Harvesting entanglement from the gravitational vacuum T. Rick Perche1 2 3 Boris Ragula1and Eduardo Mart ın-Mart ınez1 2 3 1Department of Applied Mathematics University of Waterloo Waterloo Ontario N2L 3G1 Canada

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Harvesting entanglement from the gravitational vacuum

T. Rick Perche,1, 2, 3, ∗Boris Ragula,1, †and Eduardo Mart´ın-Mart´ınez1, 2, 3, ‡

1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

2Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

We study how quantum systems can harvest entanglement from the quantum degrees of free-

dom of the gravitational ﬁeld. Concretely, we describe in detail the interaction of non-relativistic

quantum systems with linearized quantum gravity, and explore how two spacelike separated probes

can harvest entanglement from the gravitational ﬁeld in this context. We provide estimates for the

harvested entanglement for realistic probes which can be experimentally relevant in the future, since

entanglement harvesting experiments can provide evidence for the existence of quantum degrees of

freedom of gravity.

I. INTRODUCTION

Arguably, the most important unsolved question in

theoretical physics is how to give a description for the

gravitational interaction that is consistent with our un-

derstanding of quantum matter. It is well known that the

coupling of classical gravity and quantum matter is theo-

retically inconsistent [1, 2], and as such we either need a

quantum description for gravity or a complete reformula-

tion of quantum theory. However, as of today, there is no

experimental conﬁrmation of quantum behaviour of grav-

ity. Promising experimental setups, such as gravity me-

diated entanglement experiments, have been proposed to

attempt to verify quantum properties of the gravitational

ﬁeld [3–5]. Despite its promise, there is plenty of debate

regarding which quantum properties of the gravitational

interaction can be conﬁrmed by such experiments [6–12].

The core of this debate lies on how to identify genuinely

quantum degrees of freedom for the gravitational ﬁeld.

One of the most remarkable diﬀerences between theo-

ries for classical and quantum ﬁelds is their lowest energy

state. While in a classical ﬁeld theory, the ground state

corresponds to a zero-valued ﬁeld, the vacuum state of

a quantum ﬁeld theory is, arguably, not truly ‘empty’.

This gives rise to non-trivial statistics for local measure-

ments (see e.g. [13–16]). Moreover, it is well known that

the vacuum state of a quantum ﬁeld contains quantum

correlations between diﬀerent spacetime regions. This is

true even if these regions are spacelike separated [17, 18].

This fact is a fundamental feature of quantum ﬁeld the-

ory in both ﬂat and curved spacetimes [19, 20], and is

instrumental to our understanding of phenomena such

as the renormalization of the stress-energy tensor [20],

area laws in quantum ﬁeld theories [21–25] and black

hole evaporation [26–31].

This vacuum entanglement can actually be detected:

localized probes can become entangled with each other

∗trickperche@perimeterinstitute.ca

†bragula@uwaterloo.ca

‡emartinmartinez@uwaterloo.ca

through the interaction with the ﬁeld, even when they are

spacelike separated through their interaction. This is the

idea behind the protocol of entanglement harvesting [32–

34]. In recent years, the protocol has been extensively

studied in many diﬀerent scenarios [35–45], when probes

are coupled to diﬀerent ﬁeld operators [46, 47] and in

diﬀerent spacetimes [48–53].

Entanglement harvesting from spacelike separated re-

gions is only possible from a ﬁeld with quantum degrees

of freedom: a classical ﬁeld cannot contain entanglement

that can be extracted. This fact can be used to decide

whether a ﬁeld is classical or quantum. In fact, it has

been argued that an entanglement harvesting protocol

for the gravitational ﬁeld can be used to witness quan-

tum gravity (see, e.g., [12, 54]). The main goal of this

manuscript is to perform a detailed study of this setup,

and to quantify the theoretical amount of entanglement

that could be extracted from a weak gravitational quan-

tum ﬁeld.

Previous studies of entanglement harvesting which

take gravity degrees of freedom into consideration typi-

cally only couple to a scalar quantum ﬁeld [55, 56]. That

is, the eﬀect of gravity in the protocol is indirect, so that

the detectors are still coupled to the scalar ﬁeld in a clas-

sical background spacetime. However, to the authors’

knowledge no previous work has considered entanglement

harvesting directly from a quantum gravitational ﬁeld.

