# Object recognition for robotics from tactile time series data utilising
different neural network architectures

Wolfgang Böttcher ID ETH Zurich
Zurich, Switzerland
<EMAIL_ADDRESS>Pedro Machado ID Computational Neurosciences and
Cognitive Robotics Group
School of Science and Technology
Nottingham Trent University
Nottingham, UK
<EMAIL_ADDRESS>Nikesh Lama ID Computational Neurosciences
and Cognitive Robotics Group
School of Science and Technology
Nottingham Trent University
Nottingham, UK
<EMAIL_ADDRESS>T.M. McGinnity ID Computational Neurosciences and
Cognitive Robotics Group
School of Science and Technology
Nottingham Trent University
Nottingham, UK
<EMAIL_ADDRESS>
Intelligent Systems Research Centre
Ulster University
Northern Ireland, UK
<EMAIL_ADDRESS>
###### Abstract

Robots need to exploit high-quality information on grasped objects to interact
with the physical environment. Haptic data can therefore be used for
supplementing the visual modality. This paper investigates the use of
Convolutional Neural Networks (CNN) and Long-Short Term Memory (LSTM) neural
network architectures for object classification on Spatio-temporal tactile
grasping data. Furthermore, we compared these methods using data from two
different fingertip sensors (namely the BioTac SP and WTS-FT) in the same
physical setup, allowing for a realistic comparison across methods and sensors
for the same tactile object classification dataset. Additionally, we propose a
way to create more training examples from the recorded data. The results show
that the proposed method improves the maximum accuracy from 82.4 % (BioTac SP
fingertips) and 90.7 % (WTS-FT fingertips) with complete time-series data to
about 94 % for both sensor types.

###### Index Terms:

3D-CNN, CNN, LSTM, tactile sensing, object classification, BioTac, WTS-FT

##  I Introduction

The sense of touch is critical for exploring the properties of objects and
their classification. Unlike vision which is subject to many limitations, such
as illumination, background noise, occlusion etc., tactile information allows
one to directly infer material properties of the object touched. Thus, it is
crucial for modern robotics as the field advances towards autonomous grasping,
slippage detection, object detection, and dexterous manipulation. Currently
employed visual object recognition strategies are challenged, because objects
may have similar shape and appearance but different material composition. In
contrast, tactile classification strategies incorporating features of the
objects shape and material properties have proven to be promising for such
settings.

Haptic object classification systems use the material properties of the
objects as well as their shape in the classification step [1, 2, 3, 4, 5, 6,
7] as shown in Fig. 1. The topic of haptic object classification has seen a
wide variety of different data processing and classification approaches. At
the same time, it is closely related to other fields in robotics research.
Autonomous robotic systems for manipulation are often limited by the use of
visual exploration only, thus restricting the identification of the nature of
the object and impacting upon the choice of optimal grasps. Haptic exploration
at the time of grasp can therefore contribute vital information. Current
research features a variety of different approaches for haptic object
classification. However, as stated in [8], the variety of test sets and the
lack of a standard robot-hand/sensor configuration make it difficult to
perform a meaningful comparison of an algorithm’s performance.

The trend towards using deep learning in robotics [9] has indicated the
superiority of these models over traditional machine learning strategies for
haptic object classification. This paper contributes a comparison of two
different tactile sensors on the same dataset performing the classification of
objects through a unimodal haptic approach using a time series of tactile
images. It furthermore investigates the performance of 2D-CNN, 3D-CNN and LSTM
artificial neural networks used in an end-to-end manner to find an optimal
neural network architecture for haptic object recognition. Unlike many
publications, the approach utilises spatio-temporal data, facilitating high-
resolution tactile sensor data, preserved in its three-dimensional nature.
Another novelty to this field is using a three-dimensional CNN for this task
and two sensor solutions on the same object set. As opposed to the two-stage
approach in [10, 11], neural networks feature the ability to learn the
relevant features from the data and classify the object in the same training
process. Current state-of-the-art papers mostly create hand-crafted features
[11, 12] or use results of unsupervised models [10, 13] which are then fed to
a classification algorithm.

The structure of the paper is as follows: a literature review is briefly
discussed in section II, the methodology is reported in section III, results
are presented and analysed in section IV and conclusions and future work are
discussed in section V.

Figure 1: Illustration of haptic object classification steps

##  II Background and Related Works

In this section, two essential aspects of ML algorithms, namely feature
extraction and classification, are discussed. Current state-of-the-art one-
step approaches employed for the haptic object are reviewed, highlighting the
advantages of a _one-step_ approach in classification followed by challenges
in comparing and reproducing results due to lack of uniformity and consistency
in datasets, manipulators, robots and sensors.

_Two-stage classifiers_ includes those object classification solutions which
pursue an approach consisting of a first step to extract features from the
data on the object in the training period and then uses these features to
classify the object in a second step, whilst keeping these two as separate
entities. In general, two main approaches to generating features from haptic
data are normally used: manual feature design and automatic feature extraction
using machine learning methods. Manual feature extraction reduces the volume
of the raw data recorded by applying processing rules on the recorded data,
which are predefined by the authors [11, 8]. Automatic feature extraction uses
unsupervised machine learning to pre-process the raw data received from the
sensors [13, 10, 12].

_Classification_ : The data yielded from the feature extraction task
constitutes a labelled data-set. Consequently, the remaining classification
task is performed by applying a supervised learning algorithm. The most
popular classifiers in this field are $k$-nearest-neighbours ($k$-NN) and
multi-class support vector machines (SVM). Those have the advantage of being
easy to set up and computationally inexpensive when performing classification
tasks. Computational complexity substantially depends on the value for $k$
with the k-NN algorithm and on the chosen kernel for the SVM. Moreover, random
forest approaches are used as well. Those have the advantage that the most
relevant features can be extracted, which might be advantageous for improving
the robotic setup. Some papers such as [10] use an approach that returns a
histogram of different feature comparison algorithms.

###  II-A One-step classification/DeepL classification

One-step classification in the context of this paper means that the system
which performs the classification task is trained and setup as one single
entity. Hence feature extraction, representation and subsequent classification
are all performed within the same machine learning component. Data pre-
processing such as normalisation, Principle Component Analysis (PCA) or
similar are not considered a step in this definition and may be performed
before training and evaluation. Because many recent publications opted for
deep learning methods in their one-step solutions. the term _DeepL
classification_ will be used synonymously in this paper.

Schmitz et al. [9] claimed to present the first paper to use a deep learning
approach in haptic object detection and utilised an architecture of stacked
denoising autoencoder (DAE) layers which were pre-trained unsupervised using
dropout for further regularisation of their network. They performed extensive
hyperparameter optimisation (learning rate, number of hidden layers, dropout
rate, activation function, weight penalties) to optimise their solution’s
performance. The authors used a TWENDY-ONE robotic platform 111Available
online, http://www.twendyone.com, last accessed 21/4/2021 equipped with a
four-fingered hand composed of $241$ haptic pixels distributed over its
surface, as well as information on the motor positions to collect data.

Despite this being a multimodal approach, there is no difficulty in
representing the data haptic and other information in the same data vector.
The authors created a 20 household objects test set with five similarly shaped
bottles, almost certainly making it the most challenging data-set at the time.
As a comparison for their proposed classifier, they also tested a shallow
artificial neural network with prepossessing through 90% variance
PCA222Dimensionality only reduced by 8, showing almost no redundancy in the
input data which yielded only $49.3\%$ accuracy on the test set. The deep DAE
network achieved $88.4\%$ in its best configuration, showing a clear
superiority with a DeepL approach.

In contrast, Liarokapis et al. [14] utilised a random forest classifier for
haptic object classification, assuming properties such as faster processing
speed, inherent feature variable importance calculation and high accuracy of
the random forests classifier. They used a two-finger under-actuated gripper
with 8 tactile sensors on each finger and recorded 4 motor positions. As data
were acquired at two points in time per measurement, a 36-dimensional feature
vector was created and used directly for the detection process.

Data collected from 10 household items objects were recorded at different
angles and then split into two sets. One contained only object positions
called Constrained Orientation and one Free Orientation set, which has no
limitations on its angles at which the object was presented to the robot. For
the former, a 100 tree random forest yielded an accuracy of $100\%$, whereas,
for the latter, this dropped to $94.3\%$. Compared with SVMs and shallow NNs,
the chosen method is more accurate even for 10 random tree forests.

An analysis of the relevant features revealed that 12 of the 16 tactile values
could be neglected with only a minor decrease in accuracy ($<2\%$). The
relevance of shape-related data from the motors appeared to be more important
than haptic data in the given scenario. This is in contradiction to [9].
Considering the much higher commonality of the data in Schmitz et al., this
suggests that haptic data needs a certain resolution before its relevance for
classification increases.

More recent work by Gao et al. [15] compared the use of CNNs and LSTMs for
assigning property labels to the objects. Furthermore, they also incorporated
a visual means of describing the object allowing for a comparison. For the
tactile component, a two-finger setup was used. Each finger was equipped with
a sensor that delivers 19 readings related to local pressure and low-frequency
fluid pressure, high-frequency fluid vibrations, core temperature and core
temperature change.

