# Efficient Learning of a One-dimensional Density Functional Theory

M. Michael Denner  Mark H. Fischer  Titus Neupert  Department of Physics,
University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

###### Abstract

Density functional theory underlies the most successful and widely used
numerical methods for electronic structure prediction of solids. However, it
has the fundamental shortcoming that the universal density functional is
unknown. In addition, the computational result—energy and charge density
distribution of the ground-state—is useful for electronic properties of solids
mostly when reduced to a band structure interpretation based on the Kohn-Sham
approach. Here, we demonstrate how machine learning algorithms can help to
free density functional theory from these limitations. We study a theory of
spinless fermions on a one-dimensional lattice. The density functional is
implicitly represented by a neural network, which predicts, besides the
ground-state energy and density distribution, density-density correlation
functions. At no point do we require a band structure interpretation. The
training data, obtained via exact diagonalization, feeds into a learning
scheme inspired by active learning, which minimizes the computational costs
for data generation. We show that the network results are of high quantitative
accuracy and, despite learning on random potentials, capture both symmetry-
breaking and topological phase transitions correctly.

##  I Introduction

Materials with strong electronic correlations host a variety of intriguing
phenomena and quantum phases. Modeling and understanding these systems are
among the greatest challenges in theoretical condensed matter physics. For
quantitative and predictive results, numerical calculations are indispensable.
The most widely and successfully used numerical approach to the electronic
structure problem is based on density functional theory (DFT). In condensed
matter physics, DFT is often linked to band structure calculations, while it
is in principle much more powerful than that. The Hohenberg-Kohn theorems
guarantee that a (potentially correlated) many-body ground state is uniquely
determined by its energy and charge density distribution Hohenberg and Kohn
(1964). However, for practical implementations and a physical interpretation
of calculated results, the Kohn-Sham ansatz is commonly used, producing the
band structure of a different, non-interacting system with the same energy and
density Kohn and Sham (1965). The implicit assumption is that this band
structure captures the essential physics of the original system, at least if
correlations are weak enough.

A critical shortcoming of DFT is that its eponymous functional is not known;
instead, approximations on various levels of complexity are commonly employed
Mori-Sánchez  _et al._ (2008). It is important to emphasize that most of the
functional is _universal_ , representing the many-particle Schrödinger
equation. The only nonuniversal input in a DFT calculation for a crystal is
the potential landscape within the unit cell induced from the ions and core
electrons as well as the particle number, both of which do not affect the
universal part of the functional.

The recent rise of machine-learning methods used to model physical systems has
sparked hopes to use these methods for improving DFT calculations Schleder
_et al._ (2019). The approaches interject the DFT workflow at various stages,
ranging from improving the Kohn-Sham scheme by representing the exchange-
correlation functionals Lundgaard  _et al._ (2016); Kolb  _et al._ (2017); Liu
_et al._ (2017); Nagai  _et al._ (2018); Dick and Fernandez-Serra (2019);
Schmidt  _et al._ (2019) or approximating the unknown energy functional and
its derivatives Snyder  _et al._ (2012, 2013); Li  _et al._ (2016a, b); Yao
and Parkhill (2016); Seino  _et al._ (2018); Nelson  _et al._ (2019); Golub
and Manzhos (2019); Nudejima  _et al._ (2019). More recent works bypass the
Kohn-Sham-solution scheme by directly learning the mapping between material
parameters and ground-state properties Hansen  _et al._ (2013); Schütt  _et
al._ (2014); Hansen  _et al._ (2015); Schütt  _et al._ (2017); Brockherde  _et
al._ (2017); Bogojeski  _et al._ (2018); Ryczko  _et al._ (2018); Schmidt  _et
al._ (2018); Pilati and Pieri (2019); Custodio  _et al._ (2019); Ryczko  _et
al._ (2019); Zepeda-Núñez  _et al._ (2019), or constructing the ground-state
wavefunction from a corresponding density distribution Moreno  _et al._
(2019). Despite these advancements, previous approaches are either limited by
complicated, non-scalable networks, suffer from inefficient training data
generation or struggle in applications to the different physical phases of the
used models.

In this work, we take an approach that follows three guiding principles: (i)
implicit knowledge representation is the key strength of neural networks.
Therefore, we use a _neural network to implicitely represent the (minimized)
DFT functional_. (ii) We aim at solving for phases of quantum matter beyond
the band structure paradigm. To that end, we train the neural network to
directly _output correlation functions_ Moreno  _et al._ (2019), which can be
used to characterize phases and phase transitions. (iii) A balanced dataset is
the key challenge as data acquisition – theoretical or experimental – is
costly. In such settings, _active learning schemes_ Settles (2009); Gubaev
_et al._ (2019); Sivaraman  _et al._ (2019); Yao  _et al._ (2020); Teichert
_et al._ (2020) offer better results with fewer training instances. In
general, active learning describes an algorithm which can actively choose the
data it wants to learn from during training. Here we employ a procedure
inspired by active learning, to incorporate data from different system sizes
in the a priori data generation. In particular, our approach uses costly data
of larger systems only in situations where large finite-size effects are
detected.

Figure 1:  (a) Schematic of the dense neural network used to learn the map
from unit-cell potentials and filling to ground-state energy and density-
density correlators. (b) Data generation inspired by active learning allows to
check the energy deviations of different system sizes and continuing to larger
systems only if necessary. (c) Exact versus predicted ground-state energies on
the test data set of the actively learning network (ALN). The inset shows the
absolute error on the test-energy values as a density histogram $n$. (d) Mean
absolute error per correlator entry on the test data set for the ALN. (e)
Exact versus predicted ground-state energies for the network trained on a
single system size (PLN), with the inset showing the density $n$ of absolute
errors on the test data set. (f) Mean absolute error per correlator entry on
the test data set for the PLN.

Figures 1 (a) and (b) summarize our model, neural network, and workflow. We
choose to work with a one-dimensional lattice model of spinless fermions. The
hopping and interaction terms of the model define our ‘universal Schrödinger
equation’ and are therefore left unaltered throughout the study. Input to the
neural network is the problem-specific potential and particle number. Its
output is the ground-state energy $E_{\rm GS}$ as well as the density-density
correlation function. We start by introducing the employed learning scheme and
demonstrate the quantitative accuracy of network predictions, after training
it on random potentials. The active training shows superior performance
compared to conventional training. We obtain mean squared errors of the energy
of $3.08\cdot 10^{-4}$ in units of the hopping integral. Finally, we apply the
trained network to a topological and a symmetry-breaking phase transition. Our
results demonstrate a scalable architecture, able to capture interacting
lattice models, with successful applications to structured phases.

##  II Model

While density functional theory was originally formulated as a continuum
theory, it has also been successfully applied to lattice models Schönhammer
_et al._ (1995). We consider a Hamiltonian for spinless fermions on a one-
dimensional lattice with sites labelled by $i=1,\cdots,L$ under periodic
boundary conditions,

$\displaystyle\hat{H}=$
$\displaystyle-t\sum_{i}\left(\hat{c}_{i}^{\dagger}\hat{c}_{i+1}+\text{h.c.}\right)+U\sum_{i}\hat{n}_{i}\hat{n}_{i+1}$
$\displaystyle+U^{\prime}\sum_{i}\hat{n}_{i}\hat{n}_{i+2}+\sum_{i}V_{i}\hat{n}_{i},$
(1)

where $\hat{c}_{i}^{\dagger}$ and $\hat{c}_{i}$ are the fermion creation and
annihilation operators on site $i$ and
$\hat{n}_{i}=\hat{c}_{i}^{\dagger}\hat{c}_{i}$ is the corresponding density
operator. Nearest-neighbor hopping is parametrized by $t$, which will serve as
the energy unit throughout. The particles are subject to a repulsive
interaction on nearest- and next-nearest-neighbor sites which we fix to $U=1$
and $U^{\prime}=0.5$ so as to model a lattice analogue of the Coulomb
interaction. This parameter choice places the system in a metallic, but
strongly correlated phase in absence of a potential $V_{i}$, even at half
filling Markhof  _et al._ (2018).

