Continuous-variable supraquantum nonlocality

Andreas Ketterer,1, 2, ∗ Adrien Laversanne-Finot,2, † and Leandro Aolita3, 4, ‡

1Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
2Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité,

Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France
3Instituto de F́ısica, Universidade Federal do Rio de Janeiro,

Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil
4ICTP South American Institute for Fundamental Reserch Instituto de F́ısica Teórica,

UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, SP, Brazil
(Dated: February 2, 2018)

Supraquantum nonlocality refers to correlations that are more nonlocal than allowed by quantum
theory but still physically conceivable in post-quantum theories, in the sense of respecting the basic
no-faster-than-light communication principle. While supraquantum correlations are relatively well
understood for finite-dimensional systems, little is known in the infinite-dimensional case. Here, we
study supraquantum nonlocality for bipartite systems with two measurement settings and infinitely
many outcomes per subsystem. We develop a formalism for generic no-signaling black-box measure-
ment devices with continuous outputs in terms of probability measures, instead of probability distri-
butions, which involves a few technical subtleties. We show the existence of a class of supraquantum
Gaussian correlations, which violate the Tsirelson bound of an adequate continuous-variable Bell
inequality. We then introduce the continuous-variable version of the celebrated Popescu-Rohrlich
(PR) boxes, as a limiting case of the above-mentioned Gaussian ones. Finally, we perform a char-
acterisation of the geometry of the set of continuous-variable no-signaling correlations. Namely, we
show that that the convex hull of the continuous-variable PR boxes is dense in the no-signaling
set. We also show that these boxes are extreme in the set of no-signaling behaviours and provide
evidence suggesting that they are indeed the only extreme points of the no-signaling set. Our results
lay the grounds for studying generalized-probability theories in continuous-variable systems.

PACS numbers: 03.65.Ud, 03.65.Ta

I. INTRODUCTION

Bell nonlocality refers to correlations incompatible
with local hidden-variable theories [1], which explain cor-
relations between space-like separated measurement out-
comes as due exclusively to past common causes. Since
the pioneering works of Bell [1], and of Clauser, Horn,
Shimony and Holt [2], it is known that quantum mechan-
ics admits Bell nonlocality, i.e., that local measurements
on quantum entangled states produce Bell nonlocal cor-
relations. However, nonlocality is not a phenomenon ex-
clusive of quantum theory. Hypothetical supraquantum
theories satisfying the basic no-signaling principle of no-
faster-than-light communication, in consistency with spe-
cial relativity, can produce Bell correlations that are even
more nonlocal than those compatible with quantum the-
ory. This is generally referred to as supraquantum Bell
nonlocality. The first known example thereof was the
so-called Popescu-Rohrlich (PR) boxes [3]. These are
hypothetic black-box measurement devices that can vio-
late the Clauser-Horn-Shimony-Holt inequality up to its
algebraic maximum of 4, which is above the maximum

∗ andreas.ketterer@uni-siegen.de;
A.K. and A.L.-F. contributed equally.
† adrien.laversanne-finot@univ-paris-diderot.fr
‡ aolita@if.ufrj.br

value of 2
√

2 attained by quantum correlations, known
as Tsirelson’s bound [4].

Importantly, the aim of studying supraquantum nonlo-
cality is by no means to question the validity of quantum
mechanics, but, rather on the contrary, actually to gain a
better understanding of quantum nonlocality itself. For
instance, even though unphysical, PR boxes make excel-
lent units of Bell nonlocality, serving, in fact, as reference
to quantify the nonlocal weight of quantum correlations
[5, 6]. Furthermore, understanding why quantum me-
chanics is not as nonlocal as allowed by the no-signalling
principle gives us valuable insights with foundational im-
plications on the very axiomatic structure of quantum
theory. For instance, a seminal result in this direction
was the realization that the physical existence of PR
boxes would make communication complexity problems
trivial [7–9], which is a highly implausible possibility.
Hence, if one accepts that communication complexity is
not trivial as a postulate, the non-existence of PR boxes
is implied. In fact, in a similar spirit, a large effort has
been devoted to proposing physically reasonable postu-
lates from which Tsirelson’s bound can be derived from
first principles (see, e.g., Refs. [10–14]).

PR boxes have been generalized to arbitrary finite
numbers of measurement outcomes [15] and to multi-
partite systems as well [16]. What is more, in the mul-
tipartite scenario, non-trivial tight Bell inequalities are
known without a quantum violation, i.e. for which the
quantum maximum coincides with the local one and is

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2

below the no-signalling one [17]. In addition, supraquan-
tum nonlocality has been explored even in the bipartite
scenario where only one part makes measurements [18].
From a broader perspective, Bell nonlocality in gener-
alised probabilistic theories has been extensively studied
in the finite-dimensional case (see [19] and Refs. therein).
Nevertheless, in striking contrast, essentially nothing is
known about supraquantum nonlocality in continuous-
variable (CV) systems. On the one hand, this is surpris-
ing in view of the huge amount of work on CV quantum
nonlocality (see, e.g., Refs. [20–30]) and the importance
of CV systems for quantum information processing [31–
33]. On the other hand, this is at the same time un-
derstandable because, for CV systems, the set of local
correlations (as well as that of no-signalling ones) is a
generic convex set, instead of a (computationally much
tamer) convex polytope as in finite-dimensional systems
[34–36].

In this article we conduct an exploration of CV
supraquantum nonlocality. To begin with, we develop
a formalism to deal with generic no-signalling black-box
measurement devices with discrete measurement settings
(inputs) and CV measurement outcomes (outputs). The
correlations produced by such devices are described by
probability measures instead of probability distributions.
We then show the existence of a class of supraquan-
tum Gaussian PR boxes, for bipartite systems with di-
chotomic inputs and real, continuous outputs. This
is done by showing that these behaviours violate the
Calvalcanti-Foster-Reid-Drummond (CFRD) inequality
[30], which admits no quantum violation in the bipar-
tite case [29]. Next, we introduce, a limiting case of the
supraquantum Gaussian behaviours, a hierarchy of CV
PR boxes, whose ground level consists of local, deter-
ministic points and the upper levels of nonlocal, non-
deterministic ones. The CV PR boxes obtained are very
similar in structure to the finite-dimensional ones. To
end up with, we perform a characterization of the set of
CV no-signaling behaviours and show that all CV PR
boxes are extreme points of the CV no-signaling set, and
that their convex hull (i.e. the set of all finite convex
sums) is dense therein. In particular, we discuss whether
the CV PR boxes are the only extreme no-signaling be-
haviours and, along with some evidence, conjecture that
this is indeed the case.