As the dimensions of the tactile data were reduced to 4 via PCA, the system
delivered a 32-signal feature vector per time step. This was either passed to
the LSTM as a time-sequence of vectors or as a 2D matrix with features as the
first and time as the second dimension. As the task was not to identify an
object but rather to assign an unspecified number of 20 possible labels to it,
the paper used the _Area under curve_ metric to calculate the system’s
performance. Accuracy can therefore not be compared with the other papers
discussed in this section. Gao et al.[15] achieved an AUC of $72.1\%$ for the
LSTM and $83.2\%$ for the CNN. The CNN solution was able to outperform every
visual descriptor based on the Image-GoogleNet architecture; the results of
the LSTM were significantly inferior.

Table I summarises a performance comparison of the prominent research works
discussed above.

TABLE I: Main characteristics of papers employing a single-step classification
approach.

Paper | Archi- | Classes | Data | Exploratory | Accuracy
---|---|---|---|---|---
| tecture |  |  | movement |
Schmitz | stacked | 20 objects | 241 tactile | simple | $88,3\%$
et al.[9] | DAE | household | 321 feat. | grasp |
|  |  | numb. grasps |  |
Liaroka- | Random | 10 objects | 16 touch | simple |  $100\%$ constr.
pis et al. [14] | forests | household | 2 motor | grasp, | pos.
|  |  |  | 2 time |  $94.3\%$ free pos.
Gao et al. | CNN & | 20 | 32 features | hold, squeeze | CNN out-
[15] | LSTM | adjectives | 2 sensors, time | slide s/f | performs
|  |  |  |  | LSTM
Baishya & | CNN | 6 tubes | 4x4 touch | sliding |  $92.3\%$ 1 slide
Baeuml [8] |  | diff. material | 1500 time |  |  $97.5\%$ 3 slides

###  II-B Comparability of data processing solutions

The field of haptic object classification has seen various proposed systems
that rely on very different setups of sensors (high/low spatial resolution
touch sensors, one/multiple sensors) and test objects (geometrical objects,
household objects). In general, it is difficult to compare and reproduce the
results, as the approaches use different and very often unique hardware. They
lack generality or have not been tested across several platforms. Some methods
consider using only one sensor type, which means either tactile or kinesthetic
modality [13].

Therefore, accurate comparison of algorithms can only be performed using the
same environment and hardware. For example, [8] provide a comprehensive review
of classical approaches using high and low dimensional features as well as
2D-CNN neuronal networks. However, their data set is basically the same shape
although multi-material, which means it cannot be compared with, for example,
[15] who also uses LSTMs as their ML approach but exploit household objects,
which possess a different range of material and shape properties. This paper
focuses on the same ML approaches using a single dataset and robot hand but
with two different sensors, allowing for convenient and robust comparison.

##  III Methodology

###  III-A Setup

The mechanical setup for the haptic object classification comprises a human-
inspired robotic hand and high-resolution tactile sensors. To gain insight
into the influence of the sensor type on our recognition rate, two distinct
types of sensors were used: _(1.)_ The first is the curved BioTac SP sensor
which features $24$ taxels (tactile pixels), not aligned in a grid but in a
bionic inspired layout. Furthermore, this sensor provides temperature and
vibration data which have been facilitated as means of material classification
in [2]. _(2.)_ The second device is the WTS-FT sensor by Weiss robotics, which
exploits a flat surface with a slight bend towards the tip and a resolution of
$4\times 8$ taxels. These sensors were mounted to the thumb, the index and the
ring finger of an AR-10 humanoid robotic hand which provided the mechanical
actuation.

Figure 2: _Left:_ The BioTac-SP bionic inspired tactile sensor; _Right:_ The
WTS-FT tactile sensor

###  III-B Objects

A haptic object classification system is expected to extract as much
information from a single grasp as possible. Therefore, a set of objects was
composed to evaluate the method’s capabilities to infer object shapes and
detect basic material properties of the grasped objects. Practically, this
meant including a variety of elementary shapes and items with different
stiffness and elasticity but similar silhouettes. The system must
differentiate between three cylindrical objects comparable in terms of size
but distinct in their softness. Additional to the fundamental shapes of a
cylinder, a sphere and a box, two special items were designed to pose a
greater challenge to the classification algorithms: A triangular prism and an
Icosahedron which both possess the property to present a varying combination
of surfaces, edges and angles to the tactile sensors depending on the item’s
alignment in space. Classifying these objects requires the model to extract
the dynamic behaviour of the object during the grasp.

Figure 3: The set of objects which were to be classified by a single grasp:
(1) Spiky rubber ball; (2) Solid light plastic ball; (3) Empty water bottle
(sealed); (4) Thin metal pipe; (5) Solid cardboard box; (6) Box-shaped sponge;
(7) Soft cylindrical painter’s roller; (8) 3D-printed triangular prism; (9)
3D-printed icosahedron

###  III-C Data recording

To keep the data acquisition as close to a real use case as possible, while
maintaining appropriate recording intervals, the AR-10 hand was mounted on a
desk while manually feeding the object to the hand. On presentation of the
object, the hand closed its fingers/thumb. Data recording commenced as soon as
the following condition to detect a grasp was met: the haptic sensor of the
thumb and at least one of the two other fingers recognises pressure on its
taxels. If this criterion was not constantly met within the first second of
the recording, the data was ignored, and the acquisition started over again.
Approximately $\Delta t=$6\text{\,}\mathrm{s}$$ long periods of recording were
performed while the grasping programme executed finger movement to secure its
grasp. For the BioTac this resulted in 63 samples per recording and about 200
for the WTS-FT. However, for better comparability, the WTS-FT data were
downsampled to 63 samples per time series.

Each item in the object set was grasped 40 times with varying angles of attack
for the fingers. Passing the object by hand further contributes to introducing
data variance as it is practically impossible to recreate precisely the same
handover condition. This mimics the case of human-robot handover in an
industrial setting.

###  III-D Data pre-processing

#### Normalisation & Filtering

The recorded time series undergo a data prepossessing routine before passing
on to the machine learning algorithm. Firstly, the taxel values from both the
BioTac SP and WTS-FT sensors are normalised by either of the two following
methods: Map the interval from zero to the maximal value to 0…1 (from now on
referred to as _scaled_) or map the values in a way that they are arranged
like a standard Gaussian distribution ($\mu=0$, $\sigma=1$) (from now on
referred to as _normalised_). The normalisation technique used is highlighted
in the results tables III & II in the ”norm” column. Subsequently, the data is
reshaped in a suitable way for the ML method employed. Hence, all the values
from the three sensors are concatenated to one vector per time-step for LSTM
training. For the CNN models, the 2D nature of the sensor output is preserved,
and the three tactile images are joined together in the order: thumb, index,
ring finger.

As the taxels of the BioTac sensors are not aligned with a grid, these
measurements need to undergo an additional pre-processing step when used for a
CNN model. The tactile dots are mapped onto a matrix in a way that preserves
their actual distance relationships, similar to the method in Zapata-Impata et
al. [16]. As proposed in [16] there is also the option to filter over the
yielded matrix to blur the position of the taxels and create the impression of
a tactile image instead of a sparse matrix with some non-zero elements which
was not adopted in our implementation.

#### Data splitting

Recording data for a haptic dataset is cumbersome and time-consuming. There
are a limited number of ways to speed up the data capturing process as it
always involves a physical grasp. Because robotic grasping is a relatively
slow process captured multiple times per second, it is assumed that a much
lower sampling rate might actually be sufficient to build a working
classifier. Therefore, the data is down-sampled but without omitting any data
in this process. As depicted in fig. 4, the data points which would be
neglected by the downsampling are reassigned to newly created time-series.[17]
Consequently, the redundancy in the data is being used to boost the dataset
size to provide more training samples. However, these examples are likely to
be very much akin to each other. Hence, the split between training, testing
and validation dataset needs to occur before this method is applied.
Otherwise, overfitting inevitably occurs due to high similarities between both
datasets, which leads to unrealistically high validation accuracy.

Figure 4: The way time-series are split to extend the available data for
training

###  III-E Convolutional network classifier

Convolutional Neural Networks (CNN) are rapidly becoming the state-of-the-art
method for evaluating images and comparable data. Therefore, they are an
obvious choice for the tactile pictures that were recorded. Despite that, the
existence of a time dimension distinguishes it from a conventional image
classification. In [18], the authors suggest that transient processes during
the grasp may account for better classification potential, and accordingly, we
considered this aspect in the design of our network. One possibility is to
include the extra time-steps as additional channels for the CNN’s input. In
this case, the network features up to 63 individual input channels with their
own set of convolutional kernels. Another way to account for time-dependent
processes is to employ the 3D version of a CNN, which features three-
dimensional tensors instead of matrices as kernels. This design choice also
creates the opportunity to investigate whether kernels with significant
extensions in the space or the time-dimension lead to better performing
models. Assuming that bigger kernels can capture more complex patterns, this
might indicate whether the time or the spatial resolution of tactile sensing
data contribute more substantially to robust object classification.