This manuscript is organized as follows. In Section II

we review the protocol of entanglement harvesting us-

ing two spacelike separated probes. In Section III we

review the formalism of linearized quantum gravity, and

describe how non-relativistic quantum systems couple to

a weak gravitational ﬁeld. The protocol of entanglement

harvesting from the gravitational ﬁeld is described in Sec-

tion IV. We present our ﬁrst examples of entanglement

harvesting from the gravitational ﬁeld in Section V. In

Section VI we compare the results found for the grav-

itational ﬁeld with scalar model analogues. In Section

VII we study atoms coupled to quantum gravity in the

linearized regime. The conclusions of our work can be

found in Section VIII.

arXiv:2210.14921v2 [quant-ph] 18 Nov 2023

II. EXTRACTING ENTANGLEMENT FROM

QUANTUM FIELDS: THE ENTANGLEMENT

HARVESTING PROTOCOL

The goal of this section is to provide a brief review

of the protocol of entanglement harvesting from a scalar

ﬁeld, so that we can later present a model for entangle-

ment harvesting from the gravitational ﬁeld. In Subsec-

tion II A we review the formalism for an Unruh-DeWitt

(UDW) particle detector interacting with a real scalar

quantum ﬁeld. In Subsection II B, we review the proto-

col of entanglement harvesting from a scalar ﬁeld.

A. Particle detector models

In this section we review the UDW detector model [57–

59], where a localized two-level quantum system inter-

acts with a relativistic quantum ﬁeld. This model has

been extensively used in the literature in order to probe

many features of diﬀerent quantum ﬁeld theories. Among

its applications are the ability to probe the Unruh ef-

fect [57, 60, 61] and Hawking radiation [57, 62], probe the

spacetime topology and geometry [53, 63, 64], describe

communication protocols in quantum ﬁeld theory [65–

68], and, most relevant for this work, harvest entangle-

ment from a quantum ﬁeld [35–56].

We consider a massless scalar quantum ﬁeld in

(3 + 1)-dimensional Minkowski spacetime. We may ex-

press the ﬁeld in terms of a plane-wave mode expansion

ϕ(x) = 1

(2π)3

2Zd3k

p2|k|ˆakeik·x+ ˆa†

ke−ik·x,(1)

where we use inertial coordinates xµ= (t, x), so that

kµ= (|k|,k). The operators ˆa†

k, ˆakrepresent creation

and annihilation operators, respectively, for a ﬁeld mode

of momentum k. The creation and annihilation operators

satisfy the canonical commutation relations

hˆak,ˆa†

k′i=δ(3)(k−k′).(2)

In order to deﬁne the Hilbert space associated with the

quantum ﬁeld, we deﬁne the vacuum state by ˆak|0⟩= 0

for all k. The Hilbert space is then built from repeated

applications of the creation operators ˆa†

kon |0⟩, and is

usually referred to as the Fock space.

As a ﬁrst approach, we will consider our detector to

be modelled by a two-level system. This qubit detector

follows an inertial trajectory, zµ(t) = (t, x0), along which

it can interact with the quantum ﬁeld. Moreover, the free

Hamiltonian of the detector is given by

HD= Ωˆσ+ˆσ−,(3)

where Ω is the (proper) energy gap between the two lev-

els. We will denote the ground and excited states of the

detector by |g⟩and |e⟩, respectively. Then, the ladder op-

erators for our detector are ˆσ+=|e⟩⟨g|, and ˆσ−=|g⟩⟨e|.

The associated interaction between the detector and the

ﬁeld is prescribed by the following Hamiltonian in the

interaction picture

HI(t) = λχ(t)ˆµ(t)ˆ

ϕ(t, x0),(4)

where λis the coupling strength, χ(t) is the switching

function, which determines the duration of the interac-

tion, and ˆµ(t) is the time evolved monopole moment of

the detector, explicitly given by

ˆµ(t) = eiΩtˆσ++e−iΩtˆσ−.(5)

We can relax the idealization of the detector being a

pointlike system. To do so, we implement a smearing

function such that the localization of the detector is de-

ﬁned by a function f(x) centered at x0. In most physical

setups, the smearing function f(x) is given in terms of

the wavefunctions of the ground and excited state of the

system (see, e.g., [38, 69]). In fact, if the system has a

position degree of freedom ˆ

xand a canonically conjugate

momentum operator ˆ

p, one can write the interaction with

the ﬁeld as [70]