The paper is structured as follows. In Sec. II, we
set up the mathematical framework for CV no-signaling
behaviours based on probability measures. In Sec. III,
we introduce the supraquantum Bell nonlocal Gaussian
behaviours and the CV PR boxes. Sec. IV is devoted
to the geometrical characterization of the set of CV no-
signaling set. Finally, we conclude, in Sec. V, with some
final remarks and perspectives of our work.

Bob

b 2 Ra 2 R

Alice

FIG. 1. (Color online) Schematic representation of a bipartite
Bell experiment with continuous measurement outcomes in
the so-called device-independent scenario of black-box mea-
surement instruments. Two space-like separated observers,
Alice (A) and (Bob), perform local measurements on their
subsystems with dichotomic measurements choices (inputs) x
and y, respectively, and obtain continuous-variable measure-
ment outcomes (outputs) a and b.

II. PRELIMINARIES: MATHEMATICAL
REPRESENTATION OF CV BELL

CORRELATIONS

We consider a bipartite Bell experiment where two
space-like separated observers, conventionally referred to
as Alice (A) and Bob (B), make measurements. We
work in the generic device-independent scenario where
the measurement apparatuses are treated as unknown
black-box measurement devices [see Fig. 1(a)]. Alice’s
(Bob’s) device has a dichotomic input x (y) ∈ {0, 1} and a
continuous output a (b) ∈ R. That is, we are considering
infinite resolution: we want to investigate the ideal situa-
tion where the outputs can take any arbitrary real value.
The statistics produced by such devices is most conve-
niently described in terms of probability measures, which
we briefly recap in what follows. We consider probability
spaces defined by a triple {Ω,B(Ω), µ}, where Ω denotes
a sample space, B(Ω) the Borel σ-algebra of events on Ω
(i.e., the smallest σ-algebra that contains all open subsets
of Ω) and µ : B(Ω)→ [0, 1] a Borel probability measure.
In our case, the sample space is given by a product space
Ω = ΩA × ΩB, with ΩA = ΩB = R, where the first and
second factors, ΩA and ΩB, correspond to the outputs of
A and B, respectively. The probability measure µ is re-
quired to be normalized, µ(R×R) = 1, and to satisfy the
additivity property µ (

⋃
i=1Ei) =

∑
i=1 µ(Ei), for every

countable sequence {Ei}i of disjoint events Ei ∈ B(R×R),
where ∪ stands for the set union. The probability of an
event E ∈ B(R× R) is then given by P (E) := µ(E). We
denote the set of all probability measures on B(R×R) as
MR×R.

The connection between a probability measure µ and a
probability density p (with respect to the Lebesgue mea-
sure) can be made explicit in the integral representation

µ(A×B) :=

∫
A×B

dµ(a′, b′) =

∫
A

∫
B

p(a′, b′) da′ db′,

(1)



3

where A × B ∈ B(R × R), with A,B ∈ B(R), p(a′, b′)
denotes the corresponding probability density to µ, and
dµ(a, b) and da′ db′ refer to integrations with respect to
µ and the Lebesgue measure on R×R, respectively. Note
that not every probability measure can be expressed in
terms of a probability density as in Eq. (1). The ques-
tion of the existence of a probability density is answered
by the Radon-Nikodym (RN) theorem, whose statement
is briefly reviewed in Appendix A. While most assump-
tions of the RN theorem are fulfilled by any probability
measure on R× R, for us the crucial prerequisite is that
µ has to be absolutely continuous with respect to the
Lebesgue measure. However, as we will see later on, ab-
solute continuity cannot be guaranteed for all types of
probability measures which will become important when
dealing with so-called boxes describing idealized unphys-
ical outcome scenarios. Hence, all in all, it is both more
general and more convenient to work with measures, as
one needs not worry about the existence of a density.

We thus arrive at the following definition.

Definition 1 (CV Bell behaviour). A behaviour is a
joint conditional probability measure represented by a
2 × 2 matrix µ = {µx,y}x,y∈{0,1} with arbitrary proba-
bility measures [µ]x,y := µx,y ∈ MR×R as entries. The
set of all behaviours is denoted as M4

R×R.

Note that, for finite-dimensional systems, the sample
space has a finite number of events, so that joint con-
ditional probability measures reduce to the more usual
notion of joint conditional probability distributions [19].
Also as in the discrete case, since the observers are space-
like separated, µ must fulfill the no-signaling principle,
given, in this language, by the constraints:

µx,y(A× R) = µx,y(A× R) ∀ x ∈ {0, 1}, (2a)

µx,y(R×B) = µx,y(R×B) ∀ y ∈ {0, 1}, (2b)

for all A,B ∈ B(R), where y = y⊕ 1 and x = x⊕ 1, with
⊕ the sum modulo 2.

Conditions (2a) and (2b) imply respectively that Al-
ice’s and Bob’s marginal measures µx(A) := µx,y(A×R)
and µy(B) := µx,y(R×B) are independent of each others’
input, which prevents signaling. We call any µ satisfying
these conditions a no-signaling behaviour, and denote the
set of all no-signaling behaviours by MNS ⊂M4

R×R.
Quantum correlations, in turn, are those described by

the behaviours that can be expressed as

µx,y(A×B) = Tr [Mx(A)⊗My(B) %AB ] , (3)

for all A,B ∈ B(R), where ρAB is an arbitrary bipar-
tite quantum state on a Hilbert space H := HA ⊗ HB,
with HA and HB the local Hilbert spaces of Alice’s and
Bob’s systems, respectively, and Mx and My are, for all
x (y) ∈ {0, 1}, semi-spectral measures, also known as
positive-operator valued measures (POVMs) [37]. The
latter means that Mx,My : B(R) → L≥0(H) are maps
such that, for all A (B) ∈ B(R), Mx(A) ∈ L≥0(HA)
[My(B) ∈ L≥0(HB)], with L≥0(HA) [L≥0(HB)] the space

