# Scaled 360 layouts:
Revisiting non-central panoramas

Bruno Berenguel-Baeta
Instituto de Investigacion en Ingenieria de Aragon
Department of Computer Science and Systems Engineering
University of Zaragoza
Zaragoza, Spain
<EMAIL_ADDRESS>
& Jesus Bermudez-Cameo
Instituto de Investigacion en Ingenieria de Aragon
Department of Computer Science and Systems Engineering
University of Zaragoza
Zaragoza, Spain
<EMAIL_ADDRESS>
& Jose J. Guerrero
Instituto de Investigacion en Ingenieria de Aragon
Department of Computer Science and Systems Engineering
University of Zaragoza
Zaragoza, Spain
<EMAIL_ADDRESS>
Corresponding author.

###### Abstract

From a non-central panorama, 3D lines can be recovered by geometric reasoning.
However, their sensitivity to noise and the complex geometric modeling
required has led these panoramas being very little investigated. In this work
we present a novel approach for 3D layout recovery of indoor environments
using single non-central panoramas. We obtain the boundaries of the structural
lines of the room from a non-central panorama using deep learning and exploit
the properties of non-central projection systems in a new geometrical
processing to recover the scaled layout. We solve the problem for Manhattan
environments, handling occlusions, and also for Atlanta environments in an
unified method. The experiments performed improve the state-of-the-art methods
for 3D layout recovery from a single panorama. Our approach is the first work
using deep learning with non-central panoramas and recovering the scale of
single panorama layouts.

††A final version of this article can be found at
https://doi.org/10.1109/CVPRW53098.2021.00410

_Keywords_ Omnidirectional Vision $\cdot$ 3D Vision $\cdot$ Non-central
Cameras $\cdot$ Layout recovery $\cdot$ Scene understanding

##  1 Introduction

Layout recovery and 3D understanding of indoor environments is a hot topic in
computer vision research Zou et al. (2021). Most recent approaches for layout
recovery use different neural network architectures to recover the structural
elements of an indoor environment. In this context, the use of omnidirectional
images has important advantages to retrieve the shape of a whole room from a
single image. In order to handle the heavy distortions that introduce the
omnidirectional representations, we can find different approaches in the state
of the art. Dula-Net Yang et al. (2019) extracts from the equirectangular
image a perspective view of the ceiling and extrude the layout using a dual-
branched architecture. Similarly but with a different network architecture,
AtlantaNet Pintore et al. (2020) obtains the floor plan dividing the panorama
into two perspective views of ceiling and floor separately to adjust the
height of the room. On a different approach, Corners for Layouts (CFL)
Fernandez-Labrador et al. (2020) and HorizonNet Sun et al. (2019) aim to
extract the boundaries of the structural lines and corners of the room
directly from the equirectangular panorama, obtaining an up-to-scale layout in
a post-processing.

In this paper we propose a new method for layout recovery from single
panoramas. We revisit the non-central panoramas Li et al. (2004) in order to
obtain scaled 3D lines Bermudez-Cameo et al. (2017) and vertical planes from a
single image. As the equirectangular panorama, the non-central circular
panorama provides 360 information of the environment, but the image distortion
in non-central panoramas includes subtle differences allowing geometric 3D
reasoning Perdigoto and Araujo (2012). This characteristic is a clear
advantage with regard equirectangular panoramas since allows to recover the
scale of the environment without any prior knowledge. In Fig. 1 we have an
equirectangular panorama and a non-central panorama of the same scene. Even
though both images look similar allowing to recover the shape of the room,
only from the non-central circular panorama we can recover the scale without
any assumption.

Figure 1: Central (up-left) and non-central panoramas (bottom-left) have
similar appearance but there are subtle differences in favor of the second if
we want to obtain 3D information including the scale. On the right, the scaled
layout obtained from a non-central panorama in an Atlanta environment with our
solution. Figure 2: Pipeline of the proposed method. The non-central circular
panorama is processed by the fine-tuned network. The network provides the
pixel information of the structural lines and a per-column probability of a
wall-wall intersection. Then the proposed geometric pipeline, including the
new solvers, gives the final scaled layout.

In our proposal, we adapt the neural network architecture of HorizonNet Sun et
al. (2019) to non-central circular panoramas for the extraction of structural
lines of indoor environments. Besides, we propose a geometric pipeline with
two new linear solvers that jointly obtain the room height and vertical walls
location for global layout extraction in Manhattan and Atlanta world
assumptions. Our experiments show that our proposal improves the state of the
art solutions, being ours the first that obtains the scale of the layout
without using additional assumptions.

##  2 Proposed method

In order to recover the layout of a room from a single non-central circular
panorama, we propose a new pipeline, shown in Fig. 2. We use a neural network
as a line extractor from the image to reconstruct the scaled layout in a new
geometrical processing that includes two new linear solvers for non-central
projection systems.

###  2.1 Neural network as line extractor

For the first part of our pipeline, we propose a neural network as boundary
extractor for structural lines. In classical approaches, lines are extracted
with Hough transform and vanishing points, followed by an hypothesis
generation-verification algorithm Zhang et al. (2014). This approach consumes
lots of time and resources. On the other hand, neural networks have proven
that can obtain patterns on images with high accuracy in a short time and,
therefore, current approaches for layout recovery rely on the use of neural
networks.

Even though many state-of-the-art networks can handle omnidirectional
panoramas, there are not any that have considered non-central systems. We
propose to adapt the existing network architecture of HorizonNet Sun et al.
(2019) to handle non-central panoramas. The main advantage of this
architecture is that handles the information of the panorama column by column.
This is particularly interesting for the non-central circular panorama since
it is locally a central projection system in each column.

In order to adapt the network to the distortions of the non-central panoramas,
we have fine-tuned it. However, since non-central projection systems are
little used in the research community, there is no data-set available. To
overcome this difficulty, we have generated a data-set of non-central circular
panoramas with ground truth information for layout recovery in synthetic
indoor environments. This data-set is composed of around 650 different layouts
from 6 to 10 walls and more than 2500 images. The data-set will be available
under request.

###  2.2 Geometric solvers

In the second part of our pipeline, we take the pixel information provided by
the network and reconstruct the scaled layout of the room. For that purpose,
we derive two new geometric solvers to jointly obtain the room height and
vertical walls location for Manhattan or Atlanta world assumptions.

We define a wall as a set of two parallel lines contained in a vertical plane
(see Fig. 3). Let $\mathbf{L}=(\mathbf{l}^{T},\mathbf{\bar{l}}^{T})^{T}$ and
$\mathbf{M}=(\mathbf{m}^{T},\mathbf{\bar{m}}^{T})^{T}$ be the ceiling and
floor lines defined in Plücker coordinates Pottmann and Wallner (2009) and
$\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\\}$ an orthonormal basis
attached to the vertical wall. We define the closest points of the lines to
the acquisition system $\mathbf{x}_{L}$ and $\mathbf{x}_{M}$ with $h_{c}$ and
$h_{f}$, distance from the acquisition system to the ceiling and floor planes,
and $d$, distance to the wall plane, such that
$\mathbf{x}_{L}=d\mathbf{e}_{2}+h_{c}\mathbf{e}_{3}$ and
$\mathbf{x}_{M}=d\mathbf{e}_{2}+h_{f}\mathbf{e}_{3}$. Notice that with this
description we can parameterize the Plücker coordinates of the lines as
$\mathbf{l}=\mathbf{m}=\mathbf{e}_{1}$,
$\mathbf{\bar{l}}=\mathbf{x_{L}}\times\mathbf{l}=h_{c}\mathbf{e}_{2}-d\mathbf{e}_{3}$,
$\mathbf{\bar{m}}=\mathbf{x_{M}}\times\mathbf{m}=h_{f}\mathbf{e}_{2}-d\mathbf{e}_{3}$.
We also define the projecting rays that intersect the ceiling and floor lines
as $\bm{\Xi}=(\bm{\xi}^{T},\bm{\bar{\xi}}^{T})^{T}$ and
$\bm{X}=(\bm{\chi}^{T},\bm{\bar{\chi}}^{T})^{T}$ respectively.

$side(\bm{\Xi},\mathbf{L})=\bm{\xi}^{T}\left(h_{c}\mathbf{e}_{2}-d\mathbf{e}_{3}\right)+\bm{\bar{\xi}}^{T}\mathbf{e}_{1}=0$
(1)
$side(\bm{X},\mathbf{M})=\bm{\chi}^{T}\left(h_{f}\mathbf{e}_{2}-d\mathbf{e}_{3}\right)+\bm{\bar{\chi}}^{T}\mathbf{e}_{1}=0$
(2)

Known the projecting rays, given by the output of the neural network, we aim
to obtain the 3D lines that define each wall in the environment. The relation
among the projection rays and the lines is given by their intersection,
defined in equations (1) and (2). This is, in general, a non-linear problem
which is difficult to tackle directly. However, we propose two new DLT-like
approaches that allows to compute the solution for the layout as a linear
problem.

