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arXiv:2607.01351

Wigner negativity in Krylov space and emergent semiclassicality

Vijay Balasubramanian, Paweł Caputa, Onkar Parrikar, Vivek Singh

Published Jul 1, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2607.01351

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For any state in a DDD-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 DDD 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 DDD. 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)O(D)O(D) and then saturates close to its upper bound of D\sqrt{D}D​.
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.
In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space functions corresponding to the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time evolved state based on harmonic expansion of the time-dependent Wigner function.
A bstract 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 broadly shows three phases: it rises gradually for a time of O(D) O\left(\sqrt{D}\right) O(D​) O D , then hits a sharp ramp and finally saturates close to its upper bound of D \sqrt{D} D​ D .
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≪1β\ll 1β≪1. Our main analytic result is simple: to leading order in βββ 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)1+δ(β,t)1+δ(β,t), where the correction δδδ is smaller than any power of βββ. We show that δδδ is an even function of time, so there is no linear growth at t=0t=0t=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 δδδ stays below 10−1110^{-11}10−11 at t=0t=0t=0 and through the reflection, then rises slowly over a few tens of β/πβ/πβ/π, and then approaches a late-time value consistent with δ∞(β)≃0.08 e−4.4/βδ_\infty(β) \simeq 0.08\, e^{-4.4/β}δ∞​(β)≃0.08e−4.4/β for 0.25≤β≤10.25 \le β\le 10.25≤β≤1: the lower the temperature, the larger the late-time negativity. The exact survival amplitude Z(β+it)/Z(β)Z(β+it)/Z(β)Z(β+it)/Z(β) fixes the spread complexity to order t4t^4t4, 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 eS0e^{S_0}eS0​.
The growth of simple operators is essential for the emergence of chaotic dynamics and quantum thermalization. Recent studies have proposed different measures, including the out-of-time-order correlator and Krylov complexity. It is established that the out-of-time-order correlator serves as the signature of quantum many-body chaos, while the Krylov complexity provides its upper bound. However, there exist nonchaotic systems in which Krylov complexity grows exponentially, indicating that the Krylov complexity itself is not a witness of many-body chaos. In this Letter, we introduce the missing ingredient called the Krylov metric K_{mn}, which probes the size of the Krylov basis. We propose that the universal criteria for fast scramblers include (i) the exponential growth of Krylov complexity, (ii) the diagonal elements K_{nn}∼n^{h} with h∈(0,1], and (iii) the negligibility of off-diagonal elements K_{mn} with m≠n. We further show that h=ϰ/2α is a ratio between the quantum Lyapunov exponent ϰ and the Krylov exponent α. This proposal is supported by both generic arguments and explicit examples, including solvable Sachdev-Ye-Kitaev models, Luttinger liquids, and many-body localized systems. Our results provide a refined understanding of how chaotic dynamics emerge from the Krylov space perspective.