FIG. 2. (Color online) Pictorial (not rigorous) geometrical
representation of the (possible) inner structure of the setMNS

of CV no-signaling behaviours in the Bell scenario of Fig. 1.
MNS contains the setMQ of quantum behaviours, which con-
tains, in turn, the set ML of local behaviours. All three sets
are generic convex sets with infinitely many extreme points,
delimited by facets as well as curved hyper-surfaces. This is
in contrast with the finite-dimensional case, where bothMNS

andML are convex polytopes, delimited exclusively by facets
that can by characterisied by a finite number of linear Bell
inequalities. In the plot, an example of a linear Bell inequal-
ity is represented as a straight line LI. Such linear inequality
can, e.g., correspond to a Bell inequality for finite-dimensional
systems, which can be violated by CV quantum correlations
using so-called binning procedures [20–26] (see also Refs. in
[19]). Besides this, a hypothetical quantum extreme point is
shown in the figure (light-blue corner). While such points are
in principle possible, no explicit example thereof is known. In
this paper we consider a non-linear Bell inequality, the CFRD
inequality [30], represented as a curve in the plot. This in-
equality applies in the genuinely CV scenario of our inter-
est and has, additionally, the appealing feature of admitting
violations only by supraquantum behaviours (see Sec. III).
Finally, four exemplary CV PR boxes are represented as ex-
treme points of MNS (black dots).

of positive semi-definite operators on HA (HB); and that
Mx(R) = 1A (My(R) = 1B), with 11A (11B) the identity
operator on HA (HB). We call any µ satisfying Eq. (3)
a quantum behaviour, and denote the set of all quantum
behaviours by MQ. For generic Bell scenarios, the re-
lationship MQ ⊆ MNS holds. For the scenario under
consideration here, we show below that MQ ⊂ MNS.
We call any µ ∈MNS \MQ a supraquantum behaviour.

The last important class for our purposes is the one of
classical correlations, described by the behaviours pro-
duced by local hidden-variable models:

µx,y =

∫
Λ

δa(x,λ),b(y,λ) dη(λ), (4)

where λ is the hidden variable, taking values in a pa-
rameter space Λ according to a probability measure



4

η : B(Λ) → R≥0, and δa(x,λ),b(y,λ) is the CV version
of the λ-th local deterministic response function. More
precisely, δa,b denotes the Dirac measure at the point
(a, b) ∈ R2, i.e. the deterministic measure such that

δa,b(A×B) :=

{
1 if a ∈ A and b ∈ B,
0 otherwise,

(5)

for all A,B ∈ B(R). In turn, for each λ ∈ Λ, a(x, λ) and
b(y, λ) are respectively deterministic functions of x and
y, in a similar spirit to the local deterministic response
functions in finite-dimensional scenarios [19]. Since the
outputs are locally generated from each input and the
pre-established classical correlations encoded in λ, one
typically calls any µ given by Eq. (4) a local behaviour.
We denote the set of all local behaviours byML ⊆MQ.
In turn, any µ ∈MNS \ML is a nonlocal behaviour.

Finally, we emphasise that, in contrast to the finite-
dimensional case, ML does not define a polytope (i.e., a
convex set with finitely many extreme points), see Fig. 2.
This is due to the fact that Dirac measures are extreme
in MNS and ML is generated by a continuously infinite
number of them. It follows, then, that ML cannot be
characterized by a finite set of linear Bell inequalities
[34–36]. In the next section, we use a non-linear Bell
inequality to identify not only nonlocal behaviours but
supraquantum ones.

III. CONTINOUS-VARIABLE
SUPRAQUANTUM NONLOCALITY

In Ref. [30], Calvalcanti, Foster, Reid, and Drummond
derived the nonlinear Bell inequality

[〈A0B0〉 − 〈A1B1〉]2 + [〈A0B1〉+ 〈A1B0〉]2

≤ 〈A2
0B

2
0〉+ 〈A2

0B
2
1〉+ 〈A2

1B
2
0〉+ 〈A2

1B
2
1〉, (6)

where A0 and A1 (B0 and B1) are the real, continuous
outputs of Alice’s (Bob’s) box for the inputs 0 and 1, re-
spectively. Using the integral representation of Eq. (1),
the expectation values of such observables appearing in
the inequality can be recast as cross-moments of the be-
haviour elements µx,y:

〈Anax Bnby 〉 =

∫
R2

ana bnb dµx,y(a, b). (7)

Eq. (6) can be generalised to higher number of parties [30]
as well as observables per party [28]. We refer to the
bipartite dichotomic-input version of inequality, given by
Eqs. (6) and (7), as the CFRD inequality. The inequality
has a number of interesting properties [28, 30]. Specially
relevant for our purposes is the fact that it cannot be vi-
olated by any quantum bebaviour. This was first shown
in Ref. [27] for the restricted case of measurements of
(quantum) phase-space quadrature operators, and then
extended to the general case of arbitrary quantum mea-
surements in Ref. [29]. Hence, the CFRD constitutes

0 1 2 3 4 5
-1

0

1

2

3

4

FIG. 3. (Color online) (a) Density plots of a Gaussian PR box
of order 2, with centre vector characterised by a = (`,−`) = b
and width vector σ = (`/5, `/5), for the inputs (x, y) = (0, 0),
(0, 1), or (1, 0) (left) and (x, y) = (1, 1) (right). Note that,
for both plots, the projections onto the horizontal as well as
vertical axes coincide, reflecting the fact that the behaviour
is no-signalling. Each centre point may also have a different
width (or squeezing), but we do not consider that here for
simplicity. (b) Violation of the CFRD inequality, normalised
by the factor `4, by the Gaussian behaviour in question as a
function of the parameter `/σ. The CFRD inequality certifies
that the Gaussian PR box is supraquantum for the parameter

region with `/σ ≥
√

1 +
√

2 ≈ 1.55.

a non-trivial Bell inequality with no quantum violation.
Any no-signalling behaviour that violates it is thus auto-
matically certified as supraquantum, as we do next.