HI(t) = λχ(t)ˆ

ϕ(t, ˆ

x).(6)

Assuming the energy levels of the free Hamiltonian of the

detector to be discrete, the interaction Hamiltonian can

then be expanded in terms of the system’s wavefunctions

as [38, 69]

HI(t) = λχ(t)

Zd3xˆ

ϕ(t, x)|x⟩⟨x|t(7)

=λX

n,m

χ(t)

Zd3xψ∗

n(x)ψm(x)eiΩnmtˆ

ϕ(t, x)|n⟩⟨m|,

where |x⟩⟨x|tdenotes the operator |x⟩⟨x|in the interac-

tion picture, ψn(x) = ⟨x|n⟩is the eigenfunction associ-

ated with the energy eigenvalue Enand Ωnm =En−Em.

Restricting the interaction to two levels, the wave-

functions of the ground and excited states are simply

ψg(x) = ⟨x|g⟩and ψe(x) = ⟨x|e⟩.

We then notice that the diagonal terms of ˆ

HI(t) com-

mute with the detector’s free Hamiltonian. This means

that these terms would not be able to produce detector

excitations at leading order in λ, and would simply shift

the energy level of the system. That is, these terms have

no eﬀect on the detector dynamics at leading order, so

that they can be neglected. The interaction Hamiltonian

in the interaction picture then reads

H(t) =λχ(t)

Zd3xf∗(x)eiΩtˆσ++f(x)e−iΩtˆσ−ˆ

ϕ(x),

(8)

where f(x) = ψe(x)ψ∗

g(x), and Ω = Ee−Egis the energy

gap of the detector.

In order to write the time evolution of our system in

terms of an integral in spacetime, we deﬁne the interac-

tion Hamiltonian density as

H(x) = λχ(t)f∗(x)eiΩtˆσ++f(x)e−iΩtˆσ−ˆ

ϕ(x).(9)

The relationship between the interaction Hamiltonian

and the interaction Hamiltonian density in Minkowski

spacetime is explicitly given by

H(t) = Zd3xˆ

H(x).(10)

To study the interaction between the quantum ﬁeld

and the detector, we will consider the detector to start

in the ground state and the quantum ﬁeld to start in its

vacuum state. That is, we may write the initial density

operator of the detector-ﬁeld system as

ˆρ0=|g⟩⟨g|⊗|0⟩⟨0|.(11)

To obtain the ﬁnal state of the detector, we ﬁrst time-

evolve the density operator ˆρ0. The time evolution opera-

tor in the interaction picture is given by the time ordered

exponential1

UI=Texp −iZd4xˆ

HI(x),(12)

where d4x= dtd3x. Working perturbatively to second

order in λ, we may write

UI=1+U(1)

I+U(2)

I+O(λ3),(13)

where, in terms of the Hamiltonian density, we have

U(0)

I=1,(14)

U(1)

I=−iλZd4xˆ

HI(x),(15)

U(2)

I=−λ2Zd4xd4x′ˆ

HI(x)ˆ

HI(x′)θ(t−t′),(16)

and θ(t) denotes the Heaviside theta function and im-

plements the time ordering operation. Using the density

operator in Eq. (11), the time evolved density operator

may be written as a power expansion in λ,

ˆρ= ˆρ(0) + ˆρ(1) + ˆρ(2) +O(λ3),(17)

where

ˆρ(0) = ˆρ0,(18)

ˆρ(1) =ˆ

U(1)

Iˆρ0+ ˆρ0ˆ

U(1)†

I,(19)

ˆρ(2) =ˆ

U(2)

Iˆρ0+ˆ

U(1)

Iˆρ0ˆ

U(1)†

I+ ˆρ0ˆ

U(2)†

I.(20)

1Notice that for smeared detectors the notion of time-ordering can

be ambiguous but we are going to work in regimes where this is

not an issue. See [71] for details.

