An Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal Entanglement Structure Neil Dowling1and Kavan Modi1 2

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An Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal

Entanglement Structure

Neil Dowling1, ∗and Kavan Modi1, 2, †

1School of Physics & Astronomy, Monash University, Clayton, VIC 3800, Australia

2Centre for Quantum Technology, Transport for New South Wales, Sydney, NSW 2000, Australia

(Dated: February 7, 2024)

Chaotic systems are highly sensitive to a small perturbation, and are ubiquitous throughout

biological sciences, physical sciences and even social sciences. Taking this as the underlying principle,

we construct an operational notion for quantum chaos. Namely, we demand that the future state of

a many-body, isolated quantum system is sensitive to past multitime operations on a small subpart

of that system. By ‘sensitive’, we mean that the resultant states from two diﬀerent perturbations

cannot easily be transformed into each other. That is, the pertinent quantity is the complexity of

the eﬀect of the perturbation within the ﬁnal state. From this intuitive metric, which we call the

Butterﬂy Flutter Fidelity, we use the language of multitime quantum processes to identify a series of

operational conditions on chaos, in particular the scaling of the spatiotemporal entanglement. Our

criteria already contain the routine notions, as well as the well-known diagnostics for quantum chaos.

This includes the Peres-Loschmidt Echo, Dynamical Entropy, Tripartite Mutual Information, and

Local-Operator Entanglement. We hence present a uniﬁed framework for these existing diagnostics

within a single structure. We also go on to quantify how several mechanisms lead to quantum

chaos, such as evolution generated from random circuits. Our work paves the way to systematically

study many-body dynamical phenomena like Many-Body Localization, measurement-induced phase

transitions, and Floquet dynamics.

I. INTRODUCTION

Chaos as a principle is rather direct; a butterﬂy ﬂutters

its wings, which leads to an eﬀect much bigger than itself.

In other words, something small leading to a very big

eﬀect. This eﬀect arises in a vast array of ﬁelds, from

economics and ecology to meteorology and astronomy,

spanning disciplines and spatiotemporal scales.

Chaos at the microscale, on the other hand, is an ex-

ception. Quantum chaos is not well understood and lacks

a universally accepted classiﬁcation. There is a vast web

of, often inconsistent, quantum chaos diagnostics in the

literature [1], which leads to a muddy picture of what this

concept actually means. In contrast, classically chaos is

a relatively complete framework. If one perturbs the ini-

tial conditions of a chaotic dynamical system, they see

an exponential deviation of trajectories in phase space,

quantiﬁed by a Lyapunov exponent. Trying to naively

extend this to quantum Hilbert space immediately falls

short of a meaningful notion of chaos, as the unitarity

of isolated quantum dynamics leads to a preservation of

ﬁdelity with time. How then, can there possibly be non-

linear eﬀects resulting from the linearity of Schrödinger’s

equation? We will see that the structure of entanglement

holds the key to this conundrum.

∗neil.dowling@monash.edu

†kavan.modi@monash.edu

Yet, much eﬀort has been made to understand quan-

tum chaos primarily as the cause of classical chaos [2–5],

to identify the properties that an underlying quantum

system requires in order to exhibit chaos in its semiclas-

sical limit. An example of this is the empirical connection

between random matrix theory and the Hamiltonians of

classically chaotic systems [2]. Recently, with experimen-

tal access to complex many-body quantum systems with

no meaningful classical limit, and given progress in re-

lated problems such as the black hole information para-

dox [6,7] and the quantum foundations of statistical me-

chanics [8–10], quantum chaos as a research program has

seen renewed interest across a range of research commu-

nities. In this context, a complete structure of quantum

chaos, independent of any classical limit, is highly desir-

able but remains absent.

