On the Relative Distance of Entangled Systems in Emergent Spacetime Scenarios

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On the Relative Distance of Entangled Systems in

Emergent Spacetime Scenarios

Guilherme Franzmann,1, 2, ∗Sebastian M. D. Jovancic,3, †and Matthew Lawson4, ‡

1Nordita, KTH Royal Institute of Technology and Stockholm University,

Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

2Basic Research Community for Physics e.V., Mariannenstraße 89, Leipzig, Germany

3KTH Royal Institute of Technology and Stockholm University,

Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

4Savantic AB, Rosenlundsgatan 52, Stockholm, Sweden

(Dated: October 27, 2022)

Spacetime emergence from entanglement proposes an alternative to quantizing gravity and typ-

ically derives a notion of distance based on the amount of mutual information shared across sub-

systems. Albeit promising, this program still faces challenges to describe simple physical systems,

such as a maximally entangled Bell pair that is taken apart while preserving its entanglement. We

propose a solution to this problem: a reminder that quantum systems can have multiple sectors of

independent degrees of freedom, and that each sector can be entangled. Thus, while one sector can

decohere, and decrease the amount of total mutual information within the system, another sector,

e.g. spin, can remain entangled. We illustrate this with a toy model, showing that only within

the particles’ momentum uncertainty there can be considerably more entanglement than in the spin

sector for a single Bell pair. We ﬁnish by introducing some considerations about how spacetime

could be tested in the lab in the future.

I. INTRODUCTION

“Of course spacetime cannot be emergent from entanglement, as

we can place a Bell pair arbitrarily far apart from each other.”

– During a discussion session at QISS ’22 Conference.

A century has gone by since Planck discovered the

quantum-mechanical nature of our Universe. Since then,

the three fundamental interactions that mainly govern

the microscopic scales have been quantized. Together,

they compose the Standard Model of Particle Physics -

the most accurate theory ever devised by us. And gravity

does not ﬁt in it. In spite of having its ﬁrst mechanical

description introduced by Newton centuries before, the

program of quantizing gravity for arbitrarily high ener-

gies remains incomplete.

Similarly to Yukawa’s theory to describe the nuclear

force between nucleons mediated by pions, quantum gen-

eral relativity is a low-energy eﬀective ﬁeld theory [1]. As

Yukawa’s theory was superseded by quantum chromo-

dynamics, where more fundamental degrees of freedom

were introduced, the same is expected to happen with

gravity. Thus, the quantized degrees of freedom in quan-

tum general relativity, namely the spacetime ﬂuctuations

parametrized by the metric ﬁeld, will no longer be fun-

damental in the ﬁnal quantum gravity theory.

Nonetheless, most attempts to reconcile gravity with

quantum mechanics have insisted on keeping these de-

grees of freedom one way or another. String the-

ory [2], the most prevalent approach to quantum grav-

ity, introduces other spacetime degrees of freedom that

∗guilherme.franzmann@su.se

†sebjov@kth.se

‡mmlawson@ucdavis.edu

parametrize the strings’ worldsheet. Meanwhile, some

other approaches, such as loop quantum gravity [3], de-

velop new ways of quantizing the same degrees of freedom

from general relativity. Still, the single most signiﬁcant

insight since this program started came from Maldacena

in 1997 [4]. By introducing the anti-de Sitter/Conformal

Field Theory (AdS/CFT) correspondence, he showed

that a gravitational theory could be dual to a lower-

dimensional quantum mechanical theory without gravity.

Since then, evidence about the emergent nature of space-

time has piled up.

The research on spacetime emergence follows a long

thread that started with ﬁndings by Bekenstein [5], who

explored the thermodynamic properties of black holes

and related the entropy of a black hole to its surface

area. Later, Hawking [6] completed the thermodynami-

cal description by showing that black holes indeed emit

thermal radiation. Two decades later, the holographic

principle was introduced by ’t Hooft [7], stating that the

boundary of a bulk region of space encodes information

about its interior. Meanwhile, Jacobson [8] showed that

Einstein’s equation of General Relativity could be seen as

an equation of state resulting from the thermodynamic

limit of local Rindler causal horizons. Despite these de-

velopments, theories without gravity ab initio were still

lacking until Maldacena [4] proposed the AdS/CFT cor-

respondence. It established a holographic relationship

between AdS space and conformal ﬁeld theories, also re-

ferred to as the gauge/gravity duality. Then, Ryu &

Takayanagi [9] showed that in an AdS space there is a di-

rect relationship between the entanglement entropy asso-

ciated with bulk regions separated by a boundary surface

where a conformal ﬁeld theory is deﬁned and the area of

this boundary. Finally, van Raamsdonk [10] extended

this relationship by suggesting that one could relate the

arXiv:2210.14875v1 [quant-ph] 26 Oct 2022

boundary surface between bulk regions and the distance

between them in an AdS/CFT setting.