A number of ML approaches were employed to analyse the tactile data, namely
four variants of CNN networks and one LSTM network - as summarised in figures
5 and 6.

(a)

(b)

(c)

(d)

Figure 5: CNN architectures used: (a) 2D-CNN-1L (b) 2D-CNN-2L (c) 3D-CNN-1L
(d) 3D-CNN-2L

Both the 2D and 3D CNN variants were used in a single and two layer
configuration to assess an appropriate network depth for the classification
task as shown in fig. 5. Due to overfitting being considered a potential
concern with the limited data available, two dropout layers were integrated to
facilitate better model generalisation in the training process. Also, a
horizontal ($\pm 1$ taxels) and vertical shift (up to $\pm 2$ taxels) were
used as a data augmentation method during model training. Recognising the
small resolution of the tactile image, more elaborate methods of image
augmentation as for example, in Shoten and Khoshgoftaar [19] were not
employed.

###  III-F Recurrent network classifier

LSTMs have proven to be a powerful tool for machine learning on time-series
data. However, their structure does not incorporate multi-dimensional input
vectors.

Therefore, the tactile data captured at each time step was concatenated to one
vector comprising a series of 1D arrays. The LSTM layer was set up to only
pass output to the subsequent dense layer after processing the whole time
series. Dropout was activated for the LSTM nodes and recurrent dropout to the
recurrent inputs of the layer to mitigate overfitting. The dense layer was
also combined with a dropout layer. These approaches against overfitting are
more crucial for training the LSTM than for the CNN architectures. The methods
for data augmentation used on the 2D data cannot be facilitated with the 1D
data representation. Furthermore, it was decided to evaluate only a single
LSTM layer architecture (fig. 6) because a two-layer solution with the same
range of nodes as for the CNN would yield a much bigger model for an already
overfitting-prone dataset.

Figure 6: LSTM architecture used

###  III-G Network training & Optimisation

All networks were trained using the _RMSprop_ optimiser and the ReLU
activation function. The parameters of the optimiser and the number of nodes
for the CNN/LSTM layer(s) and the dense layer were assigned through
hyperparameter search using the Talos library [20]. A similar approach was
used for the dropout ratios. This strategy ensured that all network
configurations were judged by their actual model performance and not by the
quality of manually selected hyperparameters.

##  IV Results

###  IV-A Accuracy

The main objective of this paper was to evaluate the performance of different
neural network-based models on an object recognition task, comparing results
from data acquired using two robotic tactile sensors. From fig. 7, it may be
observed that for the recorded dataset on the selected group of objects, the
LSTM based network performs best for both sensor types with a best accuracy of
94.3 % of the objects being classified correctly. (BioTac,Table II) and 93.8 %
(WTS, Table III) respectively. For the CNN architectures, it may be seen the
performance of the BioTac sensors proved to be slightly better than that of
the WTS-FT sensors.

In terms of CNN based classifiers, the 2D-CNN models were generally slightly
more accurate than the 3D-CNN versions. Nevertheless, the performance results
were close between the best 3D-CNN with one CNN layer and the two 2D-CNN model
(comp. fig. 7). However, for robotic systems, the computational effort needs
to be taken into account. Thus, 3D-CNN networks may still be considered, as
the 2D-CNN models contain significantly more parameters causing higher
computational cost. That is due to having a set of filters for each channel
instead of the single set of filters for the 3D-CNN.

The confusion matrices depicted in fig. 10 provide information on the
misclassifications that occurred with the best CNN and LSTM model on the two
sensors. The CNN architecture using the BioTac sensors misclassifies the two
balls in the dataset as an icosahedron akin to a spherical object. However,
the CNN-based classifier’s major flaw with WTS-FT sensors appears to be the
distinction between the plastic bottle and the metal cylinder – a mistake seen
for the LSTM models with the same sensor type as well. Therefore it can be
inferred that this was not caused by inherent insufficiency in the model but
rather by the sensor design itself. Finally, the LSTM model with BioTac data
exhibits a different kind of model error by wrongly mapping the spiky ball’s
data to the classes of the prism or icosahedron.

TABLE II: BioTac validation accuracy 2D-CNN
---
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  scaled, filtered |  62.0 %
1 |  3 |  21 |  scaled, fit. |  70.1 %
1 |  5 |  13 |  scaled, fit. |  76.6 %
1 |  7 |  9 |  scaled, fit. |  70.9 %
1 |  1 |  63 |  scaled |  72.2 %
1 |  3 |  21 |  scaled |  72.5 %
1 |  5 |  13 |  scaled |  82.2 %
1 |  7 |  9 |  scaled |  84.0 %
2 |  1 |  63 |  scaled |  67.6 %
2 |  3 |  21 |  scaled |  80.2 %
2 |  5 |  13 |  scaled |  82.0 %
2 |  7 |  9 |  scaled |  87.7 %
3D-CNN
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  scaled |  68.5 %
1 |  3 |  21 |  scaled |  71.6 %
1 |  5 |  13 |  scaled |  78.9 %
1 |  7 |  9 |  scaled |  86.8 %
2 |  1 |  63 |  scaled |  54.6 %
2 |  3 |  21 |  scaled |  70.0 %
2 |  5 |  13 |  scaled |  69.7 %
2 |  7 |  9 |  scaled |  75.1 %
LSTM
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  scaled |  82.4 %
1 |  3 |  21 |  scaled |  90.7 %
1 |  5 |  13 |  scaled |  94.3 %
1 |  7 |  9 |  scaled |  90.0 %
TABLE III: WTS-FT validation accuracy 2D-CNN
---
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  norm. |  68.5 %
1 |  3 |  21 |  norm. |  69.8 %
1 |  5 |  15 |  norm. |  67.8 %
1 |  7 |  9 |  norm. |  70.9 %
1 |  1 |  63 |  scaled |  82.1 %
1 |  3 |  21 |  scaled |  80.9 %
1 |  5 |  15 |  scaled |  81.5 %
1 |  7 |  9 |  scaled |  77.9 %
2 |  1 |  63 |  scaled |  78.7 %
2 |  3 |  21 |  scaled |  77.8 %
2 |  5 |  15 |  scaled |  82.2 %
2 |  7 |  9 |  scaled |  79.2 %
3D-CNN
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  scaled |  71.3 %
1 |  3 |  21 |  scaled |  65.4 %
1 |  5 |  15 |  scaled |  71.5 %
1 |  7 |  9 |  scaled |  70.4 %
2 |  1 |  63 |  scaled |  64.8 %
2 |  3 |  21 |  scaled |  66.7 %
2 |  5 |  15 |  scaled |  70.2 %
2 |  7 |  9 |  scaled |  70.6 %
LSTM
Number of layers |  split ratio |  time-series length |  norma-lisation |  validation acc.
1 |  1 |  63 |  scaled |  90.7 %
1 |  3 |  21 |  scaled |  93.8 %
1 |  5 |  15 |  scaled |  88.7 %
1 |  7 |  9 |  scaled |  89.9 %

###  IV-B Performance dependence on time-series

In section III we suggested a method for splitting data to reduce the
effective sampling frequency for the benefit of the dataset size to improve
the number of training samples available.

As depicted in figures 9 and 8, the splitting ratio and the normalisation
method influenced the maximum achievable accuracy. However, the LSTM
architectures benefit from more data and requiring a remaining length of 13
(LSTM for BioTac) or 21 (LSTM for WTS-FT) to work best. The split ratio of 7,
which reduces the number of samples per series to 9, appears to be too short
to extract the necessary features to tell the objects apart. The data is less
conclusive with the CNN architectures as scaled data for the 2D-CNN on WTS-FT
sensors appears to be optimally split into 5 subsets. On the contrary, the
best CNN models for the BioTac sensor achieve peak performance for the
shortest time series length tested.

Figure 7: Maximum accuracy per architecture and sensor Figure 8: Accuracy on
the BioTac-SP sensor over time-series length

Hyperparameter optimisation for the 3D-CNN included separate parameters for
the time and spatial dimension of the convolutional kernels to determine the
optimal ratio between these dimensions. However, the results of this procedure
have remained inconclusive with regards to this property because the best ten
models were found to have quite different combinations of these two
parameters.

Figure 9: Accuracy on the WTS-FT sensors over time-series length.

(a)

(b)

(c)

(d)

Figure 10: Confusion matrices for predictions on the validation dataset: _(a)_
CNN - BioTac; _(b)_ CNN - WTS-FT; _(c)_ LSTM - BioTac; _(d)_ LSTM - WTS-FT

##  V Conclusion and Future Work

A comprehensive comparison of two tactile sensor technologies using the same
robotic hand on a common 9-class dataset was presented in this work. The
evaluation was performed using different neural network architectures to
exploit the captured information. In that regard, we found LSTM based
architectures outperformed CNN models, by a small margin. Furthermore, a
method to extend the amount of usable data from a limited number of grasps was
also proposed. The proposed method improves the maximum accuracy from 82.4 %
(BioTac) and 90.7 % (WTS-FT) with full-time series data to about 94 % for both
sensor types.