In this work we approach quantum chaos from an oper-

ational, and theory agnostic, principle: Chaos is a deter-

ministic phenomenon, where the future state has a strong

sensitivity to a local perturbation in the past. For quan-

tum processes the key ingredient will turn out to be spa-

tiotemporal entanglement. To get there, we ﬁrst identify

the underlying deﬁnition of chaos as a starting point, and

build a quantum butterﬂy ﬂutter process from this fun-

damental principle. With this, we construct a genuinely

quantum measure for chaos, based solely on this state-

ment, which we term the Butterﬂy Flutter Fidelity. This

relies on the intuition that it is the complexity induced by

a perturbation in the resultant future pure state, rather

than just orthogonality, that dictates a chaotic eﬀect. We

adapt this principle into the theory of quantum processes

arXiv:2210.14926v4 [quant-ph] 6 Feb 2024

and exploit their multitime structures. Namely, we use

a tool from quantum information theory – process-state

duality – to determine a hierarchy of necessary conditions

on meaningful notions of chaos in many-body systems.

These conditions culminate into the novel, strong metric

of the Butterﬂy Flutter Fidelity.

Fig. 1breaks up the problem of quantum chaos into

three broad components, laying out a review of the land-

scape of this multidisciplinary ﬁeld and contextualizing

our results. Panel (a) represents the mechanisms by

which quantum chaos arises. Our contribution, depicted

in Panel (b), is to identify a strong, operational criterion

for quantum chaos through sensitivity in a future state,

to the spatiotemporal quantum entanglement of the cor-

responding process. We propose that this intuitive met-

ric bridges the gap between the mechanisms in Panel (a)

and the signatures for chaos depicted in Panel (c). We

provide explicit connections between several elements of

these panels in this work, whose details we outline below.

Speciﬁcally, aﬃrming the validity of our approach, we

show that our criterion is stronger than and encom-

passes existing dynamical signatures of chaos. We show

this explicitly for the Peres-Loschmidt Echo, Dynami-

cal entropy, Tripartite Mutual Information, and Local-

Operator Entanglement.[20]That is, we identify the un-

derlying structure leading to characteristic chaotic be-

havior of each of these popular chaos diagnostics. We

oﬀer a clear hierarchy of conditions of a chaotic eﬀect,

due to a ‘butterﬂy ﬂutter’, unifying a range of (appar-

ently) inconsistent diagnostics.

Next, we show that there are several known mecha-

nisms for quantum processes that lead to quantum chaos.

In particular, we show that both Haar random unitary

dynamics and random circuit dynamics – which lead to

approximate t−design states – are highly likely to gen-

erate processes which satisfy our operational criterion

for quantum chaos. Our results also open the possibil-

ity of systematically studying other internal mechanisms

thought to generate quantum chaos, e.g. Wigner-Dyson

statistics [2], or the Eigenstate Thermalization Hypoth-

esis (ETH) [21–23].

Finally, Our approach is diﬀerent from previous works

that have usually relied on averages over Haar and/or

thermal ensembles to draw connections between some

previous signatures for quantum chaos [24–26]. We work

solely within a deterministic, pure-state setting, identi-

fying a series of conditions which stem from a sensitivity

to past, local operations, without any need to average

over operators or dynamics. Moreover, other metrics for

quantum chaos also start from the notion of a kind of a

butterﬂy eﬀect, such as the out-of-time order correlator

(OTOC) [27]. However, our sense of this intuitive idea is

diﬀerent, and does not necessarily suﬀer the same short-

falls as e.g. the OTOC which decays quickly even for

some integrable systems [28–30].

Summary of Main Result

We ﬁrst give an informal explanation of the main in-

novation of this work. We use a simpliﬁed formalism and

setup in order to convey the main ideas, with a more

detailed exposition to be given later.

Consider an isolated quantum system where a sequence

of kunitaries Axiare applied on a local subspace S, such

that the global system is deﬁned on the Hilbert space

HS⊗HE. Later we will call this sequence a butterﬂy

ﬂutter, and allow it to consist of an arbitrary sequence of

rank-one instruments (Def. 1). The outgoing state after

this protocol is

ΥRx=AxkUk⋯Ax2U2Ax1U1ψSE (1)

where Uirepresents global unitary evolution, either Flo-

quet or according to a Hamiltonian for time ti, and where

Axi≡(Axi)S⊗1E. The other choices of notation will

become apparent in the following sections.