When a quantum ﬁeld is measured, one does not have di-

rect access to the state of the ﬁeld (only to the detector’s

degrees of freedom). To ﬁnd the detector’s state after

the interaction, we compute the partial trace of ˆρwith

respect to the ﬁeld’s Hilbert space. It is then possible

to ﬁnd an expression (up to second order in λ) for the

probability of the detector transitioning from the ground

to the excited state:

L=λ2

Zd4xd4x′χ(t)χ(t′)f(x)f∗

(x′)e−iΩ(t−t′)

⟨ˆ

ϕ(x)ˆ

ϕ(x′)⟩0,

(21)

where

⟨ˆ

ϕ(x)ˆ

ϕ(x′)⟩0=1

(2π)3Zd3k

2|k|eik·(x−x′)(22)

is the two-point Wightman function of the quantum ﬁeld

in the vacuum state. Given the two-point correlator in

Eq. (22), and deﬁning the Fourier transforms of f(x)

and χ(t) as

f(k) = Zd3xf(x)eik·x,(23)

˜χ(ω) = Zdt χ(t)eiωt,(24)

we can write the excitation probability, L, as an integral

over momenta

L=λ2

(2π)3Zd3k

2|k||˜χ(Ω + |k|)|2|˜

f(k)|2.(25)

B. Entanglement harvesting protocol

In this subsection, we will review the protocol of entan-

glement harvesting using UDW detectors. Understand-

ing entanglement in QFT is a nontrivial task, given that

most of our notions of entanglement depend on a tensor

product decomposition of the Hilbert space where the

theory is deﬁned. However, in QFT, the Hilbert space of

the theory cannot easily be expressed as a tensor prod-

uct decomposition associated to localized regions of space

(see, e.g., [72] for the massive ﬁeld case). This renders

most of our methods for discussing entanglement in non-

relativistic quantum mechanics unsuitable for localized

states in QFT [25]. However, one can still use other

methods in order to quantify the ﬁeld’s quantum corre-

lations. For instance, one can quantify the entanglement

between probes that couple locally to the quantum ﬁeld,

and use these to infer the entanglement properties of the

ﬁeld. This phenomenology is generally called entangle-

ment harvesting.

In order to study the entanglement harvesting proto-

col, we consider two approximately spacelike separated

UDW detectors locally interacting with a quantum ﬁeld.

The interaction regions chosen later in this paper are

approximately spacelike separated in the sense that the

eﬀect of communication between the two regions is neg-

ligible for entanglement harvesting. See e.g., [73, 74], for

details. We consider the probes to be spacelike separated

throughout their interaction, so that they cannot com-

municate with each other. As a result, any entanglement

acquired by the probes must have come from the quan-

tum ﬁeld itself. We will label our two detectors Aand

B, where the detectors are inertial and comoving, so that

their centres of mass undergo trajectories with constant

spatial coordinates, xa=x0, and xb=x0+L, with

Lbeing the spatial separation vector between the detec-

tors. Each detector will interact with the ﬁeld according

to the interaction Hamiltonian density (9):

HI,i(x) = λiχi(t)f∗

i(x)eiΩitˆσ+

i+fi(x)e−iΩitˆσ−

iˆ

ϕ(x),

(26)

for I∈ {A,B}. Here, we have that λiis the coupling

strength, fi(x) is the smearing function localized around

the trajectory, χi(t) is the switching function, Ωiis the

energy gap, ˆσ+

i=|ei⟩⟨gi|, ˆσ−

i=|gi⟩⟨ei|are the ladder

operators of detector I, and |ga⟩,|ea⟩,|gb⟩,|eb⟩denote

the ground and excited states of the detectors.

We are interested in the case where both detectors are

initially in the ground state and the quantum ﬁeld is

initially in its vacuum state. Thus, the resulting initial

density operator of the detectors-ﬁeld system is given by

ˆρ0=|ga⟩⟨ga|⊗|gb⟩⟨gb|⊗|0⟩⟨0|.(27)

The ﬁnal interaction Hamiltonian density will then be

the sum of the individual Hamiltonian densities

HI(x) = ˆ

HI,a(x) + ˆ

HI,b(x).(28)

The unitary time evolution operator in the interaction

picture will be given by Eq. (12) with the interaction

Hamiltonian density of Eq. (28). After applying the uni-

tary time evolution operator and tracing out the quan-

tum ﬁeld’s degrees of freedom, we obtain the following

density matrix of the two-detector system at order O(λ2)

in the basis {|gAgB⟩,|gAeB⟩,|eAgB⟩,|eAeB⟩}

ˆρd=





1− Laa − Lbb 0 0 M∗

0Lbb Lba 0

0Lab Laa 0

M0 0 0





.(29)

