Random Unitaries Robustness and Complexity of Entanglement

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Random Unitaries, Robustness, and Complexity of

Entanglement

J. Odavić, G. Torre, N. Mijić, D. Davidović, F. Franchini, and S. M. Giampaolo

Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia

September 13, 2023

It is widely accepted that the dynamic of en-

tanglement in the presence of a generic circuit

can be predicted by the knowledge of the sta-

tistical properties of the entanglement spec-

trum. We tested this assumption by apply-

ing a Metropolis-like entanglement cooling al-

gorithm generated by diﬀerent sets of local

gates, on states sharing the same statistic. We

employ the ground states of a unique model,

namely the one-dimensional Ising chain with

a transverse ﬁeld, but belonging to diﬀerent

macroscopic phases such as the paramagnetic,

the magnetically ordered, and the topological

frustrated ones. Quite surprisingly, we observe

that the entanglement dynamics are strongly

dependent not just on the diﬀerent sets of

gates but also on the phase, indicating that

diﬀerent phases can possess diﬀerent types of

entanglement (which we characterize as purely

local, GHZ-like, and W-state-like) with diﬀer-

ent degree of resilience against the cooling pro-

cess. Moreover, in some circumstances, we ob-

serve a scrambling eﬀect by the algorithm that

produces a Wigner-Dyson entanglement spec-

trum statistics on a state that does not obey

a volume law for the entanglement entropy.

Our work highlights the fact that the knowl-

edge of the entanglement spectrum alone is not

suﬃcient to determine its dynamics, thereby

demonstrating its incompleteness as a charac-

terization tool. Moreover, it shows a subtle

interplay between locality and non-local con-

straints.

1 Introduction

Entanglement is the most distinctive mark of quan-

tum mechanics [1,2] and an essential resource for

many technological devices currently in develop-

ment [3–5]. Therefore, it is easy to understand the

J. Odavić: jodavic@irb.hr

G. Torre: gianpaolo.torre@irb.hr

N. Mijić: nenad.mijic@irb.hr

D. Davidović: davor.davidovic@irb.hr

F. Franchini: fabio.franchini@irb.hr

S. M. Giampaolo: salvatore.marco.giampaolo@irb.hr

growing interest in characterizing entanglement, es-

pecially in quantum many-body systems, as they rep-

resent the platform on top of which such devices are

to be implemented. Characterization of entanglement

is paramount in this context, as it is not just the sheer

amount of entanglement which plays a crucial role in

quantum applications. This is because some entan-

gled states can be described eﬃciently also through

classical resources. To provide an example, it is known

that quantum circuits starting from factorized states

diagonal in the computational basis and made by

gates from Cliﬀord’s group [6], can be eﬃciently sim-

ulated on a classical computer despite the amount

of entanglement of the output state [7,8]. The re-

sulting states are known as “stabilizer states”. How-

ever, adding gates outside Cliﬀord’s group, such as

T-gates, makes it impossible to simulate the circuits

eﬃciently on a classic computer [9–11]. The diﬀerence

between the two cases can be characterized by looking

at the statistical properties of the entanglement spec-

tra. While in the second case, the output generally

develops a Wigner-Dyson distribution in the entangle-

ment spacing statistics, in the ﬁrst one, the Poisson

distribution is always obtained [9,10,12]. This dif-

ference plays a key role in the theory of quantum in-

formation, since only circuits doped with T-gates are

capable of universal computation, as they can reach

any state in the Hilbert space independently of the

initial state [13,15].

An alternative way of analyzing this diﬀerence is

to use the concept of stochastic irreversibility (or ro-

bustness) of entanglement. The idea behind this ap-

proach is to obtain information on the entanglement

structure of a state by observing its evolution under

the action of an entanglement cooling algorithm. At

its core, this algorithm is the usual Metropolis Monte

Carlo protocol, with the cost function played not by

the energy but by an entanglement measure. A trans-

formation, chosen from a predeﬁned set, is applied to

the initial state, and the entanglement value of the

new state is determined. The new state is accepted

(retained) if the cost function has decreased and with

a certain probability otherwise. While states with a

Poissonian statistic of the entanglement spectrum are

not very resistant to this approach, i.e. after a few

steps, the total amount of entanglement in the sys-

tem tends to vanish, the cooling algorithm proves to

Accepted in Quantum 2023-09-07, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.13495v3 [quant-ph] 12 Sep 2023

be almost ineﬀective in states where this statistic fol-

lows a Wigner-Dyson distribution [9,10]. The dif-

ferent robustness of entanglement against a cooling

algorithm has been linked to a concept of “complex-

ity”, coherently with the usual picture that a Wigner-

Dyson distribution, stemming from the existence of

strong correlations between the entanglement eigen-

values, indicate a higher complexity as well as higher

robustness [14,15]. This diﬀerence in robustness has

proved to be extremely useful for distinguishing be-

tween the diﬀerent dynamical phases present in quan-

tum many-body systems [14] and falls within the re-

cent interest of the quantum many-body community

in random quantum circuits [16,17].