More recently, Cao et al. [11,12] have combined much

of this thread into a new research program. It starts from

a purely quantum-mechanical framework and its entan-

glement structure and derives classical spatial geometry

satisfying Einstein’s equation (as illustrated in Fig. 1).

In short, the program suggests that there is a mapping

between the mutual information of quantum sub-systems

and the classical geometry connecting them, giving rise

to an emergent space purely deﬁned in terms of the quan-

tum information contained in the system. This research

program builds up on several previous works [e.g 13–17].

Although this program still remains incomplete, we can

argue that space, and perhaps time, will no longer be

fundamental within this framework upon its completion.

Nonetheless, even at this stage the program faces some

challenges, as illustrated by the quote shown above. The

mutual information of quantum sub-systems relates to

the classical geometry connecting them such that the dis-

tance between sub-systems in the emergent geometry is

a monotonically decreasing function of the mutual infor-

mation (see Fig. 1). The intuition here is that if quan-

tum sub-systems are more entangled, then their locations

in the emergent geometry will be closer; if they are not

entangled, they will be as distant as allowed in the emer-

gent space. The problem is that, for instance, a maxi-

mally entangled Bell pair will always have zero distance

between its sub-components in the emergent geometry,

even though we know that its sub-components can be

arbitrarily separated. In fact, this resembles the ER =

EPR conjecture introduced in [18]. The conjecture re-

lates maximally entangled systems at the quantum level

with wormhole-like geometries at the spacetime level.

Thus, it is expected that despite the sub-components of

a Bell pair be arbitrarily far away, they would still be

connected by wormholes so that their distance is zero.

Nevertheless, the conjecture is in tension with the fact

that local observers do not see the wormhole geometry

in their labs.

Thus, our objective is to show how one can recover

the non-vanishing relative distances connecting the pair

in the emergent space. We will show how the mutual

information changes due to non-local entanglement per-

turbations and how that aﬀects the emergent spacetime

geometry. Hence, the typical notion of relative distances

across entangled systems can be recovered instead of sim-

ply recovering wormholes as suggested in [18].

The paper is divided as follows: Sec. II brieﬂy reviews

the spacetime emergence program introduced in [11,12].

In Sec. III, we model a Bell state in ever-increasing levels

of description and show one way to understand how its

parts can be arbitrarily separated. Finally, in Sec. IV

we discuss for the ﬁrst time some ideas on how spacetime

emergence can be tested in the lab, and then we conclude

with discussion in Sec. V.

𝒰a

𝒰2

𝒰b

ℳ

𝒰1

Spacetime

Emergence

ℋ

ℋ1

ℋ2

ℋa

ℋb

FIG. 1. Representation of how space emerges from quantum

mechanics. In the quantum Hilbert space H, diﬀerent quan-

tum sub-systems, Hp, such that H=NpHp, are connected

by their mutual information, which is larger for darker red

lines. In the emergent classical manifold M, the relative dis-

tance of these sub-systems, Up, decays monotonically w.r.t.

the amount of mutual information they share.

II. SPACETIME EMERGENCE AND RELATIVE

DISTANCES

The general strategy to reconstruct space from the

Hilbert space introduced in [11,12] is:

1. Start with a Hilbert space H;

2. Decompose the Hilbert space into a large numbers

of factors H=NpHp(see Appendix A 1). Every

factor is ﬁnite-dimensional1;

3. Consider only “redundancy-constrained” (RC)

states, |ψiRC, that are a generalization of states

in which the entropy of a region obeys an area law

[21];

4. Consider the weighted graph, G, with vertices rep-

resented by factors, Ap,and edges represented by

mutual information among them (see Appendix

A 2), I(Ap:Aq) = S(Ap) + S(Aq)−S(Ap:Aq).

Then, deﬁne a metric on the graph connecting the

factors;

5. Construct a metric graph, ˜

G, with smooth, ﬂat ge-

ometries by mapping G→˜

1In [19,20] is argued why we should expect that the Hilbert

spaces associated with local regions of spacetime should be ﬁnite

dimensional. Note that this implies a radical departure from

QFT, where every point in spacetime gives rise to an inﬁnite-

dimensional Hilbert space for each ﬁeld deﬁned in it.

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摘要：

OntheRelativeDistanceofEntangledSystemsinEmergentSpacetimeScenariosGuilhermeFranzmann,1,2,SebastianM.D.Jovancic,3,yandMatthewLawson4,z1Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,HannesAlfvénsväg12,SE-10691Stockholm,Sweden2BasicResearchCommunityforPhysicse.V.,Mariannenstraÿe89,Leipz...

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