Additionally, a decline in accuracy for the LSTM models with shorter time
series raises the question of obtaining the optimal sampling frequency and
grasping recording length. The relevance of the different phases of a grasp
for classification performance will be considered in future work. As the
confusion matrices for the BioTac have shown that different models exhibit
weaknesses on different object classes, ensemble techniques may be considered
for a further improvement of accuracy. For deployment purposes, techniques for
NN pruning might be helpful to reduce the computational complexity of the
proposed classifier.

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# Amplification, inference, and the manifestation of objective classical
information

Michael Zwolak<EMAIL_ADDRESS>Biophysical and Biomedical Measurement Group,
Microsystems and Nanotechnology Division, Physical Measurement Laboratory,
National Institute of Standards and Technology, Gaithersburg, MD, USA

###### Abstract

Our everyday reality is characterized by objective information—information
that is selected and amplified by the environment that interacts with quantum
systems. Many observers can accurately infer that information indirectly by
making measurements on fragments of the environment. The correlations between
the system, $\mathcal{S}$, and a fragment, $\mathcal{F}$, of the environment,
$\mathcal{E}$, is often quantified by the quantum mutual information or the
Holevo quantity that bounds the classical information about $\mathcal{S}$
transmittable by a quantum channel $\mathcal{F}$. The latter is a quantum
mutual information but of a classical-quantum state where measurement has
selected outcomes on $\mathcal{S}$. The measurement generically reflects the
influence of the remaining environment, $\mathcal{E}/\mathcal{F}$, but can
also reflect hypothetical questions to deduce the structure of
$\mathcal{S}\mathcal{F}$ correlations. Recently, Touil et al. examined a
different Holevo quantity, one from a quantum-classical state (a quantum
$\mathcal{S}$ to a measured $\mathcal{F}$). As shown here, this quantity upper
bounds any accessible classical information about $\mathcal{S}$ in
$\mathcal{F}$ and can yield a tighter bound than the typical Holevo quantity.
When good decoherence is present—when the remaining environment,
$\mathcal{E}/\mathcal{F}$, has effectively measured the pointer states of
$\mathcal{S}$—this accessibility bound is the accessible information. For the
specific model of Touil et al., the accessible information is related to the
error probability for optimal detection and, thus, has the same behavior as
the quantum Chernoff bound. The latter reflects amplification and provides a
universal approach, as well as a single-shot framework, to quantify records of
the missing, classical information about $\mathcal{S}$.

The emergence of objective, classical information from quantum systems is due
to amplification: Many pieces of the environment—e.g., many photons—each
interact with a quantum system and acquire an imprint of certain states, the
pointer states. This is the process by which select information becomes
redundant and accessible to many different observers. The framework, where the
environment decoheres systems and acts as a communication channel for the
resulting information, is known as quantum Darwinism ollivier_objective_2004 ;
ollivier_environment_2005 ; blume-kohout_simple_2005 ; blume-
kohout_quantum_2006 ; zurek_relative_2007 ; blume-kohout_quantum_2008 ;
zurek_quantum_2009 ; zwolak_quantum_2009 ; paz_redundancy_2009 ;
zwolak_redundant_2010 ; riedel_quantum_2010 ; riedel_redundant_2011 ;
riedel_rise_2012 ; zwolak_complementarity_2013 ; zurek_quantum_2014 ;
zwolak_amplification_2014 ; korbicz_objectivity_2014 ; brandao_generic_2015 ;
zwolak_amplification_2016 ; zwolak_redundancy_2017 . It is the pointer states
that survive the interaction with the environment and create “copies” of
themselves from which observers can infer the pointer state of the system.
This process has been seen experimentally in both natural unden_revealing_2019
and engineered ciampini_experimental_2018 ; chen_emergence_2019  settings, and
both theory and practical calculations are steadily progressing
balaneskovic_random_2015 ; horodecki_quantum_2015 ; giorgi_quantum_2015 ;
balaneskovic_dissipation_2016 ; knott_generic_2018 ; milazzo_role_2019 ;
campbell_collisional_2019 ; roszak_entanglement_2019 ; garcia-
perez_decoherence_2020 ; lorenzo_anti-zeno-based_2020 ; ryan_quantum_2021 ;
kicinski_decoherence_2021 ; korbicz_roads_2021 ; kwiatkowski_appearance_2021 ;
touil_eavesdropping_2022 .

Within this framework, one primary question concerns the information available
within an environment fragment as its size increases. This allows one to
quantify redundancy: If small fragments $\mathcal{F}$ of the environment
$\mathcal{E}$ all contain the same information about the system $\mathcal{S}$,
then that information is available to many observers. Given a global state,
$\rho_{\mathcal{S}\mathcal{E}}$, the accessible information

$I_{\mathrm{acc}}\left(\Pi_{\mathcal{S}}\right)=\max_{\Pi_{\mathcal{F}}}I\left(\Pi_{\mathcal{S}}:\Pi_{\mathcal{F}}\right)$
(1)

can quantify the amount of information an observer learns about
$\Pi_{\mathcal{S}}$ (a positive operator-valued measure, a POVM, on
$\mathcal{S}$) by making a measurement $\Pi_{\mathcal{F}}$ on only
$\mathcal{F}$. The quantity
$I\left(\Pi_{\mathcal{S}}:\Pi_{\mathcal{F}}\right)$ is the classical mutual
information computed from the joint probability distribution from outcomes of
$\Pi_{\mathcal{S}}$ and $\Pi_{\mathcal{F}}$. The POVM $\Pi_{\mathcal{S}}$ has
elements $\pi_{s}$ that generate an ensemble
$\left\\{\left(p_{s},\rho_{\mathcal{F}\left|s\right.}\right)\right\\}$ of
outcomes $s$ with probability
$p_{s}=\mathrm{tr}_{\mathcal{S}\mathcal{E}}\pi_{s}\rho_{\mathcal{S}\mathcal{E}}$
and conditional states
$\rho_{\mathcal{F}\left|s\right.}=\mathrm{tr}_{\mathcal{S}\mathcal{E}/\mathcal{F}}\pi_{s}\rho_{\mathcal{S}\mathcal{E}}/p_{s}=\mathrm{tr}_{\mathcal{S}\mathcal{E}/\mathcal{F}}\sqrt{\pi_{s}}\rho_{\mathcal{S}\mathcal{E}}\sqrt{\pi_{s}}/p_{s}$
on $\mathcal{F}$ (i.e., assuming the POVM acts on only $\mathcal{S}$ and an
auxiliary system but $\mathcal{F}$ is not directly affected). Allowing
$\Pi_{\mathcal{S}}$ to be arbitrary, the accessible information, Eq. (1),
depicts a situation where some auxiliary system $\mathcal{A}$, perhaps a
special observer or another part of the environment, has access directly only
to $\mathcal{S}$, makes a measurement $\Pi_{\mathcal{S}}$, and holds a record
of the outcome $s$, leaving a joint state (after tracing out the now
irrelevant $\mathcal{S}$)

$\sum_{s}p_{s}|s\rangle_{\mathcal{A}}\langle
s|\otimes\rho_{\mathcal{F}\left|s\right.}.$ (2)

An observer $\mathcal{O}$ then wants to predict the outcome $s$ by making
measurements only on $\mathcal{F}$, e.g., correlations are generated between
$\mathcal{A}$ and $\mathcal{O}$ but indirectly from separate measurements on
$\mathcal{S}$ and $\mathcal{F}$, for which Eq. (1) quantifies this capability.
One could then maximize the accessible information over all
$\Pi_{\mathcal{S}}$ to see what quantity the observer can learn most about.
This allows one to quantify the structure of correlations between
$\mathcal{S}$ and $\mathcal{F}$ induced by, e.g., a decohering interaction
between them.