$\bar{\xi}_{1}u_{x}+\bar{\xi}_{2}u_{y}-\xi_{1}v_{y}-\xi_{2}v_{x}-d\xi_{3}=0$
(3)
$\bar{\chi}_{1}u_{x}-\bar{\chi}_{2}u_{y}-\chi_{1}w_{y}-\chi_{2}w_{x}-d\chi_{3}=0$
(4)

Let the main direction of a wall be horizontal and described by the vector
$\mathbf{u}=\left(u_{x},u_{y}\right)^{T}$ such that
$\mathbf{l}=\mathbf{m}=(u_{x},u_{y},0)^{T}$. We can define vectors
$\mathbf{v}=h_{c}\mathbf{u}$ and $\mathbf{w}=h_{f}\mathbf{u}$ such that
expressions (1) and (2) become linear obtaining a set of expressions ((3) and
(4)) depending on the unknown wall homogeneous vector
$\mathbf{W}=\left(\mathbf{u}^{T},\mathbf{v}^{T},\mathbf{w}^{T},d\right)^{T}$.

$\lambda(\mathbf{v_{1}}-h_{c}\mathbf{u_{1}})=h_{c}\mathbf{u_{0}}-\mathbf{v_{0}}$
(5)
$\lambda(\mathbf{w_{1}}-h_{f}\mathbf{u_{1}})=h_{f}\mathbf{u_{0}}-\mathbf{w_{0}}$
(6)

However in this linear system $A\mathbf{W}=0$, $\mathbf{u}$, $\mathbf{v}$ and
$\mathbf{w}$ are independent variables which are non-parallel. In order to
impose the parallelism of these vectors we compute the null space of the
system with a Singular Value Decomposition (SVD) obtaining a parametric
solution which is the linear combination of the singular vectors with a set of
parameters $\lambda_{i}$. Two horizontal lines contained in a vertical plane
have 4 degrees of freedom. A minimal solution would need 2 rays for each line
of the wall, describing the null-space with three singular vectors and two
parameters $\lambda_{1}$ and $\lambda_{2}$. By solving a system of two
quadratic equations for $\lambda_{1}$ and $\lambda_{2}$ (with resultants,
action matrices or as a polynomial eigenvalue problem Kukelova et al. (2011))
we obtain a set of 4 different solutions which should be discriminated. Since
the network provides enough robust information, instead of the minimal
solution, we propose to solve the over-determined case (with a minimum of 3
rays lying to each line) with a linear combination involving two singular
vectors and single parameter $\lambda$ (such that
$\mathbf{W}=\mathbf{W}_{0}+\lambda\mathbf{W}_{1}$) obtaining two uncoupled
quadratic equations (5) and (6) respectively. These equations provide two
solutions, where only one of them sets the ceiling line above the floor line.

Figure 3: Rays and wall parameter definition. The parameters are: wall
reference system $\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\\}$;
$\bm{\Xi}$ and $\bm{X}$ define the projecting rays; $\mathbf{(l,\bar{l})}$ and
$\mathbf{(m,\bar{m})}$ are the ceiling and floor lines that define the wall;
$\mathbf{x_{L},x_{M}}$ define the closest points of the lines to the origin;
$h_{c}$, $h_{f}$ and $d$ are the ceiling and floor height and distance to the
wall respectively.

Notice that with Manhattan world assumption there is a set of walls sharing
the wall direction $\mathbf{u}=\left(u_{x},u_{y}\right)^{T}$ and the
complementary set of walls share the orthogonal direction vector
$\mathbf{u}_{\perp}=\left(-u_{y},u_{x}\right)^{T}$. Since all the walls share
the ceiling height $h_{c}$ and the floor $h_{f}$, we extend the DLT-like
fitting to the whole set of walls, by computing the null-space of
$\mathsf{A}\mathfrak{L}_{M}=0$ where $\mathfrak{L}_{M}$ is the layout vector
$\mathfrak{L}_{M}=\left(\mathbf{u}^{T},\mathbf{v}^{T},\mathbf{w}^{T},d_{1},\dotsm,d_{N}\right)^{T}$
where $N$ is the number of walls and the matrix $\mathsf{A}$ is full-filed
with relations (3) and (4). The same reasoning as in the case of a single wall
can be used to enforce parallelism among $\mathbf{u}$, $\mathbf{v}$,
$\mathbf{w}$ .

$\bar{\xi_{1}}^{\prime}+h_{c}\xi_{2}^{\prime}-d\xi_{3}^{\prime}=0$ (7)
$\bar{\chi_{1}}^{\prime}+h_{f}\chi_{2}^{\prime}-d\chi_{3}^{\prime}=0$ (8)

For Atlanta world assumption each wall can have a different horizontal
direction, therefore we have to extract each wall independently. Notice that
in this case we are not imposing that $h_{c}$ and $h_{f}$ are common to all
the walls. However, if the direction of each wall is known (for example
extracting each wall independently) we can derive a new solution for the whole
layout. Assuming that wall directions are known, we can express the projecting
rays of each wall in its own local reference system and then equations (1) and
(2) become (7) and (8) respectively, where $\bm{\Xi}^{\prime}$ and
$\bm{X}^{\prime}$ are the projecting rays in each wall reference system. Then,
we can solve the null-space of a system of linear equations
$\mathsf{A}\mathfrak{L}_{A}=\mathbf{0}$ with
$\mathfrak{L}_{A}=\left(1,h_{c},h_{f},d_{1},\dotsm,d_{N}\right)$ where
$\mathsf{A}$ is composed from equations (7) and (8). The main advantage of
this second approach is that can be used for Manhattan as well as Atlanta
world environments whenever the layout has only one ceiling and floor heights.

##  3 Experiments

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4: Examples of 3D reconstruction from the proposed method. We shown the non-central panorama and the 3D reconstruction. The green wire frame is the real 3D layout of the room. Table 1: Comparison of different methods of 3D layout recovery.  | Manhattan World assumption
---|---
| 3D IoU | 3D IoU | CEN | CE
| (u2s)
CFL Fernandez-Labrador et al. (2020) | 78.87 | - | 0.75 | -
HorizonNet Sun et al. (2019) | 82.66 | - | 0.69 | -
AtlantaNet Pintore et al. (2020) | 83.94 | - | 0.71 | -
Ours | 93.88 | 86.18 | 0.787 | 0.223
| Atlanta World assumption
HorizonNet Sun et al. (2019) | 73.53 | - | - | -
AtlantaNet Pintore et al. (2020) | 80.01 | - | - | -
Ours | 91.67 | 76.17 | 1.335 | 0.513
| higher is better | smaller is better

We have performed a set of experiments in order to evaluate our proposal and
make a comparison with the state-of-the-art methods. The metrics used for the
comparison are: 3D IoU, which refer to the 3D intersection over union of the
predicted layout and the ground truth; 3D IoU(u2s), which refer to the up-to-
scale intersection over union of the layout; CEN, which refer to the Corner
Error Normalized computed as the L2 distance of the corners divided by the
diagonal of the layout; CE, which refers to the Corner Error computed as the
L2 distance of the corners in meters.

The comparison with state-of-the-art methods is not completely fair. The
datasets used for the different methods are different, so the results can
depend on the dataset used and not only on the method. Besides, our proposal
only uses the image information in order to recover the 3D layout with the
scale while the rest of the methods in the state of the art provide up-to-
scale measures, relying on some measure in the environment for the 3D
reconstruction, e.g. the camera height. Nevertheless, a summary of this
comparison is shown in Table 1.

These results show that our proposal outperforms the state of the art methods
for Manhattan as well as Atlanta world assumptions. Besides, our method also
recovers the scale of the layout without any prior assumption. Some examples
of our results are shown in Fig. 4, where different layouts are tested. We
demonstrate that our method can handle quite challenging layouts, in different
illumination conditions and world assumptions.

##  4 Conclusions

In this paper we have proposed two new solvers for indoor layout recovery from
a single non-central circular panorama. We use a neural network for extracting
the edges of structural lines from a non-central projection system and
geometrically process the output in order to recover the 3D information of the
layout. Our experiments show that our approach with non-central circular
panoramas has better performance than state-of-the-art methods for Manhattan
and Atlanta environments. In addition, our method can extract the scale of the
room without any prior knowledge.

