# Towards Explainable Music Emotion Recognition:
The Route via Mid-level Features

###### Abstract

Emotional aspects play an important part in our interaction with music.
However, modelling these aspects in MIR systems have been notoriously
challenging since emotion is an inherently abstract and subjective experience,
thus making it difficult to quantify or predict in the first place, and to
make sense of the predictions in the next. In an attempt to create a model
that can give a musically meaningful and intuitive explanation for its
predictions, we propose a VGG-style deep neural network that learns to predict
emotional characteristics of a musical piece together with (and based on)
human-interpretable, mid-level perceptual features. We compare this to
predicting emotion directly with an identical network that does not take into
account the mid-level features and observe that the loss in predictive
performance of going through the mid-level features is surprisingly low, on
average. The design of our network allows us to visualize the effects of
perceptual features on individual emotion predictions, and we argue that the
small loss in performance in going through the mid-level features is justified
by the gain in explainability of the predictions.

##  1 Introduction

Emotions – portrayed, perceived, or induced – are an important aspect of
music. MIR systems can benefit from leveraging this aspect because of its
direct impact on human perception of music, but doing so has been challenging
due to the inherently abstract and subjective quality of this feature.
Moreover, it is difficult to interpret emotional predictions in terms of
musical content. In our quest for computer systems that can give musically or
perceptually meaningful justifications for their predictions [17], we turn to
the notion of ‘mid-level perceptual features’ as recently described and
advocated by several researchers [5, 1]. These are musical qualities (such as
rhythmic complexity, or perceived major/minor harmonic character) that are
supposed to be musically meaningful and intuitively recognizable by most
listeners, without requiring music-theoretic knowledge. It has been shown
previously that there is considerable consistency in human perception of these
features; that they can be predicted relatively well from audio recordings;
and that they also relate to the perceived emotional qualities of the music
[1].

That is the motivation for the work to be reported here. Our goal is to use
mid-level features as a basis for providing explanations of (and thus get
further insights into, or handles on) a model’s emotion predictions, by
training it to recognize mid-level qualities from audio, and predict emotion
ratings from the mid-level predictions. Further, we wish to quantify the cost
– in terms of loss of predictive performance – incurred by this detour. We
will call this the ‘cost of explainability’.

Focusing our study on a specific benchmark dataset labelled with both
perceived emotional qualities and mid-level perceptual features, we first
establish our basic VGG-style model architecture [16], showing that it can
learn the two individual prediction tasks (mid-level from audio, emotion from
mid-level) from appropriate ground-truth data, with accuracies that are at
least on par with previously published models. We then present a network with
nearly identical architecture that learns to predict emotional characteristics
of a piece by explicitly going through a mid-level feature prediction layer.
We compare this to predicting emotion directly, using an identical network
with the exception of the mid-level layer, and find that the cost of going
through the mid-level features is surprisingly low, on average. Finally, we
show that by training the network to learn to predict mid-level and emotions
jointly, the results can be further improved. A graphical overview of the
general scenario is shown in Figure 1.

The fact that in our model, emotions are predicted from the mid-level by a
single fully-connected layer, allows us to measure the effects of each of the
features on each emotion prediction, providing the basis for interpretability
and simple explanations. There are a number of application scenarios in which
we believe this could be useful; some of these will be briefly discussed in
the final section.

The remainder of this paper is structured as follows: Section 2 briefly
discusses related work on which this research is based. In Section 4 the
datasets that provide us with emotion and mid-level annotations are described.
The three different approaches to modelling emotion are summarized in Section
5. Experimental results and a demonstration of interpretability are given in
Sections 6 and 7. We discuss our findings and conclude in Section 8.

Figure 1: Three different architectures are compared for predicting emotion
from audio.

##  2 Related Work

In the MIR field, audio-based music emotion recognition (MER) has
traditionally been done by extracting selected features from the audio and
predicting emotion based on subsequent processing of these features [7].
Methods such as linear regression, regression trees, support vector
regression, and variants have been used for prediction as mentioned in the
systematic evaluation study by Huq et al [6]. Techniques using regression-like
algorithms have generally focused on predicting arousal and valence as per the
well-known Russell’s circumplex model of emotion [15]. Deep learning methods
have also been employed for predicting arousal and valence, for example [18],
that investigated BLSTM-RNNs in tandem with other methods, and [3], that used
LSTM-RNNs. Others such as [12] and [9] use support vector classification to
predict the emotion class. Aljanaki et al. [2] provide a summary of entries to
the MediaEval emotion characterization challenge and quote results for arousal
and valence prediction.

Deep neural networks are preferable for many tasks due to their high
performance but can be considered black boxes due to their non-linear and
nested structure. While in some fields such as healthcare or criminal justice
the use of predictive analytics can have life-affecting consequences [14], the
decisions of MIR models are generally not as severe. Nevertheless, also in MIR
it would be desirable to be able to obtain explanations for the decisions of a
music recommendation or search system, for various reasons (see also Section
8). Many current methods for obtaining insights into deep network-based audio
classification systems do not explain the predictions in a human
understandable way but rather design special filters that can be visualized
[13], or analyze neuron activations [8]. To the best of our knowledge, [19] is
the only attempt to build an interpretable model for MER. They performed the
task of feature extraction and selection and built models from different model
classes on top of them. The only interpretation offered is the reporting of
coefficients from their logistic regression models, without further
explanation.

##  3 Mid-level Perceptual Features

The notion of (‘mid-level’) perceptual features for characterizing music
recordings has been put forward by several authors, as an alternative to
purely sound-based or statistical low-level features (e.g., MFCCs, ZCR,
spectral centroid) or more abstract music-theoretic concepts (e.g., meter,
harmony). The idea is that they should represent musical characteristics that
are easily perceived and recognized by most listeners, without any music-
theoretical training. Research on such features has quite a history in the
fields of music cognition and psychology (see [5] for a compact discussion).
Such features are attractive for our purposes because they could provide the
basis for intuitive explanations of a MIR system’s decisions, relating as they
do to the musical experience of most listeners.

Various sets of such perceptual features have been proposed in the literature.
For instance, Friberg et al.’s set [5] contains such concepts as speed,
rhythmic clarity/complexity, articulation, dynamics, modality, overall pitch
hight, etc. In our study, we will be using the seven mid-level features
defined by Aljanaki & Soleymani [1], because they come with an openly
available set of annotated audios (see below). We recapitulate the features
and their definitions in Table 1, for convenience.

Perceptual Feature | Question asked to human raters
---|---
Melodiousness | To which excerpt do you feel like singing along?
Articulation | Which has more sounds with staccato articulation?
Rhythmic Stability |  Imagine marching along with the music. Which is easier to march along with?
Rhythmic Complexity |  Is it difficult to repeat by tapping? Is it difficult to find the meter? Does the rhythm have many layers?
Dissonance |  Which excerpt has noisier timbre? Has more dissonant intervals (tritones, seconds, etc.)?
Tonal Stability |  Where is it easier to determine the tonic and key? In which excerpt are there more modulations?
Modality (‘Minorness’) |  Imagine accompanying this song with chords. Which song would have more minor chords?
Table 1: Perceptual mid-level features as defined in [1], along with questions
that were provided to human raters to help them interpret the concepts. (The
ratings were collected in a pairwise comparison scenario.) In the following,
we will refer to the last one (Modality) as ‘Minorness’, to make the core of
the concept clearer.

##  4 Datasets

For our experiments, we need music recordings annotated both with mid-level
perceptual features, and with human ratings along some well-defined emotion
categories. Our starting point is Aljanaki & Soleymani’s _Mid-level Perceptual
Features_ dataset [1], which provides mid-level feature annotations. For the
actual emotion prediction experiments, we then use the Soundtracks dataset,
which is contained in the Aljanaki collection as a subset, and comes with
numeric emotion ratings along 8 dimensions.

###  4.1 Mid-level Perceptual Features Dataset

The Mid-level Perceptual Features Dataset [1] consists of 5000 song snippets
of around 15 seconds each annotated according to the seven mid-level
descriptors listed in Table 1. The annotators were required to have some
musical education and were selected based on passing a musical test. The
ratings range from 1 to 10 and were scaled by a factor of 0.1 before being
used for our experiments.

