# Bell-type games on deformable manifolds

David H. Oaknin  Rafael Ltd, IL-31021 Haifa, Israel  Amir Kalev  Information
Sciences Institute, University of Southern California, Arlington, VA 22203,
USA  Itay Hen  Information Sciences Institute, University of Southern
California, Marina del Rey, CA 90292, USA  Department of Physics and
Astronomy, and Center for Quantum Information Science & Technology, University
of Southern California, Los Angeles, California 90089, USA

###### Abstract

We study bipartite correlations in Bell-type games. We show that in a setup
where the information carriers are allowed to locally deform the manifold on
which the game is played, stronger correlations may be obtained than those
maximally attainable otherwise. We discuss the implications of our results in
the context of Bell’s theorem and the Einstein-Podolsky-Rosen paradox.

> ‘Theoretical physicists live in a classical world, looking out into a
> quantum-mechanical world. The latter we describe only subjectively, in terms
> of procedures and results in our classical domain.’
>
> John S. Bell Bell (1987)

##  I Introduction

Correlations measure the degree to which physical features fluctuate together.
They are responsible for the emergence of complex behavior in physical systems
at all scales, and as such play a crucial role in virtually all areas of
science Parisi (1999). In classical physics, correlations between macroscopic
features are understood as emerging from an underlying deterministic
microscopic theory, defined on an extended phase-space which includes
additional, computationally inaccessible, degrees of freedom. Averaging out
these microscopic degrees of freedom leads in turn to a seemingly
probabilistic behavior of the remaining degrees of freedom and the
correlations thereof. These principles are epitomized in classical statistical
physics, which gives rise to a relatively simple description for macroscopic
physical systems consisting of a large number of microscopic degrees of
freedom L. D. and Lifshitz (1989, 1998); Huang (1987).

Bipartite correlations between physical quantities appearing in the framework
of classical statistical physics are often subject to inequality constraints.
These constraints stem directly from a result from probabilistic logic known
as the Boole-Fréchet inequalities, which places bounds on different
combinations of probabilities concerning logical propositions or events that
are linked together Hailperin (1986); Fréchet (1935). Bell’s inequalities Bell
(1987, 1964); Fine (1982); Clauser  _et al._ (1969); Klyachko  _et al._
(2008); Greenberger  _et al._ (1989); Kochen and Specker (1967) are celebrated
examples of these types of constraints that touch upon the very nature of
physical reality as well as on the foundations of quantum mechanics through
the renowned Einstein-Podolsky-Rosen (EPR) paradox Einstein  _et al._ (1935);
Bohm (1951). Bell’s inequalities restrict the values of certain linear
combinations of correlations between pairs of binary degrees of freedom
arranged in closed loops. A well-known setup where Bell’s inequalities arise
and which is often used to illustrate the distinctive nature of quantum theory
is that devised by Clauser, Horne, Shimony and Holt (abbreviated CHSH) Clauser
_et al._ (1969), which describes a game in which two parties share pairs of
identical particles (the ‘information carriers’) and have full control over
their respective measurement devices, which detect the particles and produce
pairs of binary outcomes. In this paper, we revisit the CHSH-Bell game,
exploring a novel scenario in which the curvature of the manifold on which the
game is played is deformed by the information carriers shared between the two
parties. We find, perhaps counter-intuitively, that the latter setup allows
for stronger correlations than those attainable otherwise.

The paper is organized as follows. The general framework is described in
detail in Sec. II. In Sec. III, we consider the ‘standard’ CHSH-Bell setup,
where the degrees of freedom of the game are embedded in a two-dimensional
flat manifold. In Sec. IV, we examine the new setup, in which the curvature of
the manifold is deformed by the information carriers and in Sec. V, we discuss
the implications of our results, specifically in relation to the CHSH-Bell
inequality violations predicted by quantum theory Peres (1998). We offer
summary and conclusions in Sec. VI.

##  II The bipartite game setup

We derive our results using a bipartite CHSH-Bell-type game (or experiment)
Clauser  _et al._ (1969) played in a two-dimensional world that abides by the
rules of local and deterministic classical physics. We consider a game played
by two parties, Alice and Bob, who are located, respectively, at the two ends
$A$ and $B$ of a segment drawn on the two-dimensional flat plane. In our game,
each party has a directional detector system, which can be oriented on the
plane in a direction chosen by the party. We denote the two-dimensional unit
vectors that describe the detectors associated with Alice and Bob by the
lowercase letters, ${\vec{a}}$ and ${\vec{b}}$, respectively, and further
denote the angle between them generically by $\theta$ (we use a convention
that if $\theta\in[-\pi,\pi)$ is the angle of $\vec{b}$ with respect to
$\vec{a}$ then the relative angle of $\vec{a}$ with respect to $\vec{b}$ is
$-\theta$). We refer to these vectors as the detector’s (or party’s)
direction-vector (or simply as the detector’s vector). In each round of the
game, each party can freely and independently (in a way that will also be
precisely defined later) orient their detector’s vector along one of two
distinct (pre-chosen) directions, ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ for
Alice and ${\vec{b}}_{1}$ and ${\vec{b}}_{2}$ for Bob (see Fig. 1).

Figure 1:  Alice and Bob each choose two distinct orientations for their
respective detectors prior to the start of the game. Alice’s orientations are
denoted $\vec{a}_{1}$ and $\vec{a}_{2}$ and Bob’s are denoted $\vec{b}_{1}$
and $\vec{b}_{2}$ (solid arrows in the figure). The angles between
$\vec{b}_{j}$ and $\vec{a}_{i}$ are denoted $\theta_{ij}$. The dotted arrows
are drawn for the purpose of indicating $\theta_{ij}$.

In addition, we shall assume the existence of an oven located somewhere along
the line connecting the two parties (its precise location will not be of
importance). The oven generates pairs of aligned (unit) vectors pointing along
some random direction on the two-dimensional manifold (in a precise sense that
will be defined later) and transmits one of the copies to Alice and the other
to Bob.

The detectors at each location detect the incoming oven-generated vector, and
compare it to their own orientation in terms of the azimuthal coordinate
$\Delta\in[-\pi,\pi)$ extended from the detector (see Fig. 2). Each detector
then outputs a binary outcome, $S(\Delta)$, either $+1$ or $-1$, according to
the response function

$S(\Delta)=\mbox{sign}(\Delta)=\left\\{\begin{array}[]{ccccc}-1,&\mbox{if}&\Delta&\in&[-\pi,0)\\\
+1,&\mbox{if}&\Delta&\in&[0,\pi).\end{array}\right.$ (1)

The rule given in the above equation is deterministic and local; that is, the
output of each detection event is determined uniquely by the relative
orientation of the local detector’s vector with respect to the incoming copy
of the oven-generated vector, and it is independent of the setting of the
detector of the other party.

Upon a (joint) detection event at the two locations, the parties report the
bits of information output by their respective detectors, along with the
setting of their detector orientation, to an external referee. At the end of
the game, after many rounds, the referee uses the information received from
the parties to evaluate the bipartite correlations for each of the four joint
settings
$({\vec{a}}_{1},{\vec{b}}_{1}),({\vec{a}}_{1},{\vec{b}}_{2}),({\vec{a}}_{2},{\vec{b}}_{1})$
and $({\vec{a}}_{2},{\vec{b}}_{2})$. For any given setting of the pair of
detectors, the (empirical) bipartite correlation is defined as the difference
between the probability (or fraction of events) that the two parties report
the same output for the shared oven-generated vector and the probability (or
fraction of events) that they report opposite outcomes.

We now consider this game as it would be played in two distinct setups, and
study the structure of the bipartite correlations and the corresponding CHSH-
Bell-type constraints that emerge in each case. In the first scenario, the
game is played on the ‘usual’ flat manifold, while in the second we consider a
novel situation in which the oven-generated vectors locally curve the
(otherwise flat) manifold as they traverse it.

In both scenarios, we shall verify that the following natural symmetry
conditions are satisfied, to guarantee that neither party has a preferred
orientation and that both parties are treated on an equal footing (i.e.,
neither one is more special than the other); explicitly:

Condition 1: Both parties must observe an expected zero average of their
detection outcomes for any chosen orientation of their detector.
Condition 2: The bipartite correlations must depend only on the _relative_
orientation of their detectors.

##  III The bipartite game played on a flat manifold

Figure 2:  The bipartite game played on a plane. Alice and Bob sit at two
distinct points $A$ and $B$, respectively, on the two-dimensional plane. Each
party has two available directions to orient their detector:
${\vec{a}}\in\\{{\vec{a}}_{1},{\vec{a}}_{2}\\}$ for Alice, and
${\vec{b}}\in\\{{\vec{b}}_{1},{\vec{b}}_{2}\\}$ for Bob. (a) In each round of
the game, the oven generates two copies of a unit vectors, $\vec{\lambda}$,
one copy is sent to Alice and one for Bob. (b) Each party receives a copy of
an oven-generated vector $\lambda$ and detects its orientation relative to
their own detector. They then report one bit of information each, depending on
the detected relative azimuthal coordinates, $\Delta^{(A)}$ and $\Delta^{(B)}$
Eqs. (2) and (3).