Here, the Laa and Lbb terms are the excitation probabil-

ity of detector Aand B, respectively. The M,Lab and

Lba terms capture non-local correlations acquired by the

detectors. In the case of two qubit detectors coupled to a

scalar quantum ﬁeld, we can ﬁnd expressions for Lij and

M. Explicitly,

Lij =λiλjZd4xd4x′⟨ˆ

ϕ(x)ˆ

ϕ(x′)⟩0e−i(Ωit−Ωjt′)

×χi(t)χj(t)fi(x)f∗

j(x′),(30)

M=−λaλbZd4xd4x′⟨ˆ

ϕ(x)ˆ

ϕ(x′)⟩0θ(t−t′)

×ei(Ωata+Ωbt′

b)χa(t)χb(t′)fa(x)fb(x′)

+ei(Ωat′

a+ΩbtB)χa(t′)χb(t)fa(x′)fb(x).

(31)

We wish to quantify the entanglement between the

two detectors after they have both locally interacted

with the quantum ﬁeld. There are multiple entangle-

ment measures that can be used for this purpose. In this

manuscript, we use the negativity, which is a faithful en-

tanglement measure for a system of two qubits [75, 76]

(see, e.g.,[53] for the speciﬁc discussion in the context of

entanglement harvesting). The negativity of ˆρdis deﬁned

as N=−Pλi<0λi, where the λi’s are the eigenvalues

of the partial transpose of ˆρd. To leading order in λ, the

negativity is then found to be

N= max 0,r|M|2−(Laa − Lbb)2

4−Laa +Lbb

2!.

(32)

In particular, in the special case where the excitation

probabilities of the detectors are the same, we have that

Laa =Lbb =L, and hence the negativity becomes

N= max(0,|M| − L).(33)

This happens, for instance, in the case of identical inertial

detectors in Minkowski spacetime. In fact, from now on,

we will assume that the detectors are identical, so that

λa=λb=λ, Ωa= Ωb= Ω, and that the interaction

happens simultaneously in their frames, which implies

χa(t) = χb(t) = χ(t), and the smearings are identical

modulo a spatial translation.

Given that the quantum ﬁeld is assumed to be in its

vacuum state, we substitute Eq. (22) into Eqs. (30) and

(31), and we can write the transition probability and the

non-local term Min terms of the Fourier transform of

the smearing and switching functions as follows

Lij =λiλj

(2π)3Zd3k

2|k|˜χ∗(Ω + |k|)˜χ(Ω + |k|)˜

fi(k)˜

f∗

j(k),

(34)

M=−λaλb

(2π)3

Zd3k

2|k|Q(|k|,Ω)( ˜

fa(−k)˜

fb(k)+ ˜

fb(−k)˜

fa(k)),

(35)

where

Q(|k|,Ω) = Zdtdt′χ(t)χ(t′)ei(Ω+|k|)t′ei(Ω−|k|)tθ(t−t′),

(36)

and where we have deﬁned the Fourier transform of f(x)

and χ(t) as Eqs. (23) and (24).

The protocol of entanglement harvesting with two

UDW detectors coupled linearly to a scalar quantum

ﬁeld has been extensively studied in the literature (see

e.g. [37, 39, 47, 51, 52, 56, 73, 77–79]), and many prop-

erties of the protocol are well understood by now.

It is important to mention that there are two ways in

which two probes can become entangled via an interac-

tion with a ﬁeld [74]. The ﬁrst way only relies on com-

munication via the ﬁeld, and can be achieved through

an interaction of a quantum system with a classical ﬁeld

(see, e.g., [12]). Two probes can only become entangled

in this way if they are in causal contact, and the role

played by the ﬁeld in this case is a mere mediator that al-

lows for an exchange of information between the probes.

The second way in which detectors can become entan-

gled after an interaction with a ﬁeld is by extracting (or

harvesting) entanglement previously present in the ﬁeld.

This entangling procedure can only be performed when

the probes are coupled to a quantum ﬁeld, which has

its own local quantum degrees of freedom. Moreover,

this protocol for extracting entanglement from a quan-

tum ﬁeld can be achieved even if the detectors are not

within causal contact, and we will refer to this as entan-

glement harvesting to distinguish it from entanglement

generated by communication.

When one couples two causally connected probes to a

quantum ﬁeld, these probes become entangled both via

communication and via extracting entanglement from the

ﬁeld. This makes it a hard task to identify which part

of the entanglement acquired by the detectors was ex-

tracted from the ﬁeld and which part is due to commu-

nication [74]. For this reason, here we will be concerned

with scenarios where the probes are mostly spacelike sep-

arated and we make sure that the entanglement acquired

by the detectors is primarily extracted from the ﬁeld.