Motivated by the Hohenberg-Kohn theorems, we consider the kinetic term and the
electron-electron interactions as universal, such that the external or ionic
potential $\hat{V}_{\text{ext}}=\sum_{i}V_{i}\hat{n}_{i}$ together with the
filling uniquely determine the ground state and all of its properties. We only
consider potentials with periodicity of four sites. That is, the four values
$V_{i}$, $i=1,...,4$, completely specify the Hamiltonian for any lattice size
$L=4N_{\mathrm{uc}}$, with $N_{\mathrm{uc}}$ the number of unit cells and
$V_{i}=V_{i+4}$ for all $i$. This four-site unit cell can be thought of as the
discretized unit cell of a periodic crystal, while $V_{i}$ is the ionic
potential in this analogy. We restrict it to the range
$V_{i}\in\left[-4,4\right]$. We further denote by the real number
$0<n_{\mathrm{e}}<4$ the particle filling per unit cell. We emphasize that
despite imposing this periodicity to the potential, our approach is able to
capture phases which spontaneously break the four-site translation symmetry.

##  III Learning

The supervised-machine-learning algorithm we use bypasses the Kohn-Sham scheme
by directly learning the map from the external parameters $n_{\mathrm{e}}$ and
$V_{i}$ to the corresponding ground-state energy and density-density
correlators $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}$. The density-
density correlators are calculated for two adjacent unit cells 111In order to
also use small systems ($N_{\mathrm{uc}}=4$) with periodic boundary conditions
as part of the training data, at most two adjacent unit cells from the
interior of the density-density correlator are a valid representative.. The
chosen fully connected neural network Chollet  _et al._ (2015) consists of
four hidden layers which increase in size towards the output as depicted in
Fig. 1 (a) (see also Appendix A).

A central challenge in machine learning is unbiased and efficient data
generation; one usually deals with limited computational or experimental
resources. Here, we generate data by finite-size exact diagonalization (ED) of
systems with randomly chosen $n_{\mathrm{e}},\ V_{i}$. In order to reduce
finite size effects, these examples should naively be generated with as large
systems as possible. However, the computational cost for data generation with
ED grows exponentially with system size. For this reason, we employ a
procedure inspired by _active learning_ , performing costly large system ED,
as depicted in Fig. 1 (b), only if necessary. Using random values for
$n_{\mathrm{e}}$ and $V_{i}$ and starting with a comparison between
$N_{\mathrm{uc}}=3,4$, the scheme iteratively computes larger systems until
the finite size deviation between ground-state energies lies below a priorly
chosen threshold $\theta$. Correspondingly, the fast computation of smaller
systems is used as often as possible, while providing more accurate data in
critical cases (see Appendix B). The samples are further augmented by applying
translations within the unit cell and inversion, allowing the network to
capture the symmetries of the universal part of the Hamiltonian.

We contrast the active learning approach outlined above with a passive
learning scheme using training data generated for systems of fixed size
$N_{\mathrm{uc}}=5$, with filling $n_{\mathrm{e}}$ and on-site potentials
$V_{i}$ chosen randomly. This system size is still solvable efficiently by ED,
while sufficiently reducing finite size effects. The data are again symmetry-
augmented. Both learning procedures were run with an equal time budget to
ensure comparability.

Figure 2:  Neural network results for a transition between different
(obstructed) atomic limit insulators. (a) Compressibility $\kappa$ for various
potential strengths at quarter filling, as calculated from the actively and
passively learned neural network, exact diagonalization (ED) and density
matrix renormalization group (DMRG) of several system sizes. (b) Schematic
depiction of the potential in the four-site unit cell: depending on the
strength and sign, two obstructed atomic limits and a metallic phase can be
realized. (c) Corresponding observable $C$ as calculated from the 8x8 density-
density correlator for the same numerical methods as used for the
compressibility. The insets show the correlator as obtained from the ALN in
the first (lower left) and second atomic limit (upper right) with the unit
cell depicted in red.

A mean absolute error loss function is then optimized to obtain the weights
and biases of the actively (ALN) and passively (PLN) learning neural network.
The resulting performance is evaluated on unseen data, consisting of 20 % of
the full data set. Overfitting was avoided for both systems by suitable
hyperparameter choices (see Appendix C). The absence of significant deviations
in the energy correlation plot in Fig. 1 (c) shows that the ALN performs well
on random data, with an absolute error peaked at $1.2\cdot 10^{-2}$.
Similarly, Fig. 1 (d) shows only small errors in the correlator prediction,
with an overall mean absolute error of $1.8\cdot 10^{-3}$. The PLN performs
worse in predicting energy values and correlators, as shown in the inset of
Fig. 1 (e) and in Fig. 1 (f), with errors at least twice as large as in the
case of the ALN. This comparison shows the advantage of intelligent data
generation at fixed computational time budget.

Random potentials rarely represent a relevant physical scenario. We therefore
present in the following how the randomly trained ALN outperforms the PLN for
structured systems, where $V_{i}$ obey further symmetries. Interestingly,
including the smaller samples, which are suffering from finite-size effects,
with a reduced sample weight in the ALN training leads to a more stable model
prediction, in particular for physical systems (see Appendix B). We attribute
this feature to a regularization of the model.

##  IV Learnability of obstructed atomic limits

We consider the model introduced in Eq. (1) for a potential choice
$(V_{1},V_{2},V_{3},V_{4})=(0,V,V,0)$ at quarter filling ($n_{\mathrm{e}}=1$).
The system is metallic for $V=0$ which separates two distinct insulating
phases for $V>0$ and $V<0$. For $V>0$ ($V<0$), Wannier functions are localized
between the unit cells (in the middle of the unit cell). This corresponds to
two topologically distinct (obstructed) atomic-limit insulators, with the
intra-unit-cell hopping being effectively reduced (enhanced) compared to the
inter-unit-cell hopping. The insulating nature of these phases can be shown by
calculating the compressibility

$\kappa=\frac{1}{n^{2}_{\mathrm{e}}}\left(\frac{\partial^{2}E_{\mathrm{GS}}(n_{\mathrm{e}})}{\partial
n^{2}_{\mathrm{e}}}\right)^{-1},$ (2)

where $n_{\mathrm{e}}$ is the electron filling and
$E_{\mathrm{GS}}(n_{\mathrm{e}})$ the corresponding ground-state energy.
Figure 2 (a) shows that, as one approaches the critical metallic state around
$V=0$, $\kappa$ increases rapidly. We emphasize that since $\kappa$ is the
second derivative of the energy, it is extremely susceptible to errors. Note,
further, that the ED data show a strong even-odd effect in $N_{\mathrm{uc}}$.
We also calculated $\kappa(V)$ with the density matrix renormalization group
algorithm (DMRG) 222Calculations were performed using the TeNPy Library
(version 0.5.0), for a finite but periodic lattice with maximum bond dimension
$\chi_{\mathrm{max}}=2000$, max. truncation error $\epsilon=10\mathrm{e}{-8}$
and energy convergence criterion $\Delta E=10\mathrm{e}{-6}$.Hauschild and
Pollmann (2018) for $N_{\mathrm{uc}}=28$. Compared with these exact results,
the ALN produces a meaningful $\kappa(V)$ with a peak value $\kappa(V=0)$
interpolating between the results of even and odd $N_{\mathrm{uc}}$. On the
contrary, the PLN is worse with a less pronounced and non-symmetric peak. Even
though only trained with at most $N_{\mathrm{uc}}=6$ data, we see that the ALN
also resembles the $N_{\mathrm{uc}}=28$ DMRG result reasonably well.