We begin by considering the standard setup in which the two parties, Alice and
Bob, are located at the ends of a segment drawn on the flat two-dimensional
plane, as shown in Fig. 2. As stated in the previous section, each party has a
directional detector that can be freely oriented in either of two possible
directions, ${\vec{a}}\in\\{{\vec{a}}_{1},{\vec{a}}_{2}\\}$ and
${\vec{b}}\in\\{{\vec{b}}_{1},{\vec{b}}_{2}\\}$. In each round of the game,
the oven located somewhere between the two parties produces two identical
copies of a unit vector $({\vec{\lambda}},{\vec{\lambda}})$, which are then
transported to the parties along the path connecting them, one copy per party.
Upon receiving their respective copies of the oven-generated vector, each
party produces an outcome according to the rule specified in Eq. (1). In this
‘standard’ setup, the affine structure of the flat manifold allows one to
refer to the two copies received by the two parties as the same entity
${\vec{\lambda}}$. To compare the orientation of the incoming oven-generated
vector to the corresponding detector’s direction-vector, we use a planar polar
coordinate system, with the angle between any two vectors on the plane defined
in the interval $[-\pi,\pi)$ (see Fig. 2). Denoting the relative azimuthal
coordinate of ${\vec{\lambda}}$ and $\vec{a}$ as
$\Delta^{(A)}\coloneq\Delta({\vec{a}},{\vec{\lambda}})$ and, similarly,
between ${\vec{\lambda}}$ and ${\vec{b}}$ as
$\Delta^{(B)}\coloneq\Delta({\vec{b}},{\vec{\lambda}})$, the output bits of
the two parties are explicitly given, following the rule Eq. (1), by

$S_{A}(\Delta^{(A)})=\left\\{\begin{array}[]{ccccc}-1,&\mbox{if}&\Delta^{(A)}&\in&[-\pi,0),\\\
+1,&\mbox{if}&\Delta^{(A)}&\in&[0,\pi),\end{array}\right.$ (2)

and

$S_{B}(\Delta^{(B)})=\left\\{\begin{array}[]{ccccc}-1,&\mbox{if}&\Delta^{(B)}&\in&[-\pi,0),\\\
+1,&\mbox{if}&\Delta^{(B)}&\in&[0,\pi)\,.\end{array}\right.$ (3)

Throughout this paper we use
$\Delta(\vec{v},\vec{u})=-\Delta(\vec{u},\vec{v})$ to denote the relative
azimuthal coordinate (or coordinate difference) of one vector, $\vec{u}$, with
respect to another $\vec{v}$. We note that while on a flat manifold the
relative azimuthal coordinate between two vectors can be identified with the
_angle_ between them, this identification cannot generally be made when curved
manifolds are concerned.

We can relate the angle between the detectors, $\theta$, to the two
coordinates that determine the outcomes of the detection events,
$\Delta^{(A)}$ and $\Delta^{(B)}$, by noting that the orientation of any oven-
generated vector $\vec{\lambda}$ with respect to $\vec{a}$ equals the angle
between $\vec{\lambda}$ and $\vec{b}$ plus $\theta$, that is,

$\Delta^{(A)}=\Delta^{(B)}+\theta,$ (4)

where addition is defined here modulo $2\pi$ within the interval $[-\pi,\pi)$
(see top panel of Fig. 3 with $\varphi_{A}=\Delta^{(A)}$ and
$\varphi_{B}=\Delta^{(B)}$).

To make quantitative predictions, we turn to the symmetry conditions discussed
in the previous section. Following Condition 1, for each party and for any
direction of their detector, the expected average of the detection events
(i.e., bit outcomes) should be zero, i.e.,

$\langle S_{A}(\Delta^{(A)})\rangle=0\;\;\text{and}\;\;\langle
S_{B}(\Delta^{(B)})\rangle=0,$ (5)

for _any_ choice of $\vec{a}$ and $\vec{b}$ on the unit circle. This condition
determines the distribution of the oven-generated vectors ${\vec{\lambda}}$
over the unit circle with respect to $\vec{a}$ and $\vec{b}$. Under the
relation given in Eq. (4), to satisfy Condition 1, the oven-generated vectors
${\vec{\lambda}}$ must be uniformly distributed over the unit circle. As a
consistency check, we note that following Eq. (4), if the orientation of
$\vec{\lambda}$ with respect to $\vec{a}$, $\Delta^{(A)}$ is distributed
uniformly then, since $\theta$ is constant, so is the orientation of
$\vec{\lambda}$ with respect to $\vec{b}$, $\Delta^{(B)}$, and thus the
requirements of Eq. (5) are satisfied.

Next, let us examine the expected correlation between the two parties in this
model, defined by

$\displaystyle E_{AB}(\theta)$
$\displaystyle=\int\frac{d{\Delta^{(A)}}}{2\pi}\
S_{A}(\Delta^{(A)})S_{B}(\Delta^{(A)}-\theta)$ (6)
$\displaystyle=\int\frac{d{\Delta^{(B)}}}{2\pi}\
S_{A}(\Delta^{(B)}+\theta)S_{B}(\Delta^{(B)})\,,$

where again we make use of the relation given by Eq. (4). The correlation
between Alice and Bob’s outputs, given that their detectors are oriented along
$\vec{a}$ and $\vec{b}$, can be readily obtained by noting that the response
of Bob’s detector changes sign when the oven-generated vector
${\vec{\lambda}}$ is oriented with respect to Alice’s detector at angles
$\Delta^{(A)}=\theta$ and $\Delta^{(A)}=\pm\pi+\theta$ (modulo $2\pi$),

$\displaystyle\Delta^{(A)}=\theta\ \ $ $\displaystyle\Leftrightarrow\ \
\Delta^{(B)}=0,$ (7) $\displaystyle\Delta^{(A)}=\pm\pi+\theta\ \ $
$\displaystyle\Leftrightarrow\ \ \Delta^{(B)}=\pm\pi.$

Since the expected correlation between detectors’ outputs is given by the
probability that they report the same outcome minus the probability they
report an opposite outcome (and the probability density is $1/2\pi$ with
respect to both $\vec{a}$ and $\vec{b}$), we can write the correlation between
Alice and Bob, Eq. (6), as:

$\displaystyle E^{\rm(flat)}_{AB}(\theta)$
$\displaystyle=-\int_{-\pi}^{\theta-\pi}\frac{d\xi}{2\pi}+\int_{\theta-\pi}^{0}\frac{d\xi}{2\pi}-\int_{0}^{\theta}\frac{d\xi}{2\pi}+\int_{\theta}^{\pi}\frac{d\xi}{2\pi}$
$\displaystyle=1-\frac{2}{\pi}\theta,$ (8)

for $\theta\in[0,\pi]$, and

$\displaystyle E^{\rm(flat)}_{AB}(\theta)$
$\displaystyle=\int_{-\pi}^{\theta}\frac{d\xi}{2\pi}-\int_{\theta}^{0}\frac{d\xi}{2\pi}+\int_{0}^{\theta+\pi}\frac{d\xi}{2\pi}-\int_{\theta+\pi}^{\pi}\frac{d\xi}{2\pi}$
$\displaystyle=1+\frac{2}{\pi}\theta,$ (9)

for $\theta\in[-\pi,0]$. Thus, we find that

$E^{\rm(flat)}_{AB}(\theta)=1-\frac{2}{\pi}|\theta|$ (10)

for $\theta\in[-\pi,\pi)$. As expected, the above equation implies

$\displaystyle E^{\rm(flat)}_{AB}(0)$ $\displaystyle=1,$ (11) $\displaystyle
E^{\rm(flat)}_{AB}\Bigl{(}{\pm\frac{\pi}{2}}\Bigr{)}$ $\displaystyle=0,$
$\displaystyle E^{\rm(flat)}_{AB}(\pm\pi)$ $\displaystyle=-1\,,$

which means that the parties are fully correlated when their detectors are
aligned, uncorrelated when their detectors are perpendicular to each other and
are fully anti-correlated when their detectors are anti-aligned.

As per Condition 2, the bipartite correlation between Alice and Bob, Eq. (10),
depends only on the relative orientation (angle) of the two detectors. It is
independent of the (joint) _absolute_ orientation of the detectors (see Fig.
3). The orientations of the two detectors can be described by a ‘difference’
and a ‘sum’ (or average) of their respective angular degrees of freedom; the
former is the relative orientation of the detectors, Eq. (4), and the latter
is the joint absolute orientation of the two detector-vectors. Unlike the
relative angle, which is independent of any external reference, the joint
absolute orientation is measured relative to, and depends on, a choice of an
external reference vector (hence the bipartite correlations must not depend on
this quantity). In particular, upon choosing (for example) the oven-generated
vector as a common reference vector, the joint absolute orientation of Alice
and Bob’s detectors can be given by

$\varphi=-\frac{1}{2}\big{(}\Delta^{(A)}+\Delta^{(B)}\big{)}.$ (12)

Similar to the relative angle, we can define the absolute angle $\varphi$
between pairs of vectors located at different points on the flat manifold.
Thus, on a flat manifold the relative angle $\theta$, Eq. (4), and the joint
absolute angle $\varphi$, Eq. (12), uniquely determine $\Delta^{(A)}$ and
$\Delta^{(B)}$ (with respect to any external reference chosen to define the
absolute angle) and vice versa. In the next section we shall see that similar
arguments cannot generally be made for curved (rather, deformable) manifolds.

Figure 3:  Relative and joint absolute angles on a flat manifold. The top and
bottom parts of the figure depict two distinct orientation settings that must
yield the same bipartite correlations when the game is played on a flat
manifold. In both panels the relative angle between the detectors (denoted as
$\vec{a}$ and $\vec{b}$ in both panels for simplicity) is $\theta$. In the top
panel, the joint absolute angle of the two detectors, with respect to an
external reference vector (marked in blue) is given by
$\varphi=\frac{1}{2}(\varphi_{A}+\varphi_{B})$, whereas in the bottom panel it
is given by $\varphi=\frac{1}{2}(\varphi^{\prime}_{A}+\varphi^{\prime}_{B})$.
Correlations may depend on the relative angle $\theta$ but may not depend on
$\varphi$.