The first case that we study is a sub-class of be-
haviours that we term Gaussian PR boxes. To this end,
we first introduce two real vectors, a := (a1, . . . , ak)
and b := (b1, . . . , bk), with different components, i.e.,
such that a1 6= a2 6= . . . ak and b1 6= b2 6= . . . bk, and
one positive-real vector σ := (σ1, . . . , σk), all of length
k ∈ N. The vectors a and b determine k points (aj , bj)
where Gaussian-measure components are centred; while
the vector σ := (σ1, . . . , σk) determines their widths.
More precisely, then, we say that µ ∈MNS is a Gaussian
PR box of order k, with centre vector (a, b) and width
vector σ, if it is of the form

µ(k,a,b,σ)
x,y :=

1

k

k∑
j=1

N(aj ,b[j+x y]k ),σj , (8)

where [ ]k denotes modulo k and N(a,b),σ is the normal
(Gaussian) measure centred at (a, b) and with width σ,



5

defined through Eq. (1) with the probability density

p(a,b),σ(a′, b′) =
1

2π σ2
e−

(a−a′)2+(b−b′)2

2 σ2 . (9)

Whether a Gaussian PR box is supraquantum or not
depends on (a, b) and σ. As an example, consider next
the simple case with k = 2, a = (`,−`) = b, for some ar-
bitrary ` ∈ R6=0, and σ = (σ, σ), graphically represented
in Fig. 3 (a). It is immediate to see that the resulting
behaviour violates the CFRD inequality by the amount

max
{

8 `4 − 4
(
σ2 + `2

)2
, 0
}
. (10)

This violation is plotted in Fig. 3(b) as a function of σ/`.
Note that it grows unboundedly with `. The condition
for this Gaussian PR box to violate the CFRD inequality

is `/σ ≥
√

1 +
√

2 ≈ 1.55, as can be graphically appreci-
ated in the figure. In turn, taking, for the Gaussian PR
box above, the limit σ → 0, one obtains the behaviour
with components

µx,y =
1

2

[
δ`,(−1)xy` + δ−`,−(−1)xy`

]
, (11)

with δ the measure defined in Eq. (5). This limiting box
violates the CFRD inequality by 4`4. In fact, it is the
CV version of the original dichotomic-input dichotomic
output PR box [3].

Similarly, to define generic CV PR boxes, we take the
σ → 0 limit of the Gaussian PR boxes of Eq. (12). That
is, we say that µ(k,a,b) ∈MNS is a CV PR box of order k
and center vector (a, b), with different real components
such that a1 6= a2 6= . . . ak and b1 6= b2 6= . . . bk, if it is of
the form

µ(k,a,b)
x,y := µ(k,a,b,0)

x,y =
1

k

k∑
j=1

δaj ,b[j+x y]k . (12)

One can immediately verify that these behaviours fulfil
the no-signalling constraints (2). These boxes are the CV
version of the finite-dimensional PR boxes generalized to
arbitrarily many outputs and dichomotic inputs given in
Ref. [15]. Still, Eq. (12) does not yet describe the most
general CV PR box, because input and output relabelling
symmetries must be taken into account. For dichotomic
inputs, the possible local, reversible relabelings are given
by x → [x + 1]2 and y → [y + 1]2 [15]. The situation
is notably different, however, for the outputs, as they
are continuous. For CV outputs, the most general local,
reversible relabelings are given by a → αx(a) and b →
βy(b), where αx : Rk → Rk and βy : Rk → Rk are,

for every x, y ∈ {0, 1}, bijective maps from Rk to itself.
This amounts to reshuffling the components of the center
vectors in a reversible, input-dependent fashion, so that
the condition [αx(a)]1 6= [αx(a)]2 6= . . . [αx(a)]k and
[βy(b)]1 6= [βy(b)]2 6= . . . [βy(b)]k is always maintained.

Since the relabelings are local and reversible, all boxes
equivalent under them have the same nonlocality prop-
erties. Indeed, all the boxes given by Eq. (12), i.e. for

all different center vectors, are equivalent under input-
independent relabelings. So, any of them, i.e. for any
fixed center vector, can be taken as representative to de-
fine (modulo local, reversible, and input-dependent rela-
ballings) the entire class of all CV PR boxes. This is, in
turn, equivalent to allowing for input-dependent center
vectors (ax, by) directly in the definition:

Definition 2 (Set of CV PR boxes). We define the class
MPR as the set

MPR :=
{
µ(k,a0,a1,b0,b1) ∈MNS

}
k∈N,a0,a1,b0,b1∈Rk

,

(13)

where each behaviour component
[
µ(k,a0,a1,b0,b1)

]
x,y

is

given by a measure µ
(k,ax,by)
x,y as in Eq. (12), with a pos-

sibly different vector (ax, by) for each (x, y) ∈ {0, 1}2.

Note that for k = 1, CV PR boxes reduce to local,
deterministic behaviours, whose components are given by
Dirac delta measures. In contrast, for all k ≥ 2, Def. 2
yields non-local, non-deterministic behaviours. Here, for
simplicity, we use the term “CV PR box” for all k ∈ N
indistinctly, the distinction between local, deterministic
and non-local, non-deterministic ones being given by the
order k. In the next section, we show that every element
of MPR is an extreme behaviour of MNS and that the
convex hull of MPR is dense in MNS.

IV. CHARACTERIZATION OF THE SET OF
NO-SIGNALLING BEHAVIOURS

We start by recapping basic definitions of convex com-
binations and extremality. The convex hull Conv(M) of
an arbitrary (finite or infinite) setM of behaviours is the
set of all finite convex sums of elements of M:

Conv(M) =
{ n∑
i=1

qi µi :µi ∈M
}
qi≥0,

∑n
i=1 qi=1, n∈N

.

(14)

In turn, ifM contains an uncountably infinite number of
elements, continuous convex combinations (i.e., convex
integrals) of infinitely many elements can be considered
too but are not necessarily contained in Conv(M).

Clearly, any behaviour that admits a decomposition
in terms of a convex integral of uncountably infinitely
many behaviours, admits also a decomposition in terms
of a convex sum of finitely many behaviour. Similarly,
any behaviour that admits a decomposition in terms of a
convex sum of an arbitrary finite number of behaviours
admits also a decomposition in terms of a convex sum of
two behaviours. This leads us to the same definition of
extreme no-signaling behaviours as in discrete variables.

Definition 3 (Extreme no-signaling behaviours). We
call µ an extreme point ofMNS if, for any µ∗,µ′ ∈MNS

and 0 ≤ q ≤ 1 such that

µ = qµ∗ + (1− q)µ′ (15)



6

it holds that either q = 1 and µ∗ = µ, or q = 0 and
µ′ = µ.