## References

  * Zou et al. [2021] C. Zou, JW. Su, CH. Peng, A. Colburn, Q. Shan, P. Wonka, HK. Chu, and D. Hoiem.  Manhattan room layout reconstruction from a single 360 image: A comparative study of state-of-the-art methods.  _IJCV_ , 2021.
  * Yang et al. [2019] ST. Yang, FE. Wang, CH. Peng, P. Wonka, M. Sun, and HK. Chu.  Dula-net: A dual-projection network for estimating room layouts from a single rgb panorama.  In _CVPR_. IEEE, 2019.
  * Pintore et al. [2020] G. Pintore, M. Agus, and E. Gobbetti.  Atlantanet: Inferring the 3d indoor layout from a single 360 image beyond the manhattan world assumption.  In _ECCV_. Springer, 2020.
  * Fernandez-Labrador et al. [2020] C. Fernandez-Labrador, JM. Facil, A. Perez-Yus, C. Demonceaux, J. Civera, and JJ. Guerrero.  Corners for layout: End-to-end layout recovery from 360 images.  _IEEE Robotics and Automation Letters_ , 2020.
  * Sun et al. [2019] C. Sun, CW. Hsiao, M. Sun, and HT. Chen.  Horizonnet: Learning room layout with 1d representation and pano stretch data augmentation.  In _CVPR_. IEEE, 2019.
  * Li et al. [2004] Y. Li, HY. Shum, CK. Tang, and R. Szeliski.  Stereo reconstruction from multiperspective panoramas.  _PAMI_ , 2004.
  * Bermudez-Cameo et al. [2017] J. Bermudez-Cameo, O. Saurer, G. Lopez-Nicolas, JJ. Guerrero, and M. Pollefeys.  Exploiting line metric reconstruction from non-central circular panoramas.  _Pattern Recognition Letters_ , 2017.
  * Perdigoto and Araujo [2012] L. Perdigoto and H. Araujo.  Reconstruction of 3d lines from a single axial catadioptric image using cross-ratio.  In _ICPR_. IEEE, 2012.
  * Zhang et al. [2014] Y. Zhang, S. Song, P. Tan, and J. Xiao.  Panocontext: A whole-room 3d context model for panoramic scene understanding.  In _ECCV_. Springer, 2014.
  * Pottmann and Wallner [2009] H. Pottmann and J. Wallner.  _Computational line geometry_.  Springer Science & Business Media, 2009.
  * Kukelova et al. [2011] Z. Kukelova, M. Bujnak, and T. Pajdla.  Polynomial eigenvalue solutions to minimal problems in computer vision.  _PAMI_ , 2011.

## ACKNOWLEDGMENT

This work was supported by RTI2018-096903-B-100 (AEI/ FEDER, UE).


# An objective collapse model without state dependent stochasticity

Lotte Mertens  Institute for Theoretical Physics Amsterdam, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  Institute for
Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany  Matthijs Wesseling  Jasper van Wezel  Institute for Theoretical
Physics Amsterdam, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands

###### Abstract

The impossibility of describing measurement in quantum mechanics while using a
quantum mechanical model for the measurement machine, remains one of its
central problems. Objective collapse theories attempt to resolve this problem
by proposing alterations to Schrödinger’s equation. Here, we present a minimal
model for an objective collapse theory that, in contrast to previous
proposals, does not employ state dependent stochastic terms in its
construction. It is an explicit proof of principle that it is possible for
Born’s rule to emerge from a stochastic evolution in which no properties of
the stochastic process depend on the state being evolved. We propose the
presented model as a basis from which more realistic objective collapse
theories can be constructed.

##  I Introduction

Quantum physics gives an extremely accurate description for the dynamics of
systems consisting of up to at least $10^{6}$ atoms Bassi  _et al._ (2013);
Fein  _et al._ (2019). Contrarily, in the macroscopic realm of objects
comprised of $10^{18}$ atoms or more, it has been experimentally verified that
classical physics suffices to predict their collective dynamics Bassi  _et
al._ (2013); Marshall  _et al._ (2003); Donadi  _et al._ (2020); Vinante  _et
al._ (2017); Carlesso  _et al._ (2016); Wit  _et al._ (2019); Arndt  _et al._
(1999); Mooij and Nazarov (2006); Andrews (1997); Christian (2005). There is
an inherent conflict between these two limits which becomes apparent when
considering the measurement of microscopic particles with a macroscopic
measurement machine that itself is built from quantum constituents Komar
(1962); Wigner (1963); Adler (2003); Penrose (2014). This inconsistency is
known as the quantum measurement problem and it remains the focus of an active
field of research to this day Bassi  _et al._ (2013); Penrose (1996); Van
Wezel (2010); Schlosshauer  _et al._ (2013); Arndt and Hornberger (2014);
Carlesso  _et al._ (2022).

Proposals for solving the quantum measurement problem can be divided into
roughly two classes: interpretations and objective collapse theories. The
interpretations assume Schrödingers equation holds unaltered at all scales.
The interpretations attempt to interpret the mathematical objects appearing in
Schrödinger’s equation in a way that explains why we always perceive only a
single classical state for any macroscopic object, even if its wave function
is superposed over many such states Bassi  _et al._ (2013); Allday (2009);
Neumann  _et al._ (1955). On the other hand, objective collapse theories
assume that in between the microscopic and macroscopic worlds, at the so-
called mesoscopic scale, a transition takes place from quantum to classical
dynamics Ghirardi  _et al._ (1990); Penrose (1996); Diósi (1987). This
transition is described by a small alteration of Schrödinger’s equation that
has no measurable effect on microscopic objects, but begins to dominate the
dynamics in the mesoscopic regime Bassi  _et al._ (2013); Allday (2009);
Neumann  _et al._ (1955). The result is a reduction or ‘collapse’ of quantum
superpositions into a basis of classical states, which becomes instantaneous
in the macroscopic limit. State of the art experimental efforts have recently
begun to probe the mesoscopic region where deviations from Schrödinger’s
equation predicted by objective collapse theories may become observable Bassi
_et al._ (2013); Marshall  _et al._ (2003); Donadi  _et al._ (2020); Vinante
_et al._ (2017); Carlesso  _et al._ (2016); Wit  _et al._ (2019); Arndt  _et
al._ (1999); Mooij and Nazarov (2006); Andrews (1997); Christian (2005). These
efforts attempt to create superpositions and observe quantum interference on
mesoscopic scales, and typically involve objects of mesoscopic size, such as a
nano-mechanical resonator or a large molecule being superposed over different
vibrational modes or positions.

The precise dynamics predicted by objective collapse theories in the
mesoscopic realm depend on the details of the particular theory being
considered. In order to reproduce the observed properties of the collapse
process in the macroscopic limit however, any objective collapse theory needs
to at least contain stochastic, non-unitary Van Wezel (2010), and non-
linearMertens  _et al._ (2021) elements in its time evolution. Together, these
need to ensure that the collapse dynamics give rise to Born’s rule when
averaged over many individual collapse processes Neumann  _et al._ (1955);
Bohr (1928). In existing objective collapse theories, this outcome is
explicitly imposed in the proposed dynamical equations by multiplying the
stochastic variable with a non-linear, state dependent factor Bassi  _et al._
(2013); Ghirardi  _et al._ (1986, 1990); Penrose (1996); Diósi (1987); Pearle
(2007); Zeh (1999); Squires (1990); Wallace (2012). The origin of this
coupling remains unclear Bassi  _et al._ (2013). The resulting probability
distribution function of the combined non-linear stochastic term inevitably
depends on the state it acts on, which necessarily implies the stochastic term
has complete information on the state. Here, we show that this feature can be
avoided by explicitly constructing an objective collapse theory in which the
non-linear elements are separate from any stochastic term. The stochastic term
may still depends on state-independent properties of the system (such as the
coupling constant defining magnetic interactions between spins in a magnet),
but _not_ on properties of the state the system is in (such as the weights of
particular spin configurations in the wave function of a magnet).

Similar to conventional objective collapse theories, a relation between model
parameters is required to obtain Born’s rule. However, the model presented
here is distinguished from existing collapse models because Born’s rule
emerges without a state-dependent probability distribution function for the
stochastic variable. This proof of principle is worked out in the context of a
Heisenberg antiferromagnet acting as a measurement machine for staggered
magnetisation, and we demonstrate that it exhibits all characteristics
required of an objective collapse model. The proposed theory is not intended
to be a realistic proposal for objective collapse in antiferromagnetic
materials, but rather serves as a minimal example that can be used as the
basis for constructing more general and more realistic objective collapse
models.

The article is organised as follows. First, the requirements that should be
met by any objective collapse model are summarised. Next, a recipe is given
for constructing an objective collapse theory applying to a general two-state
measurement. Each step is then applied to the specific situation of a
Heisenberg antiferromagnet, and the properties of the ensuing collapse process
are detailed. Finally, an interpretation is given for the proposed dynamics,
and the advantages of the proposed model over existing theories are discussed.

##  II Requirements for any objective collapse theory

Regardless of their detailed constructions, all objective collapse theories
propose an alteration or correction to Schrödinger’s equation that can be
written as 111Notice that although it is formally possible to always write
corrections to Schrödinger’s equation in this form, it is not always the most
elegant or efficient formulation.:

$\displaystyle i\hbar\frac{\partial}{\partial t}\ket{\psi(t)}=(H+\epsilon
G)\ket{\psi(t)}$ (1)

Here, we (arbitrarily) use the Schrödinger picture, and explicitly include the
strength $\epsilon$ of the term $G$ generating collapse.