Finally, one can derive an inequality restricting the bipartite correlations
between Bob and Alice by concatenating the four joint settings, $\vec{a}_{1}$,
$\vec{a}_{2}$, $\vec{b}_{1}$, and $\vec{b}_{2}$, of Alice and Bob’s detectors.
Specifically, we observe that for any oven-generated vector ${\vec{\lambda}}$

$\displaystyle\Big{|}S_{A}(\Delta^{(A_{1})})\Big{[}S_{B}(\Delta^{(B_{1})})+S_{B}(\Delta^{(B_{2})})\Big{]}$
(13)
$\displaystyle+S_{A}(\Delta^{(A_{2})})\Big{[}S_{B}(\Delta^{(B_{1})})-S_{B}(\Delta^{(B_{2})})\Big{]}\Big{|}=2,$

where $\Delta^{(A_{i})}{=}\Delta(\vec{a}_{i},\vec{\lambda})$ and
$\Delta^{(B_{j})}{=}\Delta(\vec{b}_{j},\vec{\lambda})$ for $i,j{=}1,2$, since
$S_{B}(\Delta^{(B_{1})})-S_{B}(\Delta^{(B_{2})})=0$ and
$|S_{B}(\Delta^{(B_{1})})+S_{B}(\Delta^{(B_{2})})|=2$ whenever
$S_{B}(\Delta^{(B_{1})})=S_{B}(\Delta^{(B_{2})})$, while
$S_{B}(\Delta^{(B_{1})})+S_{B}(\Delta^{(B_{2})})=0$ and
$|S_{B}(\Delta^{(B_{1})})-S_{B}(\Delta^{(B_{2})})|=2$ whenever
$S_{B}(\Delta^{(B_{1})})=-S_{B}(\Delta^{(B_{2})})$. Therefore, by integration
and the use of the triangle inequality, one readily finds that the four
bipartite correlations are constrained by

$\left|E^{\rm(flat)}_{1,1}+E^{\rm(flat)}_{1,2}+E^{\rm(flat)}_{2,1}-E^{\rm(flat)}_{2,2}\right|\leq
2,$ (14)

where $E^{\rm(flat)}_{i,j}:=E^{\rm(flat)}_{AB}(\theta_{ij})$, $i,j=1,2$, and
$\theta_{ij}=\Delta({\vec{a}}_{i},{\vec{b}}_{j})$ is the angle between
$\vec{b}_{j}$ and $\vec{a}_{i}$. The above equation is the celebrated CHSH-
Bell inequality Clauser  _et al._ (1969).

At this point, it would be worth noting that Eq. (13), which is the basis for
the derivation of the CHSH-Bell inequality, is a statement about four
detection events [namely, $S_{A}(\Delta^{(A_{1})})$,
$S_{A}(\Delta^{(A_{2})})$, $S_{B}(\Delta^{(B_{1})})$, and
$S_{B}(\Delta^{(B_{2})})$] concerning one oven-generated vector,
${\vec{\lambda}}$. The statement assumes the existence of such a vector, which
can participate in all four detection events, despite the fact that in the
game itself each round involves only two detection events
[$S_{A}(\Delta^{(A_{i})})$, $S_{B}(\Delta^{(B_{j})})$, for $i,j\in\\{1,2\\}$].
As already mentioned, in flat space, there exists a global coordinate system
(or an external reference frame) according to which all vectors may be
uniquely described and which defines in a unique manner the identity of
vectors in an absolute way, making it possible to identify a vector
${\vec{\lambda}}$ appearing in say an $({\vec{a}}_{1},{\vec{b}_{1}})$
detection event with a vector appearing in an $({\vec{a}}_{2},{\vec{b}_{1}})$
detection event and so on, justifying the above statement. In the next
section, we study the CHSH-Bell game played on curved (or more precisely,
deformable) manifolds – where there is no such global coordinate system – and
study the implications of the absence of such a system to the bipartite
correlations and the CHSH-Bell inequality.

As a closing remark for this section, we note that throughout this section we
have used the detectors’ direction-vectors as a reference frame for the oven-
generated vectors. We could have instead, and without changing the physical
predictions of the model, described the game using a reference frame that is
attached to and aligned with the oven-generated vectors. From the perspective
of the oven-generated vectors, in each round of the game the two detectors’
direction-vectors are ‘generated’ far away at random angles $-\Delta^{(A)}$
and $-\Delta^{(B)}$ distributed uniformly, and then move in closer toward the
oven-generated vector located in the origin of the reference frame. The vector
is detected and a subsequent response is recorded when the detectors reach the
origin [see Fig. 2(b)].

##  IV The game played on a deformable manifold

In this section, we generalize the previous setup and consider a case wherein
the oven-generated vectors locally deform the curvature of the (otherwise
flat) manifold in which they are embedded, as they traverse it along the path
from the oven towards the detectors 111This is somewhat akin to a massive
object in general relativity inducing space-time curvature.. Specifically, we
consider the case where the oven-generated vectors break the rotational
symmetry of the manifold along the direction in which they are generated but
such that far away from the vectors the manifold remains flat (see Fig. 4 for
a schematic illustration).

In order to explore the novel features of the current framework it will be
convenient, and also illuminating, to describe the bipartite game from a
reference frame attached to and aligned with the oven-generated vectors. From
the perspective of the oven-generated vector, the manifold is curved near the
origin and becomes flatter at increasing distances from the origin. In each
round of the game, two detectors’ direction-vectors are generated far away
from the vector (where the manifold is flat) oriented at random angles (with a
distribution that we precisely describe below), and then move in closer toward
the oven-generated vector. Detector responses are recorded as they reach the
origin.

Figure 4:  The oven-generated vector locally curves space around it, breaking
rotational symmetry. The metric angle between any two vectors is defined by
their scalar product according to the local metric tensor and is depicted in
the color-coded heat map for one possible local deformation of the manifold.
While the relative azimuthal coordinate between the two dotted vectors
($\vec{u}$ and $\vec{v}$) is the same as the relative azimuthal coordinate
between the two dashed vectors ($\vec{u}^{\prime}$ and $\vec{v}^{\prime}$),
the corresponding (metric) angles are different. This figure illustrates that
in the vicinity of the oven-generated vector, the metric angle can no longer
be identified with the azimuthal coordinate.

Let us now consider the path on the manifold that connects the oven and one of
the detectors, say Alice’s. As a convention, we parameterize the path by the
continuous variable $z\in[0,\infty)$, with the oven located at $z=0$ and the
detector located (initially) at $z=\infty$, far away from the oven. Let
$\Delta$ denote the azimuthal polar coordinate on the local tangent planes
along the path. Because we describe the problem from the reference frame of
the oven-generated vector, it is convenient to choose the origin of this
azimuthal coordinate, $\Delta=0$, to be the direction of this vector (see Fig.
5); that is, the zero of the azimuthal coordinate is defined along the path as
the direction in which a parallel-transported vector, aligned at $z=0$ with
the direction of the oven-generated vector, would point. This is a natural
choice in the current model because by (locally) modifying the curvature of
the manifold in a directional way, the oven-generated vectors break the
rotational symmetry of the otherwise isotropic flat manifold, and introduce a
preferred direction on the manifold. We thus use the direction of the oven-
generated vector as a preferred reference frame (a ‘beacon’ to define the zero
direction, or zero azimuthal coordinate, everywhere along the path) such that
all other orientations are measured with respect to it.

Figure 5:  The bipartite game played on a locally deformable manifold. The
oven-generated vector locally curves space around it. The metric angle is
depicted in the color-coded heat map. (a) Away from the oven-generated vector,
at $z=\pm\infty$, space is flat and the metric angle can be identified with
the azimuthal coordinates $\Delta_{\pm\infty}$. (b) At the point of detection,
$z=0$, the metric angle is no longer synonymous with the azimuthal coordinate,
so the angle between the two detectors’ direction vectors is given by the
bijective non-linear function $\Gamma^{-1}$ acting on $\theta$, the relative
angle between the two detectors. (The observant reader will notice that in
contrast to Fig. 2, here the azimuthal coordinates are defined with respect to
the oven-generated vector.)

Before moving on, it is useful to briefly review a few basic concepts from
differential geometry, as we are discussing curved manifolds. On a general
curved manifold, vectors at any point are defined on the ‘tangent plane’
associated with that point. Comparing the orientation of two vectors in the
tangent planes associated with two distinct locations on the manifold requires
‘transporting’ them toward one another along a defined path. Thus, we shall
assume that the manifold is endowed with a ‘metric’ connection (such as the
canonical Levi-Civita connection associated with the metric tensor), which
defines the parallel-transport of (tangent) vectors along a path on the
manifold. By definition, the scalar product of any two vectors defined on the
tangent plane at a given point remains invariant as the vectors are parallel-
transported along a (piece-wise) smooth curve drawn on the manifold. Hence,
the metric angle ${\widetilde{\Delta}}({\vec{u}},{\vec{v}})$ between two
tangent unit vectors, ${\vec{u}}$, ${\vec{v}}$, which is defined as

$\cos({\widetilde{\Delta}}({\vec{u}},{\vec{v}}))=g_{i,j}u^{i}v^{j},$ (15)

where $g_{i,j}$ is the metric tensor at the location of the vectors, also
remains invariant. As mentioned in the previous section, while in flat space
one may identify the difference between the azimuthal coordinates of any two
vectors with the metric angle, this is no longer the case on a curved
manifold. In fact, in our model the metric angle between two vectors located
at a point $z$ on the path is taken to be a (non-linear) function of their
relative azimuthal coordinate, which also depends on their location on the
path, namely, ${\widetilde{\Delta}}={\cal F}(z,\Delta)$. Since the detector is
initially located in a flat region of the manifold (at $z=\infty$) the
azimuthal coordinate in this region does coincide with the metric angle.
Specifically, if the azimuthal coordinate of Alice’s direction-vector is
$\Delta_{\infty}^{(A)}$, then the metric angle
$\widetilde{\Delta}_{\infty}^{(A)}$ of her detector’s vector is simply given
by