Within the context of physical processes that give rise to quantum Darwinism,
$\Pi_{\mathcal{S}}$ is not arbitrary, however. For redundant information to be
present, there must be at least two records of some information, which, when
decoherence is the main interaction, will be the pointer information. Hence,
there must be an $\mathcal{F}$ that almost, to a degree we want to quantify,
makes a measurement of the pointer states. At the same time, the remaining
part of the environment, $\mathcal{E}/\mathcal{F}$, has already made an
effective measurement for all intents and purposes, to a degree that we can
retroactively validate. This entails that the correlations are effectively of
the form of Eq. (2) but with $\mathcal{A}=\mathcal{E}/\mathcal{F}$ or
$\mathcal{S}$ and $\Pi_{\mathcal{S}}=\hat{\Pi}_{\mathcal{S}}$ (the pointer
observable),

$\sum_{\hat{s}}p_{\hat{s}}|\hat{s}\rangle\langle\hat{s}|\otimes\rho_{\mathcal{F}\left|\hat{s}\right.},$
(3)

where $\hat{s}$ labels the pointer states (see Refs. zurek_pointer_1981 ;
zurek_environment-induced_1982  for discussion of pointer states). This form
is a consequence of “branching” blume-kohout_simple_2005  and appears in the
good decoherence limit of purely decohering models, which will be extensively
discussed below. Here, it is sufficient to note that the state, Eq. (3), is
the most relevant to quantum Darwinism. It makes little difference if one
treats the $\mathcal{A}$ as $\mathcal{E}/\mathcal{F}$ or as just the fully
decohered, or directly measured, $\mathcal{S}$, even when $\mathcal{F}$ is
extremely large in absolute terms 111Only for “global” questions, where
$\mathcal{F}$ is some sizable fraction of the environment, does it matter.
Since the environment is huge for most problems of everyday interest, such as
photon scattering, $\mathcal{F}$ can be very large—even asymptotically
large—without concern for this. However, Eq. (3) does drop exponentially small
corrections in the size of $\mathcal{E}/\mathcal{F}$ and one can not formally
take the asymptotic limit of $\mathcal{F}$ without first doing so in
$\mathcal{E}$. The degree to which asymptotic approximations work thus relies
on the balance sheet—how well records are kept in the environment components
compared to $\mathcal{E}$’s absolute size. Ref. zwolak_complementarity_2013
has dealt with retaining corrections to Eq. (3). . Hereon, I will treat the
auxiliary system $\mathcal{A}$ as if it were $\mathcal{S}$.

With states of the form in Eq. (3), the mutual information between
$\mathcal{A}=\mathcal{S}$ and $\mathcal{F}$ is the Holevo quantity

$\displaystyle\chi(\hat{\Pi}_{\mathcal{S}}:\mathcal{F})$
$\displaystyle=H\left(\sum_{\hat{s}}p_{\hat{s}}\rho_{\mathcal{F}\left|\hat{s}\right.}\right)-\sum_{\hat{s}}p_{\hat{s}}H\left(\rho_{\mathcal{F}\left|\hat{s}\right.}\right)$
(4) $\displaystyle\equiv
H_{\mathcal{F}}-\sum_{\hat{s}}p_{\hat{s}}H_{\mathcal{F}\left|\hat{s}\right.},$

where $H\left(\rho\right)=-\mathrm{tr}\rho\log_{2}\rho$ is the von Neumann
entropy for the state $\rho$. This quantity upper bounds the capacity of
$\mathcal{F}$ to transmit pointer state information (the variable $\hat{s}$ is
encoded in the conditional states $\rho_{\mathcal{F}\left|\hat{s}\right.}$).
Moreover, for an important class of interactions—purely decohering
Hamiltonians with independent environment components—the quantum Chernoff
bound determines the behavior of the optimal measurement on $\mathcal{F}$ to
extract $\hat{\Pi}_{\mathcal{S}}$ and, thus, is related to the accessible
information, Eq. (1) with $\Pi_{\mathcal{S}}=\hat{\Pi}_{\mathcal{S}}$. One can
generalize Eq. (4) by allowing one to maximize over measurements on the
system,

$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)=\max_{\Pi_{\mathcal{S}}}\chi\left(\Pi_{\mathcal{S}}:\mathcal{F}\right),$
(5)

where, when good decoherence has taken place,
$\Pi_{\mathcal{S}}=\hat{\Pi}_{\mathcal{S}}$ maximizes the Holevo quantity
zwolak_complementarity_2013 . The good decoherence limit is when
$\mathcal{E}/\mathcal{F}$ is sufficient to decohere the system and, thus, the
$\mathcal{S}\mathcal{F}$ state is exactly of the form in Eq. (3)
zwolak_redundant_2010 ; zwolak_complementarity_2013 . Here, I employ the
notation $\check{\mathcal{A}}$ of Touil et al. touil_eavesdropping_2022  to
indicate that the Holevo quantity is maximized over measurements on
$\mathcal{A}$, see also the next equation.

Touil et al. touil_eavesdropping_2022  examined an alternative Holevo quantity
with the measurement on the fragment side,

$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)=\max_{\Pi_{\mathcal{F}}}\chi\left(\mathcal{S}:\Pi_{\mathcal{F}}\right)=\max_{\Pi_{\mathcal{F}}}\left[H_{\mathcal{S}}-\sum_{f}p_{f}H_{\left.\mathcal{S}\right|f}\right],$
(6)

where the maximization is over all POVMs $\Pi_{\mathcal{F}}$ and $f$ labels
the outcomes of $\Pi_{\mathcal{F}}$ and $p_{f}$ their probabilities. In that
work, they compute the quantum mutual information, the Holevo quantity in Eq.
(4), and the alternative Holevo quantity in Eq. (6) for a “c-maybe” model of
decoherence of $\mathcal{S}$ by $\mathcal{E}$, a model that falls into the
class of purely decohering models (see below). They analytically found
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ by making use of the
Koashi-Winter monogamy relation koashi_monogamy_2004  and showed all the
mutual information quantities above approach the missing information,
$H_{\mathcal{S}}$, with a similar dependence on $\mathcal{F}$.

If one were to interpret this alternative Holevo quantity, Eq. (6), in the
typical way, then it would bound the channel capacity of $\mathcal{S}$ to
transmit information about (the optimal) $\Pi_{\mathcal{F}}$. One important
observation, however, is that, in the good decoherence limit—when the
$\mathcal{S}\mathcal{F}$ state is of the form in Eq.
(3)—$\chi\left(\mathcal{S}:\Pi_{\mathcal{F}}\right)$ lower bounds
$\chi(\hat{\Pi}_{\mathcal{S}}:\mathcal{F})$ for any $\Pi_{\mathcal{F}}$ by the
data processing inequality since $\hat{\Pi}_{\mathcal{S}}$ is already measured
on $\mathcal{S}$ by $\mathcal{E}/\mathcal{F}$. In this limit,
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ is the actual accessible
pointer information.

For an arbitrary $\mathcal{S}\mathcal{F}$ state, however, there is no strict
relation of $\chi\left(\Pi_{\mathcal{S}}:\mathcal{F}\right)$ or
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ with
$\chi\left(\mathcal{S}:\Pi_{\mathcal{F}}\right)$ or
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ 222If
$\rho_{\mathcal{S}\mathcal{F}}$ is arbitrary, then Holevo quantities with
measurements on the $\mathcal{F}$ side can not upper or lower bound quantities
with $\mathcal{S}$ side measurements. For a particular state with a given
inequality between $\mathcal{F}$ and $\mathcal{S}$ side measurements, one can
swap $\mathcal{S}$ and $\mathcal{F}$ in the state
$\rho_{\mathcal{S}\mathcal{F}}$—it’s arbitrary after all—and reverse the
inequality.. Instead, the inequality

$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)\geq
I_{\mathrm{acc}}\left(\Pi_{\mathcal{S}}\right)$ (7)

holds for any $\Pi_{\mathcal{S}}$. The measurement on the two sides of the
inequality is generically different—the measurement that maximizes
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ is not the measurement,
$\Pi_{\mathcal{F}}^{\star}$, that maximizes
$I\left(\Pi_{\mathcal{S}}:\Pi_{\mathcal{F}}\right)$ to get the accessible
information, Eq. (1). The proof of Eq. (7) is straight forward,

$\displaystyle\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$
$\displaystyle=\max_{\Pi_{\mathcal{F}}}\chi\left(\mathcal{S}:\Pi_{\mathcal{F}}\right)$
$\displaystyle\geq\chi\left(\mathcal{S}:\Pi_{\mathcal{F}}^{\star}\right)$
$\displaystyle=\chi\left(\mathcal{M}\mathcal{S}:\Pi_{\mathcal{F}}^{\star}\right)$
$\displaystyle\geq\chi\left(\Pi_{\mathcal{S}}:\Pi_{\mathcal{F}}^{\star}\right)$
$\displaystyle=I_{\mathrm{acc}}\left(\Pi_{\mathcal{S}}\right),$

where the system $\mathcal{M}$ is adjoined in a product state with
$\rho_{\mathcal{S}\mathcal{F}}$ and a unitary on $\mathcal{M}\mathcal{S}$
makes a measurement $\Pi_{\mathcal{S}}$. The fourth line follows from data
processing.

Equation (7) is an accessibility bound. Any information about $\mathcal{S}$
(i.e., that can be extracted by a direct POVM on $\mathcal{S}$) can, at best,
have $\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ amount of shared
information with $\mathcal{F}$. Then, as already noted, if the good
decoherence limit is reached, that bound becomes equality,

$\overset{\mathbf{Good\,Decoherence}}{\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)=I_{\mathrm{acc}}\left(\hat{\Pi}_{\mathcal{S}}\right),}$
(8)

for the pointer information 333This follows from the form of the state in Eq.
(3). To determine $\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ for this
state, an apparatus makes a measurement $\Pi_{\mathcal{F}}$ and records the
outcome, leaving a joint system-apparatus state
$\sum_{\hat{s},f}p_{\hat{s}}|\hat{s}\rangle\langle\hat{s}|\otimes
p_{\left.f\right|\hat{s}}\text{$|f\rangle\langle f|$}$. This is a classical-
classical state that yields, after maximizing over $\Pi_{\mathcal{F}}$, both
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$, Eq. (6), and the
accessible information, Eq. (1).. This makes
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ desirable in the context of
quantum Darwinism: It not only is a better bound on the accessible information
in the good decoherence limit—the main limit of interest for quantum
Darwinism—but it is the actual accessible information.