###  4.2 Emotion Ratings: The Soundtracks Dataset

The Soundtracks (Stimulus Set 1) dataset, published by Eerola and Vuoskoski
[4], consists of 360 excerpts from 110 movie soundtracks. The excerpts come
with expert ratings for five categories following the discrete emotion model
(happy, sad, tender, fearful, angry) and three categories following the
dimensional model (valence, energy, tension). This makes it a suitable dataset
for musically conveyed emotions [4]. The ratings in the dataset range from 1
to 7.83 and were scaled by a factor of 0.1 before being used for our
experiments. As stated above, all the songs in this set are also contained in
the Mid-level Features Dataset, so that both kinds of ground truth are
available.

##  5 Audio-to-Emotion Models

Figure 2: The same architecture from the first layer up to the ‘Adaptive
Average Pooling 2D’ layer is shared by all networks.

In the following, we describe three different approaches to modeling emotion
from audio, all based on VGG-style convolutional neural networks (CNNs). The
architectures are summarized in Figure 2. For all models, we use an Adam
optimizer with a learning rate of 0.0005 and a batch size of 8, and employ
early stopping with a patience of 50 epochs to prevent overfitting.

In terms of preprocessing, the audio samples are first converted into
149-point spectrograms calculated on randomly selected 10-second sections of
the original snippets. The audio is resampled at 22.05 kHz, with a frame size
of 2048 samples and a frame rate of 31.25 frames per second, and amplitude-
normalized before computing the logarithmic-scaled spectrogram. This results
in input vectors of size 313$\times$149\. These spectrograms are used as
inputs for the following model architectures.

###  5.1 A2E Scheme

The first model, which we term “A2E", is the most straightforward one. The
spectrograms are fed into a VGG-style CNN to directly predict emotion values
from audio. This is the leftmost path in Figure 2. This model is not
interpretable due to its black box architecture and is used as a baseline and
for computing the cost of explainability when comparing to more interpretable
but possibly worse performing models.

###  5.2 A2Mid2E Scheme

In order to obtain a more interpretable model, an intermediate step is
introduced. First, a VGG-style network is used to predict mid-level features
from audio. This model is trained on the mid-level features dataset described
in Section 4.1 above. Next, a linear regression model is trained to predict
the 8 emotion ratings in the Soundtracks dataset from the 7 mid-level feature
values that we get as an output from the mid-level predictor network. This
corresponds to a fully connected layer with 7 input units and 8 outputs and
linear (identity) activation function – see the middle path in Figure 2. We
call this scheme “A2Mid2E". A linear model is chosen because its weights can
easily be interpreted to understand the importance of each mid-level feature
in predicting the emotion ratings.

###  5.3 A2Mid2E-Joint Scheme

In an attempt to replace the step-wise training of two separate models with a
single model that, ideally, could learn an internal representation useful for
both prediction tasks, while keeping the interpretability of the linear
weights, we propose a third architechture, called “A2Mid2E-Joint" (rightmost
path in Figure 2). This network learns to predict mid-level features and
emotion ratings jointly, but still predicts the emotions directly from the
mid-level via a linear layer. This is achieved by the second last layer having
exactly the same number of units as there are mid-level features (7), followed
by a linear output layer with 8 outputs. From this network, we extract two
outputs – one from the second last layer ("mid-level layer"), and one from the
last layer ("emotion layer"). We compute losses for both the outputs and
optimize the combined loss (summation of both the losses).

##  6 Experiments

| Valence | Energy | Tension | Anger | Fear | Happy | Sad | Tender | Avg.
---|---|---|---|---|---|---|---|---|---
Mid2E (Aljanaki) | 0.88 | 0.79 | 0.84 | 0.65 | 0.82 | 0.81 | 0.73 | 0.72 | 0.78
Mid2E (Ours) | 0.88 | 0.80 | 0.84 | 0.65 | 0.82 | 0.81 | 0.74 | 0.73 | 0.79
A2E | 0.81 | 0.79 | 0.84 | 0.82 | 0.81 | 0.66 | 0.60 | 0.75 | 0.76
A2Mid2E | 0.79 | 0.74 | 0.78 | 0.72 | 0.77 | 0.64 | 0.58 | 0.67 | 0.71
A2Mid2E-Joint | 0.82 | 0.78 | 0.82 | 0.76 | 0.79 | 0.65 | 0.64 | 0.72 | 0.75
CoE${}_{\text{A2Mid2E}}$ | 0.02 | 0.05 | 0.06 | 0.10 | 0.03 | 0.02 | 0.02 | 0.08 | 0.05
CoE${}_{\text{A2Mid2E-Joint}}$ | -0.02 | 0.01 | 0.02 | 0.06 | 0.02 | 0.01 | -0.04 | 0.03 | 0.01
Table 2: Summaries of the different model performances on predicting emotion.
The last two rows show the “cost of explainability” (CoE), as the difference
between our baseline (A2E) and the newly proposed models (A2Mid2E,
A2Mid2E-Joint). A positive cost indicates a loss in performance.

The audio clips are preprocessed as described in Section 5 to obtain the input
spectrograms. During training, one random 10-second snippet from each
spectrogram is taken as input. We optimize the mean squared error, and use
Pearson’s correlation coefficient as the evaluation metric for emotion rating
prediction. Each of the paths (A2E, A2Mid2E, A2Mid2E-Joint) is run ten times
and the average correlation values are reported. Each run has a different seed
which reshuffles the train-test split.

###  6.1 Verifying our Basic Architecture

Before going any further, we first want to verify that our VGG-style network
model performs on par with comparable methods on the basic component tasks of
predicting mid-level features from audio (A2Mid) and emotions from (given)
mid-level features (Mid2E).

For the A2Mid scenario (Table 3), we train in two ways: first on the entire
Mid-level features dataset with 8% test set selected as described in [1]. We
call this A2Mid+. This is the result that should be directly compared to
column 1. Second, we train only on the songs from the Soundtracks dataset with
20% test set, and call this A2Mid. The column ‘Joint’ in Table 3 gives the
mid-level predictions produced by our A2Mid2E-Joint model. As can be seen, our
models are broadly comparable to the results reported in [1], with A2Mid+ and
Joint performing slightly better, on average.

Regarding the prediction of emotions from given mid-level feature annotations
(Table 2, first two rows), there is not much space for deviation, as the
models used by Aljanaki [1] and us are very simple.

Mid-level feature | Aljanaki | A2Mid+ | A2Mid | Joint
---|---|---|---|---
Melodiousness | 0.70 | 0.70 | 0.69 | 0.72
Articulation | 0.76 | 0.83 | 0.84 | 0.79
R. Stability | 0.46 | 0.39 | 0.39 | 0.34
R. Complexity | 0.59 | 0.66 | 0.45 | 0.46
Dissonance | 0.74 | 0.74 | 0.73 | 0.74
Tonal Stability | 0.45 | 0.56 | 0.61 | 0.63
Minorness | 0.48 | 0.55 | 0.51 | 0.57
Table 3: Correlation values for mid-level features predictions using our
models, compared with those reported by Aljanaki et al.[1].

###  6.2 Quantifying the Cost of Explainability

We now compare our three model architectures (A2E, A2Mid2E, A2Mid2E-Joint) on
the full task of predicting emotion from audio, We train on the Soundtracks
dataset as described above, with 10 runs with randomly selected train-test
80:20 splits. The A2E model serves as reference for the subsequent models with
explainable linear layers. The results can be found in Table 2.

In the case of direct emotion prediction (A2E), the final layer is connected
to 256 input nodes. However, in the A2Mid2E scheme, due to the fact that we
introduce a bottleneck (viz. the 7 mid-level predictions) as inputs to the
subsequent linear layer predicting emotions, our hypothesis is that doing so
should result in a decrease in the performance of emotion prediction. We
calculate this cost as the difference in correlation coefficients between the
two models for each emotion. The results (rows A2E and A2Mid2E in Table 2)
reflect the expected trend, but as can be seen, the decrease in performance is
quite small (less than 7% of the original correlation coefficient on average).

###  6.3 Joint Learning of Mid-level and Emotions

Further improvements in the performance of the mid-level-based network can be
obtained by training jointly on the mid-level and emotion annotations (as
described in Section 5.3). This model reduces the cost even further, as can be
seen in Table 2, row A2Mid2E-Joint. The decrease in performance is now less
than 1.5% of the correlation coefficients for the A2E case on average. We
believe this is acceptable in view of the possibility of obtaining
explanations from this network (see below).