Figure 3:  Neural network results for a spontaneous symmetry-breaking phase.
(a) Ground-state energy for $V=0$ for several electron fillings
$n_{\mathrm{e}}$ as calculated from the actively and passively learned neural
network and ED. The inset displays the noninteracting band structure. (b)
Ground-state energy as a function of the electron filling $n_{\mathrm{e}}$ in
the symmetry broken phase ($V=4$). The kink at $n_{\mathrm{e}}=1$ signals an
interaction-induced incompressible phase. The inset reveals the band
flattening of the non-interacting system. (c) Phase diagram, schematic of the
potential, and density-density correlation functions. The latter are obtained
from the ALN for $V=0$ (left) and $V=4$ (right). Off-diagonal terms, whose
inequivalence signals the symmetry-breaking phase (right) are highlighted in
red and green.

The location of Wannier centers can be used to differentiate between the two
phases. Defining
$C=\left(\langle\hat{n}_{2}\hat{n}_{2}\rangle-\langle\hat{n}_{2}\hat{n}_{3}\rangle\right)-\left(\langle\hat{n}_{4}\hat{n}_{4}\rangle-\langle\hat{n}_{4}\hat{n}_{5}\rangle\right)$
333Note that we choose $C$ to highlight that two neighboring sites are always
occupied by just one electron. That distinguishes it from the band insulator
case of filling $n_{\mathrm{e}}=2$ and a spontaneously translation symmetry
broken phase at $n_{\mathrm{e}}=1$ (both $C=0$)., the trivial atomic limit
with localization in the unit cell is obtained for $C>0$, the phase transition
happens at $C=0$ and the non-trivial atomic limit has $C<0$. Figure 2 (c)
highlights that both networks are able to capture $C$ across the transition
well, but the ALN results are markedly more accurate than the PLN results when
compared with the ED and DMRG data. This supports the statement that our
active learning scheme delivers quantitatively better results.

##  V Learnability of spontaneously symmetry-broken phases

Spontaneous breaking of translation symmetry can be triggered by introducing a
potential of the form $(V_{1},V_{2},V_{3},V_{4})=(-V,V,-V,V)$ at quarter
filling ($n_{\mathrm{e}}=1$). The symmetry-broken phase arises from the
competition between the next-nearest-neighbor interaction $U^{\prime}$ and the
increasing potential $V$. The four-site translational symmetry of Hamiltonian
(1) is broken spontaneously at $V_{c}\approx 1.8$ (see Appendix D) by the two-
site relations of the emerging degenerate ground-states. The metallic system
shows a smooth dependence of $E_{\rm GS}$ on $n_{\mathrm{e}}$ around quarter
filling [Fig. 3 (a)]. With increasing $V$, $E_{\rm GS}(n_{\mathrm{e}})$
develops a kink at $n_{\mathrm{e}}=1$, signalling the emergence of the
symmetry-broken charge-density wave (CDW) insulator [Fig. 3 (b)]. Both neural
networks represent the different phases very well, the deviations at small
fillings are attributed to the limited amount of training samples in this
limit.

Figure 3 (c) shows that the correlation in the metallic phase is short-ranged
and fast decaying, whereas the symmetry broken phase possesses a distinct
order. The corresponding order parameter $\langle
C_{\mathrm{SSB}}\rangle=\frac{1}{N_{\mathrm{uc}}}\langle\sum_{i}(-1)^{i}\hat{n}_{2i+1}\rangle$
is, however, zero, since the two degenerate ground states have opposite
imbalance in electron density between the first and third site of each unit
cell. Instead, the order can be diagnosed by computing the square of the order
parameter from the density-density correlation functions. This amounts to
$\langle C_{\mathrm{SSB}}^{2}\rangle=\frac{2}{N_{\mathrm{uc}}^{2}}\sum_{i\neq
j}\langle\hat{n}_{4i+1}\hat{n}_{4j+1}\rangle-\frac{2}{N_{\mathrm{uc}}^{2}}\sum_{i,j}\langle\hat{n}_{4i+1}\hat{n}_{4j+3}\rangle+C_{0}$,
with an overall shift
$C_{0}=\frac{1}{N_{\mathrm{uc}}^{2}}\langle\sum_{i}\hat{n}_{2i+1}\rangle$. Its
nonzero value in the symmetry broken phase is implied by the inequivalence
between the first (+) and second (-) term in the above expression, highlighted
in the density-density correlator in Fig. 3 (c) by the red (+) and green (-)
squares. This behavior is well captured by the ALN, producing quantitatively
accurate correlations in both phases.

##  VI Conclusion

We presented a supervised learning approach for lattice DFT, bypassing the
Kohn-Sham solution scheme. Employing a procedure inspired by active learning
allowed us to improve our results at fixed computational cost regarding data
generation. Focussing on correlation functions on a subsystem and taking only
the potential landscape in the unit cell and particle number as input results
in a scalable architecture. Besides verification of our algorithm on unseen
random potentials, we demonstrated that the trained networks reliably solve
for different structured phases.

Looking ahead, it is highly desirable to construct similar implicit (neural
network) representations of DFT for systems in continuous space and higher
dimensions, in particular to attack the electronic structure problem in
strongly correlated regimes. The main challenge is the generation of valid and
balanced data sets, and the incorporation of data from various sources,
including conventional DFT, Monte Carlo calculations, experiments, and future
quantum simulation devices. Two concepts on which our study relies, (1) focus
on correlation functions instead of quantum states and (2) the use of
efficient learning, should prove useful in this future venture.

###### Acknowledgements.

We thank Giuseppe Carleo and Xi Dai for insightful discussions. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (ERC-StG-
Neupert-757867-PARATOP).

##  Appendix A Network parameters

The supervised-machine-learning algorithm we propose in this paper uses dense
neural networks of identical architecture, for both the active and passive
learning scheme. The layers are connected by
Softplus$(x)=\ln{\left(1+e^{x}\right)}$ activation functions, except for the
output layer. The last layer takes into account that correlator values and
ground-state energies have different ranges, by employing a linear activation
function. Table 1 indicates the relevant parameters used in this paper.

Table 1:  Relevant parameters used to create the neural networks in this paper. Parameter | Value
---|---
Neurons Layer 1 | 50
Activation 1 | Softplus
Weight init. 1 | lecun_uniform
Neurons Layer 2 | 125
Activation 2 | Softplus
Weight init. 2 | lecun_uniform
Neurons Layer 3 | 150
Activation 3 | Softplus
Weight init. 3 | lecun_uniform
Neurons Layer 4 | 200
Activation 4 | Softplus
Weight init. 4 | lecun_uniform
Neurons Layer 5 | 65
Activation 5 | Linear
Weight init. 5 | lecun_uniform
Optimizer | Adam
Batch size | 100
Learning rate | 0.001
Epochs | 1500

##  Appendix B Training Data

The choice of training examples is crucial in a machine learning setting, as
data is precious. An intelligent data generation procedure is advantageous if
computational time is finite, removing the necessity to always calculate as
large systems as possible. Naively, the latter is the way to go in order to
avoid finite size biases in the training examples. We contrast these two
approaches as active and passive learning schemes.