$\widetilde{\Delta}_{\infty}^{(A)}={\Delta}_{\infty}^{(A)}.$ (16)

Next, let us consider the parallel transport of vectors along the path
connecting Alice’s detector and the oven. Let us denote the metric angle
between the detector’s vector and the oven-generated vector at $z=0$ (i.e.,
when the former reaches the latter) as $\widetilde{\Delta}_{0}^{(A)}$. Since
the metric angle between the detector’s vector and the oven-generated vector
is conserved along the path, we must have

$\widetilde{\Delta}_{0}^{(A)}=\widetilde{\Delta}_{\infty}^{(A)}.$ (17)

While in the flat region of the manifold we identified the metric angle with
the azimuthal coordinate, in the vicinity of the oven-generated vector, and
specifically, at $z=0$, the manifold is curved and the azimuthal coordinate is
not synonymous with the metric angle. In our model, the oven-generated vector
curves space such that the relation between the azimuthal coordinate and the
metric angle at $z=0$ is ${\widetilde{\Delta}}=\Gamma(\Delta)$, where
$\Gamma:\ [-\pi,\pi)\longrightarrow\ [-\pi,\pi)$ is an anti-symmetric
bijective function, $\Gamma(-\Delta)=-\Gamma(\Delta)$, which we shall specify
below. Therefore, if ${\Delta}_{0}^{(A)}$ is the azimuthal coordinate of
Alice’s detector direction-vector with respect to the oven-generated vector at
$z=0$, then with this definition we find that the metric angle between the two
vectors, $\widetilde{\Delta}_{0}^{(A)}$, is given by

$\widetilde{\Delta}_{0}^{(A)}=\Gamma({\Delta}_{0}^{(A)}).$ (18)

Combining Eqs. (16)-(18), we arrive at the conclusion that the azimuthal
coordinates of the detector’s direction vector at $z=\infty$ and at $z=0$
(with respect to the orientation of the oven-generated vector) are related
through the parallel-transport by

$\Delta_{\infty}^{(A)}=\Gamma(\Delta_{0}^{(A)}).$ (19)

Note that this equation does not depend on the details of the transport
(dynamics) along the the path connecting the detector and the oven, but only
on the final mapping between the azimuthal coordinates at the starting point
$z=\infty$ and the end point $z=0$.

We specify the mapping $\Gamma$ via its inverse $\Gamma^{-1}$ as follows

$\displaystyle\Gamma^{-1}(\Delta)=\left\\{\begin{array}[]{cccc}\ \
2\pi\int_{0}^{\Delta}d\xi\ \rho(\xi),&\mbox{for}&\ \Delta\in&[0,\pi),\\\ &&\\\
-2\pi\int_{\Delta}^{0}d\xi\ \rho(\xi),&\mbox{for}&\
\Delta\in&[-\pi,0),\end{array}\right.$ (23)

where $\rho(\xi)$ is some parity-symmetric,
$\rho(-\xi)=\rho(\xi)=\rho(\pi-\xi)$, non-negative density of probability over
the unit circle normalized so that $\int_{-\pi}^{\pi}d\xi\ \rho(\xi)=1$. We
further demand that $\rho(\xi)$ fulfills all the requirements which would
guarantee that $\Gamma^{-1}$ is bijective. Since by definition, $\Gamma^{-1}$
is anti-symmetric, $\Gamma^{-1}(-\Delta)=-\Gamma^{-1}(\Delta)$, $\Gamma$ is
bijective and anti-symmetric as well. Therefore, in particular, $\Gamma(0)=0$
and $\Gamma(\pm\pi)=\pm\pi$. Moreover, one can show that $\Gamma^{-1}$
preserves orthogonality,
$\Gamma^{-1}\left(\pm\frac{\pi}{2}\right)=\pm\frac{\pi}{2}$ (and similarly for
$\Gamma$). Following Eqs. (19) and (23) we obtain

$\frac{1}{2\pi}d\Delta_{0}^{(A)}=\rho(\Delta_{\infty}^{(A)})\
d\Delta_{\infty}^{(A)}.$ (24)

Equation (24) states that in a probabilistic model in which the oven generates
random vectors with no preferred direction, the azimuthal coordinate
$\Delta_{0}^{(A)}$ of the detector direction-vectors relative to the oven-
generated vector at $z=0$ will be distributed uniformly, while at $z=\infty$
the azimuthal coordinate $\Delta_{\infty}^{(A)}$ will be distributed with a
density of probability $\rho(\Delta_{\infty}^{(A)})$.

The azimuthal coordinates $\Delta_{0}^{(A)}$ and $\Delta_{\infty}^{(A)}$, can
be given a physically intuitive interpretation as follows. The former can be
understood as the ‘proper’ coordinates of the oven-generated vector. In these
coordinates the detector’s direction-vectors appear uniformly distributed,
since the direction of the sampled oven-generated vectors cannot have any
preferred orientation. The latter can be understood as the proper coordinates
of the detector’s direction-vector. In these coordinates the oven-generated
vectors are distributed with a non-uniform distribution $\rho$ as a result of
the deformation of the metric that the oven-generated vectors introduce.

As discussed in Sec. II, the binary response of the detectors is defined by
the sign of the azimuthal coordinate of the oven-generated vector with respect
to the detector direction-vector (at the point of the detection event, i.e.,
the origin $z=0$). In particular, the binary response of Alice’s detector is
then

$S_{A}(-\Delta_{0}^{(A)})=-\mbox{sign}(\Delta_{0}^{(A)})=-\mbox{sign}(\Delta_{\infty}^{(A)}),$
(25)

where there is an additional minus sign in the above equation since
$\Delta_{0}^{(A)}$ is defined to be the azimuthal coordinate of Alice’s
detector direction-vector with respect to the oven-generated vector at $z=0$,
and moreover we used the equality
$\mbox{sign}(\Delta_{0}^{(A)})=\mbox{sign}(\Delta_{\infty}^{(A)})$, since the
function $\Gamma$ is monotonically increasing and $\Gamma(0)=0$.

Of course, the discussion presented above for Alice’s detector similarly
applies to Bob’s detector. Specifically, we may parameterize the path
connecting the oven to Bob’s detector by $z\in(-\infty,0]$ where the oven-
generated vector’s location is at $z=0$ as before and the location of Bob’s
detector is initially at $z=-\infty$. Denoting the azimuthal coordinates of
Bob’s detector with respect to the oven-generated vector at $z=0$ and
$z=-\infty$ by $\Delta_{0}^{(B)}$ and $\Delta_{-\infty}^{(B)}$, respectively,
and following the same arguments we detailed above, we arrive at

$\Delta_{-\infty}^{(B)}=\Gamma(\Delta_{0}^{(B)}),$ (26)
$\frac{1}{2\pi}d\Delta_{0}^{(B)}=\rho(\Delta_{-\infty}^{(B)})\
d\Delta_{-\infty}^{(B)},$ (27)

and

$S_{B}(-\Delta_{0}^{(B)})=-\mbox{sign}(\Delta_{0}^{(B)})=-\mbox{sign}(\Delta_{-\infty}^{(B)}).$
(28)

Following Condition 1 in Sec. II, our model should satisfy a zero-mean
detection average for both Alice and Bob; that is,

$\langle S_{A}(-\Delta_{\infty}^{(A)})\rangle=0\;\;\text{and}\;\;\langle
S_{B}(-\Delta_{-\infty}^{(B)})\rangle=0,$ (29)

for any choice of the detectors’ direction-vector. This condition can
alternatively be written as

$\langle S_{A}(-\Delta_{0}^{(A)})\rangle=0\;\;\text{and}\;\;\langle
S_{B}(-\Delta_{0}^{(B)})\rangle=0.$ (30)

Equation (30) is indeed satisfied in our model since $\Delta_{0}^{(A)}$ and
$\Delta_{0}^{(B)}$ are both distributed uniformly.

To obtain the correlations between the outcomes of the two detectors, we need
to relate, similar to what we did in the previous section, the angle between
Alice and Bob’s detectors $\theta$ to the azimuthal coordinates according to
which the detection events are recorded. One could try to define this relation
as $\Delta^{(A)}_{\infty}=\Delta^{(B)}_{-\infty}+\theta$. However, this
relation would not be consistent with our previous finding that
$\Delta^{(A)}_{\infty}$ and $\Delta^{(B)}_{-\infty}$ are distributed according
to $\rho(\Delta^{(A)}_{\infty})$ and $\rho(\Delta^{(B)}_{-\infty})$,
respectively. In order to obtain a relation that complies with these findings
and that is fully consistent with Eqs. (29) and (30), we begin by writing

$\Delta^{(A)}_{0}=\Delta^{(B)}_{0}+\mbox{C}$ (31)

for some constant C that we now compute. We recall that the addition of
azimuthal coordinates is done modulo $2\pi$ within the interval $[-\pi,\pi)$,
so that Eq. (31) is consistent with variables $\Delta^{(A)}_{0}$ and
$\Delta^{(B)}_{0}$ being both uniformly distributed over the circle and they
comply with the zero-mean condition [Eqs. (29) and (30)]. Setting $C$ to
$\theta$ is also not consistent with the current model, as $\theta$ is the
relative (metric) angle between $\vec{a}$ and $\vec{b}$ while
$\Delta^{(A)}_{0}$ and $\Delta^{(B)}_{0}$ are the azimuthal coordinates in a
regime where space is curved and hence the difference in azimuthal coordinates
between two vectors does not coincide with the metric angle between them. To
determine the value of $C$ we rewrite Eq. (31) as