To proceed further—to compute the accessible information and the associated
redundancy—we need to specify a model or class of models that provide the
global states of interest. The everyday photon environment has a particular
structure where independent environment components (photons) scatter off
objects, acquire an imprint of the state, and transmit that information
onward, interacting little with each other in the process joos_emergence_1985
; joos_decoherence_2003 ; schlosshauer_decoherence_2007 ; riedel_quantum_2010
; riedel_redundant_2011 ; zwolak_amplification_2014 . This structure is
captured by purely decohering Hamiltonians by independent environment
components. I will consider this general class here. Under this evolution, the
quantum Chernoff bound (QCB) provides a universal lower bound to the
accessible information and the associated redundancy. The quantum Chernoff
result is also meaningful on its own as a single-shot result quantifying how
well an individual observer (with the best measurement apparatus) can learn
the pointer state of $\mathcal{S}$ indirectly from $\mathcal{F}$.

Pure decoherence occurs when environments select, but do not perturb, the
pointer states of $\mathcal{S}$. When the environment components do so
independently, the Hamiltonian is of the form

$\mathbf{H}=\mathbf{H}_{\mathcal{S}}+\hat{\Pi}_{\mathcal{S}}\sum_{k=1}^{{{}^{\sharp}}\mathcal{E}}\Upsilon_{k}+\sum_{k=1}^{{{}^{\sharp}}\mathcal{E}}\Omega_{k}$
(9)

with $\left[\hat{\Pi}_{\mathcal{S}},\mathbf{H}_{\mathcal{S}}\right]=0$ and the
initial state

$\rho\left(0\right)=\rho_{\mathcal{S}}\left(0\right)\otimes\left[\bigotimes_{k=1}^{{{}^{\sharp}}\mathcal{E}}\rho_{k}\left(0\right)\right].$
(10)

Here, $k$ specifies a component of the environment $\mathcal{E}$ of size
${{}^{\sharp}}\mathcal{E}$. The operators, $\Upsilon_{k}$ and $\Omega_{k}$,
are arbitrary. This class of models contains the c-maybe model of Touil et al.
touil_eavesdropping_2022 . That model has
$\hat{\Pi}_{\mathcal{S}}=0\cdot|0\rangle\langle 0|+1\cdot|1\rangle\langle 1|$
and $\exp\left[\imath\Upsilon_{k}t\right]=\sin a|0\rangle\langle 0|+\cos
a\left(|0\rangle\langle 1|+|1\rangle\langle 0|\right)-\sin a|1\rangle\langle
1|$ for all $k$, where $a$ is the angle of rotation of the “target”
environment bit after a time $t$ 444Note that all the coupling frequencies
(i.e., the energy scales divided by the reduced Planck’s constant) are
absorbed into the definition of the operators $\mathbf{H}_{\mathcal{S}}$,
$\Upsilon_{k}$ , and $\Omega_{k}$, while $\hat{\Pi}_{\mathcal{S}}$ is
dimensionless.. All other operators are 0. The collection of operators act
similarly to those in the controlled NOT gate. They just swap too, only a bit
more lazy, as here $a$’s any number, so it’s called c-maybe.

Starting from the initial product state, Eq. (10), and evolving for some time
under the Hamiltonian, Eq. (9), one can obtain the conditional states that
appear in the Holevo quantity, Eq. (4),

$\rho_{\mathcal{F}\left|\hat{s}\right.}=\bigotimes_{k\in\mathcal{F}}\rho_{k\left|\hat{s}\right.}.$
(11)

Due to the structure of the evolution, these are product states over the
components of the environment fragment. However, they need not be identically
distributed (that is, they need not be fully i.i.d.—independently and
identically distributed—states).

The structure, Eq. (11), is a manifestation of amplification. The pointer
states $\hat{s}$ leave an imprint on the environment components, of which
there are many. Observers intercepting those environment components can then
make a measurement to infer the pointer state. This is the setting of quantum
hypothesis testing. For instance, in the binary case with two pointer states
$\hat{s}=0$ or $1$, one wants to decide whether the fragment state is
$\rho_{\left.\mathcal{F}\right|0}$ or $\rho_{\left.\mathcal{F}\right|1}$ with
a minimum average probability of error,
$P_{e}=p_{\hat{s}=0}\mathrm{tr}\Pi_{\left.\mathcal{F}\right|1}\rho_{\left.\mathcal{F}\right|0}+p_{\hat{s}=1}\mathrm{tr}\Pi_{\left.\mathcal{F}\right|0}\rho_{\left.\mathcal{F}\right|1}$.
This is based on a POVM measurement, $\Pi_{\mathcal{F}}$, composed of two
positive operators $\Pi_{\left.\mathcal{F}\right|0}$ and
$\Pi_{\left.\mathcal{F}\right|1}$ (with
$\Pi_{\left.\mathcal{F}\right|0}+\Pi_{\left.\mathcal{F}\right|1}=\mathrm{I}$)
that indicate the occurrence of “0” or “1”, respectively. The first
contribution to this average error is when the actual state is
$\rho_{\left.\mathcal{F}\right|0}$, with a priori probability of occurring
$p_{\hat{s}=0}$ (where I explicitly show $\hat{s}=0$ to connect to Eq. (3))
but the measurement yielded the incorrect outcome
$\Pi_{\left.\mathcal{F}\right|1}$. Similarly for the second contribution.
Moreover, when amplification occurs, i.e., the conditional states are of the
form in Eq. (11), one is specifically interested in how the error probability
behaves as the fragment size grows. This is the setting of the QCB.

To employ the QCB, one makes use of a two-sided measurement. The first is on
$\mathcal{S}$, putting it into its pointer states (i.e.,
$\chi(\hat{\Pi}_{\mathcal{S}}:\mathcal{F})$ now provides the mutual
information between $\mathcal{S}$ and $\mathcal{F}$). This reflects the action
of $\mathcal{E}/\mathcal{F}$ and is the good decoherence limit—i.e.,
${{}^{\sharp}}\mathcal{E}\to\infty$ provided $\mathcal{S}$ and $\mathcal{E}$
have interacted for some finite time under the evolution given by Eq. (9) and
Eq. (10) 555This also requires that the coupling strength to the environment
components do not depend on ${{}^{\sharp}}\mathcal{E}$.. The second is on
$\mathcal{F}$ to access the pointer state. By Fano’s inequality
cover_elements_2006 ; nielsen_quantum_2011 ,

$\chi(\hat{\Pi}_{\mathcal{S}}:\mathcal{F})\geq
I_{\mathrm{acc}}\left(\hat{\Pi}_{\mathcal{S}}\right)\geq
H_{\mathcal{S}}-h\left(P_{e}\right)-P_{e}\ln\left[D-1\right],$ (12)

where $P_{e}$ is the error probability for extracting information about a
(sub)space of pointer states (of dimension $D$) from a measurement on
$\mathcal{F}$ 666One could replace the left hand side of this inequality with
$\chi\left(\check{S}:\mathcal{F}\right)\geq\chi(\hat{\Pi}_{\mathcal{S}}:\mathcal{F})$..
Here, I use the binary entropy,
$h\left(x\right)=-x\log_{2}x-\left(1-x\right)\log_{2}\left(1-x\right)$. The
QCB upper bound, $P_{e}^{\star}\geq P_{e}$, gives a second inequality

$\displaystyle I_{\mathrm{acc}}\left(\hat{\Pi}_{\mathcal{S}}\right)$
$\displaystyle\geq
H_{\mathcal{S}}-h\left(P_{e}\right)-P_{e}\ln\left[D-1\right]$
$\displaystyle\geq
H_{\mathcal{S}}-h\left(P_{e}^{\star}\right)-P_{e}^{\star}\ln\left[D-1\right],$
(13)

which is part way to the final QCB result zwolak_amplification_2014 ;
zwolak_amplification_2016 .