The naive approach generates data with exact diagonalization of a five unit
cell system, with electron fillings $n_{\mathrm{e}}$ and potentials $V_{i}$ in
the unit cell chosen randomly. The potentials are in the range $|V_{i}|\leq
4$, and the filling can assume values between $0.6\leq n_{\mathrm{e}}\leq 3.4$
electrons per unit cell. Data augmentation is then used to enlarge the dataset
and allow the neural network to capture the underlying symmetries of the
physical problem. This means applying translations $V_{i}\rightarrow V_{i+1}$
and inversion within the unit cell, where the former leads to a shift of the
correlator data. The resulting dataset consists of 12.020 pairs of fillings
and external potentials, mapped to the corresponding ground-state energies and
density-density correlators. The split in training, validation and test sets
is illustrated in Tab. 2.

Table 2:  Data for the passively trained neural network. Parameter | Value
---|---
Total number of samples | 12020
Number of training samples | 7212
Number of validation samples | 2404
Number of test samples | 2404
Number of samples from 20 site ED | 12020
Sample weight | 1

However, large system diagonalization is not always necessary to obtain
accurate data. This is the basis for an active learning scheme, which performs
the diagonalization of large systems only if strong finite size effects are
detected. The input to this procedure is the random choice of electron
fillings $n_{\mathrm{e}}$ and potentials $V_{i}$ in the unit cell. The former
assuming values between $0.5\leq n_{\mathrm{e}}\leq 3.5$ electrons per unit
cell and the latter being chosen in the range $|V_{i}|\leq 4$. The training
data for both networks was chosen as $|V_{i}|\leq 4$ in order to ensure that
the transition to the symmetry broken phase lies within the trained potential
range.

If the ground-state energy for a $N_{\mathrm{uc}}=3$ and $N_{\mathrm{uc}}=4$
unit cell system, calculated with exact diagonalization, deviates more than a
priorly chosen threshold $\theta$, a system with one additional unit cell is
being calculated. This procedure is repeated up to $N_{\mathrm{uc}}=6$ unit
cells if necessary. This means that the neural network can query a larger
system whenever the deviation of ground-state energies exceeds the chosen
threshold $\theta$, thereby reducing finite size effects. Naively one would
remove the small inaccurate samples from the dataset, however we found that
including them with a reduced sample weight yields similar results on random
test data. However, we obtain better and more stable results on physical
phases, meaning that the presence of the less accurate data points acts as a
regularization of the model. As we strive for meaningful results on physical
phases and not on random potentials, we consider this approach to be more
promising. Data augmentation is again used not only to enlarge the dataset to
74.500 input-output data pairs (see Tab. 3), but also to allow the network to
capture the underlying symmetries.

Table 3:  Data for the actively trained neural network. Parameter | Value
---|---
Total number of samples | 74500
Number of training samples | 44700
Number of validation samples | 14900
Number of test samples | 14900
Number of samples from 16 site ED | 68850
Number of samples from 20 site ED | 5500
Number of samples from 24 site ED | 150
Sample weight 16 site data | 1444This weight is increased to 3 if no 20 or 24 site system had to be calculated for this sample.
Sample weight 20 site data | 2555This weight is increased to 3 if no 24 site system had to be calculated for this sample.
Sample weight 24 site data | 3

The choice of the threshold $\theta$ is a crucial parameter of the active
learning scheme. We therefore investigate the implications of different
thresholds (see Tab. 4), and test the performance of the trained models on
unseen examples. These examples are not part of the training set and were
generated with random fillings $n_{\mathrm{e}}$ and potentials $V_{i}$ in the
range $|V_{i}|\leq 4$ for systems of $N_{\mathrm{uc}}=5,6$.

Table 4:  Number of samples for different choices of the threshold $\theta$, calculated with exact diagonalization and an equal time budget for each parameter choice. $N_{\mathrm{uc}}$ | $\theta=0.0$ | $\theta=0.001$ | $\theta=0.0025$ | $\theta=0.005$ | $\theta=0.01$ | $\theta=1.5$
---|---|---|---|---|---|---
4 | 1350 | 4200 | 13500 | 18850 | 81075 | 83250
5 | 1335 | 575 | 1030 | 790 | 850 | 0
6 | 245 | 10 | 30 | 15 | 20 | 0

The resulting mean absolute error is presented in Fig. 4, indicating that the
error decreases with increasing $\theta$. When considering random potentials,
it is therefore advantageous to use training data generated with as large
$\theta$ as possible, resulting in the largest possible dataset. Consequently,
this means larger systems with $N_{\mathrm{uc}}=5,6$ are never calculated.

The random data generation with potentials in the range $|V_{i}|\leq 4$ causes
however mostly situations where the electrons are localized in the potential
landscape, due to large potentials $V_{i}$. Finite size deviations are however
negligible in such a case. This means that a good performance on this data
does not necessarily correspond to a good elimination of finite size effects
in a realistic physical potential. Figure 4 highlights this with the
evaluation of a dataset with potentials $|V_{i}|\leq 1$, with the minimum of
the obtained mean absolute error shifting towards smaller values of $\theta$.

Consequently, a trade-off is needed between the total number of training
samples and the number of samples with $N_{\mathrm{uc}}=5,6$. The best
compromise is reached for $\theta=0.0025$, with a large enough size of the
dataset, while containing significant information about larger systems. This
parameter was therefore used to generate the final dataset used to train the
actively learning neural network in this paper.

Figure 4: Mean absolute error of neural network predictions for various
datasets, trained on examples generated with different threshold $\theta$ as
illustrated in Tab 4. Random potentials were used to generate the evaluation
data, with a $N_{\mathrm{uc}}=5,|V_{i}|\leq 4$ dataset consisting of 5600
samples, $N_{\mathrm{uc}}=6,|V_{i}|\leq 4$ containing 220 datapoints and 1205
$N_{\mathrm{uc}}=5,|V_{i}|\leq 1$ input - output pairs. The evaluation sets
were not used in the training process of the neural networks.

##  Appendix C Training Performance

The actively and passively constructed datasets are split into training,
validation and test sets. Achieving consistent performance on the optimized
and unseen data indicates that the network has not been overfitted. The
corresponding results for both training schemes are presented in Tab. 5 and 6,
highlighting that the intended mapping from electron fillings and potentials
to ground-state energy and density-density correlator is well captured.