In the present paper, we test how strong is the re-

lationship between stochastic irreversibility and the

entanglement spectrum statistic, by placing a partic-

ular emphasis on the choice of the initial states. In

previous works (e.g. see Ref. [14]), the states to which

the cooling algorithm was applied were randomly gen-

erated. On the contrary, our starting states are the

ground states of the one-dimensional quantum Ising

model in its diﬀerent macroscopic phases. As we will

show, the entanglement spectrum of all these states

follows a Poisson distribution for the level-spacing

statistics, once some degeneracies are properly dealt

with. Therefore, the action of the cooling algorithm

could be expected to be independent of the initial

state phase. Instead, quite surprisingly, our results

show a very diﬀerent picture.

The most peculiar behavior we observe is associ-

ated with the ground states of models in a topologi-

cally frustrated phase [20–25]. They are obtained by

imposing frustrated boundary conditions (that is, pe-

riodic boundary conditions with an odd number of

sites) on systems with antiferromagnetic interactions.

The resulting ground states can be largely character-

ized as a linear superposition of single-dressed kink

states [26,27] (topological solitons) and this repre-

sentation allows us to describe, both qualitatively and

quantitatively, the behavior of various physical quan-

tities even in the presence of integrability breaking

terms [26–29]. In these states, after the cooling algo-

rithm, we observe a stabilization of the entanglement

to a reduced baseline value (ﬁnite, but strongly de-

pendent on the size of the system), and any further

reduction appears to be statistically unlikely.

In the present work, we consider two sets of opera-

tions within the cooling algorithm. They are made of

one- and two-body gates, with the latter acting only

on neighboring spins. In this sense, our algorithm can

be considered made of local gates. In the ﬁrst set,

we include only operations preserving the parity sym-

metry of the Hamiltonian which, as a consequence,

cannot explore every state in the Hilbert space, thus

generating a violation of ergodicity. On the contrary,

in the second, the set of operations is extended to a

complete set to ensure, at least in principle, access to

any state.

While the end values of the entanglement obtained

by the cooling algorithm starting from states with

topological frustration are not qualitatively inﬂuenced

by the choice of employed gates, this is not true for the

ground states of the paramagnetic phase. On them,

the action of the cooling algorithm is practically neg-

ligible if the set of operations is limited to the one in

which elements commute with the parity of the Hamil-

tonian, while their entanglement is quickly reduced if

the larger set of gates is considered. This dependence

on the gate set also allows distinguishing the para-

magnetic phase from the ferromagnetic one, since on

the latter the cooling algorithm is unable to destroy

most of their entanglement, regardless of the gate set

taken into consideration.

The diﬀerence between the two cases lies in the

presence of entanglement of purely local nature in the

paramagnetic case, while both the ground states in

the ordered and topologically frustrated phases have

long-range quantum correlations [26,30]. Long-range

entanglement is less aﬀected by the action of local

gates, and even less so as the system size increases. On

the other side, when the entanglement is local, local

gates can easily reduce it, although the impossibility

of exploring the whole Hilbert space with a reduced

set of gates can still prevent its complete suppression.

As noted above, the entanglement spectrum statis-

tics of the initial states are always (mostly) Pois-

sonian, and indeed this is also the case at the end

of the cooling if the non-universal gate set is em-

ployed. However, when the universal one is used and

when there is suﬃcient local entanglement to act on,

the ﬁnal states display a Wigner-Dyson (WD) entan-

glement spectrum statistics. Before our work, WD

statistics has always been observed in states with vol-

ume law entanglement, but indeed we can conﬁrm

that this is not the case for our states.

The paper is organized as follows: In Sect. 2we in-

troduced the model used to generate the input states

of our cooling algorithm. Then, in Sect. 3we describe

in detail the cooling algorithm and the diﬀerent sets

of local gates. Afterward, in Sect. 4, we describe the

results obtained, with a particular focus on how the

size of the system, its quantum phase, and the gate set

aﬀects the evolution of the diﬀerent states under the

action of the cooling algorithm. In Sect. 5we draw

our conclusions.

2 The Model

As highlighted in the introduction, our goal is to apply

an entanglement cooling algorithm (see also Sect. 3)

to states that are ground states of the same Hamil-

tonian but in diﬀerent macroscopic phases. We focus

on the ground states of the one-dimensional spin-1/2

transverse ﬁeld Ising model (TFIM), since it is a pro-

totypical example of a many-body system possessing

Accepted in Quantum 2023-09-07, click title to verify. Published under CC-BY 4.0. 2

−5.0−2.5 0.0 2.5 5.0

J/h

0.0

0.5

1.0

1.5

frustrated phase

AFM

PARA

N= 5

N= 9

N= 13

N= 17

N= 21

QPT

−5.0−2.5 0.0 2.5 5.0

0.0

0.1

0.2

0.3

concurrence

Figure 1: Half-chain Rényi-2 entropy of the transverse-ﬁeld

Ising model. The quantum phase transitions (black dashed

lines) separate the paramagnetic phase (|J/h|<1) from the

ferromagnetic (J/h < −1) and the frustrated antiferromag-

netic one (J/h > 1). The excess entanglement in the frus-

trated AFM regime is an increasing function of the system

size, which saturates at the thermodynamic limit [26]. In the

inset, we plot the nearest-neighbor concurrence [33], where

the excess concurrence in the frustrated phase decreases with

the chain length. In the thermodynamic limit, the curves

would be symmetric around the origin.

diﬀerent macroscopic phases. It is deﬁned by the fol-

lowing Hamiltonian,

H=J

l=1

σx

lσx

l+1 −h

l=1

σz

l,(1)

where the σαwith α=x, y, z are the Pauli operators.