Now we similarly introduce a strictly diﬀerent set of

kunitaries, labeled by the list y. We take these uni-

taries to be orthogonal to the ﬁrst choice, in the Hilbert-

Schmidt sense such that tr[A†

xiAyi]=0for all i∈[1, k].

Note that we impose no such constraint on operations

for diﬀerent times, Axicompared to Axjwith i≠j. We

later loosen this condition such that these can be col-

lectively, approximately orthogonal operations. The out-

going state is deﬁned analogously to Eq. (1), with the

same global dynamics Uiand subsystem decomposition

HS⊗HE, but diﬀerent unitary ‘perturbations’. We can

then ask, how much do these two resultant states, ΥRx

and ΥRy, diﬀer?

This question is a direct translation to quantum me-

chanics, of the principle of chaos as a sensitivity pertur-

bation. The task is to deﬁne exactly what we mean by

this sensitivity. As mentioned in the introduction, ﬁdelity

is preserved under unitary evolution. Further, as we dis-

cuss in Section III A and Appendix E, the ﬁdelity cannot

be the full story: most dynamics irrespective of integra-

bility will lead to a small ﬁdelity ΥRxΥRy2. We will

show that this orthogonality translates into an entropic

condition on the underlying process for this perturbation

protocol, namely that a genuinely chaotic system should

necessarily have a volumetrically scaling spatiotemporal

entanglement.

We instead strengthen this by deﬁning a new metric to

compare these states, which we call the Butterﬂy Flutter

Fidelity (Def. 2). This compares how diﬀerent the two

ﬁnal states are in a complexity sense, and measures the

ﬁdelity after what we call a correction unitary V

ζ∶=sup

V∈RΥRxVΥRy2.(2)

This quantity is depicted graphically in Fig. 4(a). Here,

a)b)c)

Quantum ChaosInternal Mechanisms Dynamical Signatures

FIG. 1. A schematic of the causes, structure, and eﬀects of quantum chaos. a) Internal mechanisms of chaos are the intrinsic properties

of the dynamics that lead to chaotic eﬀects. For example, properties of the Hamiltonian such as (i) level spacing statistics and (ii) the

Eigenstate Thermalization Hypothesis (ETH), or properties of the quantum circuit describing the dynamics such as (iii) whether it forms

a unitary design. b) In this work we will identify general quantum butterﬂy ﬂutter protocol, and from this argue that chaos reduces

to a hierarchy of conditions on the process describing the dynamics, including the volume-law spatiotemporal entanglement structure.

This principle forms the stepping stone between causal mechanisms of chaos and observable diagnostics of chaos. We remark that we

only conjecture that level spacing statistics and ETH (panels a.i and a.ii) lead to quantum chaos as formalized in this paper, and that

these relationships form an interesting open question. c) Operational diagnostics for quantum chaos. Some popular probes include (i) The

Peres-Loschmidt Echo, also known as Fidelity Decay or Loschmidt echo, which is the measure of the deviation between states, for evolution

under a perturbed compared to an unperturbed Hamiltonian [11,12]; (ii) The Dynamical Entropy, which quantiﬁes how much information

one gains asymptotically from repeatably measuring a subpart of a quantum system [3,13–15]; and (iii) Local Operator Entanglement,

which measures the complexity of the state representation of a time evolved Heisenberg operator [16–18]. Another example which we

analyze in this work (not shown) is the Tripartite Mutual Information, which measures entanglement properties of a state representation

of a local input space of a channel together with a bipartition of the output space [19].

Ris a restricted set of unitaries on HS⊗HE, which for

now can be considered to be the set of simple (low-depth)

circuits. Intuitively, this measure (2) determines whether

the orthogonality between ΥRxand ΥRyis complex or

not. That is, is the sensitivity stemming from past per-

turbations (local unitaries) easily correctable? Based on

our operational criteria for quantum chaos, we argue that

the dynamics are chaotic if this not easily correctable –

when ζ≈0for an appropriately deﬁned set of corrections

R, and for any choice of butterﬂy ﬂutters. This notion of

chaos then allows us to identify a connection with entan-

glement properties of the underlying process describing

the ‘butterﬂy ﬂutter’ protocol.