Table 5: Performance of the passively trained neural network. The last table section shows the performance on the ALN test dataset. Parameter | Value
---|---
MAE on $E_{\mathrm{GS}}^{\mathrm{test}}$ | 2.8e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{test}}$ | 1.5e-3
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{test}}$ | 3.6e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{test}}$ | 5.9e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{validation}}$ | 2.6e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{validation}}$ | 1.3e-3
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{validation}}$ | 3.5e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{validation}}$ | 4.9e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{training}}$ | 2.5e-3
MSE on $E_{\mathrm{GS}}^{\mathrm{training}}$ | 1.2e-3
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{training}}$ | 3.2e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{training}}$ | 4.2e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{ALNtest}}$ | 3.7e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{ALNtest}}$ | 2.7e-3
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{ALNtest}}$ | 4.5e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{ALNtest}}$ | 7.0e-5
Table 6: Performance of the actively trained neural network. The last table section shows the performance on the PLN test dataset. Parameter | Value
---|---
MAE on $E_{\mathrm{GS}}^{\mathrm{test}}$ | 1.2e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{test}}$ | 2.4e-4
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{test}}$ | 1.8e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{test}}$ | 1.8e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{validation}}$ | 1.2e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{validation}}$ | 2.3e-4
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{validation}}$ | 1.8e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{validation}}$ | 1.5e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{training}}$ | 1.2e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{training}}$ | 2.3e-4
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{training}}$ | 1.8e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{training}}$ | 1.2e-5
MAE on $E_{\mathrm{GS}}^{\mathrm{PLNtest}}$ | 1.8e-2
MSE on $E_{\mathrm{GS}}^{\mathrm{PLNtest}}$ | 5.2e-4
MAE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{PLNtest}}$ | 2.9e-3
MSE on $\langle\hat{n}_{i}\hat{n}_{j}\rangle_{\mathrm{GS}}^{\mathrm{PLNtest}}$ | 3.6e-5

##  Appendix D Determination of the point of spontaneous symmetry-breaking

The four-site translational symmetry of the considered extended Hubbard model
can be spontaneously broken by the introduction of an external potential
$\hat{V}_{\text{ext}}=\sum_{i=1}^{4}V_{i}\hat{n}_{i}$. Choosing a potential of
the form $V_{1}=-V_{2}=V_{3}=-V_{4}=-V$ causes a competition between next-
nearest neighbor repulsion $U^{\prime}$ and the potential $V$ at quarter
filling. As a result, a symmetry-breaking phase with two degenerate ground-
states emerges, corresponding to ordering the electrons either to site one or
to site three in each unit cell.

We consider two approaches in order to identify the potential strength at
which the spontaneous symmetry-breaking occurs. For each of these we
investigate a finite size scaling plot, to derive the extrapolated point of
the phase transition in the thermodynamic limit.

Figure 5:  (a) Finite size scaling plot of the critical potential $V_{c}$,
obtained with a small potential $\delta V$ to break the ground-state
degeneracy. Density matrix renormalization group (DMRG) calculations of open
boundary conditions are combined with the exact diagonalization (ED) of a 16
site system. (b) Schematic of the potential landscape in the unit cell. (c)
Finite size scaling plot of the critical potential $V_{c}$, obtained with an
additional half unit cell and electron to break the ground-state degeneracy.
Density matrix renormalization group (DMRG) calculations of open boundary
conditions are combined with the exact diagonalization (ED) of an 18 site
system. (d) Schematic of the potential landscape with additional two lattice
sites.

As stated above, the symmetry broken phase possesses a distinct order in the
density-density correlator, occupying only sites with negative potential. By
adding a small potential on one site, e.g. $V_{1}=-V-\delta V,~{}\delta V\ll
V$, one of the degenerate ground-states is favoured. Consequently, the
occupation $\langle\hat{n}_{i}\rangle_{\mathrm{GS}}$ will concentrate on the
first lattice site of each unit cell. We therefore investigate the difference
in the occupation of site one and three,
$C=\langle\hat{n}_{1}\rangle_{\mathrm{GS}}-\langle\hat{n}_{3}\rangle_{\mathrm{GS}}$.
In the metallic phase for a vanishing potential $V=0$, all sites are equally
occupied, despite the small offset $\delta V$. This changes around the point
of the phase transition, where explicitly one of the symmetry-breaking ground
states is selected. The position of this jump in $C$ can then be studied with
various system sizes and extrapolated to the thermodynamic limit. Since exact
diagonalization already reaches computational boundaries for quite small
systems, we additionally employ the density-matrix renormalization group
algorithm (DMRG) 666Calculations were performed using the TeNPy Library
(version 0.5.0), for a finite lattice with maximum bond dimension
$\chi_{\mathrm{max}}=1000$, max. truncation error $\epsilon=10\mathrm{e}{-10}$
and energy convergence criterion $\Delta E=10\mathrm{e}{-6}$.Hauschild and
Pollmann (2018). In order to ensure convergence also for large systems, open
boundary conditions are considered. The influence of the boundary can however
be neglected when considering density-density correlations in the middle of
the system. Figure 5(a) indicates the transition point as extrapolated from
several system sizes, leading to a potential of $V_{c}=1.744$.

Additionally, one can study the emergence of the spontaneously symmetry-
breaking phase in the energy spectrum. The restriction to systems with open
boundary conditions for DMRG calculations leads to a degeneracy of ground
states of the order of the number of unit cells, compared to two states in the
case of periodic boundaries. We therefore enlarge the system by two lattice
sites and add one additional electron for the entire system, breaking the
ground-state degeneracy. The transition to the symmetry broken phase can
consequently be probed by calculating the gap $E_{\mathrm{GS}}-E_{\mathrm{1}}$
for several system sizes. The critical potential $V_{c}$ corresponds to the
point of gap opening. The corresponding results are presented in Fig. 5(c),
with an extrapolated transition point at $V_{c}=1.864$. Taking the mean
between both results gives a critical potential $V_{c}\approx 1.8$.
Correspondingly, choosing the range of training potentials as
$V_{i}\in\left[-4,4\right]$ secures that the neural networks can capture the
phase transition.

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# Wigner’s Friend Depends on Self-Contradictory Quantum Amplification

Andrew Knight<EMAIL_ADDRESS>
###### Abstract

In a recent paper, Żukowski and Markiewicz showed that Wigner’s Friend (and,
by extension, Schrödinger’s Cat) can be eliminated as physical possibilities
on purely logical grounds. I validate this result and demonstrate the source
of the contradiction in a simple experiment in which a scientist S attempts to
measure the position of object $\ket{O}=\ket{A}_{S}+\ket{B}_{S}$ by using
measuring device M chosen so that $\ket{A}_{M}\approx\ket{A}_{S}$ and
$\ket{B}_{M}\approx\ket{B}_{S}$. I assume that the measurement occurs by
quantum amplification without collapse, in which M can entangle with O in a
way that remains reversible by S for some nonzero time period. This assumption
implies that during this “reversible” time period,
$\ket{A}_{M}\neq\ket{A}_{S}$ and $\ket{B}_{M}\neq\ket{B}_{S}$ – i.e., the
macroscopic pointer state to which M evolves is uncorrelated to the position
of O relative to S. When the scientist finally observes the measuring device,
its macroscopic pointer state is uncorrelated to the object in position
$\ket{A}_{S}$ or $\ket{B}_{S}$, rendering the notion of “reversible
measurement” a logical contradiction.

Schrödinger’s Cat [28] just won’t die. Its conscious counterpart, Wigner’s
Friend [30], has also made a recent resurgence in the physics academy [6, 7,
10, 24, 25]. Ordinarily, physicists treat such issues as empirical questions,
relegating armchair philosophy to the humanities. That’s why, as a legal
scholar, I was surprised and delighted that Physical Review Letters published
[32], in which Żukowski and Markiewicz argue that the Wigner’s Friend (”WF”)
paradox can be eliminated on a priori logical grounds because it depends on
the “inherently self-contradictory notion” that WF measures outcomes. In
essence, the authors show that a reversible measurement is a logical
contradiction. If they are correct, their work may have implications not only
for the physical viability of any macroscopic quantum superposition “beyond
the Heisenberg cut,” including Schrödinger’s Cat (“SC”), but also for physical
limits to quantum computing and whether decoherence [16] causes objective wave
function collapse.