The first requirement that any objective collapse model written in the form of
Eq. (1) should obey, is that the time evolution it generates should reduce to
standard quantum mechanics when acting on microscopic objects. This is ensured
by assuming $\epsilon$ to be suitably small, so that any deviations from the
usual quantum dynamics become apparent only at very late times scaling with
$1/\epsilon$ (which could even lie beyond the current age of the universe).
Despite the smallness of $\epsilon$, however, any successful collapse theory
should also predict macroscopic objects that are somehow forced into a
superposition of classically distinguishable states (f.e. by instantaneously
coupling them to a microscopic quantum system), to very quickly collapse
towards just one such state Feldmann and Tumulka (2012); Toroš  _et al._
(2017). This happens when the operator $G$ scales with some measure $N$ of the
system size, such as the number of particles it involves, its mass, its
volume, or the value of its classical order parameter, which distinguishes
different classical states Bassi  _et al._ (2013); van Wezel (2008). The
collapse time scale $\tau_{c}\propto 1/(N\epsilon)$ can then become
arbitrarily small in the thermodynamic limit, while remaining arbitrarily
large for microscopic objects van Wezel (2008).

With this requirement satisfied, the time evolution implied by Eq. (1) can be
used to describe quantum measurement by splitting it into two stages Neumann
_et al._ (1955). First, a microscopic object is coupled to a macroscopic
measurement device in such a way that part of the device, traditionally
referred to as the ‘pointer’, becomes entangled state with the microscopic
object. This is often called pre-measurement. It is then in a superposition of
classically distinguishable states, or pointer states Zurek (1981). Secondly,
this superposition will almost instantaneously reduce to just a single pointer
state, after which the measurement outcome can be read off. The pre-
measurement is dominated by the coupling between the measurement device and
the microscopic object, which is encoded in $H$, while $G$ induces the latter
part of the collapse to a single pointer state.

This description of measurement brings to the fore a second requirement for
any objective collapse theory: it should cause stable quantum state reduction.
That is, when acting on a macroscopic system, $G$ should cause it to be
reduced to a single pointer state. Moreover, once the measurement device is
localised in a single pointer state, it should not be able to spontaneously
evolve out of it. The requirement of reduction to a single pointer state turns
out to be readily fulfilled for example by realising that the dynamics
generated by the usual Schrödinger equation (with $G=0$) is unstable Van Wezel
(2010). That is, if $G$ is not Hermitian and couples to an order parameter of
any symmetry-breaking system (such as the position of a crystal or the
magnetization of a magnet), it has been shown that even an infinitesimal value
for $\epsilon$ suffices to instantaneously collapse sufficiently large
macroscopic systems into a pointer basis Van Wezel (2010).

Notice that although $G$ being non-Hermitian implies non-unitary time
evolution, this does not present any fundamental problems with regard to wave
function normalisation. Upon redefining expectation values as
$\langle{O}\rangle\equiv{\bra{\psi}{O}\ket{\psi}}/{\bra{\psi}\ket{\psi}}$, all
of the physical predictions of quantum mechanics can be recovered even with a
time-dependent norm Van Wezel (2010); Mertens  _et al._ (2021).

The possible lack of energy conservation under non-unitary time evolution
similarly does not pose a problem 222Notice that technically, as long as the
time evolution is invariant under time translation, there is always an
associated Noether charge that should be defined as representing a conserved
energy Beekman  _et al._ (2019). This energy, however, is not generally equal
to $\langle H\rangle$.. The conservation of $\langle H\rangle$ is ensured in
the thermodynamic limit if the non-Hermitian field couples (with non-vanishing
strength) only to the order parameter, such as the centre of mass position or
magnetization, and not to any internal degrees of freedom such as sound or
spin wave excitations. This way, the non-Hermitian field causes transitions
only between states that are degenerate in the thermodynamic limit, and hence
have no effect on the total energy Van Wezel (2010). For mesoscopic systems,
small fluctuations in $\langle H\rangle$ provide one possible way to
experimentally distinguish the predictions of objective collapse theories from
those of unitary interpretations. Van Wayenburg (2020).

Finally, the third requirement on the predicted dynamics of the pointer state
in any objective collapse theory, is that it should reproduce Born’s rule.
That is, the relative frequency of any particular measurement outcome should
equal the squared weight of the corresponding component in the initial pointer
state wave function. This requirement also implies that in general, a given
initial state needs to be able to collapse to different pointer states
representing different measurement outcomes. This variation in possible end
states necessarily implies the presence of a stochastic variable in the time
evolution generator, which changes value from one collapse process to the
next, and possibly even within a single process. It has recently been shown
that besides a stochastic component, it is also necessary for the generator to
contain a non-linear component (independent of normalisation) to be able to
satisfy Born’s rule Mertens  _et al._ (2021).

In summary, the ingredients necessary for any objective collapse theory are
that its time evolution operator should be non-unitary, should scale with the
system size, should couple only to the order parameter of the system in the
thermodynamic limit, and should contain both non-linear and stochastic terms.

##  III Constructing an objective collapse theory

Here, we explicitly construct an objective collapse theory by systematically
including all necessary ingredients identified above. We will construct the
theory in the context of an antiferromagnet acting as a measurement device van
Wezel (2008), but the procedure is readily generalised. The result differs in
a crucial aspect from the many existing flavours of objective collapse theory
Ghirardi  _et al._ (1990, 1986); Pearle (2007); Diósi (1987); Penrose (2014):
it gives rise to Born’s rule without the stochastic contribution to the
evolution having any knowledge of the state of the system. This way, Born’s
rule is not imposed on the dynamics by the way the theory is formulated, but
rather emerges spontaneously in the thermodynamic limit.

###  III.1 Introducing the pointer basis

Consider a macroscopic system governed by the Hamiltonian $H$, consisting of
many internal degrees of freedom. If the system represents a measurement
device, it will be described by a classically accessible collective variable
describing properties of the system as a whole, such as its centre of mass or
total magnetization, which we call the ‘pointer’ Van Wezel (2010); van Wezel
(2008). The possible states for the pointer correspond to different symmetry-
broken configurations of the system. The pointer states could be different
positions along a dial of an actual pointer, they could be different
configurations of text on a computer screen displaying measurement outcomes,
or they could be any other set of classically distinguishable configurations.
To be specific, we will here consider an antiferromagnet whose pointer states
consist of different orientations of the spins making up the antiferromagnet.

In fact, it suffices to consider only initial states of the antiferromagnet
superposed over two states, with opposite staggered magnetisation. If the
objective collapse theory does not reproduce all aspects of measurement for
two-state superpositions, it follows by induction that it can also not work
for any more complicated initial state. Formulating collapse dynamics for two-
state superpositions thus provides a minimal model for an objective collapse
theory. Moreover, we will focus only on the pointer states of the
antiferromagnet and ignore all of its internal degrees of freedom because
these are the only degrees of freedom involved in spontaneous symmetry
breaking Mertens  _et al._ (2021).

To be specific, we consider a Heisenberg antiferromagnet, with positive
isotropic coupling $J$ between neighbouring quantum spins of size 1/2:

$\displaystyle
H^{\text{AF}}=\sum_{j=1}^{N}J\,{\mathbf{S}}_{j}\cdot{\mathbf{S}}_{j+1}.$

The spins on neighbouring sites will anti-align, forming two sub-lattices $A$
(even sites) and $B$ (odd sites) with opposite spins. Upon taking the Fourier
transform it becomes clear that the collective parts of the Hamiltonian
$H_{0}$ (with $k=0,\pi$) decouple from the internal spin wave degrees of
freedom (with $k\neq 0,\pi$) Beekman  _et al._ (2019):

$\displaystyle
H_{0}=H^{\text{AF}}-H^{\text{internal}}=\frac{4J}{N}{\mathbf{S}}_{A}\cdot{\mathbf{S}}_{B}.$
(2)

$H_{0}$ is the so-called Lieb Mattis Hamiltonian Lieb  _et al._ (1961).

Notice that in the thermodynamic limit, all states with different values of
the total spin $\mathbf{S}=\mathbf{S}_{A}+\mathbf{S}_{B}$ become degenerate
with the ground state, while maintaining only a vanishing contribution to the
free energy of the antiferromagnet Beekman  _et al._ (2019).

The usual theory of spontaneous symmetry breaking (SSB) describes the
emergence of stable states that are not invariant under a symmetry of the
Hamiltonian. For the Lieb-Mattis Hamiltonian of Eq. (2) the spin-rotational
symmetry of the antiferromagnet can be broken by adding an infinitesimal
symmetry breaking field to the Hamiltonian:

$\displaystyle H=H_{0}-B(S_{A}^{z}-S_{B}^{z}).$ (3)

depending on the sign of $B$, it represents a magnetic field that points
either up or down along the $z$-direction on the $A$-sublattice, while it
points in the opposite direction on the $B$-sublattice. The pointer state with
staggered magnetisation $S_{A}^{z}-S_{B}^{z}=\pm N/2$ is singled out in the
thermodynamic limit as the unique ground state by even a vanishingly small
field $B=\pm|B|$ Beekman  _et al._ (2019). These states are stable in the
limit of vanishing $B$ despite the fact that they break the spin-rotational
symmetry of $H_{0}$. They are the pointer states of the antiferromagnetic
measurement machine considered here.