##  7 Obtaining Explanations

Since the mapping between mid-level features and emotions is linear in both
proposed schemes (A2Mid2E, A2Mid2E-Joint), it is now straightforward to create
human-understandable explanations. Linear models can be interpreted by
analyzing their weights: increasing a numerical feature by one unit changes
the prediction by its weight. A more meaningful analysis is to look at the
effects, which are the weights multiplied by the actual feature values [10].
An effects plot shows the distribution, over a set of examples, of the effects
of each feature on each target. Each dot in an effects plot can be seen as the
amount this feature contributes (in combination with its weight) to the
prediction, for a specific instance. Instances with effect values closer to 0
get a prediction closer to the intercept (bias term). Figure 3 shows the
effects of the model A2Mid2E-Joint.

First we will show how this can be used to provide model-level explanations
and then we will explain a specific example at the song level.

###  7.1 Model-level Explanation

Before a model is trained, the relationship between features and response
variables can be analyzed using correlation analysis. The pairwise
correlations between mid-level and emotion annotations in our data are shown
in Figure 4. When we compare this to the effect plots in Figure 3, or the
actual weights learned for the final linear layer (Figure 5) it can be seen
that for some combinations (e.g., valence and melodiousness, happy and
minorness) positive correlations go along with positive effect values and
negative correlations with negative effect values, respectively. This is not a
general rule, however, and there are several examples (e.g., tension and
dissonance, energy and melody) where it is the other way around. The
explanation for this is simple: correlations only consider one feature in
isolation, while learned feature weights (and thus effects) also depend on the
other features and must hence be interpreted in the overall context. Therefore
it is not sufficient to look at the data in order to understand what a model
has learned.

To get a better understanding, we will look at each emotion separately, using
the effects plot given in Figure 3. In addition to the direction of the effect
– which we can also read from the learned weights in Figure 5 (but only
because all of our features are positive) – we can also see the spread of the
effect which tells us more about the actual contribution the feature can have
on the prediction, or how different combinations of features may produce a
certain prediction.

###  7.2 Song-level Explanations

Effect plots also permit us to create simple example-based explanations that
can be understood by a human. The feature effects of single examples can be
highlighted in the effects plot in order to analyze them in more detail, and
in the context of all the other predictions. To show an interesting case we
picked two songs with similar emotional but different midlevel profiles. To do
so we computed the pairwise euclidean distances between all songs in emotion
($d_{\text{E}}$) and midlevel space ($d_{\text{Mid}}$) separately, scaled both
to the range $[0,1]$ and combined them as
$d_{\text{comb}}=d_{\text{E}}-(1-d_{\text{Mid}})$. We then selected the two
songs from the Soundtracks dataset that maximised $d_{\text{comb}}$. The
samples are shown in Figure 3 as a red square (song #153) and a blue dot (song
#322). The reader can listen to the songs/snippets by downloading them from
the Soundtracks dataset
page111https://www.jyu.fi/hytk/fi/laitokset/mutku/en/research/projects2/past-
projects/coe/materials/emotion/soundtracks.

| predicted | annotated
---|---|---
| #153 | #322 | #153 | #322
valence | 0.28 | 0.39 | 0.38 | 0.46
energy | 0.37 | 0.50 | 0.37 | 0.54
tension | 0.40 | 0.46 | 0.50 | 0.56
anger | 0.28 | 0.23 | 0.15 | 0.22
fear | 0.41 | 0.27 | 0.18 | 0.28
happy | 0.17 | 0.21 | 0.17 | 0.17
sad | 0.20 | 0.23 | 0.27 | 0.28
tender | 0.18 | 0.23 | 0.10 | 0.10
Table 4: Emotion prediction profiles for the two example songs #153 and #322.

As can be seen from Figure 3 and from the emotion prediction profile of the
two songs (see Table 4), both songs have relatively high predicted values for
tension and energy, but apparently for different reasons: song #322 more
strongly relies on “minorness" and “articulation" for achieving its “tense"
character; on the other hand, its rhythmic stability counteracts this more
strongly than in the case of song #153. The higher score on the “energy"
emotion scale for #322 seems to be primarily due to its much more articulated
character (which can clearly be heard: 153 is a saxophone playing a chromatic,
harmonically complex line, 322 is an orchestra playing a strict, staccato
passage).

Figure 3: Effects of each mid-level feature on prediction of emotion. The
boxplots show the distribution of feature effects of the model ‘A2Mid2E-Joint’
helping us to understand the model globally. Additionally, two example songs
(blue dots, red squares) are shown to provide song-level explanations (see
Section 7.2 for a discussion). Figure 4: Pairwise correlation between mid-
level and emotion annotations. Figure 5: Weights from the linear layer of the
‘A2Mid2E-Joint’ model.

##  8 Discussion and Conclusion

Model interpretability and the possibility to obtain explanations for a given
prediction are not ends in themselves. There are many scenarios where one may
need to understand why a piece of music was recommended or placed in a certain
category. Concise explanations in terms of mid-level features would be
attractive, for example, in recommender systems or search engines for ‘program
music’ for professional media producers, where mid-level qualities could also
be used as additional search or preference filters222A demonstration can be
found in https://shreyanc.github.io/ismir_example.html. As another example,
think of scenarios where we want a music playlist generator to produce a music
program with a certain prevalent mood, but still maintain musical variety
within these limits. This could be achieved by using the mid-level features
underlying the mood/emotion classifications to enforce a certain variability,
by sorting or selecting the songs accordingly.

There are several obvious next steps that need to be taken in the research.
The first is to extend this analysis to a larger set of diverse datasets and
emotion-related dimensions.333In fact, we do have preliminary results on a
second dataset – the _MIREX-like Mood Dataset)_ of [11], which is also covered
by the Mid-level Perceptual Features Dataset of [1] and differs from
Soundtracks in that is comes with discrete mood labels. The results confirm
the general trends reported in the present paper, but because of the different
emotion/mood encoding scheme, further optimisations on our models may still
improve the results further.

Second, we plan to extend the models and sets of perceptual features. One
rather obvious (and obviously relevant) perceptual dimension that is
conspicuously missing from our (Aljanaki & Soleymani’s) set of mid-level
features is perceived speed (which is not the same as tempo). Adding this
intuitive musical dimension is an obvious next step towards improving our
model. Of course, this will require an appropriate ground truth for training.

Generally, the relation between the space of musical qualities (such as our
mid-level features) and the space of musically communicated emotions and
affects deserves more detailed study. A deeper understanding of this might
even give us means to control or modify emotional qualities in music by
manipulating mid-level musical properties.

##  9 Acknowledgments

This research has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under grant agreement No. 670035 (project “Con Espressione”).

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# Compasslike states in a thermal reservoir and fragility of their
nonclassical features

Naeem Akhtar  0000-0002-6756-2898 Department of Physics, Jiangsu University,
Zhenjiang 212013, China  Xiaosen Yang  0000-0003-2140-7849 Department of
Physics, Jiangsu University, Zhenjiang 212013, China  Muhammad Asjad
0000-0001-6895-3332 Mathematics Department, Khalifa University of Science and
Technology, Abu Dhabi 127788, United Arab Emirates  Jia-Xin Peng
0000-0002-0119-0139 Quantum Institute for Light and Atoms, State Key
Laboratory of Precision Spectroscopy, Department of Physics, School of Physics
and Electronic Science, East China Normal University, Shanghai 200062, China
Gao Xianlong  0000-0001-6914-3163<EMAIL_ADDRESS>Department of Physics,
Zhejiang Normal University, Jinhua 321004, China  Yuanping Chen
<EMAIL_ADDRESS>Department of Physics, Jiangsu University, Zhenjiang 212013,
China

###### Abstract

Superposed photon-added and photon-subtracted squeezed-vacuum states exhibit
sub-Planck phase-space structures and metrological potential similar to the
original compass states (superposition of four coherent states) but are more
closely tied to modern experiments. Here, we observe that these compasslike
states are highly susceptible to loss of quantum coherence when placed in
contact with a thermal reservoir; that is, the interaction with the thermal
reservoir causes decoherence, which progressively suppresses the capacity of
these states to exhibit interference traits. We focus on the sub-Planck
structures of these states and find that decoherence effects on these features
are stronger with increasing the average thermal photon number of the
reservoir, the squeezing parameter, or the quantity of added (or subtracted)
photons to the squeezed-vacuum states. Furthermore, we observe that the sub-
Planck structures of the photon-subtracted case survive comparatively longer
in the thermal reservoir than their counterparts in the photon-added case, and
prolonged contact with the thermal reservoir converts these compasslike states
into a classical state.