Logic-based algorithms do occasionanlly sneak into debates about the nature of
the physical world. For example, [14] argues that the case for scalable
quantum computing rests on a logical contradiction, [4, 26] argue that the
consciousness-causes-collapse hypothesis is logically immune from empirical
falsification, while [18] argues that a logical contradiction infests
literature on the characterization of quantum superpositions. Can a purely
logical argument tell us something about the physical possibility of SC/WF? No
one has ever created SC/WF, despite the confusing use of “cat state” to
describe certain quantum superpositions [31]. The assertion that SC/WF is a
physical possibility is itself a logical inference arising from an assumption
of quantum amplification without collapse. If that assumption can be shown to
be self-contradictory on logical grounds, then indeed a logical argument can
tell us something about SC/WF.

Many of the big problems in foundational physics, including the measurement
problem [11, 21] and SC/WF, arise from a universality assumption (“U”) that
the quantum mechanical wave function always evolves linearly or unitarily or
reversibly, an assumption that seems to be rarely challenged in the
literature.111The authors of [8, 12, 15, 17, 23] are among these maverick
challengers. According to the mathematics of quantum mechanics, when an object
in a quantum superposition interacts with another system, such as a measuring
device, the two “entangle” or “correlate” as if the system inherits the
superposition. Unless the superposition objectively collapses (contrary to U),
it gets inherited by other systems with which the system interacts, leading to
an amplification of the original superposition.

Consider a cat in an isolated box. The box contains a radioisotope (which is
in a quantum superposition over states “decayed” and “undecayed”) and a device
that releases a poison if the radioisotope decays. U implies that the device
inherits the radioisotope’s superposition, then the cat inherits the device’s
superposition, leading to that confusing hypothetical SC state in which the
cat seems to be in a superposition over states “dead” and “alive.” Quantum
amplification without collapse, which I’ll describe in more detail below, is
crucial for the possibility of SC.

A scientist S (initially in state $\ket{S}$) wants to measure the position of
a tiny object O, which is in a superposition of macroscopically distinct
semiclassical position eigenstates corresponding to locations A and B
separated by some distance $d$. Neglecting normalization constants,
$\ket{O}=\ket{A}+\ket{B}$. To measure it, he uses a measuring device M
(initially in state $\ket{M}$) configured so that a measurement of the
position of O will correlate M and O such that M will then evolve over time to
a corresponding macroscopic pointer state, denoted $\ket{M_{A}}$ or
$\ket{M_{B}}$. Device M in state $\ket{M_{A}}$, for example, indicates “A”
such as with an arrow-shaped indicator pointing at the letter “A.” In other
words, M is configured so that if it measures O at location A then M will,
through a semideterministic causal chain that amplifies the measurement,
evolve to some state that is obviously different to scientist S than the state
to which it would evolve had it measured O at location B.

Let’s say that the experiment is set up at time $t_{0}$; then at time $t_{1}$
M “measures” O via some initial correlation event, after which M then evolves
in some nonzero time $\Delta t$ to a correlated macroscopic pointer state;
then at $t_{2}$ S reads the device’s pointer. Time $\Delta t$ is small, so in
the following equation I’ll ignore it. The scientist in state $\ket{S_{A}}$,
for example, (believes he) has measured the position of O at position A by
nature of having observed the device in state $\ket{M_{A}}$. The standard
narrative in quantum mechanics, following from the assumption of U, is this
von Neumann [22] amplification chain:

$\displaystyle t_{0}:$ $\displaystyle\ket{O}\ket{M}\ket{S}$ (1)
$\displaystyle=$ $\displaystyle\Bigl{(}\ket{A}+\ket{B}\Bigr{)}\ket{M}\ket{S}$
$\displaystyle\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}$
$\displaystyle\Bigl{(}\ket{A}\ket{M_{A}}+\ket{B}\ket{M_{B}}\Bigr{)}\ket{S}$
$\displaystyle\stackrel{{\scriptstyle t_{2}}}{{\longrightarrow}}$
$\displaystyle\ket{A}\ket{M_{A}}\ket{S_{A}}+\ket{B}\ket{M_{B}}\ket{S_{B}}$

According to Eq. 1, at time $t_{1}$, object O and device M are entangled but
scientist S remains separable, while at time $t_{2}$, all three are entangled.
What this means physically is that between times $t_{1}$ and $t_{2}$, S can –
at least “in principle” – reverse the correlation event between O and M by
performing a properly designed interference experiment. The state at $t_{1}$
is the elusive “cat state” in which M seems to be in a superposition over
state $\ket{M_{A}}$, in which it macroscopically indicates “A,” and state
$\ket{M_{B}}$, in which it indicates “B.” The possibility of producing SC
depends on the assumption that a quantum object can entangle with a cat in a
way that remains physically reversible by an external observer for some
nonzero time. SC cannot exist, even in principle, unless Eq. 1 is correct. Is
it? Eq. 1 implies:

  * •

There is an event between O and M at $t_{1}$, and

  * •

S can physically reverse this event at any time before $t_{2}$.

However, if S can reverse the event, then in what sense, by whose observation,
and by whose clock does the event occur?

At $t_{1}$, can S correctly claim that the event occurred? No. At $t_{1}$, S
is uncorrelated to M and O and has absolutely no evidence that they
interacted. He can’t have any evidence, because evidence would correlate them.
He can’t say, at $t_{1}$, that he will have evidence at $t_{2}$, because he
can reverse the event before $t_{2}$. In the reversible period between $t_{1}$
and $t_{2}$, there is not, from the perspective of S, an event between O and
M.

At $t_{1}$, does the event occur from the perspective of M? Again, no. Ref.
[32] shows that “Outcomes of premeasurements… are an internally inconsistent
notion (and therefore are not ‘facts’).” And there is another reason. What
does time $t_{1}$ even mean? By what standard? Well, by a clock, of course! A
clock is just an object that physically ticks off time – a wristwatch, a
cellular clock, a vibrating piece of quartz, etc. If you were to take an
isolated system containing, say, an old cat and a grandfather clock, and
perfectly reversed the entire system222And I mean perfectly. The trajectory of
every atom would have to be reversed with perfect precision. It’s not
possible, because even the largest objects in the universe (black holes) are
swayed by unpredictable quantum chaos [5], but it helps with the thought
experiment. until the cat was young, then, from the cat’s perspective, was the
cat ever old? No.333See interesting discussions in [2, 3, 20]. By perfectly
reversing the system, every piece of evidence that might indicate the passage
of time is retroactively erased. The clock, of course, indicates no passage of
time, but neither does the cat’s body or brain. If you perfectly reverse an
isolated system, you perfectly reverse its time, too.444Deutsch [9] argues
that WF can be produced without perfectly reversing time by having WF write a
letter to Wigner that is uncorrelated to the outcome that he “observed.” Salom
[27] correctly points out Deutsch’s error.

On this basis, Eq. 1 is false because it implies two contradictory statements.
How can there be an event at $t_{1}$ if S can, before $t_{2}$, physically
reverse it as if it never happened? Ref. [32] is correct: a reversible
measurement is a contradiction. The von Neumann amplification chain of Eq. 1,
on which the in-principle possibility of SC rests, is internally inconsistent
and thus false.