Now, we know that every µ ∈ MPR has a finite num-
ber of outcomes with non-zero probability and belongs
to MNS. That is, µ is either an extreme point of MNS

or it can be decomposed as the convex sum of at most
finitely many points in MNS. However, the fact that
finite-dimensional PR boxes are no-signalling extreme
implies that the former is the case. This follows from
the fact that finite-dimensional PR boxes are given by
an equivalent expression to that in Eq. (12) where Kro-
necker deltas are in the place of the Dirac ones [15]. This
proves, then, that all CV PR boxes are no-signaling ex-
treme:

Observation 1 (Extremality of MPR). All elements of
MPR are extreme points of MNS.

Observation 1 constitutes, in turn, a generalisation to
the CV realm of the result of Ref. [38], where it is shown
that any extreme point of the no-signaling set with a
given finite number of inputs and outputs is also ex-
treme in the no-signaling set with any higher (but still
finite) number of inputs and outputs. In addition, since
MPR is not finite, the observation also directly implies
that MNS is not a polytope. On the other hand, the
fact that MNS contains behaviours with infinitely many
outcomes with non-zero probability (e.g., the Gaussian
PR boxes of the previous section) automatically implies
that MNS 6⊆ Conv(MPR), in striking contrast with the
finite-dimensional case. This is due to the fact that ev-
ery behaviour in Conv(MPR) necessarily has only finitely
many outcomes with non-zero probability. Nevertheless,
we show in App. B thatMNS is approximated arbitrarily
well by Conv(MPR), in the formal sense of there existing,
for all µ ∈ MNS, a sequence of elements in Conv(MPR)
that converges to µ. This proves the following.

Theorem 1 (Conv(MPR) dense in MNS). The clo-

sure Conv(MPR) of Conv(MPR) equals MNS. In other
words, Conv(MPR) is a dense subset of MNS.

The theorem is proven in detail in App. B. Let us
sketch the proof idea here. We consider first the case
of behaviours defined on a compact domain [−K,K]2.
There, we can use standard techniques from measure the-
ory to show that for any no-signaling behaviour µ one
can find a sequence of convex sums of CV PR boxes µn
that converges to it. The main idea is then to define the
considered sequence in such a way that its components
become good approximations of the components of µ, in
the limit of large n. This procedure can be seen as a gen-
eralization to the approximation of a function by piece-
wise constant functions as it is used in integration theory.
Next, one generalizes this further to an infinite sequence
of compact intervals which, in the infinite-length limit,
covers the whole space R× R.

Even though MPR consists exclusively of extreme
points ofMNS, the fact that Conv(MPR) is a strict sub-
set of MNS in principle leaves room for other extreme

points in MNS that are not contained in MPR. In the
following, we approach this problem systematically by fo-
cusing first on behaviours with compact support. In this
case, a related problem was addressed by D. Milman,
who proved that, given a compact convex subset C of a
locally convex space E (see [39] for a definition of locally

convex) and another set T ⊂ C such that Conv(T ) = C,
it follows that all extreme points of C are in the closure
of T [39]. The space M[−K,K]2 of probability measures

with bounded domain [−K,K]2 ⊂ R2, is a compact sub-
set of the locally convex space of all measures on the same
domain. The same holds also for the set of behaviours
M4

[−K,K]2 . Moreover, the set of no-signaling behaviours

on [−K,K]2 is a closed subset ofM4
[−K,K]2 and thus also

compact, which enables us to use Milman’s theorem to
characterize its extreme points. In what follows, we deal
with no-signaling and PR box behaviours on a compact
domain. To emphasize this, we equip the correspond-
ing no-signaling set and the set of CV PR boxes with a

superscript K, i.e. M(K)
NS and M(K)

PR . Consequently, we
arrive at the corollary:

Corollary 1 (Characterization ofM(K)
NS ). Every extreme

point of M(K)
NS belongs to the closure of M(K)

PR .

Further on, it is interesting to investigate if the closure

of M(K)
PR contains behaviours that are extreme as well.

If this was not the case, it would prove that all extreme

points ofM(K)
NS are inM(K)

PR . We thus have to answer the
question if PR boxes of infinite order, i.e. in the limit
k →∞ (see Eq. (12)), are also extreme. In Appendix C
we provide evidence suggesting that this is not the case.
More precisely, we provide an examplary sequence of PR
boxes whose limiting behaviour is not extreme, thus im-

plying thatM(K)
PR is not a closed set. This evidence leads

us to the following conjecture.

Conjecture 1 (Characterization of M(K)
NS ). Every ex-

treme point of M(K)
NS belongs to M(K)

PR .

Even though, the preceding discussion was restricted
to behaviours with outcomes on a compact set, we have
reasons to believe that the conjecture holds also in the
general case of unbounded support. Namely, in probabil-
ity theory it is a rather standard result that all extreme
points of the set of probability measures are given by
Dirac measures (see Eq. (5)). In particular, this is the
case for probability measures defined on R. Similarly,
the extreme no-signaling behaviours may have also only
finite support, which would suggest our Conjecture 1 also
in the general case of behaviours defined on R. A proof of
Conjecture 1 would however require more involved argu-
ments which go beyond the scope of the present article.

Let us finish with some final clarifications on the
boundary and the boundedness of MNS. In the finite-
dimensional case, the boundary between the no-signaling
behaviours and behaviour-like objects that still sat-
isfy the no-signaling constraints but involve non-positive



7

probability distributions is given by the subset of all con-
vex combinations of no-signaling extreme points result-
ing in non strictly-positive behaviours (i.e., whose (x, y)-
th components are probability measures assigning zero
probability to some event). Consequently, the set of no-
signaling behaviours has a nonempty interior. In con-
trast, for infinite dimensional behaviours, the boundary
of MNS is actually MNS itself showing that its interior
is empty. The latter can be proven using convergence
arguments similar to those used in the proof of Theorem
1, i.e. every no-signaling behaviour is arbitrarily close
(in the weak-convergence sense) to a non-positive no-
signaling beahviour. This may at first blush seem bizarre,
but it is actually a typical property of compact convex
sets in infinite-dimensional spaces. Indeed, the sets of
probability distributions or quantum sates for infinite-
dimensional systems display exactly the same property
(see, e.g., Ref. [40]).