##  I Introduction

The prototypical Schrödinger cat state, a superposition of macroscopically
distinct quantum states, originated from the well-known gedanken experiments
[1], in which a cat appeared to be both alive and dead simultaneously. This
notion morphed into a macroscopic cat state [2, 3], which is the superposition
of two distinguishable coherent states, and then into multicomponent cat
states [4, 5]. Such states are of great interest because they may hold
particular non-classical features such as non-Gaussian interference
characteristics [6, 7], sub-Planck phase-space structures [4], and have both
fundamental and practical implications in the field of continuous-variable
quantum information processing. These implications span from understanding the
fundamental physics of quantum decoherence [8] to practical implications in
quantum metrology [9, 10, 11], quantum teleportation, cryptography [12, 13,
14], and so on.

The addition (or subtraction) of photons to the squeezed-vacuum states (SVSs)
leads to the quantum states having Wigner phase-space features similar to
those of cat states [15, 16, 17]. Similarly, when photons are added to or
subtracted from the superposition of SVS, one may obtain compasslike states
[18] that have comparable sub-Planck phase-space structures as the original
compass states [4]. Sub-Planck structures have been explored in various
contexts [19, 20, 21, 22, 23, 24, 25, 26, 27, 28], and it has been found that
such structures are highly sensitive to environmental decoherence [29] and
play a crucial role in the sensitivity of a quantum state against phase-space
displacements [30, 31, 32]. Both theoretically and experimentally, the
addition (or subtraction) of photons to the SVSs has been explored [33, 34,
35, 36, 37], and it has been found that these methods are quite effective in
producing larger cat states for quantum computing [38].

The interaction of a quantum system with its surrounding environment leads to
the loss of its quantum features, a so-called decoherence phenomena [8, 39].
Superposed states, such as macroscopic catlike states, are theoretically
attainable but difficult to achieve in practical applications due to their
extreme sensitivity to environmental decoherence [40, 41]. In the case of
macroscopic systems, the interaction with the environment can never be avoided
because the decoherence rate is proportional to the “macroscopic separation”
between the two states [42, 43, 44, 45], and when propagating through damping
channels, such states rapidly lose their non-classical properties and the
corresponding negative oscillations of their Wigner functions [46, 47, 48, 49,
50, 51, 52]. Consequently, it has been discovered that although such
superpositions are theoretically possible, they are difficult to observe in
actual experiments as a macroscopic object like a cat cannot be completely
isolated from its surroundings [53]. The recent rapid advancement in the
theory of quantum information processing, which involves the protection of
quantum states and their quantum dynamics after contact with an environment
generating decoherence, is a highly important subject [54, 55, 56].

In this work, we theoretically investigate the interaction between the quantum
states of our interest and a thermal reservoir that acts as a cause of
decoherence; in particular, we examine how the decoherence deforms the non-
classical phase-space assets of our states. The Fokker-Planck equation that
controls the temporal evolution of the relevant quantum system can be obtained
directly from the master equation for the density operator in the Born-Markov
approximations [40, 41]. Using the Wigner quasidistribution formalism [57], we
solve the Fokker-Planck equations to obtain the temporal evolution of the
associated Wigner functions under a thermal reservoir [41]. We pick
compasslike states for this interaction, which were recently obtained by
adding to or subtracting photons from two superposed SVSs [18]. As previously
mentioned, the Wigner functions of these states contain sub-Planck structures,
which makes them a potential substitute for the original compass states [4].
Moreover, compared to the original compass states, these states may be more
appropriate for contemporary experiments [36, 37].

We mainly focus on analyzing the impact of the given thermal reservoir on the
sub-Planck structures of the stated compasslike states. Visualizing plots of
the corresponding Wigner functions, we observe that the sub-Planck structures
of these compasslike states are highly influenced by the thermal reservoir;
that is, the decoherence aroused by this interaction may eventually smear out
these structures from the phase space. A numerical examination of these Wigner
function plots reveals that the corresponding sub-Planck structures decay more
quickly as the average thermal photon number of the reservoir, the quantity of
added (or subtracted) photons to the squeezed-vacuum states, or the squeezing
parameter increase. Compared to the corresponding photon-subtracted case, the
sub-Planck structures related to the photon-added case decay faster in the
thermal reservoir. Furthermore, we observe that the long-term contact of these
compasslike states with the thermal reservoir eventually converts them into a
thermal state.

The structure of our paper is as follows: §II contains an overview of the
basic concepts that are employed throughout this manuscript. We address the
interaction between the quantum states of our interest and a thermal reservoir
in §III. The §IV contains the main conclusions and the detailed physical
explanations of our findings.

##  II Theoretical framework

This section provides an overview of the main ideas and findings from earlier
research that are relevant to the present work. We organized this section as
follows: First, we review the basic concept of the phase-space representation
of a quantum state via the Wigner function in §II.1, and then this concept is
further reviewed for the quantum states of our interest in §II.2. In §II.3, we
revisit the model of thermal reservoir that will be employed later on in this
work to interact with the quantum states of §II.2.

###  II.1 Phase-space analysis

The Wigner quasiprobability distribution is the visualization of quantum
mechanical states in phase space [57, 58, 59, 60]. The Wigner function for a
generic quantum state $\hat{\rho}$ is written as an expectation value of the
parity kernel [59]

$\displaystyle
W\left(\alpha\right):=\text{tr}\left[\hat{\rho}\hat{\Delta}(\alpha)\right]~{}\text{with}~{}\alpha\in\mathbb{C},$
(1)

where

$\displaystyle\hat{\Delta}(\alpha):=2\hat{D}(\alpha)\hat{\Pi}\hat{D}^{\dagger}(\alpha),\,\hat{\Pi}:=\left(-1\right)^{\hat{a}^{\dagger}\hat{a}}$
(2)

being the displaced parity operator, and $\hat{a}^{\dagger}$ ($\hat{a}$) are
raising (lowering) operators that satisfy the commutation relation
$[\hat{a},\hat{a}^{\dagger}]=1$.

The quantum uncertainty principle [60, 61, 53] arising from commutator
relations $\left[\hat{x},\hat{p}\right]=\text{i}$ of the position operator
$\hat{x}$ and the momentum operator $\hat{p}$ obeys

$\displaystyle\Delta x\Delta p\geq\frac{1}{2},$ (3)

where

$\Delta C^{2}:=\braket{\hat{C}^{2}}-\braket{\hat{C}}^{2}$ (4)

is the uncertainty of any operator $\hat{C}$, and we set $\hbar=1$ hereafter.

The single-mode SVS can be written as

$\displaystyle\ket{\psi}=\hat{S}(r)\ket{0}$ (5)

with

$\displaystyle\hat{S}(r):=$
$\displaystyle\exp\bigg{[}\frac{r}{2}\left(\hat{a}^{\dagger
2}-\hat{a}^{2}\right)\bigg{]},$ (6)

is the squeezing operator [62], with $r$ being the real squeezing parameter.
The Wigner function of the SVS is

$\displaystyle W(\alpha)=\frac{2}{\pi}\exp\left(-2|\bar{\alpha}|^{2}\right),$
(7)

where

$\displaystyle\bar{\alpha}=\alpha\cosh(r)-\alpha^{*}\sinh(r)~{}\text{with}~{}\alpha:=\nicefrac{{(x+\text{i}p)}}{{\sqrt{2}}}.$
(8)

The Wigner function exhibits a Gaussian distribution, indicating that SVS is a
Gaussian state that deviates from classicality when these states are squeezed
[62]. Non-Gaussian operations such as photon addition (PA) or subtraction (PS)
applied to a Gaussian SVS lead to non-Gaussian SVSs [17, 63, 64, 15]; that is,
the Wigner functions of the resulting states may attain non-Gaussian terms.
The non-classical nature of such states is indicated by the negative value of
their Wigner function, apart from the squeezing in their quadrature, which is
another indicator of the aforementioned nature.