$\Gamma^{-1}\big{(}\Delta^{(A)}_{\infty}\big{)}=\Gamma^{-1}\big{(}\Delta^{(B)}_{-\infty}\big{)}+\mbox{C}.$
(32)

Now, consider the case where the oven-generated vector is aligned with Bob’s
detector vector, i.e., when $\Delta^{(B)}_{-\infty}=0$. In this case, ${\rm
C}=\Gamma^{-1}\big{(}\Delta^{(A)}_{\infty}\big{)}$, and moreover, since
Alice’s detector has a relative angle $-\theta$ with respect to Bob’s
detector, $\Delta^{(A)}_{\infty}=-\theta$. Therefore, we find that

$\Delta^{(A)}_{0}=\Delta^{(B)}_{0}-\Gamma^{-1}(\theta),$ (33)

where the addition of coordinates is defined modulo $2\pi$ in the interval
$[-\pi,\pi)$. Equation (33) can be equivalently expressed as

$\Delta^{(A)}_{\infty}=\Gamma\big{(}\Gamma^{-1}\big{(}\Delta^{(B)}_{-\infty}\big{)}-\Gamma^{-1}(\theta)\big{)}.$
(34)

Since $\Gamma$ takes a uniformly distributed random variable in $[-\pi,\pi)$
to a zero-mean random variable distributed according to $\rho$, we conclude
that Eq. (29) is fulfilled under Eq. (34) for any angle $\theta$, that is, for
any choice of Alice and Bob’s detector orientations.

We can now obtain the correlation between Alice and Bob for our model. We
already showed that $\Delta^{(A)}_{0}$ and $\Delta^{(B)}_{0}$ are distributed
uniformly. The correlations between the two parties can then be obtained by
noting that, following Eq. (33),

$\displaystyle\Delta^{(A)}_{0}=-\Gamma^{-1}(\theta)\ \ $
$\displaystyle\Leftrightarrow\ \ \Delta^{(B)}_{0}=0,$ (35)
$\displaystyle\Delta^{(A)}_{0}=\pm\pi-\Gamma^{-1}(\theta)\ \ $
$\displaystyle\Leftrightarrow\ \ \Delta^{(B)}_{0}=\pm\pi,$

i.e., the response of Bob’s detector changes when
$\Delta^{(A)}_{0}=-\Gamma^{-1}(\theta)$ and
$\Delta^{(A)}_{0}=\pm\pi-\Gamma^{-1}(\theta)$. We can therefore write the
expected correlation between the outcomes of the two parties, Eq. (6), after
following the same steps taken in the previous section [c.f., Eq. (10)], as

$\displaystyle
E^{\rm(deform)}_{AB}(\theta)=1-\frac{2}{\pi}\big{|}\Gamma^{-1}(\theta)\big{|}=1-4\Big{|}\int_{0}^{\theta}\
d\xi\ \rho(\xi)\Big{|}\,.$ (36)

Note that we were careful to express the correlations as a function of the
relative angle $\theta$ between the two detectors, as defined by the bipartite
experiment. As desired, the bipartite correlations satisfy

$\displaystyle E^{\rm(deform)}_{AB}(0)$ $\displaystyle=1,$ (37) $\displaystyle
E^{\rm(deform)}_{AB}\Bigl{(}{\pm\frac{\pi}{2}}\Bigr{)}$ $\displaystyle=0,$
$\displaystyle E^{\rm(deform)}_{AB}(\pm\pi)$ $\displaystyle=-1.$

similar to the correlations on a flat manifold.

Note also that by properly defining $\rho(\Delta)$, our model is capable of
reproducing any bipartite correlation function as long as it fulfills the
symmetry constraint
$E^{\rm(deform)}_{AB}(\theta)=E^{\rm(deform)}_{AB}(-\theta)=-E^{\rm(deform)}_{AB}(\pi-\theta)$.
For example, by considering $\rho(\xi)=1/2\pi$, that is, a uniform
distribution over the unit circle, we recover the bipartite correlations on
the flat manifold, which was considered in Sec. III and which satisfies the
original CHSH-Bell inequality, Eq. (14). Alternatively, by considering the
probability density distribution

$\rho(\xi)=\frac{1}{4}\left|\sin\left(\xi\right)\right|,$ (38)

we are led to bipartite correlations of the form

$E^{\rm(deform)}_{AB}(\theta)=\cos(\theta),$ (39)

which is the correlation dependence predicted by quantum mechanics for the
CHSH-Bell experiment with pairs of maximally entangled particles. In
particular, for the specific choice of relative angles,
$-\theta_{11}=\theta_{12}=\theta_{21}=\theta_{22}/3=\pi/4$, we obtain that

$\left|E^{\rm(deform)}_{1,1}+E^{\rm(deform)}_{1,2}+E^{\rm(deform)}_{2,1}-E^{\rm(deform)}_{2,2}\right|=2\sqrt{2},$
(40)

in violation of the CHSH-Bell inequality on a flat manifold, Eq. (14), where
$E^{\rm(deform)}_{i,j}=E^{\rm(deform)}_{AB}(\theta_{ij})$ and $\theta_{ij}$ is
the angle between $\vec{a}_{i}$ and $\vec{b}_{j}$.

##  V Discussion

The violation of the CHSH-Bell inequality, Eq. (14), in the model introduced
above implies that the inequality does not apply for deformable manifolds. As
noted in Sec III, the CHSH-Bell inequality is derived under the premise of an
equivalence relation between vectors at different locations on the manifold on
which the game is played, which allows one to refer to a vector as a single
entity independently of the path along which it is transported. While this
assumption is well justified in flat space, we now show that in the current
model, where the oven-generated vectors locally curve the manifold on which
the game is played, this equivalence does not hold. Specifically, we show that
in the current model one cannot use ‘the same’ oven-generated vector to define
uniquely and unambiguously four detection events (corresponding to the four
settings of the detectors) as one does in the derivation of the CHSH-Bell
inequality, c.f., Eq. (13).

For concreteness, let us consider the specific choice of relative angles
$-\theta_{11}=\theta_{12}=\theta_{21}=\theta_{22}/3=\pi/4$ that was discussed
above. Consider a detection event where Alice and Bob’s detectors are
described by the azimuthal coordinates $\Delta^{(A_{2})}_{0}$ and
$\Delta^{(B_{1})}_{0}$ with respect to the oven-generated vector. According to
Eq. (33), these azimuthal coordinates are related by

$\Delta^{(A_{2})}_{0}=\Delta^{(B_{1})}_{0}-\Gamma^{-1}(\pi/4).$ (41)

Now suppose that Alice switches her detector’s setting from orientation
$\vec{a}_{2}$ to $\vec{a}_{1}$. Since
$\Delta^{(A_{1})}_{0}=\Delta^{(B_{1})}_{0}+\Gamma^{-1}(\pi/4)$, we obtain that

$\Delta^{(A_{2})}_{0}=\Delta^{(A_{1})}_{0}-2\Gamma^{-1}(\pi/4).$ (42)

Let Bob now change his detector’s orientation from $\vec{b}_{1}$ to
$\vec{b}_{2}$. This leads us to the relation

$\Delta^{(A_{2})}_{0}=\Delta^{(B_{2})}_{0}-3\Gamma^{-1}(\pi/4).$ (43)

Finally, if Alice changes her detector’s orientation back to $\vec{a}_{2}$, we
arrive at

$\displaystyle\Delta^{(A_{2})}_{0}$
$\displaystyle=\Delta^{(A_{2}\\!\circlearrowleft)}_{0}-3\Gamma^{-1}(\pi/4)+\Gamma^{-1}(3\pi/4)$
(44) $\displaystyle=\Delta^{(A_{2}\\!\circlearrowleft)}_{0}-\delta,$

where $\circlearrowleft$ is used here to denote the new azimuthal coordinate
after the sequence of mappings has been applied, and
$\delta=3\Gamma^{-1}(\pi/4)-\Gamma^{-1}(3\pi/4)$. Of course $\delta$ will be
nonzero in the general case.

Equation (44), also illustrated in Fig. 6, implies that the four concatenated
joint two-detector settings considered in the CHSH-Bell inequality [c.f., Eq.
(13)] can no longer be consistently described simultaneously on a deformable
manifold due to a holonomy. From the point of view of the reference frames
attached to the detector direction-vectors, the oven-generated vectors acquire
a geometric phase $\delta$ over the closed concatenation of detector settings
that enter into the CHSH-Bell inequality. This non-zero geometric phase is
however physically permissible because it preserves the statistical
distribution $\rho$ of the oven-generated vectors over the unit circle, and at
each round of the experiment, only one of the four possible settings of the
two detectors is experimentally accessible.

Figure 6:  Holonomy in the bipartite game played on a deformable manifold. The
four concatenated settings of the two detectors considered in the bipartite
game played on deformable manifold give rise to a holonomy. When a given oven-
generated vector is shared between the four settings,
$(\vec{a}_{2},\vec{b}_{1})$, $(\vec{b}_{1},\vec{a}_{1})$,
$(\vec{a}_{1},\vec{b}_{2})$, and $(\vec{b}_{2},\vec{a}_{2})$ it will
accumulate a geometric phase $\delta$, as schematically shown in the figure.