Lastly, we stress that in the present work we did not
touch the question of whether the setMNS is bounded or
not. Doing so would require to introduce an appropriate
metric and, as we are dealing with infinite dimensional
spaces, the boundedness of the set MNS might depend
on its particular choice. For instance, with respect to
the Lévy-Prokhorov metric, which is a metric on the set
of probability measures associated to the weak topology,
the set of all probability measures is bounded. Hence,
for this metric also the no-signaling set is bounded, since
the components of behaviours are by definition always
probability measures.

V. FINAL DISCUSSION

We have studied supraquantum Bell correlations in a
genuinely CV regime, i.e., without discretisation proce-
dures such as binning [20–26]. To the best of our knowl-
edge, this is the first such investigation reported. Here,
genuine CV supraquantumness was witnessed by the vio-
lation of the CFRD inequality [30], which, for the bipar-
tite case, is known not to admit any quantum violation
[27, 29]. We found a class of supraquantum Gaussian PR
boxes, whose zero-width limit gives the CV PR boxes.
Here, we have explicitly checked the supraquantumness
of both Gaussian and CV PR boxes of order k = 2. Inter-
estingly, due to symmetries in the CFRD inequality, no
violation can be found for k = 3, but supraquantumness
of CV PR boxes of higher orders is guaranteed by the
supraquantumness of the equivalent boxes in finite di-
mensions. In turn, the supraquantumness of finite-width
Gaussian PR boxes of higher order can be verified violat-
ing – via some appropriate binning – finite-dimensional
Bell inequalities above their quantum limit; but this is
outside the scope of this paper.

In addition, we have characterised the set of CV no-
signaling correlations from a geometrical point of view.
To this end, we devised a mathematical framework to
deal with arbitrary CV no-signaling behaviours, based on

conditional probability measures instead of conditional
probability distributions. With this, we have shown that,
for CV systems, the convex hull (i.e. the set of all fi-
nite convex sums) of all CV PR boxes is dense in the
no-signaling set, instead of equal to it as in finite dimen-
sional systems. In particular, this result tells us that
every no-signalling behaviour can be approximated ar-
bitrary well by a sequence of behaviours with a finite
number of non-zero probability outcomes. Consequently,
the nonlocality of every CV no-signaling behaviour can
always be detected with discrete Bell inequalities in com-
bination with a binning procedure, for sufficiently large
number of bins.

Since every CV PR box assigns a non-zero probability
to a finite number of outcomes, being thus in one-to-
one correspondence with a discrete PR box in the usual
finite-dimensional scenario, it is not surprising that ev-
ery CV PR box is extreme in the no-signaling set. In
contrast, the possibility that all extreme points of the
no-signaling set are given by CV PR boxes, as suggested
by Conjecture 1, appears as more surprising. Indeed,
it would evidence a qualitative difference between the
structure of quantum theory and that of generic proba-
bility theories compatible with the no-signaling principle,
a question that has been previously considered in other
scenarios too [41]. Namely, in quantum theory we know
about the existence of behaviours with an uncountably
infinite number of non-zero probability outcomes which
are extreme in the set of CV quantum correlations. The
latter quantum behaviours can be built, e.g., with ex-
treme quantum POVMs with a continuous spectrum [42–
44] acting on pure CV entangled states. We leave the
proof (or disproof) of this conjecture as an open ques-
tion for future investigations.

Another interesting question for future investigations
is how to formalize the notion of tightness [34] for CV
Bell inequalities; and, in particular, whether the CFRD
inequality is tight or not. In finite dimensions, non-trivial
tight Bell inequalities without a quantum violation exist
in the multipartite scenario [17], but no equivalent exam-
ple is known for bipartite systems. If the CFRD inequal-
ity were tight, our results would give it the status of the
first known example of a non-trivial tight Bell inequality
with no quantum violation in the bipartite setting.

To end up with, far from being just a mere abstract
exercise, studying supraquantum non-locality helps us
understand quantum non-locality itself. Efficient tools
to study non-locality for discrete systems –such as semi-
definite or linear programming– no longer apply for CV
systems; so that the characterization of non-local corre-
lations is a much harder task. We thus hope that our
findings can be useful for future research, such as, e.g.,
searching for novel CV Bell inequalities or, more gener-
ally, studying generalized-probability theories in CV sys-
tems.



8

ACKNOWLEDGMENTS

LA thanks D. Cavalcanti for useful comments and the
International Institute of Physics (IIP) Natal, Brazil,
where parts of this work were carried out, for the hos-
pitality. LA’s work is financed by the Brazilian min-
istries MEC and MCTIC and Brazilian agencies CNPq,
CAPES, FAPESP, FAPERJ, and INCT-IQ. ALF and AK
acknowledge CAPES-COFECUB project Ph-855/15 and
CNPq for financial support. AK acknowledges financial
support from the ERC (Consolidator Grant 683107/Tem-
poQ), and the DFG. A special thank goes to T. Cabana,
S. Kaakai and C. Ketterer for the clarification of mathe-
matical details, and to J. P. Pellonpää for valuable discus-
sions about extremality problems in quantum mechanics.

Appendix A: Radon-Nikodym Theorem

A measurable space is given by a pair (X,Σ), where
X is some nonempty set and Σ denotes a σ-algebra on
X. We define a measure ν on X to be σ-finte if X is a
countable union of measurable sets Xi with finite mea-
sure ν(Xi) < ∞. Note that every probability measure
µ on R is also σ-finite since, on the one hand, R can be
expressed as countable union of measurable set and, on
the other hand, we have by definition that µ(A) < 1, for
all A ⊂ R. Furthermore, a measure ν is called absolutely
continuous with respect to µ, if from ν(A) = 0 it follows
µ(A) = 0, for every measurable set A ⊂ X. Now, we are
in the position to state the Radon-Nikodym Theorem.

Theorem 2 (Radon-Nikodym). Given a σ-finite mea-
sure ν on (X,Σ) that is absolutely continuous with re-
spect to a σ-finite measure µ on (X,Σ), then it exists a
measurable function f : X → [0,∞), referred to as the
Radon-Nikodym derivative, such that

ν(A) =

∫
A

f dµ, (A1)

for any measurable subset A ⊂ X.