###  III.2 Introducing non-unitary dynamics

Time evolution generated by the Schrödinger equation is necessarily
invertible, but as claimed before Van Wezel (2010), it is also unstable to
non-unitary perturbations. Applying the same principles of equilibrium
spontaneous symmetry breaking, but now in the setting of non-equilibrium time
evolution then generates the spontaneous breakdown of invertible time
evolution in the presence of even a vanishingly small non-Hermitian term.

We will explicitly consider a non-Hermitian version of the symmetry-breaking
perturbation given by writing $i\epsilon B(S_{A}^{z}-S_{B}^{z})$, with
$\epsilon$ vanishingly small and $B$ finite. In addition, we break invertible
time evolution on the level of the collective Hamiltonian by introducing the
infinitesimal Wick rotation: $H_{0}\to(1+\epsilon i)H_{0}$. This perturbation
can be interpreted as the leading order contribution arising from a full-
fledged non-unitary theory for physics beyond Schrödinger’s equation, and we
neglect all higher order contributions.

Both of the sublattice spin operators $\mathbf{S}_{A/B}$ appearing in the
collective Hamiltonian scale with the system size $N$. The total strength of
the non-unitary contribution to the pointer dynamics therefore scales with
$N\epsilon$, and displays the non-commuting limits typical of spontaneous
symmetry breaking Beekman  _et al._ (2019):

$\displaystyle\lim_{\epsilon\to 0}\lim_{N\to\infty}N\epsilon=\infty$
$\displaystyle\lim_{N\to\infty}\lim_{\epsilon\to 0}N\epsilon=0.$ (4)

This implies that the speed of the collapse process induced by the non-unitary
term depends on the size of the pointer. If $N$ is small, the non-unitary part
is negligible for any sufficiently small $\epsilon$. In the thermodynamic
limit however, the number of spins making up the collective anti-ferromagnetic
state of the pointer is so large that any infinitesimal coupling suffices to
qualitatively influence the pointer dynamics Van Wezel (2010); van Wezel
(2008).

###  III.3 Introducing an external stochastic field

Once an object in the thermodynamic limit is in a state with a single,
definite value for the order parameter, it does not unitarily evolve out of
that state within any measurable time, even if the direction of the symmetry-
breaking field changes. There is therefore no reason to constrain ${B}$ to be
a static external field, and we will allow it to vary in time. In fact, since
it represents an infinitesimal symmetry breaking field originating from
sources beyond the control of any feasible experiment or observation, we
assume the direction of $\mathbf{B}(t)$ to vary randomly in time, with
correlation time $\tau_{r}$. At the start of a measurement process, the
direction of the symmetry breaking field is then fully random and has equal
probability of being aligned with any particular pointer state.

Notice that at this point we have two different non-Hermitian contributions to
the Hamiltonian. The first originated from Wick rotating the original
symmetric Hamiltonian, and yields spontaneous non-unitary dynamics for the
system even without any external influences. The second term is a non-
Hermitian version of a symmetry breaking field. This is due to the interaction
with an (as yet undefined) external actor and introduces a stochastic
contribution to the evolution. The interplay between the two non-Hermitian
terms will determine the collapse dynamics and measurement outcomes.

###  III.4 Introducing non-linearity

In order to reproduce Born’s rule, the time evolution operator necessarily
needs to contain a non-linear term Mertens  _et al._ (2021). As argued before,
the non-linearity needs to go beyond normalization of the wave function, as
that is already made redundant by the definition
$\langle{O}\rangle\equiv{\bra{\psi}{O}\ket{\psi}}/{\bra{\psi}\ket{\psi}}$. One
way to introduce such a non-linearity is by considering in the Hamiltonian
only the effective, collective influence of all microscopic degrees of freedom
on the order parameter. For general symmetry-breaking systems taken out of
equilibrium it is well-known that the Gross-Pitavskii equations (a form of
dynamical mean field theory) give a good approximation of the dynamics of the
order parameter Gross (1961); Pitaevskii (1961); Bogoliubov (1947); Pitaevskii
and Stringari (2016); Okulov (2013).

In our case, the Gross-Pitaevskii approach consists of approximating the
interaction between sub-lattices by each spin being exposed to the magnetic
field generated by the spins in the opposing sub-lattice:
${\mathbf{S}}_{A}\cdot{\mathbf{S}}_{B}\approx\langle{\mathbf{S}}_{A}\rangle\cdot{\mathbf{S}}_{B}+{\mathbf{S}}_{A}\cdot\langle{\mathbf{S}}_{B}\rangle$.
These terms can be evaluated at each time step, giving a set of differential
equations that define the time evolution starting from a given initial state.

The Gross-Pitaevskii approach yields a good approximation for the collective
dynamical response seen in isolated systems without ensemble averaging over
multiple experiments Bogoliubov (1947); Pitaevskii and Stringari (2016);
Okulov (2013). Thus the approximate form of the self-interaction, containing
factors like $\bra{\psi}{\mathbf{S}}_{A}\ket{\psi}$, which have the appearance
of expectation values, actually do not imply ensemble averages or measurements
being performed. They can be considered as the instantaneous collective
magnetisation of all spins in one sub-lattice influencing the spins in the
other within a single experiment.

###  III.5 Assembling the objective collapse theory

Putting everything together, the time evolution of equation (1) for the
antiferromagnetic pointer state can be written as:

$\displaystyle i\hbar\frac{\partial}{\partial t}\ket{\psi(t)}$
$\displaystyle=\left[\frac{4J}{N}(1+i\epsilon)\left(\frac{\bra{\psi(t)}{\mathbf{S}}_{A}\ket{\psi(t)}}{\bra{\psi(t)}\ket{\psi(t)}}\cdot\mathbf{S}_{B}+\frac{\bra{\psi(t)}\mathbf{S}_{B}\ket{\psi(t)}}{\bra{\psi(t)}\ket{\psi(t)}}\cdot\mathbf{S}_{A}\right)+{i\epsilon\mathbf{B}(t)}\cdot({\mathbf{S}}_{A}-{\mathbf{S}}_{B})\right]\ket{\psi(t)}$
(5)

As noted before, it is a necessary requirement for any objective collapse
theory to describe the quantum state reduction of a measurement machine that
is initially superposed over just two pointer states. We therefore consider
the simplest possible case of a state composed of two eigenstates of the order
parameter:

$\displaystyle\ket{\psi(t)}=n(t)e^{i\xi(t)/2}$
$\displaystyle\left(e^{i\phi(t)/2}\cos(\theta(t)/2)\ket{\uparrow\downarrow}\right.$
$\displaystyle+\left.e^{-i\phi(t)/2}\sin(\theta(t)/2)\ket{\downarrow\uparrow}\right).$
(6)

Here, $n(t)$ represents the norm of the wave function, $\xi(t)$ its overall
phase, $\phi(t)$ the relative phase between pointer states, and $\theta$ is
the angle determining their relative weights. The states
$\ket{\uparrow\downarrow}$ and $\ket{\downarrow\uparrow}$ are arbitrarily
chosen to lie along the $z$-axis.

The direction of the symmetry-breaking field $\mathbf{B}(t)$ in equation (5)
does not necessarily align with direction of the collective staggered
magnetisations in the state $\ket{\psi(t)}$. Because of the rigidity
associated with the spontaneously broken symmetry of the antiferromagnetic
state, however, any components of the field orthogonal to the magnetisation
direction will take a time proportional to the system size to have any
significant effect Beekman  _et al._ (2019). Without loss of generality, we
can therefore approximate the interaction terms as
$\mathbf{B}\cdot\mathbf{S}_{A,B}\approx B_{0}\cos(\chi)S^{z}_{A,B}$, with
$B_{0}$ the amplitude of the symmetry-breaking field, and $\chi$ the angle
between $\mathbf{B}$ and the $z$-axis. Choosing a random direction for the
three-dimensional vector $B$ then corresponds to randomly sampling a value for
$\chi$ from the probability density function $f=\sin(\chi)/2$. Following the
same reasoning, we can also approximate the Gross-Pitaevskii terms by their
projections onto the $z$-axis:
$\langle\mathbf{S}_{A,B}\rangle\cdot\mathbf{S}_{B,A}\approx
S_{A,B}^{z}S_{B,A}^{z}$.

Inserting all definitions into the Hamiltonian and calculating the time
dependence of $\theta$ from equation (5) as described in appendix A finally
yields:

$\displaystyle\dot{\theta}$
$\displaystyle=-\frac{JN\epsilon}{\hbar}\sin(\theta)\left(\cos(\theta)-\frac{B_{0}}{J}\cos(\chi(t))\right).$
(7)

This equation shows the dynamics of the relative weights parameterised by
$\theta(t)$ to be independent of the variables $n(t)$, $\xi(t)$, and $\phi(t)$
so that we only need to consider the relative weights when modelling the
objective collapse process. In particular, it justifies the claim that the
overall norm $n(t)$ can be safely absorbed into a redefinition of the
expectation value without affecting any of the collapse dynamics.