From the point of view of the reference frame attached to the oven-generated
vector, Eq. (44) reveals that in contrast to the case of the flat manifold
where one may define an absolute frame of reference with respect to which all
vectors on the manifold may be measured against, in the present model, the
deformation of the manifold spontaneously breaks the rotational symmetry
associated with joint absolute rotations of the two detectors. Since the
absolute orientation of the detectors with respect to the oven-generated
vectors acquires a global geometric phase through cyclic coordinate
transformations, it is not a well-defined physical quantity. Rather, it is a
spurious ‘gauge’ degree of freedom that cannot play any role in the physical
predictions of the model, as dictated by Condition 2. We note that this
spontaneous breaking of rotational symmetry is caused by the information
carriers and, therefore, in their absence the symmetry is of course present.
Hence, the parties retain the usual symmetries needed to define the settings
of the detectors’ direction-vectors prior to the start of the game.

In the presence of a pair of oven-generated vectors traversing the manifold
toward the two detectors, breaking the rotational symmetry associated with
joint absolute rotations of the two detectors, each party is left with its own
set of coordinates, related to each other via Eq. (34). As already noted, all
these sets of coordinates are symmetrically equivalent to each other as none
of them is preferred over any other.

##  VI Summary and conclusions

In this work, we studied the bipartite correlations that emerge in two-party
Bell-type games. We showed that by allowing the information carriers shared by
the parties (our oven-generated vectors) to locally deform the curvature of
the manifold on which the game is played, spontaneously breaking the isotropy
of space, classical correlations that violate the usual Bell inequalities
associated with the game may appear. We examined a specific example of a local
and deterministic Bell-type game in which the bipartite correlations match
those predicted by quantum mechanics for actual CHSH-Bell experiments. We
showed that nonetheless our model does not contradict CHSH-Bell’s theorem
Clauser  _et al._ (1969) since a setup in which the manifold is deformed by
the shared information carriers does not satisfy the assumptions underlying
the theorem.

Our model is founded upon the simple realization that the bipartite
correlations predicted by quantum mechanics for Bell states depend only on the
relative orientation between the two detectors involved in the game, while the
angle associated with joint absolute rotations (with respect to the
orientation of the information carriers) is a spurious gauge degree of freedom
that plays no role in the prediction of physical observables. This gauge
degree of freedom allows, in our model, for the appearance of a non-zero
geometric phase through cyclically concatenated coordinate transformations,
which in turn gives rise to bipartite correlations that evade the constraints
imposed by Bell’s theorem. Our local and deterministic model thus proposes a
physically intuitive resolution to the EPR paradox and the correlations
associated with quantum entanglement.

We note that while we have formulated our model on a two-dimensional surface,
our results could have similarly been derived for a higher-dimensional
manifold where the planes in which the oven-generated and detector vectors
reside are orthogonal to the path along which they are connected. The latter
setup would have more closely resembled actual CHSH-Bell experiments wherein
the paired photons shared between the parties travel along paths that are
perpendicular to the polarization plane in which the information is encoded.

One may wonder whether a ‘static’ curved manifold would have sufficed to
reproduce the results of the model presented above. In the model studied here,
in each round of the experiment, the oven-generated vector spontaneously
breaks the isotropy of the manifold, yet on average, over the course of many
rounds, the symmetry of the manifold is recovered. This important feature
would be absent from any model with static curvature as the latter would
either have a broken rotational symmetry at each round of the game (as well as
after averaging over many rounds) or – if the static curved manifold does not
break isotropy – not lead to a violation of the CHSH-Bell inequality.

We believe our work also opens the door to novel interesting research avenues
that have the potential to deepen our understanding of physical reality. The
framework presented here offers a physically intuitive explanation to the
experimentally confirmed violations of the renowned Bell inequalities.
Notwithstanding, our model does not single out the ‘choice of nature’ for the
bipartite correlations predicted by quantum mechanics. Expressed differently,
while our mathematical framework allows the bipartite correlations to go
beyond the _classical_ limit, it also allows to reach the maximal
algebraically allowed violations of the inequalities and, hence, it does not
explain the origin of the Tsirelson bound Cirel’son (1980) imposed by the
quantum formalism. We leave the exploration of this topic to future work.

The formalism presented in this study generalizes to all other Bell-type
experiments with pairs of maximally entangled particles. In future studies, we
plan to explore additional Bell-type games that involve the correlations
encountered in three-particle systems such as Greenberger-Horne-Zeilinger
(GHZ) Bell games. Also worth exploring are experiments that may be sensitive
to the curvature of space introduced by the information carriers in actual
Bell experiments, as posited in our model.

The relation between the framework developed here and the fundamental
principles underlying General Relativity is of great interest as well. Thus
another line of research that we believe is worth pursuing is the formulation
of our mathematical framework in pseudo-Riemannian Lorentzian manifolds,
relating our derivation above directly to the general theory of relativity.

Finally, this work suggests that, very generally, when formulating a
statistical model of a physical system involving gauge degrees of freedom, one
should take into account the possibility that the system is prone to having
the symmetries of its macroscopic phase space spontaneously broken, allowing
for the occurrence of geometric phases associated with the gauge degrees of
freedom.

## References

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††thanks: 0000-0002-5209-872X††thanks: 0000-0002-1235-0505

# A Modern Look at the Oscillation Physics Case for a Neutrino Factory

Peter B. Denton<EMAIL_ADDRESS>High Energy Theory Group, Physics Department,
Brookhaven National Laboratory, Upton, NY 11973, USA  Julia Gehrlein
<EMAIL_ADDRESS>Physics Department, Colorado State University,
Fort Collins, CO 80523, USA

(July 2, 2024)

###### Abstract

The next generation of neutrino oscillation experiments, JUNO, DUNE, and HK,
are under construction now and will collect data over the next decade and
beyond. As there are no approved plans to follow up this program with more
advanced neutrino oscillation experiments, we consider here one option that
had gained considerable interest more than a decade ago: a neutrino factory.
Such an experiment uses stored muons in a racetrack configuration with
extremely well characterized decays reducing systematic uncertainties and
providing for more oscillation channels. Such a machine could also be one step
towards a high energy muon collider program. We consider a long-baseline
configuration to SURF using the DUNE far detectors or modifications thereof,
and compare the expected sensitivities to the three-flavor oscillation
parameters to the anticipated results from DUNE and HK. We find that a
neutrino factory can improve our understanding of CP violation and also aid in
disentangling the complicated flavor puzzle.

##  I Introduction

The next generation of neutrino oscillation experiments, including JUNO
Abusleme  _et al._ (2022), DUNE Abi  _et al._ (2020), and HK Abe  _et al._
(2015) are expected to make significant headway in addressing three of the
remaining known unknowns in particle physics. Notably, they will have
excellent $\gg 5\sigma$ sensitivity to the mass ordering (the sign of $\Delta
m^{2}_{31}$), they will have good $\theta_{23}$ octant determination
capabilities provided that it isn’t too close to maximal, and both DUNE and HK
will each individually provide $>5\sigma$ sensitivity to CP violation for
significant regions of parameter space. In addition, they will provide some
resolution on measuring $\delta$ (we use the standard parameterization of the
PMNS Pontecorvo (1957); Maki  _et al._ (1962) matrix Denton and Pestes
(2021)). Atmospheric neutrino experiments such as HK, the IceCube-upgrade
Ishihara (2021), and KM3NeT-ORCA Adrian-Martinez  _et al._ (2016) will also
provide complementary information. For a recent overview of this picture, see
Denton  _et al._ (2022).

While the outlook is hopefully rosy, it is important to carefully examine
what, if anything, should come after these experiments in the neutrino
oscillation program. Several experiments have been proposed with varying
levels of detail. One is using a water-based liquid scintillator detector
THEIA which combines the advantages of water and liquid scintillator, likely
in place of one of the four LArTPC modules for DUNE Askins  _et al._ (2020,
2022). Another is an additional HK-like tank in Korea experiencing a different
portion of the same beam that HK experiences to target the second oscillation
maximum in appearance Abe  _et al._ (2018). There is also discussion about
building a very large (much larger than HK) water Cherenkov detector in Sweden
combined with additional beam upgrades of the ESS to create the ESSnuSB also
to target the second oscillation maximum in Alekou  _et al._ (2021, 2022);
Abele  _et al._ (2023).

In this article we discuss a different option, a neutrino factory. Neutrino
factories were seriously considered as an option to probe CP violation in the
first decade of this millennium largely due to the expectation of small
$\theta_{13}<1^{\circ}$ which severely limits the possible size of the CP
signal De Rujula  _et al._ (1999). In general, a neutrino factory is a muon
storage ring in a racetrack configuration with long straight sections pointing
towards the far detector and provides powerful beams of both muon and electron
neutrinos of opposite charges Geer (1998). The initial spectrum of this decay
is extremely well characterized unlike the initial spectrum from traditional
fixed target neutrino oscillation experiments which primarily produce
neutrinos from light meson decays with a range of energies. See Bogacz  _et
al._ (2022) for a recent review of the history of neutrino factories as well
as a discussion of the non-oscillation physics program at such a machine.

A neutrino factory may also make sense as a first step on the path to a multi-
TeV high energy muon collider to probe the electroweak sector, see e.g.
Accettura  _et al._ (2023). This first step could be chronologically as a part
of a technology demonstrator, a part of the accelerator chain, or both. In
addition, the recent 2023 P5 report suggested consideration of neutrino
physics to be done with a muon accelerator p5_ . A neutrino factory may also
be a necessary successor to the upcoming experiments if there is a
disagreement in the determination of the oscillation parameters among the
different experiments; a situation that the neutrino field is well accustomed
to.

In this paper we will discuss the oscillation physics at a neutrino factory
considering the current knowledge of the oscillation parameters and address
how a modern look at the oscillation physics case for a neutrino factory
compares to previous studies. We then derive numerical sensitivities for the
precision on $\delta$ for a potential setup of a neutrino factory for two
different baselines and energies, before we conclude.