The problem with Eq. 1, as I will explain below with a simple example, is that
the appearance of the nonzero time period ($t_{2}-t_{1}$) is an illusion
caused by failure to keep track of what the letters “A” and “B” actually refer
to in each of the terms. In the above example, $\ket{A}$ is an eigenstate of
the position operator corresponding to localization at position A. But Galileo
and Einstein might point out that “position A” only has meaning relative to
other objects. For scientist S using measuring device M, is position A
relative to S or to M? Is the object O in Eq. 1 in state
$\ket{A}_{S}+\ket{B}_{S}$ or $\ket{A}_{M}+\ket{B}_{M}$? Ordinarily, it would
not matter. After all, when would the scientist and his measuring device
disagree on the location of position A?

Imagine two measuring devices that are in a superposition relative to each
other [1, 13, 19, 33]. The first one might see position A as a sharp point
while the second might see it “blurred.” If the blurred view still allows the
second measuring device to distinguish position A from position B, then
there’s no problem. But if the width of the “blurriness” exceeds the distance
$d$ between A and B, then “position A” as measured by the first device will
not correlate to “position B” as measured by the second. So if the scientist
and his measuring device were spatially “blurry” relative to each other, they
might disagree about “position A,” in which case the state of O would have to
be written as either $\ket{A}_{S}+\ket{B}_{S}$ or $\ket{A}_{M}+\ket{B}_{M}$.

In reality, it’s hard to see how this could happen. Quantum dispersion is a
natural process that makes all objects spatially fuzzy relative to each other
over time. A consequence of quantum uncertainty, loosely defined as $\Delta
x(m\Delta v)\geq\hbar/2$, dispersion is inversely related to mass as well as
tempered by decohering interactions with particles zipping around the universe
[16]. The larger an object is, the more slowly it disperses and the more
opportunity it has to be localized by another object.555Tegmark [29], for
example, calculates that the coherence length of a $10\mu m$ dust particle,
due to decoherence by cosmic microwave background radiation ubiquitous
throughout the universe, is $10nm$, a full three orders of magnitude smaller
than the dust particle itself. Physically, this means that such a dust
particle could not, in the actual universe, disperse adequately to subject it
to a double-slit interference experiment. So while the scientist might in
principle be delocalized from the measuring device by some miniscule amount,
that amount is much, much smaller than the distance $d$ distinguishing
positions A and B. As long as position A relative to the scientist is
approximately the same as position A relative to the device, then
$\ket{O}=\ket{A}_{S}+\ket{B}_{S}\approx\ket{A}_{M}+\ket{B}_{M}$.666$\ket{A}_{S}\approx\ket{A}_{M}$,
for instance, does not mean that location A is the same distance from S as
from M. Rather, it means that location A relative to S is well correlated to
location A relative to M; there is zero (or inconsequential) quantum fuzziness
between the locations; S and M “see” see point A as essentially the same
sharply defined location.

Let’s look again at the implications of Eq. 1 and how its internal
contradiction arises in a simple example in which the distinction between
states $\ket{A}_{S}$ and $\ket{A}_{M}$, for example, is respected. In Fig. 1a,
object O is shown at time $t_{0}$ in a superposition of position eigenstates
$\ket{A}_{S}$ and $\ket{B}_{S}$ corresponding to locations at $A_{S}$ and
$B_{S}$ relative to scientist S, where O is shown crosshatched to represent
its superposition relative to S. Measuring device M has slots (a) and (b) and
is configured so that detection of object O in slot (a) will, due to a semi-
deterministic causal amplification chain, cause it to evolve over nonzero time
$\Delta t$ to a macroscopic pointer state in which a large arrow indicator
points to the letter “A,” and vice versa for detection of object O in slot
(b). S intends to measure the object’s location relative to him and wants the
device’s indicator to correlate to that measurement. Therefore, because the
device’s detection of O in slot (a) actually corresponds to measurement of O
at location $A_{M}$ relative to M (and vice versa for slot (b)), S configures
M at time $t_{0}$ so that $A_{M}\approx A_{S}$ and $B_{M}\approx B_{S}$. The
experiment is designed so that the initial correlation event between O and M
occurs at time $t_{1}$, M evolves to its macroscopic pointer state by time
$t_{1}+\Delta t$, and S reads the device’s pointer at $t_{2}$.

Ignoring Eq. 1 for the moment, Figs. 1b and 1c show how the scientist might
want the system to evolve. In Fig. 1b (at time $t_{1}$), as long as
$A_{M}\approx A_{S}$ and $B_{M}\approx B_{S}$, the device’s detection of O in
slot (a) correlates to the object’s location at $A_{S}$.777No collapse or
reduction of the wave function is assumed in this example or anywhere else in
this paper. Fig. 1b simply shows the correlation between detection in slot (a)
and position $A_{S}$; there would also be a correlation (not shown) between
detection in slot (b) and position $B_{S}$. Then, in Fig. 1c (at time
$t_{1}+\Delta t$), M has evolved so that the indicator now points to letter
“A,” correlated to the device’s detection of O in slot (a). Then, when S looks
at the indicator at time $t_{2}$, he will observe the indicator pointing at
“A” if O was localized at $A_{S}$ and “B” if it was localized at $B_{S}$,
which was exactly his intention in using M to measure the object’s position.

Figure 1: Fig. 1a. An object is shown at time $t_{0}$ in a superposition of
position eigenstates $\ket{A}_{S}$ and $\ket{B}_{S}$ corresponding to
macroscopically distinct positions $A_{S}$ and $B_{S}$ relative to scientist
S, who uses measuring device M to measure the object’s position. Figure 2:
Fig. 1b. The object is measured at time $t_{1}$ by M so that detection in slot
(a), which corresponds to localization at position $A_{M}$ relative to M, also
correlates to position $A_{S}$ relative to S. The situation of detection in
slot (b), corresponding to localization at position $B_{M}$, is not shown.
Figure 3: Fig. 1c. By time $t_{1}+\Delta t$, M evolves to a macroscopic
pointer state in which an indicator points to letter “A,” read by S at time
$t_{2}$.

In Fig. 1b, O is indeed localized relative to M, and since $A_{M}\approx
A_{S}$ and $B_{M}\approx B_{S}$, it is also localized relative to S. The
scientist does not know, of course, whether object O was detected in slot (a)
or (b), and Fig. 1b only shows the first possibility, but it is in slot (a) or
(b), with slot (a) correlated to $A_{S}$ and slot (b) correlated to $B_{S}$
(both of which are also localized relative to S). Because O is localized
relative to S at $t_{1}$, S and O are correlated, in which case S cannot do an
interference experiment to show O or M in superposition, which conflicts with
the requirements of Eq. 1.

Let’s imagine a slight variation to Fig. 1b, shown in Fig. 2, in which S has
misplaced M so that its measurement of O at position $A_{M}$ (by nature of
detecting the object in slot (a)) correlates to position $B_{S}$ relative to
the scientist. When the scientist later sees the indicator point to the letter
“A,” the result is certain to be wrong. However, certainty implies
correlation; if the scientist ever realizes that he misplaced the device, he
will know that the measurement outcome was actually B.

Figure 4: Fig. 2. In a variation on Fig. 1b, M has been misplaced so that
detection in slot (a), which corresponds to localization at position $A_{M}$
relative to M, correlates to position $B_{S}$ relative to S. The situation of
detection in slot (b), corresponding to localization at position $B_{M}$, is
not shown.