The sub-Planck structures are the phase-space features having dimensions far
smaller than the bound given by the uncertainty relation (3), and such
structures have achieved significant attention in quantum metrology [30, 31,
32] due to their high sensitivity to environmental decoherence [29]. The PA
and PS cases of the superposed SVSs have been analyzed previously, and the
Wigner functions of these states are shown to have sub-Planck structures [18].
Specifically, it has been found that an excess amount of the photons added to
or subtracted from the superposition of SVSs can cause sub-Planck structures
in the phase space. We revisit the sub-Planck structures contained by the
Wigner functions of these non-Gaussian SVSs next.

Figure 1: The Wigner function of the PA and PS cases of the superposed SVSs:
(a) and (b) show the corresponding PA cases with $m=5$ and $m=12$,
respectively, while (b) and (c) show the corresponding PS cases with $m=5$ and
$m=12$, respectively. For all cases, $r=0.8$.

###  II.2 Compasslike states

This section provides an overview of the recently introduced compasslike
states [18], which were obtained by either adding photons to or subtracting
photons from the superposition of two SVSs. It has been shown that when a
significant quantity of photons is added (or subtracted), the Wigner functions
of these states are shown to have sub-Planck phase-space features. Let us
first review the case when $m$ photons are added to the superposed SVSs, which
can be expressed as follows:

$\displaystyle\ket{\psi_{\text{PA}}}:=N_{\text{PA}}\hat{a}^{\dagger
m}\left(\hat{S}(r)\ket{0}+\hat{S}(-r)\ket{0}\right),$ (9)

where

$\displaystyle N_{\text{PA}}=$
$\displaystyle\Bigg{[}\frac{2}{\sqrt{\cosh(2r)}}\left(\frac{\cosh(r)}{\sqrt{\cosh(2r)}}\right)^{m}m!P_{m}\left(\frac{\cosh(r)}{\sqrt{\cosh(2r)}}\right)$
$\displaystyle+2\cosh^{m}(r)m!P_{m}\big{(}\cosh(r)\big{)}\Bigg{]}^{-\frac{1}{2}},$
(10)

is the normalization coefficient. The Wigner function of the state
$\ket{\psi_{\text{PA}}}$ is calculated as [18]

$\displaystyle
W_{\ket{\psi_{\text{PA}}}}(\alpha)=N^{2}_{\text{PA}}\left[W_{\bigoplus}(\alpha)+2\operatorname{Re}\big{(}W_{\bigotimes}(\alpha)\big{)}\right].$
(11)

where the first term is

$\displaystyle W_{\bigoplus}(\alpha)=W^{+}(\alpha)+W^{-}(\alpha),$ (12)

where

$\displaystyle W^{\pm}(\alpha)=$
$\displaystyle\aleph_{\pm}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\left[\mp
2\coth(r)\right]^{l}}{l!\left[(m-l)!\right]^{2}}$
$\displaystyle*\left|H_{m-l}\left[-\mathrm{i}\sqrt{\pm
2\coth(r)}\bar{\alpha}_{\pm}\right]\right|^{2},$ (13)

with $H_{m}$ representing the Hermite polynomial, and

$\displaystyle\aleph_{\pm}:=\frac{\text{e}^{\theta_{\pm}}\left[\pm\sinh(2r)\right]^{m}}{\pi
2^{2m}},\bar{\alpha}_{\pm}:=\alpha\cosh(r)\mp\alpha^{*}\sinh(r),$ (14)

where

$\displaystyle\theta_{\pm}:=\pm\sinh(2r)\left(\alpha^{*2}+\alpha^{2}\right)-2|\alpha|^{2}\cosh(2r).$
(15)

The second term in Eq. (11) is

$\displaystyle
W_{\bigotimes}(\alpha):=O^{-}_{m}\sum^{m}_{l=0}\frac{(m!)^{2}\left[-2\text{i}\coth(r)\right]^{l}}{[(m-l)!]^{2}}H_{m-l}\left[\text{i}\Omega\alpha_{-}\right]$
$\displaystyle*H_{m-l}\left[-\Omega\alpha^{*}_{+}\right],O^{\pm}_{m}=\frac{\text{e}^{\theta}\left[\pm\text{i}\tanh(2r)\right]^{m}}{\pi
2^{2m}\cosh(r)\sqrt{1+\tanh^{2}(r)}},$ (16)

where

$\displaystyle\Omega:=\frac{\sqrt{\tanh(2r)}}{\sinh(r)},$ (17)

and

$\displaystyle\theta:=-\tanh(2r)\left(\alpha^{2}-\alpha^{2*}\right)-2|\alpha|^{2}\operatorname{sech}(2r)$
(18)

with

$\displaystyle\alpha_{\pm}:=\alpha^{*}\sinh(r)\pm\alpha\cosh(r).$ (19)

We plot the Wigner function of the state $\ket{\psi_{\text{PA}}}$ in Figs.
1(a) and 1(b) by varying photon number $m$. This implies that as the number of
added photons increases, the area of the resulting sub-Planck structures
decreases. The four Gaussian-like peaks and the chessboard-like pattern are
produced by the term $W_{\bigoplus}(\alpha)$, where the tiles in the
chessboard pattern are sub-Planck structures in phase space. On the other
hand, oscillations that occur far from the phase-space center are caused by
$W_{\bigotimes}(\alpha)$.

In a similar way, subtracting $m$ photons from the superposition of two SVSs
yields

$\displaystyle\ket{\psi_{\text{PS}}}:=N_{\text{PS}}\hat{a}^{m}\left(\hat{S}(r)\ket{0}+\hat{S}(-r)\ket{0}\right),$
(20)

where

$\displaystyle N_{\text{PS}}:=$
$\displaystyle\Bigg{[}\frac{2}{\sqrt{\cosh(2r)}}\left(\frac{\sinh(r)}{\sqrt{\cosh(2r)}}\right)^{m}m!P_{m}\left(\frac{\sinh(r)}{\sqrt{\cosh(2r)}}\right)$
$\displaystyle+2\big{(}-\mathrm{i}\sinh(r)\big{)}^{m}m!P_{m}\big{(}\mathrm{i}\sinh(r)\big{)}\Bigg{]}^{-\frac{1}{2}},$
(21)

is the normalization coefficient, and the corresponding Wigner function of
this state $\ket{\psi_{\text{PS}}}$ is calculated as [18]

$\displaystyle
W_{\ket{\psi_{\text{PS}}}}(\alpha)=N^{2}_{\text{PS}}\left[W_{\bigoplus}(\alpha)+2\operatorname{Re}\big{(}W_{\bigotimes}(\alpha)\big{)}\right].$
(22)

With

$\displaystyle W^{\pm}(\alpha)=$
$\displaystyle\aleph_{\pm}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\left[\mp
2\tanh(r)\right]^{l}}{l!\left[(m-l)!\right]^{2}}$
$\displaystyle*\left|H_{m-l}\left[-\mathrm{i}\sqrt{\pm
2\tanh(r)}\bar{\alpha}_{\pm}\right]\right|^{2},$ (23)

the first term of Eq. (22) is expressed as

$\displaystyle W_{\bigoplus}(\alpha)=W^{+}(\alpha)+W^{-}(\alpha).$ (24)

The second term of Eq. (22) is

$\displaystyle W_{\bigotimes}(\alpha):=$ $\displaystyle
O^{+}_{m}\sum^{m}_{l=0}\frac{(m!)^{2}\left[2\text{i}\tanh(r)\right]^{l}}{[(m-l)!]^{2}}H_{m-l}\left[-\omega\alpha_{-}\right]$
$\displaystyle*H_{m-l}\left[-\text{i}\omega\alpha^{*}_{+}\right],\omega:=\frac{\sqrt{\tanh(2r)}}{\cosh(r)}.$
(25)

The relevant Wigner function for the state $\ket{\psi_{\text{PS}}}$ is shown
in Figs. 1(c) and 1(d) for different values of $m$, and similar to the PA
case, we observe that the area of the sub-Planck structures decreases when the
number of subtracted photons is increased. Similarly as in the PA case, the
term $W_{\bigoplus}(\alpha)$ produces Gaussian-like peaks and a central
pattern resembling a chessboard consisting of sub-Planck structures in phase
space. And the interference pattern that appears far from the phase-space
origin is caused by the term $W_{\bigotimes}(\alpha)$. Note that an excess
amount of the photons, that is, $m\gg 1$, is required to induce the sub-Planck
structures in the Wigner phase space of the corresponding states. Figure 1
illustrates that the PA case contains smaller sub-Planck structures than the
equivalent PS case. Note that, when photons are added to or subtracted from
the SVS, the average photon number of the resultant state increases, and the
subsequent PA case has a higher average photon number than the PS for the same
number of photons applied to the SVS [65, 66]. Additionally, the PA case of
SVSs has more potential for quantum metrology [18, 65] and is more non-
classical than the corresponding PS case [67].