For example, one could choose two butterﬂy ﬂutters

as a sequence of kPauli Xgates on a single qubit of a

many-body system at ktimes, and the other to be a series

of identity maps (do nothing). With free, global evolu-

tion occurring between each gate, the Butterﬂy Flutter

Fidelity Eq. (2) would then indicate that the dynamics

is chaotic if the ﬁdelity between the ﬁnal states is small,

ζ≈0, even after trying to align the two ﬁnal states using

any small depth, local circuit V. This quantity is the

main focus of this work.

The rest of the paper is structured as the following: We

present a review of the appropriate tools with which we

need to analyze the Butterﬂy Flutter Fidelity Eq. (2)

in Section II. This predominantly includes the theory

of multitime quantum process [31–33], allowing us to

describe all possible perturbations and resultant eﬀects

within a single quantum state. Then in Section III we

present a set of increasingly stronger, necessary condi-

tions on a dynamical process for which ζ≈0in Eq. (2).

These conditions are all motivated from the principle of

chaos as a sensitivity to perturbation, and start with a

minimal sense of what a large eﬀect could be, stemming

from a past, local perturbation. This main results sec-

tion then culminates in the Butterﬂy Flutter Fidelity, as

the strongest condition in this hierarchy. We conclude

this section by comparing Butterﬂy Flutter Fidelity to

the classical ideas of chaos, and detailing how one could

in-principle measure it in experiment.

In Section IV, we support the proposed conditions by

showing how a range of previous dynamical signatures

of chaos agree with them, as depicted in Fig. 1. We

summarize these connections in Fig. 2, which serves as a

summary of this work and the related work of Ref. [30].

Finally, in Section Vwe discuss mechanisms of chaos that

Volume-law

Spatiotemporal

Entanglement (C2)

Small Buttery

Flutter Fidelity

(C3)

Max entanglement

in B:R splitting (C1)

Linear growth of

Local Operator

Entanglement

Exponentially

decaying

OTOC

Small Peres-

Loschmidt Echo

Non-zero Dynamical

Entropy

Maximally negative

Tripartite Mutual

Information

(i)

(v) (vi)

(ii)

(iii)

(iv)

FIG. 2. A summary of the results of this work, where directed ar-

rows mean implication. The shaded region with pink boxes is the

hierarchy of conditions on quantum chaos as a sensitivity to per-

turbation proposed in this work, (C1)-(C3). (i) Volume-law spa-

tiotemporal entanglement of Υis strictly stronger than maximum

entanglement in the single bipartition B∶R. (ii) A small Butterﬂy

Flutter Fidelity (Def. 2) necessarily implies the volume-law spa-

tiotemporal entanglement of Υ(Prop. 3), with equivalence when

the initial state is area-law (Prop. 4). (iii) The (trotterized) Peres-

Loschmidt Echo constitutes the particular case of an asymptotically

many-time, weak unitary butterﬂy ﬂutter (Section IV A), while an

extensive Dynamical entropy is equivalent to an extensive entan-

glement scaling in the splitting B∶R(Prop. 1and Section IV B).

(iv) For a single-time butterﬂy ﬂutter, volume-law spatiotemporal

entanglement directly implies a (near) maximally negative Tripar-

tite mutual information of the corresponding channel (Prop. 7). (v)

For a single-time butterﬂy ﬂutter, if the Butterﬂy Flutter Fidelity

is small for any initial state, then for an operator entanglement

complexity measure the Local Operator Entanglement grows lin-

early with time (Thm. 8). (vi) If the Local Operator Entanglement

grows linearly with time, then general OTOCs necessarily decay ex-

ponentially [30].

lead to the operational eﬀects we propose. In particular,

we prove that random dynamics – both fully Haar ran-

dom and those generated by unitary designs – typically

lead to chaos.