Now, suppose we demand, consistent with Eq. 1, that at time $t_{1}$, scientist
S can, with an appropriately designed interference experiment, demonstrate
that O and M are in a superposition relative to S. That requirement implies
that the object’s location, as measured by M via the correlation event at
$t_{1}$, does not correlate to the object’s location relative to S. The
device’s detection at $t_{1}$ of O in, for example, slot (a), which
corresponds to its measurement of the object at $A_{M}$, cannot correlate to
the object’s location at $A_{S}$. Thus, to ensure that O remains unlocalized
relative to S when the correlation event at $t_{1}$ localizes O relative to M,
that location which M measures as $A_{M}$ (by detection of object O in slot
(a)) cannot correlate to location $A_{S}$. By combining Figs. 1b and 2, this
situation is shown in Fig. 3a in which both O and M are shown in a
superposition of position eigenstates relative to S.

Figure 5: Fig. 3a. Combining Figs. 1b and 2, both O and M are shown at time
$t_{1}$ in a superposition of position eigenstates relative to S. Detection in
slot (a) by M in position $M_{A}$ correlates to O in position $A_{S}$ while
detection in slot (a) by M in position $M_{B}$ correlates to O in position
$B_{S}$. The situation of detection in slot (b) is not shown.

In Fig. 3a, the object’s crosshatching, like in Fig. 1a, represents its
superposition over locations $A_{S}$ and $B_{S}$. Analogously (but without
crosshatching), device M localized at $M_{A}$ is shown superimposed with
device M localized at $M_{B}$. Importantly, $M_{A}$ is the position of M,
relative to S, that would measure the position of object O at $A_{M}$ (by
detecting it in slot (a)) as position $A_{S}$, while $M_{B}$ is the position
of M that would measure the position of object O at $A_{M}$ (by detecting it
in slot (a)) as position $B_{S}$. Because O remains uncorrelated to S at
$t_{1}$ (as demanded by Eq. 1), the device M that detects O in slot (a) must
also be uncorrelated to S at $t_{1}$. (The same is true for device M that
detects O in slot (b), which is not shown.) What is demonstrated in Fig. 3a is
that the correlation event at $t_{1}$ between M and O that localizes O
relative to M requires that M is not correlated to S, thus allowing S to
demonstrate M in a superposition, as required by Eq. 1. Then, in Fig. 3b, at
time $t_{1}+\Delta t$, M has evolved so that the indicator now points to
letter “A,” correlated to the device’s detection of O in slot (a).

Figure 6: Fig. 3b. By time $t_{1}+\Delta t$, device M evolves to a macroscopic
pointer state in which an indicator points to letter “A.” Because the device’s
output is completely uncorrelated to localization at $A_{S}$ or $B_{S}$, it
failed to perform the desired measurement.

However, now we have a problem. In Fig. 3b, the pointer indicating “A,” which
is correlated to the device’s localization of O at $A_{M}$, is not correlated
to the object’s localization at $A_{S}$. When S reads the device’s indicator
at time $t_{2}$, it is not that the output is certain to be wrong, but rather
that the output is certain to be uncorrelated to the measurement he intended
to make. It’s not merely that the output is unreliable – sometimes being right
and sometimes being wrong – it’s that the desired measurement simply did not
occur. The correlation event at time $t_{1}$ did not correlate the scientist
to the object’s location relative to him. That is only possible if that
location which M would measure at $t_{1}$ as $A_{M}$ could be measured by S as
either $A_{S}$ or $B_{S}$, which is only possible if $A_{M}\neq A_{S}$.

To recap: At $t_{0}$, S sets up the experiment so that $A_{M}\approx A_{S}$
and $B_{M}\approx B_{S}$, which is what S requires so that M actually measures
what it is designed to measure. If Eq. 1 is correct, then at time $t_{1}$,
$A_{M}\neq A_{S}$ and $B_{M}\neq B_{S}$.

This has two consequences. First, we need to explain how M could become
adequately delocalized relative to S so that $A_{M}\neq A_{S}$ and $B_{M}\neq
B_{S}$ in the time period from $t_{0}$ to $t_{1}$. We are not talking about
relative motion or shifts; we are talking about $A_{M}$ and $A_{S}$ becoming
decorrelated from each other so as to be in a location superposition relative
to each other. In other words, how do $A_{M}$ and $A_{S}$, which were well
localized relative to each other at time $t_{0}$, become so “fuzzy” relative
to each other via quantum wave packet dispersion that $A_{M}\neq A_{S}$ at
time $t_{1}$? They don’t. As long as the relative coherence length between two
objects is small relative to the distance $d$ separating distinct location
eigenstates, then the situation in which $A_{M}\neq A_{S}$ cannot happen over
any time period [29].

Second, the quantum amplification in a von Neumann chain depends on the
ability of measuring devices to measure what they are intended to measure.
State $\ket{A}_{S}+\ket{B}_{S}$ can only be amplified through entanglement
with intermediary devices if those terms actually become correlated to states
$\ket{A}_{S}$ and $\ket{B}_{S}$, respectively. But if, as required by Eq. 1,
there is some nonzero time period ($t_{2}-t_{1}$) in which S can demonstrate M
and O in a superposition, then the device’s measurement of the object’s
location relative to it cannot correlate to the object’s location relative to
S, in which case future states of S cannot be correlated to either
$\ket{A}_{S}$ or $\ket{B}_{S}$. Eq. 1 is internally inconsistent and is
therefore false. There is no time period in which S can measure M and O in a
superposition.

This is not a subtle or merely semantic contradiction I am pointing out. Let
us place a particle in a superposition in which one eigenstate localizes the
particle on the left side of a huge auditorium (in front of a giddy audience
of physicists), while the other eigenstate localizes it on the right side. I
then use a measuring device – I must use a measuring device, of course, since
we are talking about a particle too tiny to see – configured so that if it
measures the particle on the left side, it displays a huge “A” visible for all
to see, and vice versa for the right side. Now, if Eq. 1 is correct, then when
we look at the device and see a huge “A” on it, it does not tell us that the
particle was detected on the left side of the room! Rather, it tells us that
the device measured it at location $A_{M}$ which is uncorrelated – and thus
irrelevant – to our intended measurement. If Eq. 1 is correct, then the device
does not measure.

In conclusion, we start with the simplest and most fundamental of
measurements, in which scientist S attempts to measure the position of object
$\ket{O}=\ket{A}_{S}+\ket{B}_{S}$ by using measuring device M chosen so that
$\ket{A}_{M}\approx\ket{A}_{S}$ and $\ket{B}_{M}\approx\ket{B}_{S}$. We then
assume that the measurement occurs by quantum amplification without collapse,
as described by Eq. 1, in which M can entangle with O in a way that remains
reversible by S for some nonzero time period. This implies that at time
$t_{1}$, $\ket{A}_{M}\neq\ket{A}_{S}$ and $\ket{B}_{M}\neq\ket{B}_{S}$ – i.e.,
the macroscopic pointer state to which M evolves is uncorrelated to the
position of O relative to S. Then at time $t_{2}$, the scientist in state
$\ket{S_{A}}$ has observed device M in state $\ket{M_{A}}$ and the scientist
in state $\ket{S_{B}}$ has observed device M in state $\ket{M_{B}}$, neither
of which is correlated to the position of object O at $A_{S}$ or $B_{S}$.
Thus, Eq. 1 cannot yield the desired measurement. If measurement proceeds by
the quantum amplification of Eq. 1, then there is no measurement.

Eq. 1 is internally inconsistent because it implies that a measuring device
used to distinguish states $\ket{A}_{S}$ and $\ket{B}_{S}$ cannot do so. A
reversible measurement is indeed a contradiction. Therefore, if the in-
principle possibility of Schrödinger’s Cat depends on whether measurement
proceeds according to quantum amplification without collapse, as characterized
by Eq. 1, then the cat is truly dead.

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