Limiting our analysis to systems with periodic bound-

ary conditions (σα

N+1 =σα

1) made by an odd number

of spins (N= 2M+1 with M∈N), the so-called frus-

trated boundary conditions (FBC), the model admits

three distinct phases. When the local ﬁeld dominates

over the interaction term between spins, the system

is in the paramagnetic phase (PARA), characterized

by a gapped spectrum and a vanishing spontaneous

magnetization in the longitudinal directions. On the

contrary, when the interaction term dominates over

the local ﬁeld (|J|>|h|), we can realize two inequiva-

lent phases depending on the sign of the interaction J.

If the interaction is ferromagnetic (FM), i.e. when it

favors parallel alignments (J < 0), the system shows

a gapped magnetically ordered phase, with a ﬁnite

magnetization in the x- direction [31]. On the other

hand, in the case of an antiferromagnetic interaction

(AFM, J > 0) the system is in a gapless topological

frustrated phase [20–25], where the spontaneous mag-

netization is destroyed by the presence of a delocalized

kink excitation. We remark that the TFIM is also in-

tegrable. We do not use this feature in our analysis,

and we do not expect it to inﬂuence our results: to

check this assumption, in Sec. 4we will also add an

integrability breaking term and apply the same algo-

rithm to the resulting ground state.

A way to discriminate among the three model’s

phases is by looking at the various kinds of bipar-

tite entanglement [3,32] that can be quantiﬁed by

the Rényi entropies, deﬁned as,

Sα(ρA) = 1

1−αlog2Tr [ρα

A],(2)

which depends on the parameter α∈[0,1) ∪(1,∞].

In eq. (2),ρA≡TrB|Ψ⟩⟨Ψ|is the reduced density

matrix obtained by tracing out from the ground state

|Ψ⟩all the degrees of freedom of spins that lie outside

the subset A. In the limit α→1+, the Rényi entropies

reduce to the von Neumann entropy [3]: S1(ρA) =

−Pkλklog2λk, where {λk}is the set of eigenvalues

of the reduced density matrix ρA.

In Fig. 1we present the behavior of S2(ρA)as a

function of the ratio J/h for h > 0in the case in

which Ais made by (N−1)/2contiguous spins. For

the sake of simplicity, we refer to the string made by

(N−1)/2contiguous spins as the half chain. Dif-

ferently from the other phases, in the topologically

frustrated one, we observe a relevant dependence of

the entanglement on the size of the chain that can

be explained by taking into account that in such a

phase the ground states are characterized by a delo-

calized excitation that increases the total amount of

entanglement [26].

While the half-chain Rényi-αentanglement entropy

captures the non-local nature of entanglement, the

concurrence [33] measures its local contribution. It is

deﬁned starting from the reduced density matrix ob-

tained by tracing out the degrees of freedom of every

site of the chain but two, which in this case we take

as neighboring sites. In the inset of Fig. 1, we observe

that contrary to what happens to S2, the contribution

of the delocalized excitation in the frustrated phase

decreases as the chain length increases and vanishes

in the thermodynamic limit. As a consequence, for

large systems, the concurrences of the FM and the

AFM phases coincide. However, due to the complex-

ity of the cooling algorithm, our system will always

be well below such a limit. Hence, for the analyzed

cases, the amount of local entanglement of a ground

state in the presence of a topological frustrated sys-

tem will be relatively higher than the one coming from

the FM phase.

Despite the diﬀerences in the entanglement en-

tropies discriminating the various phases of the model,

the statistical properties of the entanglement are al-

most the same. To highlight this fact, we focus on the

probability density function of the consecutive level

spacing ratio. Ordering the set of eigenvalues {λk}of

the reduced density matrix ρAfrom the smallest to

the biggest, the set of the consecutive level spacing

ratios are

rk=λk+1 −λk

λk−λk−1

, k = 2,3,4, ..., 2⌊N/2⌋−1.(3)

The elements of this set are distributed accordingly

with diﬀerent statistics, see Appendix B. The results

obtained are shown in Fig. 2for four diﬀerent pa-

rameter choices. We observe a signiﬁcant deviation

Accepted in Quantum 2023-09-07, click title to verify. Published under CC-BY 4.0. 3

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RandomUnitaries,Robustness,andComplexityofEntanglementJ.Odavić,G.Torre,N.Mijić,D.Davidović,F.Franchini,andS.M.GiampaoloRuđerBoškovićInstitute,Bijeničkacesta54,10000Zagreb,CroatiaSeptember13,2023Itiswidelyacceptedthatthedynamicofen-tanglementinthepresenceofagenericcircuitcanbepredictedbytheknowledgeo...

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