II. TOOLS: MULTITIME QUANTUM

PROCESSES AND SPATIOTEMPORAL

ENTANGLEMENT

Many of the results of this work rely on the application

of ideas from entanglement theory to multitime quantum

processes, in order to interpret the overarching problem

of chaos in isolated many-body systems. We here give

only an overview of the relevant facets of this topic, and

refer the reader Appendix Afor more information, and

to Refs. [33,34] for a more complete introduction to the

process tensor framework.

Consider a ﬁnite dimensional quantum system. A

quantum process is a quantum dynamical system under

the eﬀect of multitime interventions on some accessible

local space HS. These interventions are described by

instruments, which trace non-increasing quantum maps.

The dynamics between interventions can then be dilated

to a system-environment HS⊗HE, such that the total

isolated state on HS⊗HEevolves unitarily on this ex-

tended space. A k−step process tensor is the mathemat-

ical description of a such a process, encoding all possible

spatiotemporal correlations in a single object; analogous

to how a density matrix encodes single-time measure-

ments.

In this work we will generally consider rank-one instru-

ments, such as unitary matrices and projective measure-

ments (including the outgoing state). In this case, we are

able to write down the full pure state on HS⊗HEat the

end of this process,

ψ′

SE =UkAxkUk−1. . . U1Ax1ψSE ,(3)

where we have rewritten this as the conditional state of a

subpart of process Υ, and will explain exactly what this

means below. Axican be arbitrary norm non-increasing

operators, with ∑xiA†

xiAxi=1. That is, anything that

maps pure states to (possibly subnormalized) pure states.

This includes e.g. unitary operators or projective mea-

surements. We stress that Axiare considered to act lo-

cally on HS, such that Axi≡A(S)

xi⊗1(E). As every-

thing is pure here, there is no need to consider super-

operators or density matrices, and left multiplication by

matrices is a suﬃcient description (see Appendix Afor

the mixed-state extension of this). ΥRxcould be a

sub-normalized pure state, for example if the instruments

chosen to be a series of projective measurements

ψ′

SE =pxΥRx.(4)

Here, Axi=xixi,pxis the probability of observ-

ing this outcome, and where we have neglected the (un-

observable) global phase. We will usually consider the

(normalized) conditional state ΥRxwhen investigating

chaotic eﬀects, as we will be concerned with the resultant

state rather than the probability that it is produced.

Rather than choosing a particular instrument Axifor

each intervention, one can instead feed in half of a max-

imally entangled state from an ancilla space, as shown

in Fig. 3. This results in the pure state Υ, encoding

both any interventions on the multitime space in the past

which we call HB, and the ﬁnal pure state on the global,

isolated system, on the space HR. This is the generalized

Choi–Jamiołkowski Isomorphism (CJI) [31,32], shown in

Fig. 3. Alternatively to this ancilla-based construction,

the pure process tensor can be deﬁned succinctly as

Υ∶=Uk∗ ⋅ ⋅ ⋅ ∗ U1∗ψSE (t1),(5)

where ∗is the Link product, corresponding to compo-

sition of maps within the Choi representation [35], and

is essentially a matrix product on the HEspace, and a

tensor product on the HSspace. A ket of a rank-one

FIG. 3. Tensor network diagram of the protocol producing the

Choi state of a pure process tensor through the generalized Choi-

Jamiołkowski isomorphism [31,36]. This means that input indices

are put on equal footing with output indices, through appending

a maximally entangled ancilla system ϕ+at each time, and in-

serting half of this state into the process. The ﬁnal output state

of this protocol encodes all multitime spatiotemporal correlations:

a pure process tensor. A multitime expectation value can then

be computed in this representation by ﬁnding the Hilbert Schmidt

inner product between this (normalized) Choi state and the (su-

pernormalized) Choi state of a multitime instrument. The system

HSdenotes the singletime space where instruments act, and the

environment HEthe dilated space such that all dynamics are uni-

tary. Here the independent Hilbert spaces are labeled such that

(ℓ)i((ℓ)o) is the input (output) system space HSat time tℓ, show-

ing that the ﬁnal output Υcorresponds to a (2k+2)−body pure

quantum state.

instrument Acorresponds to the single-time Choi state

A∶=(A⊗1)ϕ+,(6)

by the usual single-time CJI: channel-state duality [34].