##  IV Collapse dynamics

Having formulated a model for the quantum state reduction of a single
superposition of pointer states, we will discuss the collapse dynamics it
gives rise to. We will determine the outcomes of the predicted evolution for
different regimes of pointer size, the frequencies with which particular
outcomes are obtained, and we will check that the model meets the minimal
requirements for objective collapse theories formulated above.

###  IV.1 Microscopic region

In the microscopic limit, the pointer itself is a quantum system that
typically consists of a small number of constituent particles (roughly, fewer
than $10^{6}$ atoms Bassi  _et al._ (2013)). Taking the strength $\epsilon$ of
the non-unitary perturbation to be infinitesimally small such that
$N\epsilon\to 0$, then renders the entire dynamics described by equation (7)
negligible on any measurable time scale. As expected, only the regular,
unitary quantum dynamics governed by Schrödinger’s equation remains in this
limit.

###  IV.2 Macroscopic region

To describe a macroscopic pointer we consider the thermodynamic limit
$N\to\infty$. The collapse dynamics of equation (7) then dominates the time
evolution for any non-zero value of $\epsilon$, thus satisfying the first
requirement for objective collapse theories. The collapse time $\tau_{c}$ in
this limit will be far shorter than any correlation time of the random
variable $\chi(t)$. The angle $\chi$ is therefore approximately constant
during the collapse process, and the evolution of $\theta(t)$ can be depicted
in a flow diagram like the one in figure 1. Here, $\dot{\theta}$ is depicted
as a function of the instantaneous value of $\theta$ itself, for different
values of the random variable $\chi$ ranging from zero in black (top curve) to
$\pi$ in orange (bottom). The qualitative properties of the dynamics can be
directly seen in the plot. For positive values of $\dot{\theta}$ the angle
$\theta$ will increase over time. Since the black (top) line for $\chi=0$ lies
entirely above the $\dot{\theta}=0$ axis, the value of $\theta$ will increase
over time regardless of its initial value. The state will thus flow towards
$\ket{\downarrow\uparrow}$ and stay there forever. The flow is in the opposite
direction for $\chi=\pi$, while for intermediate values of $\chi$ the outcome
of the collapse dynamics depends on the initial value of $\theta$.

The value of $\chi$ determines the initial value of $\dot{\theta}$, and its
sign will not change during time evolution, thus driving the state towards
either of the two components in the initial superposition, where it remains
forever after. This is the manifestation of the stability of quantum state
reduction in the present model, which was the second requirement for objective
collapse theories. We show in subsection IV.3 that variations in the value of
$\chi$ after the state has collapsed do not affect this stability, and the
macroscopic antiferromagnet will not spontaneously evolve into a superposition
of pointer states.

Figure 1: Flow diagram for the pointer state dynamics. The rate of change
$\dot{\theta}$ is depicted as a function of the instantaneous angle $\theta$
for different values of the random variable $\chi$, ranging from $\chi=0$
(dark, top) to $\chi=\pi$ (light, bottom). Here, we used $J=B_{0}$. The arrows
show the direction of flow, while the dots indicate fixed points.

To determine the probability of finding the final state
$\ket{\uparrow\downarrow}$ as the outcome of an individual collapse process,
we integrate the probability density for finding a particular value of $\chi$
over all values that lead to the desired measurement outcome. Since the
function $\sin(\theta)$ is always positive for $\theta\in[0,\pi]$, the sign of
$\dot{\theta}$, and thus the result of the collapse dynamics, is determined
entirely by the sign of $\cos(\theta(0))-(B_{0}/J)\cos(\chi)$. The probability
for finding the final state $\ket{\uparrow\downarrow}$ is thus given by:

$\displaystyle P_{\ket{\uparrow\downarrow}}$
$\displaystyle=\int_{0}^{\pi}\frac{\sin(\chi)}{2}\,\Theta\left(\cos(\theta_{0})-\frac{B_{0}}{J}\cos(\chi)\right)d\chi$
$\displaystyle=\int_{\arccos(\frac{J}{B_{0}}\cos(\theta_{0}))}^{\pi}\frac{\sin(\chi)}{2}d\chi$
$\displaystyle=\frac{1}{2}\left(1+\frac{J}{B_{0}}\cos(\theta_{0})\right)$ (8)

Here $\Theta(x)$ is the Heaviside step function, and we defined
$\theta(0)=\theta_{0}$. In the second line the step function is converted into
a lower bound on the integral. The result in the final line equals
$\cos^{2}(\theta_{0}/2)$ if and only if $J=B_{0}$. This would correspond to
Born’s rule, as can be read of from Eq. 6, and thus the third requirement on
objective collapse theories being satisfied. We discuss possible physical
mechanisms that may result in such a relation between $J$ and $B_{0}$ in
section V

###  IV.3 Mesoscopic region

For mesoscopic systems, in which the number of constituent particles is
neither small nor approaching the thermodynamic limit, there will be a range
of system sizes such that the collapse time $\tau_{c}$ is larger than the
correlation time $\tau_{r}$ of the random variable but still finite. The value
of $\chi$ will then vary in time during the collapse process.

The probability of obtaining a given measurement outcome in this regime can in
principle be written as a path integral in a straightforward generalisation of
equation (8). Here, we take the complementary approach of evaluating the
probabilities numerically by sampling random values for $\chi(t)$ from a
distribution that enforces both its correlation time $\tau_{r}$ and its long-
time uniform probability density profile Wesseling (2020); Deserno (2002).

Taking very large values for the ratio $B_{0}/J$ while keeping
$\tau_{r}/\tau_{c}$ finite, the stochastic term in equation (7) dominates the
dynamics and the measurement outcome is determined primarily by the sign of
$\cos(\chi(t))$ at early times. Since $\cos(\chi(t))$ starts from a flat
distribution, the probability of finding either pointer state is one half,
independent of the initial state.

In the opposite extreme of very small $B_{0}/J$ the stochastic term becomes
negligible and the collapse statistics is determined entirely by the initial
sign of $\cos(\theta)$. The probability for finding the measurement outcome
$\ket{\uparrow\downarrow}$ is then one for $\theta<\pi/2$, and zero otherwise.

Figure 2: Reproducing Born’s rule. The relative frequency of evolutions coming
within $\delta\theta$ of the state $\ket{0}$ is plotted as a function of the
initial weight $\sin^{2}(\theta_{0}/2)$. Dots represent calculated values,
while lines are included as a guide to the eye only. The diagonal dashed line
indicates Born’s rule, while the differently coloured data sets show different
values of the parameter $B_{0}$, ranging from $B_{0}=1.0$ at the top through
$2.0$, $5.0$, $7.0$, $10$, $50$, and $B_{0}=100$ at the bottom. We set $J=1$
throughout. For each set of parameter values, $10^{4}$ instances of the
dynamics are calculated for a maximum of $20^{3}$ time steps. Taking
$\tau_{r}$ small compared to $\tau_{c}$ and the size of time steps, the random
variable $\chi\in[0,\pi]$ is sampled each time step from the probability
density function $\sin(\chi)/2$, which yields a flat distribution for the
value of $\cos(\chi)$.

For intermediate values of the ratio $B_{0}/J$ the collapse statistics may be
expected to smoothly interpolate between these extreme behaviours, suggesting
the possibility of reproducing Born’s rule for fine-tuned parameter values, as
shown in figure 2 for the example of small $\tau_{r}/\tau_{c}$. Indeed, we
numerically established that for any value of the parameters $N$, $J$, and
$\tau_{r}$ in equation (7), there is a value of $B_{0}$ which results in
Born’s rule being obeyed by the collapse statistics. For the objective
collapse model of equation (7) the parameters yielding Born’s rule in the
mesoscopic regime are found to approximately follow the relation:

$\displaystyle J\approx 2B_{0}^{2}N\tau_{r}$ (9)

Notice that for the macroscopic regime with $\tau_{r}>\tau_{c}$, we saw before
that Born’s rule is obeyed if $J=B_{0}$. At the point
$\tau_{r}=\tau_{c}=1/(2JN)$ connecting the two regimes, the mesoscopic
relation of equation (9) connects continuously to the macroscopic one.

Notice that a correlation between parameters is not an assumption of Born’s
rule, but rather an indication of an underlying physical relation between the
parameters. A example of similar correlations occurs in Einstein’s use of
fluctuation-dissipation to explain why the diffusion and drift terms in
Brownian motion must be fine-tuned with respect to one another Einstein
(1905); Kubo (1966). The fine-tuning seen here similarly signals the fact that
the physical observables represented by these parameters are related.