The QCB upper bounds the error probability, $P_{e}^{\star}\geq P_{e}$, for
both the $D=2$ case audenaert_discriminating_2007 ; audenaert_asymptotic_2008
; nussbaum_chernoff_2009  or the $D>2$ cases li_discriminating_2016 . There is
no fundamental difference between these cases, it is only the closest two
states that determine the asymptotic decay of $P_{e}$ when $D>2$. I will
restrict to $D=2$ from hereon to make a correspondence with Touil et al.
touil_eavesdropping_2022 . The error probability (bound) is

$P_{e}^{\star}=\min_{0\leq c\leq
1}p_{1}^{c}p_{2}^{1-c}\prod_{k\in\mathcal{F}}\mathrm{tr}\left[\rho_{\left.k\right|1}^{c}\rho_{\left.k\right|2}^{1-c}\right].$
(14)

For pure $\mathcal{S}\mathcal{E}$ states in the purely decohering scenario,
Eq. (9) and Eq. (10), $c$ can be any value between 0 and 1 within the
generalized overlap contribution,
$\mathrm{tr}\left[\rho_{\left.k\right|1}^{c}\rho_{\left.k\right|2}^{1-c}\right]$,
and it will give the exact overlap
$\left|\langle\psi_{\left.k\right|1}|\psi_{\left.k\right|2}\rangle\right|^{2}=\left|\gamma_{k}\right|^{2}$
(which is also the decoherence factor $\gamma_{k}$ squared for this case of
pure states). Touil et al. touil_eavesdropping_2022  consider the homogeneous
case where $\gamma_{k}=\gamma$ for all $k$, which I will also consider (see
Refs. zwolak_amplification_2014 ; zwolak_amplification_2016  for inhomogeneous
results).

For pure states, therefore, only the prefactor needs optimizing over $c$ as
the generalized overlap gives
$\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}$ for all $0\leq c\leq 1$ and
with ${{}^{\sharp}}\mathcal{F}$ the number of components in $\mathcal{F}$. The
prefactor is optimal at one of the two boundaries ($c=0$ or $c=1$), giving

$P_{e}^{\star}=\min\left[p_{1},p_{2}\right]\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}.$
(15)

I use slightly different notation here than Ref. touil_eavesdropping_2022  to
keep the correspondence with prior work. Opposed to pure states, for mixed
$\mathcal{S}\mathcal{E}$ states within the pure decohering scenario, Eq. (9)
and Eq. (10), the error probability (bound) is
$\sqrt{p_{1}p_{2}}\prod_{k\in\mathcal{F}}\mathrm{tr}\left[\rho_{\left.k\right|1}^{1/2}\rho_{\left.k\right|2}^{1/2}\right]$
for both spin and photon models zwolak_amplification_2014 ;
zwolak_amplification_2016  (i.e., $c=1/2$ is optimal). Either prefactor,
$\min\left[p_{1},p_{2}\right]$ or $\sqrt{p_{1}p_{2}}$, will give a bound for
the pure state case. Letting the prefactor to be just some $C$, the QCB result
for pure, homogeneous $\mathcal{S}\mathcal{E}$ is

$I_{\mathrm{acc}}\left(\hat{\Pi}_{\mathcal{S}}\right)\geq
H_{\mathcal{S}}-h\left(C\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right)\equiv\mathcal{X}_{QCB},$
(16)

where I stress that this is a classical-classical information about random
variable $\hat{s}$ (pointer states on $\mathcal{S}$) with measurement outcomes
on $\mathcal{F}$. If we want general $\mathcal{S}\mathcal{E}$ states, but
still the pure decoherence model, Eq. (9) and Eq. (10), we have exactly the
same form as Eq. (16) but the decoherence factor (the pure state overlap) is
replaced by the generalized measure of overlap,
$\mathrm{tr}[\rho_{\left.k\right|1}^{1/2}\rho_{\left.k\right|2}^{1/2}]$, see
Ref. zwolak_amplification_2016  for these expressions in terms of generic
angles (between conditional states) and lengths on the Bloch sphere for spins
and Ref. zwolak_amplification_2014  for photons.

The QCB is a universal result. The bound Eq. (14) is true for all models of
pure decoherence by independent spins or the standard photon model, all
dimensions in between (qutrits, qudits, etc.), inhomogeneous models, pure and
mixed $\mathcal{S}\mathcal{E}$ states, and ones with individual self-
Hamiltonians on $\mathcal{E}$. The only stipulation for Eq. (14) and the lower
bound $H_{\mathcal{S}}-H\left(P_{e}^{\star}\right)$ is that one is
distinguishing within a two-dimensional subspace of $\mathcal{S}$ pointer
states. For higher dimensional subspaces, the number of pointer states, $D$,
appears in Eq. (13) and the exponent in the decay of $P_{e}^{\star}$ requires
a pair-wise minimization of the generalized overlap over conditional states
(as well as a different prefactor outside of the exponential).

The most important aspect of the compact form, Eq. (16), and its
generalization to higher $D$, is that the right hand side reflects actual,
inferable information about the pointer states that the observer can retrieve
by interaction with just $\mathcal{F}$ in a single shot. Moreover, while the
QCB is traditionally cast as an asymptotic result, we have not actually used
any asymptotic limits to obtain Eq. (16). Both of these aspects—single shot
and finite $\mathcal{F}$—provide a natural setting for our world, where
observers are “agents” within these regimes. One can then ask questions about
resources of observers (for instance, global versus local measurements on
$\mathcal{F}$ subcomponents kincaid_phases_nodate  or the ability to perform
coherent measurements blume-kohout_quantum_2013 ) that further refine the
results but do not change the fundamental framework of single-shot, finite
$\mathcal{F}$ inference.

Let’s return to the c-maybe model and the Holevo quantities. Touil et al.
touil_eavesdropping_2022  present results for the quantum mutual information,
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$, and
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$. In the good decoherence
limit, the latter two are

$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)=-\frac{1}{2}\log_{2}\left[p_{1}p_{2}\left(1-\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right)\right]-\sqrt{1-4p_{1}p_{2}\left(1-\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right)}\mathrm{Arctanh}_{2}\left[\sqrt{1-4p_{1}p_{2}\left(1-\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right)}\right]$
(17)

and

$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)=H_{\mathcal{S}}+\frac{1}{2}\log_{2}\left[p_{1}p_{2}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right]+\sqrt{1-4p_{1}p_{2}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}}\mathrm{Arctanh}_{2}\left[\sqrt{1-4p_{1}p_{2}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}}\right]$
(18)

in the form as they appear in their main text but using the notation here (Eq.
(17) and Eq. (20) in Ref. touil_eavesdropping_2022 ). Rewriting these in terms
of binary entropy gives

$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)=h\left[\frac{1}{2}\left(1+\sqrt{1-4p_{1}p_{2}\left(1-\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}\right)}\right)\right],$
(19)

corresponding to the good decoherence expressions in Ref.
zwolak_redundant_2010 , and

$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)=H_{\mathcal{S}}-h\left[\frac{1}{2}\left(1+\sqrt{1-4p_{1}p_{2}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}}\right)\right].$
(20)

We see that Eq. (16) and Eq. (20) have a similar structure. Indeed, in the
good decoherence limit and for pure conditional states, the accessible
information, which is equivalent to Eq. (18) or Eq. (20), is equal to
$H_{\mathcal{S}}-h\left(P_{e}\right)$. Here,
$P_{e}=\frac{1}{2}\left(1-\mathrm{tr}\left|p_{1}\rho_{\left.\mathcal{F}\right|1}-p_{2}\rho_{\left.\mathcal{F}\right|2}\right|\right)$
is the optimal error probability, which is given by the Helstrom measurement
helstrom_quantum_1976 , for single shot state discrimination of the
conditional fragment states levitin_optimal_1995 ;
fuchs_distinguishability_1995 ; ban_upper_1996 . This is not true for mixed or
for higher dimensional pointer subspaces sasaki_accessible_1999 ;
shor_number_2000 ; shor_adaptive_2003 ; barnett_quantum_2009 . It can be
verified by a direct computation of the error probability from the optimal
measurement for the pure conditional states. For
$\rho_{\mathcal{F}\left|\hat{s}\right.}$ pure, the trace distance in the
Helstrom expression just requires diagonalizing an operator in a two-
dimensional subspace, giving

$P_{e}=\frac{1}{2}\left(1-\sqrt{1-4p_{1}p_{2}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}}\right)$
(21)

(this readily generalizes to the inhomogeneous case: The factor
$\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}$ just needs to be replaced by
$\prod_{k\in\mathcal{F}}\left|\gamma_{k}\right|^{2}$). This result makes no
use of the fact that the environment components were spins, and thus it is
directly applicable to (pure state) photon scattering off an object in a two
dimensional superposition, more directly supporting the connection discussed
in Touil et al. touil_eavesdropping_2022  and extending it to
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ in the good decoherence
limit 777Moreover, as with the QCB result, the form of the accessible
information for pure $\mathcal{S}\mathcal{E}$ states,
$H_{\mathcal{S}}-h\left(P_{e}\right)$ , with the optimal $P_{e}$ from Eq. (21)
holds regardless of the environment components. They can be spins, qudits, or
photons. Furthermore, the connection with hypothesis testing allows for even
more general statements about models that are not purely decohering. For
instance, for projection-valued measurements and pure $\mathcal{S}\mathcal{E}$
states, one obtains the same accessible information,
$H_{\mathcal{S}}-h\left(P_{e}\right)$, but the error probability just has the
overlap between the conditional fragment states, which can behave in a manner
that is not exponentially decaying with ${{}^{\sharp}}\mathcal{F}$..

While specific to the case of $D=2$ and pure $\mathcal{S}\mathcal{E}$ states
evolving under Eq. (9) and Eq. (10), the connection provides a window into the
behavior of different ways to quantify correlations. The alternate Holevo
quantity, $\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$, becomes the
inferable information in this specific setting. However, inferable information
has a universal form that goes beyond this specific setting of dimensionality
and purity.