Here, we have gathered the multitime Hilbert space

where the full multitime instruments act on a space with

the single label,

HB≡Hio

S(tk−1)⋅ ⋅ ⋅ ⊗ Hio

S(t1)⊗Hio

S(t0)(7)

called the ‘butterﬂy’ space HB, where Hio

S(tj)≡Hi

S(tj)⊗

S(tj).Hirepresents the input space to the process,

while Horepresents the output. The ‘remainder’ space

HR– the full ﬁnal state on the system plus environment

at the end of the protocol, where the ‘butterﬂy’ does not

act – is

HR≡Ho

S(tk)⊗Ho

E(tk).(8)

All of these are clearly labeled in Fig. 3. It will become

apparent in the following section why we name these

spaces as such.

From Eq. (5) we can determine the outgoing (possibly

sub-normalized) state Eq. (4) from projections on this

state,

ψ′

SE =xΥ.(9)

For independent instruments at each intervention time,

we have that

x∶=xk⊗ ⋅ ⋅ ⋅ ⊗ x1,(10)

where each single-time state is constructed as in Eq. (6).

Alternatively, one could trace over the ﬁnal state on HR,

and the reduced state on HB,ΥB, is the process ten-

sor [31–33], as we describe in Appendix A.

The key point here is that through the CJI we have

reduced all possible correlations of a dynamical multi-

time experiment to a single quantum state, Υ. This

means that all the machinery from many-body physics is

available to describe multitime eﬀects. A subtle diﬀer-

ence from the single-time case is that the normalization

of these Choi states do not exactly correspond to the

normalization of states and projections. Instruments are

taken to be supernormalized, while processes have unit

normalization and so constitute valid quantum states

ΥΥ=1,and xx≤d2k

S,(11)

where the inequality is saturated for deterministic instru-

ments: CPTP maps. This normalization ensures that

one gets well deﬁned probabilities in Eq. (4).

Therefore, dynamical properties of a process such as:

non-Markovianity [32,33,37–39], temporal correlation

function equilibration [40,41], whether its measurement

statistics can be described by a classical stochastic pro-

cess [42–44], multipartite entanglement in time [45], and

other many-time properties [46], can all be clearly de-

ﬁned in terms of properties of the quantum state Υ.

However, the spatiotemporal entanglement structure of

this multitime object is largely unexplored, and we will

show that this has vast implications for understanding

quantum chaotic versus regular dynamics.

Any pure quantum state ψAB on HA⊗HBcan be

decomposed across any bipartition A∶Bvia the Schmidt

decomposition,

ψAB =



i=1

λiαiAβiB,(12)

where αiαj=δij =βiβj.χis called the bond di-

mension or Schmidt rank, dictating intuitively how much

of the subsystems are entangled with each other. The

bond dimension is equal to one if and only if the state is

separable across A∶B.

Using this decomposition (12), one can successively in-

crease the size of the subsystem HA, and determine how

the bond dimension scales. We will deal with one spatial

dimension systems when discussing spatiotemporal en-

tanglement in this work, as characteristic entanglement

scaling depends on the underlying geometry [47]. Our re-

sults should generalize in a straightforward way to higher

spatial dimensions. If χis bounded by min{dA, D}for

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AnOperationalMetricforQuantumChaosandtheCorrespondingSpatiotemporalEntanglementStructureNeilDowling1,∗andKavanModi1,2,†1SchoolofPhysics&Astronomy,MonashUniversity,Clayton,VIC3800,Australia2CentreforQuantumTechnology,TransportforNewSouthWales,Sydney,NSW2000,Australia(Dated:February7,2024)Chaoticsyste...

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An Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal Entanglement Structure Neil Dowling1and Kavan Modi1 2

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