Given the fine-tuned relation between model parameters, equation (7) is an
objective collapse model that reproduces Born’s rule, assuming that the final
states obtained are stable. That this is not obvious in the presence of a
time-varying stochastic parameter is clear from the observation that such a
term always destabilises the solutions of linear time evolution equations
Mertens  _et al._ (2021). In the present case, however, the non-linear nature
of the collapse dynamics protects the stability of the pointer states. This
can be easily seen by noticing that $d\theta/dt$ near one of the poles of the
Bloch sphere is directed towards the pole for almost all values of the random
parameter.

###  IV.4 Lower bound on the correlation time

The small but non-zero chance of a state evolving away from the poles can be
used to further constrain the model parameters. First, approximate the time-
varying stochastic term as a process in which $\cos(\chi(t))$ is constant for
intervals equal to the correlation time $\tau_{r}$ and chosen randomly from a
flat distribution in every new interval. Starting from an equal-weight
superposition, the state of the system then comes within roughly
$\delta\theta\sim\exp(-\tau_{r}/\tau_{c})\sim\exp(-JN\epsilon\tau_{r})$ of a
pole of the Bloch sphere within a single correlation time $\tau_{r}$. We can
consider the collapse of an object of size $N$ stable if after collapsing into
one of these regions, it may be expected to stay there for at least the age of
the observable universe ($\tau_{u}\sim 4.4\,\cdot 10^{17}$ s). The probability
of finding a value for $\chi$ that would take the system out of the interval
$[0,\delta\theta]$ is equal to $\sin^{2}(\delta\theta/2)$. Roughly, the
collapse is stable within a time equal to the age of the universe if
$\tau_{r}/\sin^{2}(\delta\theta/2)>\tau_{u}$, or equivalently if
$2JN\epsilon\tau_{r}\gtrsim\ln(\tau_{u}/(4\tau_{r}))$. More generally,
observing an object of size $N$ to remain collapsed for a time $\tau_{u}$ thus
provides the value of $\epsilon\tau_{r}$ with a lower bound. This allows
existing methods constraining objective collapse theories Feldmann and Tumulka
(2012) to be applied to the current model.

Further experimental predictions may be deduced from the dependence of the
predicted collapse time on the system size. These can for example result in
bounds on parameter values using experiments similar to those proposed in the
context of other theories, which track quantum interference effects for
superpositions of ever heavier or larger objects Carlesso and Donadi (2019).

Similarly, further constraints on the model may be obtained by excluding the
possibility of super-luminal communication Gisin (1989); Ferrero  _et al._
(2004).

##  V Discussion

The objective collapse model constructed here for an antiferromagnet
superposed over two pointer states meets all minimal requirements for a theory
of quantum measurement. It predicts a negligible effect on the dynamics of
microscopic systems, which thus evolve purely according to Schrödinger’s
equations. At the same time, it predicts macroscopic systems to
instantaneously collapse towards a single classical state. Moreover, owing to
the combination of stochastic and non-linear ingredients, the classical end
states are stable and the frequencies of outcomes obey Born’s rule.

The construction of the model employed a series of generic steps that suggest
the possibility of constructing similar objective collapse theories in more
general settings. To wit, because the pointer was assumed to have a
spontaneously broken symmetry, the order parameter dynamics separated from the
effects of internal degrees of freedom and we could focus the model on only
the collective properties of the antiferromagnet. The required stochastic
nature of the collapse process is included in a natural way by assuming that
the infinitesimal symmetry-breaking field is beyond the control of any
experiment. Finally, its necessary non-unitary character is introduced in the
form of an infinitesimal Wick rotation of the time axis, while the theory is
rendered effectively non-linear by writing its dynamics in the form of Gross-
Pitaevskii equations.

Although all of the steps above straightforwardly generalise to any symmetry-
breaking pointer state and any initial state configuration, we find the fine-
tuning of parameter values required for obtaining Born’s rule not to be
straightforwardly reproducible in more general settings. We therefore do not
argue that the current model presents a realistic theory of nature. Rather, we
present it as a proof of principle establishing that Born’s rule can emerge
from a consistent objective collapse theory without assuming it in the
definition of the model.

This should be contrasted with existing objective collapse models, in which
the probability density function governing the values attained by the
stochastic component depend on the instantaneous state of the system. This is
done either explicitly by using a probability density function that coincides
with the instantaneous real-space wave function Pearle (2007), or implicitly
by considering a Wiener process whose probability density profile is
multiplied by a state-dependent factor Ghirardi  _et al._ (1986, 1990); Bassi
_et al._ (2013).

In the well-known QMUPL model for example Bassi  _et al._ (2013); Diósi
(1989), all necessary ingredients for an objective collapse theory –a
stochastic term, non-unitarity, and a non-linear term– are present. But the
non-linear term multiplies the stochastic process and hence renders the
probability density function of the stochastic variable dependent on the state
of the system being measured. The stochastic term determines the probabilities
of measurement outcomes, and the fact that its distribution scales precisely
with the expectation value that one hopes to find as a result of the
measurement dynamics amounts to introducing Born’s rule in the definition of
the model. Moreover, assuming the properties of the stochastic influence to
depend on the object being measured severely complicates any physical
interpretation for its origin.

In contrast, the model proposed here has separate non-linear and stochastic
terms, which do not depend on one another in any way. The stochastic variables
are therefore drawn from a state-independent probability density profile, and
can be interpreted as originating from a universal, dynamically fluctuating,
non-unitary noise field.

In conclusion, we established the possibility of constructing a model that
satisfies all requirements of an objective collapse theory, without the
presence of Born’s rule being implied by a state-dependent stochastic term. We
thus provide proof of principle for the possibility of Born’s rule
spontaneously emerging in quantum measurement, and suggest the presented model
as a possible starting point in the search for further objective collapse
theories that do not impose Born’s rule.

