arXiv:2607.04065 · Physical review. D/Physical review. D.

The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS$_3$

Ritam Basu

hep-thcond-mat.stat-mechquant-phSubmitted Jul 5, 2026• Announced Jul 7, 2026• 0 citations

Abstract

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 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 \emph{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 $\sinh^{4Δ}(πt/β)$, while the raw, seed-normalized state negativity saturates, as dictated by the $O(\sqrt{D})$ bound. Using the exact negative binomial statistics of the Krylov chain and the momentum dictionary of Caputa et al.~\cite{Caputa:2024}, the rate of the negativity scales as the growth rate of the Krylov variance at late time asymptotics \emph{if and only if} $Δ=1$, the Breitenlohner-Freedman saturating dimension in $AdS_3$. This dimensionality is special in that the negativity rate is the product of the proper radial position and momentum, $\mathcal{R} \propto \mathcal{C} P_ρ$, i.e., the rate of the tidal stretch of nearby geodesic falling into the horizon. We comment on the 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.

DOI: 10.1103/6pn8-sd9f

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Complexity study of the Hartle-Hawking state in JT gravity

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The Information Content of Krylov Observables: A Machine Learning Approach

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Wigner negativity, random matrices and gravity

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Wigner negativity in Krylov space and emergent semiclassicality

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Spread complexity in AdS/CFT: connecting quantum information and bulk dynamics

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Complexity Growth and the Krylov-Wigner function

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arXiv:2607.04065 · Physical review. D/Physical review. D.

The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS $_3$ The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS$_3$

Ritam Basu

hep-thcond-mat.stat-mechquant-phSubmitted Jul 5, 2026• Announced Jul 7, 2026• 0 citations

arXiv HTML PDF

Abstract

\sinh^{4Δ}(πt/β)

, while the raw, seed-normalized state negativity saturates, as dictated by the

O(\sqrt{D})

bound. Using the exact negative binomial statistics of the Krylov chain and the momentum dictionary of Caputa et al.~\cite{Caputa:2024}, the rate of the negativity scales as the growth rate of the Krylov variance at late time asymptotics \emph{if and only if}

Δ=1

, the Breitenlohner-Freedman saturating dimension in

AdS_3

. This dimensionality is special in that the negativity rate is the product of the proper radial position and momentum,

\mathcal{R} \propto \mathcal{C} P_ρ

, i.e., the rate of the tidal stretch of nearby geodesic falling into the horizon. We comment on the 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.

DOI: 10.1103/6pn8-sd9f

Related papers

Complexity study of the Hartle-Hawking state in JT gravity

Ritam Basu

The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. In this paper, we study the growth of Wigner negativity of the Hartle-Hawking state of Jackiw-Teitelboim gravity under time evolution. We work in the gravitational length basis, at genus zero, and at high temperature, $β\ll 1$ . Our main analytic result is simple: to leading order in $β$ the state is a Gaussian wavepacket of minimum uncertainty, sitting at rest at the turning point of the Liouville wall. Its Wigner function is therefore positive, and the negativity is $1+δ(β,t)$ , where the correction $δ$ is smaller than any power of $β$ . We show that $δ$ is an even function of time, so there is no linear growth at $t=0$ , and that once the packet has reflected off the wall its leading-order evolution is a Clifford shear, which cannot change the negativity. We also numerically study the time evolution of the Wigner function and its negativity. We observe that $δ$ stays below $10^{-11}$ at $t=0$ and through the reflection, then rises slowly over a few tens of $β/π$ , and then approaches a late-time value consistent with $δ_\infty(β) \simeq 0.08\, e^{-4.4/β}$ for $0.25 \le β\le 1$ : the lower the temperature, the larger the late-time negativity. The exact survival amplitude $Z(β+it)/Z(β)$ fixes the spread complexity to order $t^4$ , and it also fixes the seed-normalised Wigner diagnostic of our earlier work (arXiv:2607.04065, arXiv:2607.17346). We take this as evidence that the length basis is ideally suited for a dual, semi-classical description of the dynamics at genus zero. Beyond genus zero the length basis is overcomplete, and we make no statement about finite $e^{S_0}$ .The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. In this paper, we study the growth of Wigner negativity of the Hartle-Hawking state of Jackiw-Teitelboim gravity under time evolution. We work in the gravitational length basis, at genus zero, and at high temperature, $β\ll 1$. Our main analytic result is simple: to leading order in $β$ the state is a Gaussian wavepacket of minimum uncertainty, sitting at rest at the turning point of the Liouville wall. Its Wigner function is therefore positive, and the negativity is $1+δ(β,t)$, where the correction $δ$ is smaller than any power of $β$. We show that $δ$ is an even function of time, so there is no linear growth at $t=0$, and that once the packet has reflected off the wall its leading-order evolution is a Clifford shear, which cannot change the negativity. We also numerically study the time evolution of the Wigner function and its negativity. We observe that $δ$ stays below $10^{-11}$ at $t=0$ and through the reflection, then rises slowly over a few tens of $β/π$, and then approaches a late-time value consistent with $δ_\infty(β) \simeq 0.08\, e^{-4.4/β}$ for $0.25 \le β\le 1$: the lower the temperature, the larger the late-time negativity. The exact survival amplitude $Z(β+it)/Z(β)$ fixes the spread complexity to order $t^4$, and it also fixes the seed-normalised Wigner diagnostic of our earlier work (arXiv:2607.04065, arXiv:2607.17346). We take this as evidence that the length basis is ideally suited for a dual, semi-classical description of the dynamics at genus zero. Beyond genus zero the length basis is overcomplete, and we make no statement about finite $e^{S_0}$.