##  II Neutrino Factory Setup

At a neutrino factory the neutrinos originate from the decays of positively or
negatively charged muons, leading to a beam of (anti-)electron neutrinos and
(anti-)muon neutrinos. There are a few main differences for the neutrino beam
of a neutrino factory compared to conventional neutrino beams from fixed
target experiment:

  * •

The achievable maximal neutrino energy is higher at a neutrino factory as the
produced muon neutrinos have energies up to the muon energy. Indeed the muon
neutrino energy spectrum rises with energy instead of falling like at a fixed
target neutrino oscillation experiment, see fig. 1.

  * •

The composition and the expected energy of the neutrino beam is well known as
radiative corrections to muon decay are small Greub  _et al._ (1994); Laing
and Soler (2008); Tomalak (2022). Furthermore, no $\nu_{\tau}$ get produced in
the source, allowing for clean searches for $\nu_{\tau}$ appearance channels.

  * •

The neutrino beam at a neutrino factory is composed of equally many neutrinos
as anti-neutrinos, albeit of different flavors. In particular, the flux
consists of 50% $\nu_{e}$ which allows for $\nu_{e}\to\nu_{\tau}$ searches,
unlike at fixed target oscillation experiments where the $\nu_{e}$ flux is
designed to be small.

  * •

The neutrino energy is tunable and flexible, which allows for physics studies
at different, well-known energies.

The first studies of a neutrino factory studies were focused on measuring a
small $\theta_{13}$, $\delta$, and the mass ordering and identified problems
with degeneracies in the oscillation analysis Barger  _et al._ (2002); Freund
_et al._ (2001); Huber  _et al._ (2006). For small values of $\theta_{13}$,
charge ID (CID) of muons or electrons was crucial to suppress the beam
background for the appearance channels and to measure $\delta$. For this
reason, many studies assumed a 100 kT magnetized iron detector Choubey  _et
al._ (2011).

With the current and envisioned knowledge of all oscillation parameters and
the present experimental landscape, the task of a future neutrino factory is
no longer to discover the oscillation parameters but more likely to make
precise measurements and/or potentially resolve any discrepancies identified
in previous measurements. Therefore, with a different physics goal and a more
modern experimental landscape, the experimental requirements on a neutrino
factory also necessarily evolve.

Figure 1: The initial neutrino flux composition at DUNE and a neutrino factory
from protons on a fixed target (left axis) or muon decays at the energy
optimized for the FNAL-SURF baseline: $E_{\mu}=5$ GeV (right axis).

We consider two possible neutrino factory scenarios: the neutrino source is
Fermilab and, to make use of existing facilities, the far detector to be
located at SURF, resulting in a baseline of 1284.9 km. Alternatively, we also
consider the neutrino source at Brookhaven National Laboratory, to make use of
the AGS/RHIC/EIC accelerator complex, which leads to baseline of 2542.3 km to
SURF. We assume as matter density for both configurations 2.848
g/$\text{cm}^{3}$. We assume that the far detector is based on a LArTPC
configuration with a total fiducial target mass of 40 kT consistent with the
envisioned DUNE detector. Based on previous exploratory studies of neutrino
factories we assume $10^{21}$ muon (or anti-muon) decays per year Choubey  _et
al._ (2011).

To simulate a neutrino factory we make use of DUNE’s far detector design Abi
_et al._ (2021) in the GLoBES Huber  _et al._ (2005) framework. We also
consider oscillated tau neutrinos events with a tau neutrino CC cross section
from Yaeggy (2023). We include tau events as background only Donini  _et al._
(2011) and don’t consider tau signal events for this three-flavor oscillation
study due to the large uncertainties on the signal efficiency at LAr
detectors; including this channel would improve the sensitivity to measure the
oscillation parameters, although likely at a subleading level due to the lower
statistics because the $\nu_{\tau}$ cross section is suppressed at these
energies. As the initial neutrino flux is well known we assume a normalization
uncertainty of 2% on the electron signal and background, 5% on the muon signal
and background, 10% on the NC background, 20% on the tau background.

##  III Sensitivities

For our simulations we consider priors on the oscillation parameters other
than $\delta$ from Gonzalez-Garcia  _et al._ (2021) and we fix the MO to NO,
as the MO as well as all oscillation parameters will be determined via
multiple independent methods with future experiments and our results are
largely independent of the true MO. Therefore we do not study the sensitivity
of a neutrino factory on these parameters but instead study the possible
improvements a neutrino factory can bring on the precision on $\delta$. This
provides an estimate of the ability of a neutrino factory to differentiate
among differing measurements from upcoming experiments.

We studied the sensitivity to the oscillation parameters as a function of muon
energy as well as the ratio of muon to anti-muon running time. For the FNAL-
SURF (BNL-SURF) baseline we find that a muon energy of $E_{\mu}=5$ (8) GeV
with equal muon and anti-muon running time allows for the most precision on
the determination of $\delta$ for most cases. The higher energy is necessary
for the longer baseline to get the appearance maximum to occur within the muon
decay spectrum at that baseline. Depending on the specific details of the
experimental configuration as well as progress in measuring the oscillation
parameters at NOvA and T2K, SK and IceCube, as well as DUNE and HK, may lead
to slight modifications on these optimal energies.

In fig. 2 we show the expected $1\sigma$ precision on $\delta$ as a function
of CID and exposure for the FNAL-SURF setup for
$\delta_{\text{true}}=0,-90^{\circ}$. We include the anticipated information
from 10 years each of DUNE and HK. For DUNE we assume 480 kT-MW-year which
corresponds to 5 years of each neutrino running and anti-neutrino with 1.2 MW
proton beam and with a total fiducial volume of 40 kT of LAr. For HK we assume
190 kT water detector, 1.3 MW beam running for 10 years with
$\nu:\bar{\nu}=1:3$ from Scott (2021). We simulate both experiments within the
GLoBES framework.

Since the beam at a neutrino factory consists of equal amounts of neutrinos
and anti-neutrinos, there is an inherent large contribution to the final
states coming from the other charge, i.e. the processes $\nu_{e}\to\nu_{\mu}$
and $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ lead to the same final state although
with a different energy spectrum due to different initial spectra and
different oscillation probabilities. Differentiating the events from the
disappearance channel in the analysis of the disappearance channel has been
identified as an important step towards a successful neutrino factory early on
De Rujula  _et al._ (1999). Here we study the impact of CID efficiency
$\epsilon_{CID}$ in terms of either electron or muon charge identification. We
define the CID parameter $\epsilon_{CID}$ as

$\displaystyle N_{\nu_{\text{f,obs}}}$
$\displaystyle=\frac{\epsilon_{f}}{2}\left[(1+\epsilon_{CID})N_{\nu_{f}}+(1-\epsilon_{CID})N_{\bar{\nu}_{f}}\right]$
(1) $\displaystyle N_{\bar{\nu}_{\text{f,obs}}}$
$\displaystyle=\frac{\epsilon_{f}}{2}\left[(1+\epsilon_{CID})N_{\bar{\nu}_{f}}+(1-\epsilon_{CID})N_{\nu_{f}}\right]$
(2)

with the detection efficiency $\epsilon_{f}$ and $N_{\nu}$ is the number of
(anti-)neutrinos of flavor $f$ we get after taking oscillations of the initial
flux into account. DUNE could potentially differentiate a muon final state
from an anti-muon final state with $\epsilon_{CID}\approx 72\%$ Ternes  _et
al._ (2019) from muon capture on argon. In addition, inelasticity can provide
some CID information as well.

Larger values of $\epsilon_{CID}$ could be achieved with magnetized detectors,
an idea pursued by the INO collaboration. They estimate a CID efficiency of
$\epsilon_{CID}\gtrsim 95\%$ for muons with GeV energy Ahmed  _et al._ (2017).
Electron CID at GeV energies has been studied in the scenario of a totally
active scintillator detector in a magnetic field Bross  _et al._ (2008);
Asfandiyarov  _et al._ (2014), and a magnetized LAr detector Rubbia (1977,
2001, 2004, 2009). Given that the final DUNE detector modules remain
undesigned, a discussion of the possible benefit of working to improve CID is
quite timely. In Huber and Schwetz (2008) it has been pointed out that CID can
also be achieved statistically at some level, provided that the energy
resolution is sufficiently good.

Our results show that CID can improve the precision on $\delta$ by 15-20%. We
have also studied the effect of assuming muon and electron CID at the same
time and the results are similar. We find that electron CID has a bigger
impact on the improvements in precision on $\delta$. This is likely due to the
fact that the detected $\nu_{\mu}\to\nu_{e}$ spectrum is more similar to the
detected $\bar{\nu}_{e}\to\bar{\nu}_{e}$ spectrum than the detected
$\nu_{e}\to\nu_{\mu}$ spectrum to the detected
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ spectrum, see appendix A. We find that the
sensitivity of a neutrino factory comes somewhat more from the
$\nu_{e}\to\nu_{\mu}$ golden channel De Rujula  _et al._ (1999); Cervera  _et
al._ (2000) and less from the $\nu_{\mu}\to\nu_{e}$ channel which will be
leveraged by future proton accelerator neutrino experiments. The oscillation
probabilities in these two channels are the same up to the change
$\delta\to-\delta$, so each channel serves as an important cross check of the
other, but provides no fundamentally new insights into the nature of CP
violation or conservation. They do, however, provide a probe of CPT invariance
because a neutrino factory is sensitive to four different appearance channels
which CP, T, and CPT conjugates of each other. Thus a neutrino factory,
especially when combined with DUNE and HK, will provide a valuable unique test
of CPT invariance.

Overall, the improvements on precision on $\delta$ with the BNL-SURF setup are
larger than at the FNAL-SURF setup largely because in the former setup the
neutrinos experience more matter effects Wolfenstein (1978a) due to the higher
neutrino energy which increases the appearance probability.