_Redundancy_. The decay to the classical plateau—the missing information
$H_{\mathcal{S}}$ about the system—for the quantities in Eq. (19), Eq. (20),
and Eq. (16) all are controlled by the $\mathcal{F}$-induced decoherence
factor, $\gamma^{2{{}^{\sharp}}\mathcal{F}}$. Ultimately, though, we are
interested in the redundancy of information. This requires introducing a
control, the information deficit $\delta$, that reflects the fact that one can
not generally obtain perfect knowledge from a finite-size fragment
$\mathcal{F}$. This is typically taken as

$\mathcal{X}\left(\mathcal{F}\right)\geq
H_{\mathcal{S}}\left(1-\delta\right),$ (22)

where $\mathcal{X}\left(\mathcal{F}\right)$ is some mutual information
(quantum mutual information, Holevo, accessible information, etc.). This is
the form I will employ here. However, both the form of the QCB and the form of
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ (in the good decoherence
limit) suggest employing the information deficit as an entropic quantity when
thresholding entropic measures of information,

$\mathcal{X}\left(\mathcal{F}\right)\geq
H_{\mathcal{S}}-H\left[\delta\right].$ (23)

This allows $\delta$ to be a factor reflecting distinguishability of
conditional states and allows for non-asymptotic computations to proceed for
the redundancy (it removes the transcendental form of the equations). I will
not use this in what follows.

The approach to the plateau and the redundancy (to within $\delta$) have
simple asymptotic results regardless of quantity used to compute them. The
decay exponent to the plateau, $\xi$, of some information theoretic quantity
$\mathcal{X}\left(\mathcal{F}\right)$, such as Eq. (19), Eq. (20), or Eq.
(16), is

$\xi=-\lim_{{{}^{\sharp}}\mathcal{F}\to\infty}\frac{1}{{{}^{\sharp}}\mathcal{F}}\ln\left[H_{\mathcal{S}}-\mathcal{X}\left(\mathcal{F}\right)\right].$
(24)

For the pure, homogeneous c-maybe model, all three decay to the plateau with
exponent

$\xi=-\ln\left|\gamma\right|^{2}.$ (25)

That is universality in a nutshell. Moreover, the exponent is the leading
order of the redundancy,

$R_{\delta}\simeq{{}^{\sharp}}\mathcal{E}\frac{\xi}{\ln
1/\delta}={{}^{\sharp}}\mathcal{E}\frac{\ln\left|\gamma\right|^{2}}{\ln\delta}.$
(26)

This is the essence of the QCB: The exponent—the quantum Chernoff information,
$\xi_{QCB}$, or its inhomogeneous counterpart, $\bar{\xi}_{QCB}$—controls the
redundancy, see Refs. zwolak_amplification_2014 ; zwolak_amplification_2016
for additional discussion and results. For the pure c-maybe model, this
exponent is the same whether using Eq. (19), Eq. (20), or Eq. (16). The
quantum mutual information also yields the same decay and redundancy in the
good decoherence limit, as it is the same as
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ from Eq. (19) 888In order
to apply Eq. (24) for the quantum mutual information, one needs
${{}^{\sharp}}\mathcal{E}\to\infty$. As already mentioned in a prior footnote,
though, this will entail good decoherence provided some finite interaction
between $\mathcal{S}$ and $\mathcal{E}$ components has taken place. . In other
words, all the information theoretic quantities provide the same decay and
redundancy, which the asymptotic calculation, Eq. (24), makes apparent in a
non-empirical manner.

Figure 1: Approach to the plateau. Information measures $\mathcal{X}$ versus
fragment size ${{}^{\sharp}}\mathcal{F}$ for $p_{1}=1/4$ and $\gamma=7/8$. All
three quantities, $\mathcal{X}=\mathcal{X}_{QCB}$ (green line),
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ (orange line), and
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ (blue line), rapidly rise
to the classical plateau, $H_{\mathcal{S}}$, as the fragment size
${{}^{\sharp}}\mathcal{F}$ increases. The quantum mutual information,
$I\left(\mathcal{S}:\mathcal{F}\right)$ (not shown), is equivalent to
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ when good decoherence is
present. The QCB result, $\mathcal{X}_{QCB}$, lower bounds the other two, but
is close to $\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$. The inset
shows the decay to the plateau. All three measures decay with the same
exponent. The $\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ does, though,
deviate from the other two quantities, as the latter two have a prefactor that
depends on ${{}^{\sharp}}\mathcal{F}$ (both with the same functional form).
This offset does not influence the redundancy asymptotically (i.e., as a
relative correction, it itself decays).

Figure 1 shows the approach to the plateau for the three information measures.
The quantity $\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ is a weaker
bound to the accessible information. Yet, the separation between the decay
curves is unimportant for passing the threshold in Eq. (22):
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ passes it sooner than the
other quantities, but this only gives a relative correction to Eq. (26) that
goes to zero asymptotically (${{}^{\sharp}}\mathcal{F}$ and $-\ln\delta$ have
to simultaneously go to infinity), albeit weakly as $1/\ln\delta$ 999To
clarify this statement, let
$R_{\delta}=R_{\delta}^{\circ}+R_{\delta}^{\prime}$, with $R_{\delta}^{\circ}$
from the right hand side of Eq. (26) and $R_{\delta}^{\prime}$ the
corrections. The relative correction, $R_{\delta}^{\prime}/R_{\delta}^{\circ}$
decays as $1/\ln\delta$ for $\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$
and as $\ln\left(\ln 1/\delta\right)/\ln\delta$ for
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$ and $\mathcal{X}_{QCB}$ as
$\delta\to\infty$. In other words, $R_{\delta}^{\prime}\sim
1/\left(\ln\delta\right)^{2}$ asymptotically. The very weak prefactor,
$\ln\left(\ln 1/\delta\right)$, for the latter two cases is due to the
presence of ${{}^{\sharp}}\mathcal{F}$ in the prefactor in Eqs. (28) and
(29).. The leading order contribution to the decay for
$\chi\left(\check{\mathcal{S}}:\mathcal{F}\right)$ is

$\frac{p_{1}p_{2}\log_{2}\frac{p_{2}}{p_{1}}}{p_{2}-p_{1}}\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}$
(27)

or with a prefactor of $1/2\ln 2$ when $p_{1}=p_{2}=1/2$. For
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$, the decay is

$p_{1}p_{2}\log_{2}\left[\frac{e}{p_{1}p_{2}}\left|\gamma\right|^{-2{{}^{\sharp}}\mathcal{F}}\right]\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}$
(28)

and, for the QCB result,

$C\log_{2}\left[\frac{e}{C}\left|\gamma\right|^{-2{{}^{\sharp}}\mathcal{F}}\right]\left|\gamma\right|^{2{{}^{\sharp}}\mathcal{F}}$
(29)

with $C=\min\left[p_{1},p_{2}\right]$ or $\sqrt{p_{1}p_{2}}$ depending on
whether we take the pure state result or generically take the mixed state
bound. These forms show the same exponential decay but the latter two have a
weak dependence of the prefactor on ${{}^{\sharp}}\mathcal{F}$.

To summarize, quantum Darwinism clarifies the role of the proliferation of
information in the quantum-to-classical transition. Here, I examined the
quantity introduced by Touil et al. touil_eavesdropping_2022 ,
$\chi\left(\mathcal{S}:\check{\mathcal{F}}\right)$, where an (optimal)
measurement is made on the fragment, reminiscent of the quantum Chernoff
bound. It provides an appealing approach to finding the redundancy of
information, as it is an accessibility bound that becomes the accessible
information in the limit of good decoherence. For the special case of a pure
$\mathcal{S}\mathcal{E}$ state, the accessible information is directly related
to the optimal error probability for distinguishing conditional states on the
environment (i.e., hypothesis testing or inference), of which an exact
expression (including the prefactor) can be computed. Moreover, this
connection immediately generalizes the result to any pure, $D=2$ model (spin
environments, qudit environments, photon environments, etc.) and to
inhomogeneous environments (including ones with self-Hamiltonians, as in Eq.
(9)). That decay, as expected, has the same exponent as the QCB, as the QCB
promises (and only promises) to yield the right asymptotic decay, not the
prefactor. Asymptotic analysis provides a non-empirical way to show that all
quantities give the same redundancy—due to the same exponent—to leading order
(and that corrections are small) and makes universality of the plateau
approach manifest. Since the QCB applies more generally, its universal bound
should further help shed light on future results that yield exact entropic
quantities or alternative bounds. Its importance—the QCB’s importance—goes
beyond this, though, as it provides a single shot, finite $\mathcal{F}$
framework for understanding how we observers learn in a quantum Universe.

I would like to thank W. H. Zurek for inspiration, many years of entertaining
and enlightening discussions, and moneywine bets on various scientific topics.
I would also like to thank J. Elenewski and A. Touil for helpful comments on
this manuscript.

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