The superposed photon-added and photon-subtracted SVSs presented above can be
an alternative to the original compass states [4], which have the same phase-
space features but may have a connection with more feasible experiments. For
example, it has been demonstrated previously that the superpositions of
coherent states could be constructed deterministically by using third-order
Kerr nonlinearity [3]. However, this method requires the availability of Kerr
nonlinearity for an order, which is actually not applicable to currently
available Kerr media. Also, states of this type are particularly prone to
loss, and because absorption cannot be ignored in currently available Kerr
media, the capacity to extract coherent-state superpositions before they lose
their quantum properties is severely limited [68]. Hence, adding or
subtracting photons from Gaussian SVSs seems to be a more feasible approach
for creating coherent-state superpositions [36, 37].

###  II.3 Thermal channels

The influence of damping on the quantum properties of systems was originally
discussed in [69, 70, 71, 72, 73, 74, 41, 40, 42], and it has been found that
such damping channels have a strong impact on the quantum properties held by a
system [74, 42]. This section discusses the model of a finite-temperature
thermal reservoir, which in the present work is referred to as a thermal
channel. The master equation describes the time evolution of a single-mode
state denoted by the density operator $\hat{\rho}$ in this thermal channel is
[41, 40]

$\displaystyle\frac{\text{d}\hat{\rho}}{\text{d}t}=$
$\displaystyle\kappa\left(\bar{n}+1\right)\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right)$
$\displaystyle+\kappa\bar{n}\left(2\hat{a}^{\dagger}\hat{\rho}\hat{a}-\hat{a}\hat{a}^{\dagger}\hat{\rho}-\hat{\rho}\hat{a}\hat{a}^{\dagger}\right),$
(26)

where $\kappa$ is the decay rate, and $\bar{n}$ denotes the average number of
thermal photons in the cavity. The first term of the right-handed side in Eq.
(II.3) describes the transfer through the decay of photons from the quantum
system to the thermal reservoir, while the second term corresponds to the
transfer of excitation from the non-zero temperature thermal reservoir to the
quantum system. For $\bar{n}=0$, Eq. (II.3) leads to the case describing the
decay of the quantum state to a zero-temperature reservoir (also known as the
photon-loss channel) [50].

The temporal evolution of the Wigner function of a quantum state in the
thermal channel given by Eq. (II.3) can be obtained as [49]

$\displaystyle
W(\zeta,t)=\frac{2}{\overline{T}}\int\frac{d^{2}\alpha}{\pi}W(\alpha)\exp\left(\frac{-2\left|\zeta-\alpha\text{e}^{-\kappa
t}\right|^{2}}{\overline{T}}\right),$ (27)

where $\zeta=\nicefrac{{(x+\text{i}p)}}{{\sqrt{2}}}$ and
$\overline{T}=(1+2\bar{n})T$ with $T=1-\text{e}^{-2\kappa t}$. Note that in
the following, we utilized a scaled version of the time, $\tau=\kappa t$, for
our convince.

The subsequent sections provide a theoretical analysis of the interactions
between the thermal reservoir covered in this section and the compasslike
states of §II.2. To determine the time evolution of the system prepared in
these compasslike states, we use the exact solution of the Fokker-Planck
equation described through the master equation in (II.3) for the Wigner
quasidistributions, which are easily obtained by solving Eq. (27). This
solution clearly describes the impact of the thermal channel on the quantum
dynamics of the compasslike states under consideration. Here, we primarily
study the decoherence effects resulting from this interaction on the sub-
Planck structures of these compasslike states.

The thermal reservoir given by Eq. (II.3) has been shown to have a substantial
effect on the non-classical features retained by a quantum state in phase
space; that is, cavity damping causes decoherence, which swiftly eliminates
the oscillations of the Wigner functions. For example, this phenomenon has
been previously discussed for other quantum states as well [46, 47, 48, 49,
50, 51, 52]. The above-mentioned photon-added case of superposed SVSs has been
found to be richer than the equivalent photon-subtracted case in terms of
holding non-classical nature, and more metrological potential is now being
subjected to the thermal reservoir to see how these distinctions will impact
their behavior against environmental decoherence.

##  III Interaction with environment

In this section, we address the interaction between the compasslike states of
§II.2 and the thermal reservoir provided in §II.3. We discuss in detail how
the interaction with the given thermal reservoir deforms the non-classical
phase-space regions contained by these compasslike states, specifically their
sub-Planck structures. To accomplish this, we compute the temporal evolution
of the Wigner functions associated with these states in the thermal reservoir
by using Eq. (27), and our analysis is structured in the following sections:
First, in §III.1, we provide the mathematical terms involved in the temporal
evolution of the corresponding Wigner functions of our compasslike states, and
then the discussion about these temporal evolutions of the Wigner functions is
provided in §III.2. The impact of the thermal reservoir on the sub-Planck
structures contained by these compasslike states is covered in §III.3, and
then finally, in §III.4, we estimate the time at which these sub-Planck phase-
space structures disappear.

###  III.1 Multitude of mathematics

To make our discussion more comprehensible, in this section, we separately
introduce intricate mathematical terms that will be used in the temporal
evolution of the corresponding Wigner functions of the PA and PS cases of the
stated compasslike states.

Let us first provide the terms related to the PA case, that is,

$\displaystyle W^{\pm}(\zeta,\tau)=$
$\displaystyle\frac{\left[\pm\sinh(2r)\right]^{m}\text{e}^{\frac{B^{\pm}_{1}}{A_{1}}-\frac{2|\zeta|^{2}}{\overline{T}}}}{\pi~{}2^{2m-1}\overline{T}\sqrt{A_{1}}}\sum^{m}_{l=0}\sum^{m-l}_{k=0}$
$\displaystyle\frac{(m!)^{2}[\mp
2\coth(r)]^{l}(-G^{\pm}_{1})^{k}\chi_{1}^{m-l-k}}{k!l!\big{[}(m-l-k)!\big{]}^{2}(-A_{1})^{m-l}}$
$\displaystyle*H_{m-l-k}\left(\text{i}\Theta_{1}C^{\pm}_{1}\right)H_{m-l-k}\left(\text{i}\Theta_{1}D^{\pm}_{1}\right),$
(28)

where

$\displaystyle\bar{\zeta}^{\pm}=\zeta\cosh(r)\mp\zeta^{*}\sinh(r),$ (29)
$\displaystyle A_{1}=\frac{4\text{e}^{-2\kappa
t}}{\overline{T}^{2}}\left(\text{e}^{-2\tau}+2\overline{T}\cosh(2r)+\overline{T}^{2}\text{e}^{2\tau}\right),$
(30) $\displaystyle
B^{\pm}_{1}=\frac{8\text{e}^{-4\tau}}{\overline{T}^{3}}\Big{(}|\zeta|^{2}+\overline{T}\text{e}^{2\tau}|\bar{\zeta}^{\mp}|^{2}\Big{)},$
(31) $\displaystyle C^{\pm}_{1}=\frac{8\text{i}\text{e}^{-3\tau}\sqrt{\pm
2\coth
r}}{\overline{T}^{2}}\Big{(}\bar{\zeta}^{\pm*}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{\mp*}\Big{)},$
(32) $\displaystyle D^{\pm}_{1}=-\frac{8\text{i}\text{e}^{-3\tau}\sqrt{\pm
2\coth(r)}}{\overline{T}^{2}}\Big{(}\bar{\zeta}^{\pm}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{\mp}\Big{)},$
(33) $\displaystyle
E_{1}=\frac{16\text{e}^{-2\tau}\cosh^{2}(r)}{\overline{T}},$ (34)
$\displaystyle G^{\pm}_{1}=\pm 16\coth
r\left(1+\frac{\text{e}^{-2\tau}\cosh(2r)}{\overline{T}}\right),$ (35)
$\displaystyle\chi_{1}=E_{1}-A_{1},$ (36)
$\displaystyle\Theta_{1}=\frac{1}{2\sqrt{A_{1}\chi_{1}}}.$ (37)