Figure 2: Precision on $\delta$ in degrees as a function of neutrino factory
exposure and CID. The neutrino factory has a baseline and muon energy of
1284.9 km and 5 GeV and also includes 10 years of both DUNE and HK. The shaded
regions are for electron CID and the lines are for muon CID; electron CID does
slightly better in most cases. The horizontal lines show possibly achievable
muon CID. The true values of $\delta$ are $0$ (left) and $-90^{\circ}$
(right). We assume as ratio of neutrino to anti-neutrino running time of 1:1
for the neutrino factory and DUNE 1:1 and 1:3 for HK.

While increased precision on $\delta$ and other oscillation parameters for the
sake of precision is an important goal of particle physics, we also provide
some context for these sensitivities. Because $\delta$ is the last remaining
oscillation parameter that is largely unconstrained, it plays a key role in
flavor model predictions Girardi  _et al._ (2015); Everett  _et al._ (2021),
along with other parameters such as $\theta_{12}$ and $\Delta m^{2}_{21}$
Gehrlein  _et al._ (2022) which are expected to become measured with much more
precision in the coming years by JUNO Abusleme  _et al._ (2022). While there
are many conceivable predictions on neutrino parameters from many different
structures (see Denton and Gehrlein (2024) for a recent survey), here we focus
on charged lepton corrections as they predict the mixing parameters, but not
the masses. In fig. 3 we show the predictions of $\cos\delta$ from four
popular charged lepton correction structures as discussed in Everett  _et al._
(2021) alongside the expected $1\sigma$ precision on $\cos\delta$ (taking the
wider allowed ranges in all cases due to the sign degeneracy since long-
baseline experiments predominantly measure $\sin\delta$ from their appearance
channels, although they also probe $\cos\delta$ from their disappearance
measurements Denton (2023)). We consider both the precision from the
combination of DUNE and HK as well as the expected improvement from including
the neutrino factory at the FNAL baseline. In table 1 we document the expected
precision on $\delta$ for various experiments.

Figure 3: The predicted regions of $\cos\delta$ due to several popular
structures within the charged-lepton correction framework, adapted from
Everett  _et al._ (2021); GR1 and GR2 are two flavor model predictions based
on the golden ratio Datta  _et al._ (2003); Everett and Stuart (2009);
Rodejohann (2009); Adulpravitchai  _et al._ (2009); Feruglio and Paris (2011);
Ding  _et al._ (2012), TBM is the tribimaximal flavor model prediction
Wolfenstein (1978b); Harrison  _et al._ (2002); Harrison and Scott (2002);
Xing  _et al._ (2006); He and Zee (2003), and HEX is based on a hexagonal
structure Albright  _et al._ (2010). Along the top are the expected $1\sigma$
sensitivities to $\cos\delta$ at the various central values predicted by the
models. DUNE and HK are both at 5+5 years as described in the text, as is
neutrino factory with the FNAL baseline.

We see that while after DUNE and HK there will be some modest discriminating
capabilities in some cases, the addition of neutrino factory will considerably
increase the capability to differentiate flavor models. Combined with JUNO’s
constraint on solar parameters should lead to further discrimination
capabilities as well.

##  IV Discussion and Recommendations

Our results show that CID plays a smaller role than emphasized in studies a
decade ago. This is primarily due to the fact that LArTPC detectors have
sufficiently good energy resolution to distinguish signal from background as
well as identify the different flavors. In fact, the LArTPC energy resolution
could be even better than assumed here, see e.g. De Romeri  _et al._ (2016);
Friedland and Li (2019).

A neutrino factory would also improve the precision on other oscillation
parameters, in particular $\theta_{23}$ and $\Delta m^{2}_{31}$ which have
important connections to flavor models as well as probes of the absolute
neutrino mass scale such as cosmology and neutrinoless double beta decay.
Indeed, a neutrino factory could decrease the 1$\sigma$ uncertainty on
$\theta_{23}$ by factor of two (three) from 480 kT-MW-yr of DUNE with a total
exposure of $80\times 10^{23}\mu$-kT-year for a baseline FNAL(BNL)-SURF.
However a neutrino factory has only limited sensitivity to the solar
parameters, just like DUNE Denton and Gehrlein (2023).

Using a similar detector setup for a neutrino factory as for near-future
experiments could further reduce the uncertainties on the cross section and
detector efficiencies, as there is an ongoing program to measure the neutrino
cross section on Ar and advancing the LAr technology Machado  _et al._ (2019).

A neutrino factory could be an appealing possible future neutrino oscillation
experiment should the results of HK and DUNE disagree. If the tension between
NOvA and T2K in the CP violation measurements Denton  _et al._ (2021);
Chatterjee and Palazzo (2021) persists, DUNE will be able to probe it Denton
_et al._ (2023). However if the tension requires further oscillation studies
at a higher neutrino energy, longer baseline, and overall smaller flux
uncertainties than fixed target oscillation experiments a neutrino factory is
a favorable setup for future studies.111Note that atmospheric neutrino
experiments can also provide oscillation studies at higher energies and longer
baselines however their flux uncertainty and flavor composition is more
uncertain than at a neutrino factory. Furthermore, at a neutrino factory the
neutrino energy is flexible and tunable Delahaye  _et al._ (2013). Finally, at
a neutrino factory six oscillation channels and their CP conjugate ones are
accessible with similar large number of events as the initial neutrino beam
consists equally of $\nu_{e},\nu_{\mu}$.

Table 1: Resolution on $\delta$ for combinations of different experiments. For DUNE we assume 480 kT-MW-yr, for HK we assume 2.47 MT-MW-yr, for the neutrino factory exposure we assume 10 years for a 40 kT LAr detector and $10^{21}$ muon decays per year.  $\delta=(-90^{\circ},~{}0)$ | no CID | 100% eCID | 100% $\mu$CID
---|---|---|---
HK | $(20.8^{\circ},5.6^{\circ})$ | – | –
DUNE | $(17.7^{\circ},9.3^{\circ})$ | – | –
DUNE+HK | $(13.9^{\circ},4.8^{\circ})$ | – | –
DUNE+HK+NF(FNAL) | $(10.6^{\circ},3.8^{\circ})$ | $(8.3^{\circ},3.1^{\circ})$ | $(8.9^{\circ},3.3^{\circ})$
DUNE+HK+NF(BNL) | $(9.2^{\circ},3.9^{\circ})$ | $(7.9^{\circ},3.3^{\circ})$ | $(8.5^{\circ},3.4^{\circ})$

##  V Conclusions

Motivated by the recent P5 report, we study the physics potential of a
neutrino factory in improving the precision on the determination of the
remaining undetermined neutrino oscillation parameters with a focus on the
complex phase $\delta$ which governs the amount of CP violation in the
neutrino sector. We consider two different baselines and muon energies with a
LAr detector based at SURF. Similar baselines and energies were considered as
a low-energy neutrino factory Geer  _et al._ (2007); Bross  _et al._ (2008)
appropriate to measure CPV for large values of $\theta_{13}$ Tang and Winter
(2010); Dighe  _et al._ (2012); Ballett and Pascoli (2012); Christensen  _et
al._ (2013). Such a neutrino factory setup could be achieved with a proton
beam with 2-4 MW Hol (2000); Oza (2001); Albright  _et al._ (2004); Geer and
Zisman (2007); Ankenbrandt  _et al._ (2009).

We find that a neutrino factory has several important benefits for the
community:

  * •

Improved precision on several fundamental parameters including the amount of
CP violation,

  * •

Improved flavor model differentiation capabilities,

  * •

A technological stepping stone on the way to a high energy muon collider to
measure the electroweak sector with increased precision.

Such a machine may also be necessary in the event of a tension in the future
oscillation data, a scenario that currently exists with, albeit at low
significance. As there are numerous valuable benefits to such a machine, we
feel that it would be a strong addition to the neutrino oscillation
experimental landscape moving forward.

We also investigated the role of charge identification of electrons or muons
(or both). We found that CID can lead to some improvement in the sensitivity
to CP violation. The impact is largest with larger statistics as CID helps to
cut through systematics which become relevant at high statistics.

A neutrino factory is a natural successor to the upcoming accelerator based
long-baseline neutrino oscillation program to begin to move the neutrino
sector towards the levels of precision already achieved in other sectors of
particle physics.

###### Acknowledgements.

PBD acknowledges support by the United States Department of Energy under Grant
Contract No. DE-SC0012704.

##  Appendix A Number of events

Figure 4: Number of events at a neutrino factory and DUNE for the BNL-SURF and
FNAL-SURF baseline with muon energy 8 GeV and 5 GeV for 5+5 years of running.
We show the number of events from the $\nu_{e}\to\nu_{\mu}$ channel in red for
varying values of $\delta$ and the dominant background from
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$.

In fig. 4 we show the number of events at a neutrino factory and DUNE. The
background for the FNAL baseline is larger whereas the signal at the BNL
baseline is where the background has a dip. We also note that the shape of the
disappearance spectra is quite different than that from a proton fixed target
source due to the very different spectral shape, see fig. 1.

##  Appendix B Results for BNL baseline

In fig. 5 we show the resolution of $\delta=0,-90^{\circ}$ for the baseline
between BNL-SURF. We see that a BNL based neutrino factory would lead to
significant improvement in the precision on $\delta$, especially for
$\delta=-90^{\circ}$, with only a modest benefit from CID.

Figure 5: Resolution on $\delta$ as a function of exposure of a neutrino
factory with baseline 2542.3 km, corresponding to the distance between BNL-
SURF combined with DUNE and HK. We also show the results for 100% eCID and
100% $\mu$CID as well as without CID.

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