hep-thCross-listedarXiv:2610.00844Oct 2, 2026

The Information Content of Krylov Observables: A Machine Learning Approach

Ritam Basu

We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity $C(t)$ , the discrete Wigner negativity $N(t)$ , and the normalized negativity $χ(t)=N(t)/|S(t)|$ , where $S(t)$ is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable $SL(2,R)$ Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients $b_n=α\sqrt{n(n+2h-1)}$ . To introduce chaos we add a GUE random matrix to the integrable chain, $H(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}$ , where $R_0$ is its spectral half-width. Either $C(t)$ or $N(t)$ alone determines the thermofield temperature, with $R^2\simeq0.999$ . However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: $χ$ predicts $C$ better than $C$ predicts $χ$ . This gap grows from $+0.33$ to $+0.77$ , while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and $χ(t)$ carries it while $C(t)$ can still be recovered from $χ(t)$ . We derive a lower bound on this gap, set by the fraction of the variance of $χ$ that comes from $S$ .We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity $C(t)$, the discrete Wigner negativity $N(t)$, and the normalized negativity $χ(t)=N(t)/|S(t)|$, where $S(t)$ is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable $SL(2,R)$ Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients $b_n=α\sqrt{n(n+2h-1)}$. To introduce chaos we add a GUE random matrix to the integrable chain, $H(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}$, where $R_0$ is its spectral half-width. Either $C(t)$ or $N(t)$ alone determines the thermofield temperature, with $R^2\simeq0.999$. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: $χ$ predicts $C$ better than $C$ predicts $χ$. This gap grows from $+0.33$ to $+0.77$, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and $χ(t)$ carries it while $C(t)$ can still be recovered from $χ(t)$. We derive a lower bound on this gap, set by the fraction of the variance of $χ$ that comes from $S$.

hep-thCross-listedarXiv:2607.17346Jul 21, 20260 citations

Wigner negativity, random matrices and gravity

Ritam Basu, Pratyusha Chowdhury, Anirban Ganguly +3 more

A bstract Given a choice of an ordered, orthonormal basis for a D -dimensional Hilbert space, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents any quantum state as a real, normalized function on a discrete phase space. The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. Further, Wigner negativity also gives a lower bound on an entropic measure of spread complexity. In this paper, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. In [1], a perturbative argument was given to show that the Krylov basis minimizes the early time growth of Wigner negativity in the large- D limit. Using tools from random matrix theory, here we show that for a generic choice of basis, the Wigner negativity for a classical initial state becomes exponentially large in an O (1) amount of time evolution. On the other hand, we show that in the Krylov basis the negativity grows at most as a power law, and becomes exponentially large only at exponential times. We take this as evidence that the Krylov basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large- D at sub-exponential times. For the Gaussian unitary ensemble, this effective description is the q → 0 limit of q -deformed JT gravity.A bstract Given a choice of an ordered, orthonormal basis for a D -dimensional Hilbert space, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents any quantum state as a real, normalized function on a discrete phase space. The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. Further, Wigner negativity also gives a lower bound on an entropic measure of spread complexity. In this paper, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. In [1], a perturbative argument was given to show that the Krylov basis minimizes the early time growth of Wigner negativity in the large- D limit. Using tools from random matrix theory, here we show that for a generic choice of basis, the Wigner negativity for a classical initial state becomes exponentially large in an O (1) amount of time evolution. On the other hand, we show that in the Krylov basis the negativity grows at most as a power law, and becomes exponentially large only at exponential times. We take this as evidence that the Krylov basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large- D at sub-exponential times. For the Gaussian unitary ensemble, this effective description is the q → 0 limit of q -deformed JT gravity.