Now we provide the terms

$\displaystyle W_{\bigotimes}(\zeta,\tau)=$
$\displaystyle\frac{[-\tanh(2r)]^{m}\text{e}^{\frac{B^{+}_{2}}{A_{2}}-\frac{2|\zeta|^{2}}{\overline{T}}}}{\pi\overline{T}\sqrt{A_{2}}\cosh(r)2^{2m-1}\sqrt{1+\tanh^{2}(r)}}\sum^{m}_{l=0}$
$\displaystyle\sum^{m-l}_{k=0}\frac{(m!)^{2}[2\text{i}\coth(r)]^{l}(-G_{2})^{k}\chi_{2}^{m-l-k}}{k!~{}l![(m-l-k)!]^{2}(-A_{2})^{m-l}}$
$\displaystyle*H_{m-l-k}\left(\text{i}\Theta_{2}C_{2}\right)H_{m-l-k}\left(\text{i}\Theta_{2}D_{2}\right),$
(38)

where

$\displaystyle\Lambda^{\pm}=\pm
2\operatorname{sech}(2r)|\zeta|^{2}+\big{(}\zeta^{2}-\zeta^{*2}\big{)}\tanh(2r),$
(39) $\displaystyle
A_{2}=\frac{4}{\overline{T}}\left(\frac{\text{e}^{-4\tau}}{\overline{T}}+2\text{e}^{-2\tau}\text{sech}(2r)+\overline{T}\right),$
(40) $\displaystyle
B^{\pm}_{2}=\frac{\text{e}^{-4\tau}}{\overline{T}^{3}}\left(8|\zeta|^{2}\pm
4\overline{T}\text{e}^{2\tau}\Lambda^{\pm}\right),$ (41) $\displaystyle
C_{2}=-\frac{8\text{i}\text{e}^{-3\tau}\Omega}{\overline{T}^{2}}\left(\bar{\zeta}^{+*}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{-*}\right),$
(42) $\displaystyle
D_{2}=\frac{8\text{e}^{-3\tau}\Omega}{\overline{T}^{2}}\left(\bar{\zeta}^{-}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{+}\right),$
(43) $\displaystyle
E_{2}=\frac{4\text{e}^{-2\tau}\Omega^{2}\sinh(2r)}{\overline{T}},$ (44)
$\displaystyle
G_{2}=-\frac{8\text{i}\Omega^{2}}{\overline{T}}\big{(}\text{e}^{-2\tau}+\overline{T}\cosh(2r)\big{)},$
(45) $\displaystyle\chi_{2}=E_{2}-A_{2},$ (46)
$\displaystyle\Theta_{2}=\frac{1}{2\sqrt{A_{2}\chi_{2}}}.$ (47)

We now provide the terms related to the temporal evolution of the Wigner
function associated with the PS case of the superposed SVSs. These terms are

$\displaystyle W^{\pm}(\zeta,\tau)=$
$\displaystyle\frac{[\pm\sinh(2r)]^{m}\text{e}^{\frac{B^{\pm}_{1}}{A_{1}}-\frac{2|\zeta|^{2}}{\overline{T}}}}{\pi~{}2^{2m-1}\overline{T}\sqrt{A_{1}}}\sum^{m}_{l=0}\sum^{m-l}_{k=0}$
$\displaystyle\frac{(m!)^{2}[\mp
2\tanh(r)]^{l}(-G^{\pm}_{3})^{k}\chi_{3}^{m-l-k}}{k!l!\big{[}(m-l-k)!\big{]}^{2}(-A_{1})^{m-l}}$
$\displaystyle*H_{m-l-k}\left(\text{i}\Theta_{3}C^{\pm}_{3}\right)H_{m-l-k}\left(\text{i}\Theta_{3}D^{\pm}_{3}\right),$
(48)

and

$\displaystyle W_{\bigotimes}(\zeta,\tau)=$
$\displaystyle\frac{[-\text{i}\tanh(2r)]^{m}\text{e}^{\frac{B^{-}_{2}}{A_{2}}-\frac{2|\zeta|^{2}}{\overline{T}}}}{\pi\overline{T}\sqrt{A_{2}}\cosh(r)2^{2m-1}\sqrt{1+\tanh^{2}(r)}}\sum^{m}_{l=0}$
$\displaystyle\sum^{m-l}_{k=0}\frac{(m!)^{2}[2\text{i}\tanh(r)]^{l}(-G_{4})^{k}\chi_{4}^{m-l-k}}{k!~{}l![(m-l-k)!]^{2}(-A_{2})^{m-l}}$
$\displaystyle*H_{m-l-k}\left(\text{i}\Theta_{4}C_{4}\right)H_{m-l-k}\left(\text{i}\Theta_{4}D_{4}\right),$
(49)

where

$\displaystyle C^{\pm}_{3}=\frac{8\text{i}\text{e}^{-3\tau}\sqrt{\pm
2\tanh(r)}}{\overline{T}^{2}}\Big{(}\bar{\zeta}^{\pm*}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{\mp*}\Big{)},$
(50) $\displaystyle D^{\pm}_{3}=-\frac{8\text{i}\text{e}^{-3\tau}\sqrt{\pm
2\tanh(r)}}{\overline{T}^{2}}\Big{(}\bar{\zeta}^{\pm}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{\mp}\Big{)},$
(51) $\displaystyle
E_{3}=\frac{16\text{e}^{-2\tau}\sinh^{2}(r)}{\overline{T}},$ (52)
$\displaystyle G^{\pm}_{3}=\pm
16\tanh(r)\left(1+\frac{\text{e}^{-2\tau}\cosh(2r)}{\overline{T}}\right),$
(53) $\displaystyle\chi_{3}=E_{3}-A_{1},$ (54)
$\displaystyle\Theta_{3}=\frac{1}{2\sqrt{A_{1}\chi_{3}}},$ (55) $\displaystyle
C_{4}=-\frac{8\text{i}\text{e}^{-3\tau}\omega}{\overline{T}^{2}}\left(\bar{\zeta}^{-*}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{+*}\right),$
(56) $\displaystyle
D_{4}=\frac{8\text{e}^{-3\tau}\omega}{\overline{T}^{2}}\left(\bar{\zeta}^{+}+\overline{T}\text{e}^{2\tau}\bar{\zeta}^{-}\right),$
(57) $\displaystyle
E_{4}=\frac{4\text{e}^{-2\tau}\omega^{2}\sinh(2r)}{\overline{T}},$ (58)
$\displaystyle
G_{4}=-\frac{8\text{i}\omega^{2}}{\overline{T}}\big{(}\text{e}^{-2\tau}+\overline{T}\cosh(2r)\big{)},$
(59) $\displaystyle\chi_{4}=E_{4}-A_{2},$ (60)
$\displaystyle\Theta_{4}=\frac{1}{2\sqrt{A_{2}\chi_{4}}}.$ (61)

Now, we will include the equations presented above in our investigation of the
corresponding Wigner functions.

Figure 2: The Wigner function of the PA case in the thermal reservoir, where
$m=12$ and $r=0.8$. (a) $\bar{n}=0$ and $\tau=0.01$; (b) $\bar{n}=0.5$ and
$\tau=0.01$; (c) $\bar{n}=0.5$ and $\tau=0.03$; (d) $\bar{n}=1$ and
$\tau=0.1$. Figure 3: The Wigner function of the PS case in the thermal
reservoir, where $m=12$ and $r=0.8$. (a) $\bar{n}=0$ and $\tau=0.01$; (b)
$\bar{n}=0.5$ and $\tau=0.01$; (c) $\bar{n}=0.5$ and $\tau=0.03$; (d)
$\bar{n}=1$ and $\tau=0.1$.

###  III.2 Fragility of nonclassical features

In this section, we examine the interaction between the thermal reservoir and
the compasslike states introduced in the previous sections. The temporal
evolution of the Wigner functions associated with the PA and PS cases of these
compasslike states is presented here by incorporating their components of
§III.1. Here, the main focus of our observations is to examine the impact of
decoherence on the non-classical phase-space structures included by the Wigner
functions of these compasslike states.