## References

  * Bassi  _et al._ (2013) A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. of mod. phys 85, 471–527 (2013).
  * Fein  _et al._ (2019) Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, Nature Physics 15, 1242 (2019).
  * Marshall  _et al._ (2003) W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys. Rev. Lett. 91 (2003).
  * Donadi  _et al._ (2020) S. Donadi, K. Piscicchia, C. Curceanu, L. Diósi, M. Laubenstein, and A. Bassi, Nature Physics (2020), 10.1038/s41567-020-1008-4.
  * Vinante  _et al._ (2017) A. Vinante, R. Mezzena, P. Falferi, M. Carlesso, and A. Bassi, Physical Review Letters 119 (2017), 10.1103/physrevlett.119.110401.
  * Carlesso  _et al._ (2016) M. Carlesso, A. Bassi, P. Falferi, and A. Vinante, Physical Review D 94 (2016), 10.1103/physrevd.94.124036.
  * Wit  _et al._ (2019) M. D. Wit, G. Welker, K. Heeck, F. M. Buters, H. J. Eerkens, G. Koning, H. V. D. Meer, D. Bouwmeester, and T. H. Oosterkamp, Review of Scientific Instruments 90, 015112 (2019).
  * Arndt  _et al._ (1999) M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. V. D. Zouw, and A. Zeilinger, Nature 401, 680–682 (1999).
  * Mooij and Nazarov (2006) J. E. Mooij and Y. V. Nazarov, Nature Physics 2, 169–172 (2006).
  * Andrews (1997) M. R. Andrews, Science 275, 637–641 (1997).
  * Christian (2005) J. Christian, Physical Review Letters 95 (2005), 10.1103/physrevlett.95.160403.
  * Komar (1962) A. Komar, Physical Review 126, 365–369 (1962).
  * Wigner (1963) E. P. Wigner, Am. J. Phys. 31, 6–15 (1963).
  * Adler (2003) S. L. Adler, Stud. Hist. Phil. Science B 34, 135 (2003).
  * Penrose (2014) R. Penrose, Found. Phys. 44, 557 (2014).
  * Penrose (1996) R. Penrose, Gen. Relativ. Gravit. 28, 581 (1996).
  * Van Wezel (2010) J. Van Wezel, Symmetry 2, 582 (2010).
  * Schlosshauer  _et al._ (2013) M. Schlosshauer, J. Kofler, and A. Zeilinger, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44, 222–230 (2013).
  * Arndt and Hornberger (2014) M. Arndt and K. Hornberger, Nature Physics 10, 271 (2014).
  * Carlesso  _et al._ (2022) M. Carlesso, S. Donadi, L. Ferialdi, M. Paternostro, H. Ulbricht, and A. Bassi, Nature Physics 18, 243 (2022).
  * Allday (2009) J. Allday,  _Quantum reality: theory and philosophy_ (CRC Press, 2009).
  * Neumann  _et al._ (1955) J. Neumann, R. T. Beyer, E. P. Wigner, and R. Hofstadter,  _Mathematical Foundations of quantum mechanics_ (Princeton University Press, 1955).
  * Ghirardi  _et al._ (1990) G. Ghirardi, A. Rimini, and P. Pearle, Phys. Rev. A 42, 78 (1990).
  * Diósi (1987) L. Diósi, Physics Letters A 120, 377–381 (1987).
  * Mertens  _et al._ (2021) L. Mertens, M. Wesseling, N. Vercauteren, A. Corrales-Salazar, and J. van Wezel, Physical Review A 104, 052224 (2021).
  * Bohr (1928) N. Bohr, Nature 121, 580–590 (1928).
  * Ghirardi  _et al._ (1986) G. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 580 (1986).
  * Pearle (2007) P. Pearle, Open Systems and Measurement in Relativistic Quantum Theory , 195–234 (2007).
  * Zeh (1999) H. D. Zeh,  _The Physical Basis of the Direction of Time_ (Springer, Heidelberg, 1999).
  * Squires (1990) E. J. Squires, Phys. Lett. A 145, 67–68 (1990).
  * Wallace (2012) D. Wallace, in  _Many Worlds?: Everett, Quantum Theory, and Reality_ , edited by S. Saunders, J. Barrett, A. Kent, and D. Wallace (Oxford University Press, 2012).
  * Note (1) Notice that although it is formally possible to always write corrections to Schrödinger’s equation in this form, it is not always the most elegant or efficient formulation.
  * Feldmann and Tumulka (2012) W. Feldmann and R. Tumulka, Journal of Physics A: Mathematical and Theoretical 45, 065304 (2012).
  * Toroš  _et al._ (2017) M. Toroš, G. Gasbarri, and A. Bassi, Physics Letters A 381, 3921–3927 (2017).
  * van Wezel (2008) J. van Wezel, Phys. Rev. B 78, 054301 (2008).
  * Zurek (1981) W. H. Zurek, Physical Review D 24, 1516–1525 (1981).
  * Note (2) Notice that technically, as long as the time evolution is invariant under time translation, there is always an associated Noether charge that should be defined as representing a conserved energy Beekman  _et al._ (2019). This energy, however, is not generally equal to $\langle H\rangle$.
  * Van Wayenburg (2020) B. Van Wayenburg, “1.3 million euros to catch quantum collapse,” https://www.universiteitleiden.nl/en/news/2020/02/1.3-million-euros-to-catch-quantum-collapse (2020).
  * Beekman  _et al._ (2019) A. J. Beekman, L. Rademaker, and J. van Wezel, SciPost Phys. Lect. Notes , 11 (2019).
  * Lieb  _et al._ (1961) E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961).
  * Gross (1961) E. P. Gross, Il Nuovo Cimento 20, 454–477 (1961).
  * Pitaevskii (1961) L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).
  * Bogoliubov (1947) N. Bogoliubov, J. Phys 11, 23 (1947).
  * Pitaevskii and Stringari (2016) L. Pitaevskii and S. Stringari,  _Bose-Einstein condensation and superfluidity_ , Vol. 164 (Oxford University Press, 2016).
  * Okulov (2013) A. Okulov, Journal of Low Temperature Physics 171, 397 (2013).
  * Wesseling (2020) M. Wesseling,  _An objective collapse theory using spontaneous symmetry breaking_ , Master’s thesis, University of Amstedam (2020).
  * Deserno (2002) M. Deserno, Department of Chemistry and Biochemistry UCLA, USA (2002).
  * Einstein (1905) A. Einstein, Annalen der physik 4 (1905).
  * Kubo (1966) R. Kubo, Reports on progress in physics 29, 255 (1966).
  * Carlesso and Donadi (2019) M. Carlesso and S. Donadi, Springer Proceedings in Physics , 1–13 (2019).
  * Gisin (1989) N. Gisin, Helv. Phys. Acta 62, 363 (1989).
  * Ferrero  _et al._ (2004) M. Ferrero, D. Salgado, and J. Sánchez-Gómez, Physical Review A 70, 014101 (2004).
  * Diósi (1989) L. Diósi, Physical Review A 40, 1165–1174 (1989).
  * Mertens and van Wezel (2023) L. Mertens and J. van Wezel, “Environment-assisted invariance does not necessitate born’s rule for quantum measurement,”  (2023).

##  Appendix A Time evolution on the Bloch sphere

The time evolution of quantum states can be written as:

$\displaystyle\frac{\partial}{\partial t}\ket{\psi(t)}$
$\displaystyle=\frac{i}{\hbar}\hat{F}(t)\ket{\psi(t)}$ (10)

with $F$ the (possibly non-Hermitian) time evolution generator. Specializing
to the time evolution of the two-state superpostion of antiferromagnetic
states discussed in the main text, we parameterize the state in its most
general form as:

$\displaystyle\ket{\psi(t)}$
$\displaystyle=n(t)e^{i\xi(t)/2}\left(e^{i\phi(t)/2}\cos(\theta(t)/2)\ket{\uparrow\downarrow}+e^{-i\phi(t)/2}\sin(\theta(t)/2)\ket{\downarrow\uparrow}\right)$
$\displaystyle\Rightarrow~{}~{}~{}\frac{\partial}{\partial t}\ket{\psi(t)}$
$\displaystyle=\frac{1}{2}\Big{(}2\dot{n}\cos(\theta/2)-n\dot{\theta}\sin(\theta/2)+i(\dot{\xi}+\dot{\phi})n\cos(\theta/2)\Big{)}e^{i\xi/2}e^{i\phi/2}\ket{\uparrow\downarrow}$
$\displaystyle+\frac{1}{2}\Big{(}2\dot{n}\sin(\theta/2)+n\dot{\theta}\cos(\theta/2)+i(\dot{\xi}-\dot{\phi})n\sin(\theta/2)\Big{)}e^{i\xi/2}e^{-i\phi/2}\ket{\downarrow\uparrow}.$
(11)

Here, we introduced the time-dependent norm $n(t)$, overall phase $\xi(t)$,
relative phase $\phi(t)$ and the angle $\theta(t)$ determining the relative
weights of the wave function components. This expression gives the left hand
side of equation 10. To find the right hand side, we recall the Hamiltonian
for the antiferromagnet with non-unitary modifications to Schrödingers
equation introduced in the main text:

$\displaystyle\hat{H}\ket{\psi(t)}$
$\displaystyle=\left[\frac{4J}{N}(1+i\epsilon)\left(\frac{\bra{\psi(t)}{\hat{S}}^{z}_{A}\ket{\psi(t)}}{\bra{\psi(t)}\ket{\psi(t)}}\hat{S}^{z}_{B}+\frac{\bra{\psi(t)}\hat{S}^{z}_{B}\ket{\psi(t)}}{\bra{\psi(t)}\ket{\psi(t)}}\hat{S}^{z}_{A}\right)+i\epsilon
B_{0}\cos(\chi)(\hat{S}^{z}_{A}-\hat{S}^{z}_{B})\right]\ket{\psi(t)}$
$\displaystyle=-\frac{N}{2}\left[J(1+i\epsilon)\cos(\theta)-i\epsilon
B_{0}\cos(\chi)\right]ne^{i\xi/2}\left(e^{i\phi/2}\cos(\theta/2)\ket{\uparrow\downarrow}-e^{-i\phi/2}\sin(\theta/2)\ket{\downarrow\uparrow}\right),$
(12)

where in the second line we again focused on the two-state superposition of
equation (11), and wrote the non-linear contributions to the Hamiltonian as
${\bra{\psi(t)}\hat{S}^{z}_{A,B}\ket{\psi(t)}}/{\bra{\psi(t)}\ket{\psi(t)}}=\pm{N}/{4}(\cos^{2}(\theta/2)-\sin^{2}(\theta/2))=\pm{N}\cos(\theta)/4$.
We also used the fact that each sublattice has $N/2$ spin halves to write
$\hat{S}^{z}_{A,B}\ket{\uparrow\downarrow}=\pm{N}/{4}\ket{\uparrow\downarrow}$.

Substituting equation (11) for the left hand side and equation (12) for the
right hand side of (10), we can separately equate the real and imaginary parts
of the components for each of the two basis states. This gives four equations,
which can be solved for the four parameters characterising the two-state
superposition. The imaginary parts yield:

$\displaystyle\dot{\xi}=0$ $\displaystyle\dot{\phi}=-JN\cos(\theta)/\hbar.$
(13)

These are equal to the usual mean-field expressions for unitary time evolution
a Lieb-Mattis antiferromagnet. The real parts of equation contain the non-
unitary contributions and yield:

$\displaystyle\frac{\dot{n}}{n}$
$\displaystyle=\frac{JN\epsilon}{2\hbar}\cos(\theta)\left(\cos(\theta)-\frac{B_{0}}{J}\cos(\chi)\right)$
$\displaystyle\dot{\theta}$
$\displaystyle=-\frac{JN\epsilon}{\hbar}\sin(\theta)\left(\cos(\theta)-\frac{B_{0}}{J}\cos(\chi)\right).$
(14)

The overall phase $\xi$ and normalisation $n$ do not appear in the equations
for $\dot{\theta}$ and $\dot{\phi}$. This indicates that they remain
unobservable even under the non-unitary time evolution, and can ignored when
analysing the flow of $(\theta,\phi)$ on the Bloch sphere. Notice that if they
could be observed there would immediately be super-luminal communication,
explained in detail in Zurek (1981); Mertens and van Wezel (2023).