arXiv:2506.02110Journal of High Energy PhysicsJan 16, 20263 citations

Wigner negativity in Krylov space and emergent semiclassicality

Vijay Balasubramanian, Paweł Caputa, Onkar Parrikar +1 more

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large- $N$ , complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity -- a measure of the complexity of classical simulation -- under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an $O(1)$ constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large $O(1)$ times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase ( $t^{1/2}$ growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-$N$, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity -- a measure of the complexity of classical simulation -- under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an $O(1)$ constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large $O(1)$ times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase ($t^{1/2}$ growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.

arXiv:2607.01351Jul 1, 20260 citations

Spread complexity in AdS/CFT: connecting quantum information and bulk dynamics

R. McDonald

This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.

Jan 21, 20260 citations

Complexity Growth and the Krylov-Wigner function

Ritam Basu, Anirban Ganguly, Nath, Souparna +1 more

For any state in a $D$ -dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function -- a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large $D$ limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large $D$ . We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity rises gradually for a time of $O(D)$ and then saturates close to its upper bound of $\sqrt{D}$ .For any state in a $D$-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function -- a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large $D$ limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large $D$. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity rises gradually for a time of $O(D)$ and then saturates close to its upper bound of $\sqrt{D}$.

arXiv:2402.13694Feb 21, 20240 citations

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

_3

\sinh^{4Δ}(πt/β)

, while the raw, seed-normalized state negativity saturates, as dictated by the

O(\sqrt{D})

Δ=1

, the Breitenlohner-Freedman saturating dimension in

AdS_3

. This dimensionality is special in that the negativity rate is the product of the proper radial position and momentum,

\mathcal{R} \propto \mathcal{C} P_ρ

SU(1,1)

discrete series.

β\ll 1

. Our main analytic result is simple: to leading order in

β

the state is a Gaussian wavepacket of minimum uncertainty, sitting at rest at the turning point of the Liouville wall. Its Wigner function is therefore positive, and the negativity is

1+δ(β,t)

, where the correction

δ

is smaller than any power of

β

. We show that

δ

is an even function of time, so there is no linear growth at

t=0

, and that once the packet has reflected off the wall its leading-order evolution is a Clifford shear, which cannot change the negativity. We also numerically study the time evolution of the Wigner function and its negativity. We observe that

δ

stays below

10^{-11}

t=0

and through the reflection, then rises slowly over a few tens of

β/π

, and then approaches a late-time value consistent with

δ_\infty(β) \simeq 0.08\, e^{-4.4/β}

for

0.25 \le β\le 1

: the lower the temperature, the larger the late-time negativity. The exact survival amplitude

Z(β+it)/Z(β)

fixes the spread complexity to order

t^4

, and it also fixes the seed-normalised Wigner diagnostic of our earlier work (arXiv:2607.04065, arXiv:2607.17346). We take this as evidence that the length basis is ideally suited for a dual, semi-classical description of the dynamics at genus zero. Beyond genus zero the length basis is overcomplete, and we make no statement about finite

e^{S_0}

We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity

C(t)

, the discrete Wigner negativity

N(t)

, and the normalized negativity

χ(t)=N(t)/|S(t)|

, where

S(t)

is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable

SL(2,R)

Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients

b_n=α\sqrt{n(n+2h-1)}

. To introduce chaos we add a GUE random matrix to the integrable chain,

H(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}

, where

R_0

is its spectral half-width. Either

C(t)

N(t)

alone determines the thermofield temperature, with

R^2\simeq0.999

. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears:

χ

predicts

C

better than

C

predicts

χ

. This gap grows from

+0.33

+0.77

, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and

χ(t)

carries it while

C(t)

can still be recovered from

χ(t)

. We derive a lower bound on this gap, set by the fraction of the variance of

χ

that comes from

S

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-

N

, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity -- a measure of the complexity of classical simulation -- under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an

O(1)

constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as

t^{1/2}

at large

O(1)

times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase (

t^{1/2}

growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.

For any state in a

D

-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function -- a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large

D

limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large

D

. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity rises gradually for a time of

O(D)

and then saturates close to its upper bound of

\sqrt{D}