Experimentally, this kind of setup can be used to test

whether a ﬁeld possesses quantum degrees of freedom:

only a quantum ﬁeld allows spacelike separated locally

coupled detectors to become entangled.

III. LINEARIZED QUANTUM GRAVITY AND

MATTER: A GRAVITATIONAL DETECTOR

MODEL

The goal of this section is to summarize the description

of the interaction of a non-relativistic quantum system

with linearized quantum gravity. First, we review the

quantization of linearized perturbations of the metric in

Subsection III A. Then, in Subsection III B we describe

the coupling of a particle detector with the quantized

perturbations of the gravitational ﬁeld.

A. Quantum ﬂuctuations in a Minkowski

background

Einstein’s theory of general relativity describes space-

time as a four dimensional manifold with a Lorentzian

metric gµν that satisﬁes Einstein’s equations:

Gµν = 8πℓ2

pTµν ,(37)

where Gµν is the Einstein tensor, Tµν is the stress-energy

momentum tensor and ℓpis the Planck length (recall that

in natural units ℓp=√G).

Einstein’s equations are highly non-linear due to the

dependence of Gµν on the metric. In order to describe

small ﬂuctuations of the gravitational ﬁeld around a

given background, we can consider perturbations γµν of

the spacetime metric, so that the eﬀective metric be-

comes gµν +γµν and Eq. (37) deﬁnes an equation of mo-

tion for γµν . Considering |γµν | ≪ 1 one can then expand

these equations to linear order in γµν to obtain a linear

equation of motion for the metric perturbations. This

approach is commonly called linearized gravity, and the

resulting theory for the ﬁeld γµν is invariant under gauge

transformations of the form γµν 7−→ γµν +∇µξν+∇νξµ,

where ξµis the inﬁnitesimal generator of such transfor-

mations.

In this manuscript we will be concerned with perturba-

tions around Minkowski spacetime, without the presence

of matter (Tµν = 0). In this case, the equations of motion

for γµν simplify, so that in inertial coordinates (t, x, y, z)

the ﬁrst order variation of the Einstein tensor can be

written as

G(1)

µν (γ) (38)

=∂(µ∂αγν)α−1

2□γµν −1

2∂µ∂νγ−1

2ηµν (∂α∂βγαβ −□γ),

where G(1)(γ) denotes the Einstein tensor to linear or-

der in γµν ,□=∂α∂αis the D’Alembertian operator,

γ=ηµν γµν and ηµν is the Minkowski metric, which we

use to lower and raise indices. In the absence of matter,

the linearized Einstein’s equations for the gravitational

perturbations then read

G(1)

µν (γ)=0,(39)

which deﬁnes equations of motion for the linearized met-

ric perturbations whose solutions are typically called

gravitational waves. The theory for γµν can also be

thought of as a theory for a tensor ﬁeld associated with

the action

S[γ] = 1

16πℓ2

pZd4x(40)

−1

2∂αγµν ∂αγµν +1

2∂λγ∂λγ−∂βγ∂αγαβ +∂νγνα∂µγµα.

The action above can be obtained by evaluating the met-

ric in the Einstein-Hilbert action at ηµν +γµν , and only

considering the leading order terms in γµν . In partic-

ular, extremizing this action with respect to γµν yields

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HarvestingentanglementfromthegravitationalvacuumT.RickPerche,1,2,3,∗BorisRagula,1,†andEduardoMart´ın-Mart´ınez1,2,3,‡1DepartmentofAppliedMathematics,UniversityofWaterloo,Waterloo,Ontario,N2L3G1,Canada2PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,N2L2Y5,Canada3InstituteforQuantumComputing...

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Harvesting entanglement from the gravitational vacuum T. Rick Perche1 2 3 Boris Ragula1and Eduardo Mart ın-Mart ınez1 2 3 1Department of Applied Mathematics University of Waterloo Waterloo Ontario N2L 3G1 Canada.pdf

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Harvesting entanglement from the gravitational vacuum T. Rick Perche1 2 3 Boris Ragula1and Eduardo Mart ın-Mart ınez1 2 3 1Department of Applied Mathematics University of Waterloo Waterloo Ontario N2L 3G1 Canada

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