Appendix B: Proof of Theorem 1

Before turning to the proof of Theorem 1 we provide
some preliminary notions of the type of convergence that
we will use in the following, i.e. the weak convergence.
We say that a sequence of measures (µn)n∈N ∈MΩ con-
verges weakly towards same µ ∈MΩ, with n→∞, if:∫

Ω

fdµn →
∫

Ω

fdµ, (B1)

for all f ∈ Cb(Ω), where Cb(Ω) denotes the set of
bounded and continuous functions f : Ω → R. In what
follows, if not stated differently, we will always implicitly
assume the use of weak convergence for sequences of mea-
sures. Moreover, since we often consider behaviours (i.e.

matrices with entries given by probability measures), we
say that a sequence of behaviours µn weakly converges
to µ if [µn]x,y → [µ]x,y,∀x, y.

Weak convergence is a natural choice in the present
context because it is directly applicable to sequences
of measures without resorting to a specific distributions
in terms of some random variables. Other, possibly
stronger, notions of convergence do exist but are not re-
quired here. Furthermore, as we will see shortly, weak
convergence is also meaningful with respect to physical
considerations since, from experiments, one usually ex-
tracts some statistical moments of a probability measure
instead of the measure itself.

Further on, as stated also in the main text, in order
to prove that Conv(MPR) is dense in MNS we need to
show that for every behaviour µ ∈ MNS one can find
a sequence in Conv(MPR) that converges weakly to µ.
In order to keep the proof of Theorem 1 as instructive
as possible, we will first provide a proof for the case
of behaviours with compact support meaning that their
components are probability measures on Ω = [−K,K]2.
A generalization of the proof to the most general case
Ω = R× R will then be ensued afterwards.

Then, the following Lemma holds.

Lemma 1 (Conv(M(K)
PR ) dense in M(K)

NS ). The closure

Conv(M(K)
PR ) of Conv(M(K)

PR ) equals M(K)
NS . In other

words, Conv(M(K)
PR ) is a dense subset of M(K)

NS .

Note that, according to the introduced nomencla-
ture in the main text we equiped the corresponding no-
signaling set and the set of CV PR boxes with a super-

script K, i.e. M(K)
NS and M(K)

PR .

Proof of Lemma 1. Without loss of generality we can re-
strict the following proof to the case K = 1, i.e. Ω =
[−1, 1]2. The strategy consists of explicitly constructing,

for every arbitrary µ ∈ M(1)
NS, a sequence of behaviours

µn ∈ Conv(M(1)
PR) that weakly converges to µ. The proof

is divided in three steps: First, for every µ ∈ MNS, we
define a sequence of behaviours that weakly converges to
µ. Second, we show that each element of this sequence is
indeed a no-signaling behaviour. Third, we show that all
such elements can be expressed as a convex sum of CV
PR boxes.

For the first step, we divide the interval [−1, 1] in n ≥ 1
segments of the same length, denoting each one by In
(note that a generalization of the following proof to arbi-
trary K’s would simply involve a rescaling of the defined
intervals In.). Next, we define µn as follows:

[µn]x,y =

n∑
k,l=1

[µ]x,y(Ik × Il) δak,bl ,∀x, y, (B2)

where (ak, bl) is a point located in the interval Ik × Il
and the Dirac measure is defined according to Eq. (5).
The behaviours µn have the same weight as µ on each
of the squares Ik × Il, but concentrated on a single point



9

(ak, bl). In this way, µn becomes better and better ap-
proximations of µ, with increasing n.

To prove that µn is indeed weakly converging to µ,
it suffices to prove that each of its component is weakly
converging to the components of µ. Let f be a bounded
and continuous function defined on the domain [−1, 1]×
[−1, 1]. Integrating f with respect to µx,yn , yields:

∫
[−1,1]2

fd[µn]x,y =
∑
k,l

f(ak, bl)[µ]x,y(Ik × Il). (B3)

The sum on the right-hand side of Eq. (B3) can be
bounded from below and above in the following way:

∑
k,l

[µ]x,y(Ik × Il) min
Ik×Il

f ≤
∫

[−1,1]×[−1,1]

fd[µn]x,y (B4)

≤
∑
k,l

[µ]x,y(Ik × Il)max
Ik×Il

f,

(B5)

where min
Ik×Il

(max
Ik×Il

) denotes the minimum (maximum) of

the function f over the cell Ik × Il. The same inequality
holds if we integrate f with respect to [µ]x,y, and, since f
is continuous, this proves that

∫
fd[µn]x,y →

∫
fd[µ]x,y

and that [µn]x,y → µ,∀x, y. It follows that µn → µ.

As for the second step, we now prove that µn is no-
signaling for all n. For a given n > 0 and x, y ∈ {0, 1}n,

the marginal of [µn]x,y on Bob’s side is given by:

[µn]x,y([−1, 1]×B) =

n∑
k,l=1

[µ]x,y(Ik × Il)δak,bl([−1, 1]×B)

=
∑
l

δbl(B)
∑
k

[µ]x,y(Ik × Il)

=
∑
l

δbl(B)[µ]x,y([−1, 1]× Il),

(B6)

where δbl is the Dirac measure located at bl in the lth in-
terval. Since we know that µ is a no-signalling behaviour
it follows that [µ]x,y([−1, 1] × Il) does not depend on x
(compare with Eq. (2a)). The same argument holds for
the Alice’s marginal and proves that the µn’s are no-
signalling behaviours.

The third and last step to complete the proof is to show
that µn can be written as a convex sum of finitely many
CV PR boxes. For this we note that the µn’s are no-
signalling behaviours with a finite number of outcomes
(the centers of the intervals Ik,l) and support [−1, 1]2.
However, we know form the finite-dimensional case that
all behaviours with only finitely many outcomes with
non-zero probability can be expressed as a convex com-
bination of finitely many PR boxes. Taking instead their
continuous-variable generalizations (12), yields the de-
sired decomposition.

