly through phase-space integration and replica functionals to find this growth rate.

Mira: They also describe the state evolving as an SU(one one) coherent state characterized by a squeezing parameter A(t), where success probability A(t) squared is related to how much the state is squeezing, which monotonously increases towards unity <ref:2607.04065#pg1,SU(1, 1) coherent state>.

Lev: The fact that the conformal dimension doesn't even appear in the expression for the squeezing parameter A(t) initially, but only appears later as r = two in occupation statistics, points to a subtle interplay between those parameters <ref:2607.04065#pg1>.

Kai: So, they are using these mathematical tools—the SU(one one) algebra and the negative binomial distribution for occupation probabilities—to build this physical picture of spreading complexity <ref:2607.04065#pg0>.

The paper's improvements: Kai: The authors point out that their primary improvement is establishing this specific matching condition between two growth rates: comparing the normalized negativity rate with the growth rate of the Krylov wavepacket variance.

Mira: They show that both of these rates depend on a single scaling variable, xi(t), and by matching them as power laws, they arrive at that critical dimension = one <ref:2607.04065#pg0>.

Lev: This matching condition is mathematically rigorous because it forces the two different ways of measuring growth—one from negativity and one from variance—to be consistent under specific conditions.

Kai: And once they hit that critical point, they get a clean geometric reading: the rate becomes R proportional to C P rho, which is explicitly defined as the rate of tidal stretching of neighboring infalling geodesics.

Mira: This is where they solidify the holographic dictionary; it suggests that at this specific dimension, we have a direct proportionality between quantum state evolution and classical gravitational dynamics.

Lev: For error correction, this means that if we can engineer a system whose dynamics exhibit this behavior near the critical point, we might be able to use it as a benchmark for understanding geometric effects on information flow.

Kai: They also touch on the bulk string interpretation by noting that the shared quadratic Casimir between the boundary Krylov chain and the string Hilbert space suggests a deep structural link between these two descriptions.

Mira: This Casimir matching implies that the transverse string size operator and our boundary Krylov chain possess identical quantum mechanical properties, suggesting a unitary equivalence between them.

Lev: That unitary equivalence is significant because it means we can use tools developed for one system to make predictions about the other, even if they look very different on the surface.

Conclusion: Kai: So, looking at "The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS three" we see that this negativity rate is a precise diagnostic for the second moment of the boundary state's spread during infalling dynamics <ref:2607.04065#pg1>.

Mira: Ultimately, the paper establishes a connection where the late-time behavior of this rate linearizes at = one providing a clean geometric reading tied to tidal stretching in AdS space <ref:2607.04065#pg0>.

Lev: From an error correction viewpoint, this suggests that we should look for systems exhibiting this specific scaling behavior near that critical dimension if we want to see how entanglement structure relates to geometry.

Kai: It’s a framework that links the algebraic structure of Krylov operators, the SU(one one) symmetry, and concrete gravitational concepts like geodesic stretching in a mathematically consistent way <ref:2607.04065#pg0>.

Mira: The implication is that we have found a specific signature—the negativity rate—that probes geometric effects beyond what simpler measures of state complexity can provide.

Lev: I think the work on this paper opens up avenues for thinking about how fundamental symmetries, like su(one one), might constrain the possible dynamics of quantum systems in a way that mimics gravity <ref:2607.04065#pg0>.

Kai: It’s a very rich piece of work that connects quantum information theory directly to the geometry we see in string theory and gravity models.

Mira: We're definitely excited about how this framework could inform our understanding of holographic duality and the deep structure of quantum states.

Tata Institute of Fundamental Research

hep-th, cond-mat.stat-mech, quant-ph

Submitted: 2026-07-05

Updated: 2026-10-06

Comments: 41 pages , 5 figures

DOI: 10.1103/6pn8-sd9f

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: In spread complexity, the average position of an operator along its Krylov chain recovers the right radial momentum of an infalling particle in AdS, yet it is a measure of the first moment,

Key concepts

Spread Complexity
This concept relates to how an operator's position along its Krylov chain recovers the right radial momentum of an infalling particle in Anti-de Sitter (AdS) space. However, it only measures the first moment, not how much the wavepacket spreads away from its classical path.
Normalized Krylov-Wigner Negativity Rate
This rate is defined by dividing out trivial decay to probe the second moment of a boundary state. It serves as a diagnostic tool for measuring how the wavepacket spreads in relation to its classical trajectory, linking it to holographic dictionaries.
su(1, 1) Module
The dynamics are governed by an underlying symmetry described by a semi-infinite hopping chain that cannot be organized by finite symmetry. This structure is precisely the simplest realization of an su(1, 1) module, which dictates the irreducible representation of the Krylov chain.
Critical Dimension ($Δ = 1$)
A specific dimension where a crucial matching condition is met. At this point, the ratio between the negativity rate and the variance growth rate becomes flat. This condition yields a clean geometric reading where the rate equals the product of proper radial position and momentum ($R ≈ C P ho$).