With Lemma 1, we next prove Theorem 1.
Proof of Theorem 1. Now, we consider the case Ω =
R × R. Again, we consider a µ ∈ MPR and want to
prove that there exists a sequence µn ∈ Conv(MPR) for
which each component converges weakly to the compo-
nents of µ. To do so, we divide [−n, n], with n ≥ 1, in
2n2 subintervals of length 1/n and denote them by In as
before. Furthermore, we define the components of µn as
follows:

[µn]x,y =

2n2∑
k,l=1

[µ]x,y(Ik × Il)δak,bl + [νn]x,y, (B7)

where (ak, bl) is a point located in the square Ik × Il,
δak,bl is the Dirac measure. The first term of Eq. (B7)
corresponds to the same construction as in the com-
pact case treated in Lemma 1, whereas the second term
νn is merely necessary to ensure the no-signaling condi-
tions (2a) and (2b) on R× R. It reads as follows:



10

[νn]x,y =

n∑
l=−n

(
[µ]x,y

(
]n,∞[× Il

)
δn+1,bl + [µ]x,y

(
]−∞,−n[× Il

)
δ−(n+1),bl

)
+

n∑
k=−n

(
[µ]x,y

(
Ik × ]n,∞[

)
δak,(n+1) + [µ]x,y

(
Ik × ]−∞,−n[

)
δak,−(n+1)

)
+ [µ]x,y

(
]n,∞[× ]n,∞[

)
δn+1,n+1 + [µ]x,y

(
]−∞,−n[× ]n,∞[

)
δ−(n+1),n+1

+ [µ]x,y
(
]n,∞[×]−∞,−n[

)
δn+1,−(n+1) + [µ]x,y

(
]−∞,−n[× ]−∞,−n[

)
δ−(n+1),−(n+1), (B8)

where ]a, b[ refers to an open interval bounded by a and b,
respectively. Note that, in contrast to the compact case
treated in Lemma 1, the measures [µn]x,y are defined on
different intervals for different n’s. We will now complete
the proof of Theorem 1 by showing the weak convergence
of this sequence in the general case. The other parts of
the proof remain unchanged.

Let f ∈ Cb(R2) and ε ∈ [0, 1], we want to prove that
there exists an n0 ∈ N such that |

∫
R2 fdµn−

∫
R2 fdµ| < ε

for all n > n0, where this inequality should be understood
as component wise inequality. Since µ is a set of proba-
bility measures and f is a bounded function, there exists
an n1 ∈ N such that:

[µ]x,y(R2 \ [−n1, n1]) < min

(
ε,

ε

maxR2 |f |

)
, (B9)

for all (x, y) and all n > n1. It follows that:∣∣∣ ∫
R2

fd[µn]x,y −
∫

R2

fd[µ]x,y

∣∣∣
<
∣∣∣ ∫

R2\[−n1,n1]2
fd[µn]x,y −

∫
R2\[−n1,n1]2

fd[µ]x,y

∣∣∣
+
∣∣∣ ∫

[−n1,n1]2
fd[µn]x,y −

∫
[−n1,n1]2

fd[µ]x,y

∣∣∣. (B10)

While the first term on the right and side of inequality
(B10) becomes:∣∣∣ ∫

R2\[−n1,n1]2
fd[µn]x,y −

∫
R2\[−n1,n1]2

fd[µ]x,y

∣∣∣
≤
∣∣∣ ∫

R2\[−n1,n1]2
fd[µn]x,y

∣∣∣+
∣∣∣ ∫

R2\[−n1,n1]2
fd[µ]x,y

∣∣∣
≤ maxR2 |f |[µn]x,y(R2 \ [−n1, n1]2) + ε

= maxR2 |f |[µ]x,y(R2 \ [−n1, n1]2) + ε

≤ 2ε, (B11)

the second term contains an integration over a compact
area, which allows us to use the statement of Lemma 1.
Hence, we can conclude that this term is smaller than
ε for sufficiently large n. Note that Lemma 1 does
not apply directly here since the considered sequence of
behaviours is not no-signaling on the compact domain

[−n1, n1]2, but rather on R2. However, dropping the no-
signaling condition does not contradict with the conver-
gence of this sequence. By combining inequalities (B10)
and (B11) we finally arrive at∣∣∣ ∫

R2

fd[µn]x,y −
∫

R2

fd[µ]x,y

∣∣∣ < 3ε, (B12)

for n sufficiently large. This quantity goes to zero as ε
goes to zero and thus µn weakly converges to µ.

Appendix C: Concerning Conjecture 1

Here we construct a specific example of a sequence of
CV PR boxes, with increasing order k, whose limit is not
an extreme no-signaling behavior anymore. This suggests
that one cannot obtain extreme no-signaling behaviors as
limits of a sequences of CV PR boxes when the order k
goes to infinity. We will restrict ourselves to measures on
[0, 1]2 but it can be straightforwardly extended to R2.

Proof. We prove that there is a sequence µn ∈M(1)
PR that

converges to an element µ that is outside ofM(1)
PR. Let µ

be the set of measures where the two outcomes are always
perfectly correlated for all settings: µx,y(a, b) = δ(a− b).
µ is clearly no-signaling, but not extreme.

We define µn as follows:

µx,yn =

{
1
n

∑n
k=0 δ kn ,

k
n
, for x · y = 0,

1
n

[∑n−1
k=0 δ k

n ,
k+1
n

+ δ1,0

]
, for x · y = 1,

(C1)

which yields∫∫
[0,1]2

f(a, b)µx,yn (a, b) (C2)

=

{
1
n

[∑n
k=0 f

(
k
n ,

k
n

)]
, for x · y = 0,

1
n

[∑n−1
k=0 f

(
k
n ,

k+1
n

)
+ f(1, 0)

]
, for x · y = 1,

(C3)

where f ∈ Cb([0, 1]2). Now, by a apply-
ing standard integration theory it follows that[

1
n

∑n−1
k=0 f

(
k
n ,

k+1
n

)
+ f(1, 0)

]
→

∫
[0,1]

f(a, a) =



11∫∫
[0,1]2

f(a, b)µx,y(a, b). We thus proved that µn con-

verges to an element that is outside ofM(1)
PR (since µ has

an infinite number of outcomes contrary to all elements

of M(1)
PR).

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	Continuous-variable supraquantum nonlocality
	Abstract
	I Introduction
	II Preliminaries: mathematical representation of CV Bell correlations
	III Continous-variable supraquantum nonlocality
	IV Characterization of the set of no-signalling behaviours
	V Final discussion
	 Acknowledgments
	A Radon-Nikodym Theorem
	B Proof of Theorem ??
	C Concerning Conjecture ??
	 References