Terminology

Summary

In spread complexity, the average position of an operator along its Krylov chain recovers the right radial momentum of an infalling particle in AdS, yet it is a measure of the first moment, irrespective of the spread of the wavepacket away from its classical trajectory.

The core diagnostic and its physical meaning

The paper proposes that 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. This normalized negativity is defined by dividing out the trivial decay of the return amplitude, allowing it to probe the holographic dictionary via the second moment, beyond the classical trajectory. The key finding is that at late times, this normalized negativity becomes a fixed power of the second moment of the Krylov wavepacket, N (t) ∝ Var(NˆKrylov) ∆, for every dimension. This relation linearizes precisely at a critical dimension: "only at this dimension does the negativity rate track the growth rate of the Krylov variance, and there, through the momentum dictionary of Caputa et al. [1], the rate becomes the product of the proper radial position and momentum, R ∝ C Pρ."

The algebraic structure underpinning spread complexity

The dynamics are rooted in an underlying symmetry: "a semi-infinite hopping chain whose off-diagonal entries grow like n as n → ∞ can never be organized by a compact, finite dimensional symmetry. Instead, it is precisely the simplest realization of a lowest-weight su(1, 1) module. The Krylov chain operators are associated with the Lie algebra su(1, 1), where the position being promoted to a Cartan generator and ladder operators implement hopping dynamics. This structure ensures that the Krylov chain carries a single irreducible representation, characterised by ∆ alone and by nothing else about the microscopic Hamiltonian, as evidenced by the constant quadratic Casimir: C(K)2 = ∆(∆ − 1)."

The distribution of occupation amplitudes

The evolved state is described as an SU(1, 1) (Perelomov) coherent state, characterized by a squeezing parameter A(t). The probabilities of occupying each rung are governed by the negative binomial distribution: the probability distribution given by (3.1) is the NB(2∆, A(t)2), where the success probability, p = A(t)2, is the measure of squeezing, monotonously increasing towards unity as the state spreads down the chain. The time dependence of this parameter is determined by the regularised thermal two-point function, leading to a compact expression for A(t) that shows that the conformal dimension ∆ does not enter into the expression for the squeezing parameter at all, it only enters as the parameter r = 2∆ of the occupation statistics.

The matching condition and critical dimension

The paper establishes a crucial link between boundary observables by comparing two growth rates: the normalized negativity rate (4.11) with the growth rate of the Krylov wavepacket variance (5.1). Both rates are functions of a single scaling variable, ξ(t). The comparison reduces to matching two power laws: "N (t) ∝ [Var(NˆKrylov)]∆, sinh2(πt/β) ≫ ε0, ∆ > 1/2. This matching condition yields the critical dimension: The ratio R/(d Var/dt) is flat only at the critical value ∆ = 1, where the negativity rate acquires its clean geometric reading of R ∝ C Pρ—the rate of tidal stretching of neighbouring infalling geodesics."

The holographic interpretation

In a speculative bulk string interpretation, the identification conjecture Nˆstr ≃ NˆKrylov is motivated by the shared Casimir: "The discrete-series sector of the string Hilbert space Hstr relevant to the infalling dynamics carries a representation of su(1, 1) whose quadratic Casimir equals ∆(∆ − 1), coinciding with C(K)2 = ∆(∆ − 1) of the Krylov chain. This suggests that the transverse string size operator and the boundary Krylov chain possess the same quadratic Casimir, implying a unitary equivalence between them, which would map the normalized negativity rate as the rate of growth of the squared transverse string area (7.5). This identification identifies the normalized negativity rate as the proper momentum dressed by C2∆−1, whereas the variance rate (6.2) is dressed by 1 + C/∆, coinciding precisely when ∆ = 1 and yielding R(t) ∝ C(t) Pρ."

How it works

The analysis proceeds through several key steps:

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, which establishes a deep connection between quantum information theory (Krylov complexity/Wigner negativity) and holographic physics (AdS3 black hole dynamics) via the SU(1, 1) algebra.

Here are the specific improvements that can be made to AI systems by integrating these findings:

The improved AI system can perform the following tasks:

  1. Find and Diagnose Non-Classical Structures in High-Dimensional Quantum States (Quantum State Diagnostics).

  2. Predict and Characterize the Dynamics of Infalling/Scrambling Systems in Holographic Duals.

  3. Establish Universal Scaling Laws for Operator Growth in Chaotic Quantum Systems.

Specific Improvements:

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 4Δ(πt/β). But the raw normalized-state negativity saturates to an O(1) constant, well below the O(sqrt 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) proportional to[Var(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 proportional to 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.

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