First, we include the discussion related to the interaction between the PA
case of our compasslike states and the thermal reservoir, which is made
through the temporal evolution of the corresponding Wigner functions. This
temporal evolution of the Wigner function for the PA case with terms
$W^{\pm}(\zeta,\tau)$ and $W_{\bigotimes}(\zeta,\tau)$ provided in the
expressions (III.1) and (III.1), respectively, with

$\displaystyle W_{\bigoplus}(\zeta,\tau)=W^{+}(\zeta,\tau)+W^{-}(\zeta,\tau)$
(62)

represents the temporal evolution of the central chessboard-like pattern and
the terms denoted by $W_{\bigotimes}(\zeta,\tau)$ represent the temporal
evolution of the interference pattern appearing far away from the phase-space
origin, and hence, the total temporal evolution of the Wigner function of PA
case yields

$\displaystyle
W_{\ket{\psi_{\text{PA}}}}(\zeta,\tau)=N^{2}_{\text{PA}}\left[W_{\bigoplus}(\zeta,\tau)+2\operatorname{Re}\left(W_{\bigotimes}(\zeta,\tau)\right)\right],$
(63)

which we plot in Fig. 2.

Similarly, the temporal evolution of the Wigner function associated to the PS
case is written as

$\displaystyle
W_{\ket{\psi_{\text{PS}}}}(\zeta,\tau)=N^{2}_{\text{PS}}\left[W_{\bigoplus}(\zeta,\tau)+2\operatorname{Re}\left(W_{\bigotimes}(\zeta,\tau)\right)\right],$
(64)

and it is plotted in Fig. 3. The corresponding terms
$W_{\bigoplus}(\zeta,\tau)=W^{+}(\zeta,\tau)+W^{-}(\zeta,\tau)$ with
$W^{\pm}(\zeta,\tau)$ are provided in Eq. (III.1), which shows temporal
evolution of the central chessboard pattern. The second term
$W_{\bigotimes}(\zeta,\tau)$ in Eq. (64) is supplied in Eq. (III.1) depicts
the temporal evolution of the interference originating far away from the phase
space.

Figures 2 and 3 present the temporal evolution of the Wigner functions in the
given thermal reservoir for PA and PS cases of our compasslike states,
respectively. Here, we include the situations for a zero value of average
thermal photon number ($\bar{n}=0$), also referred to as the photon-loss
channel [50], and a non-zero value of average thermal photon number,
$\bar{n}\not=0$. We analyze the temporal evolution of corresponding Wigner
functions in the thermal reservoir for a short- and long-range of time, i.e.,
for small and large $\tau$ values. Note that $\bar{n}=0$ and $\tau\approx 0$
represent the case when there is no interaction between our states and the
thermal reservoir, and consequently, the Wigner functions of corresponding PA
and PS cases may exhibit similar forms, as illustrated in §II.2.

First, let us consider the scenario when $\bar{n}=0$ and $\tau=0.01$. We
present the corresponding Wigner distributions associated with PA and PS cases
in Figs. 2(a) and 3(a), respectively. By comparing these Wigner distributions
with their equivalent non-interacting cases, which are shown in Figs. 1(b) and
1(d), we observe that the interaction of the thermal reservoir with these
states results in the decay of the non-classical parts of the corresponding
Wigner functions. As we can see, this decay of non-classical features is
readily observable for the sub-Planck structures contained in the central
chessboard-like pattern. Figures 2(b) and 3(b) represent the PA and PS cases,
respectively, in which the average thermal photon number of the reservoir is
raised to $\bar{n}=0.5$ while the interaction time remains the same as in the
previous case, i.e., $\tau=0.01$. We find that the non-classical phase-space
structures are much more suppressed than in the preceding situation of zero-
average thermal photon number. This shows that in the case of non-zero values
of the average thermal photon number, the non-classical assets existing in the
stated compasslike states degrade significantly faster than in the case of a
zero-average thermal photon number.

As illustrated in Figs. 2(c) and 3(c), for PA and PS cases, respectively,
monitoring the interaction for a longer duration, that is, setting $\tau=0.03$
and maintaining the average thermal photon number $\bar{n}=0.5$ as in the
preceding cases, leads to the sub-Planck structures being smeared out of the
phase space, and other non-classical structures are also weakened well here.
Eventually, we raised the interaction duration to $\tau=0.1$ and the average
thermal photon number of the reservoir to $\bar{n}=1$, as illustrated in Figs.
2(d) and 3(d), respectively, for the PA and PS cases of the corresponding
compasslike states. The associated positive peak Wigner distributions for
these examples show that the higher values of interaction time and the average
thermal photon number of the thermal reservoir entirely remove the non-
classical features from the phase space.

Moreover, for the case when the interaction time is very high, i.e.,
$\tau\rightarrow\infty$, the corresponding compasslike states are simply
transformed into thermal states with the subsequent Wigner function, that is,

$\displaystyle
W_{\text{th}}(\zeta,\infty)=\frac{1}{\pi(2\bar{n}+1)}\exp\left(-\frac{2|\zeta|^{2}}{(2\bar{n}+1)}\right).$
(65)

As we can see, the corresponding Wigner function of thermal states follows a
Gaussian distribution centered at the phase-space origin and is unaffected by
the squeezing parameter $r$ or the photon number $m$. For $\bar{n}=0$, this
Wigner function yields the Wigner function of a vacuum state.

To sum up, we examined the time evolution of the Wigner functions related to
the compasslike states under investigation in the thermal reservoir. We
observed these Wigner functions at zero and non-zero values of the average
thermal photon number for both short and long timeframes. Our finding
indicates that the non-classical phase-space structures found in the Wigner
functions of these states are dispersed by this interaction, leading to a
quicker decay of these characteristics at non-zero-average thermal photon
number in the reservoir compared to the reservoir of zero-average thermal
photon number. After a very long interaction, these states finally become
thermal states, indicating that the given quantum states have lost all of
their non-classical components due to the decoherence sparked by this
interaction. For our purposes, we will focus more on the sub-Planck structures
that are enclosed by the chessboard-like pattern and see how the interaction
with the specified heat reservoir affects these tiny structures. The cavity
parameters, such as the decay time $\tau$ and the average thermal photon
number $\bar{n}$ present in the reservoir, were previously adopted as
$\tau=10^{-2}$ in experiments to achieve Fock states with significantly
smaller values of $\bar{n}$ [75, 76]. A more thorough explanation of how the
thermal channel affects the sub-Planck structures of the present compasslike
states is covered in the next section.

Figure 4: Decay of the central sub-Planck structure: (a) $m=5$, $r=0.5$ and
$\bar{n}=0$, (b) $m=11$, $r=0.5$ and $\bar{n}=0$, (c) $m=11$, $r=0.5$ and
$\bar{n}=0.5$, and (d) $m=11$, $r=0.8$ and $\bar{n}=0.5$.

###  III.3 Deformation of core interference fringes

In the previous section, we discussed the temporal evolution of the Wigner
distributions for the PA and PS scenarios of our compasslike states in the
stated thermal reservoir. The non-classical phase-space features included by
these Wigner functions degrade after the interaction with this thermal
reservoir, as we can see by visualizing the relevant Wigner distributions.

In the present section, we now study the influence of this thermal reservoir
on the sub-Planck structures contained by these PA and PS cases of the
compasslike states. It has been found that sub-Planck structures of such
states are the crucial and notable component [4, 18], so it is necessary to
find out how to preserve these features in order to prevent them from being
destroyed after interacting with such heat channels. Here, we now explore the
effects of the added (or subtracted) photons ($m$), the average thermal photon
number ($\bar{n}$), and the squeezing parameter ($r$) on the decay rate of
these sub-Planck structures.

First, let us consider the function $f(\tau)$ that evaluates the rate at which
the central sub-Planck structure of our compasslike states decays over time,
that is,

$f(\tau)=\frac{W_{\bigoplus}(0,\tau)}{\left|W_{\bigoplus}(0,0)\right|},$ (66)

where $\left|W_{\bigoplus}(0,0)\right|$ denotes the initial height of the
central peak. In Fig. 4, we plot $\log\big{(}|f(\tau)|\big{)}$ for both the PA
and PS cases of these states. We vary $m$, $r$, and $\bar{n}$ and discuss how
$f(\tau)$ behaves over $\tau$. The red dashed line represents the PA cases,
and the solid blue line plot represents the PS cases.

First, we discuss the effect of increasing amounts of added (or subtracted)
photons on the decay rate of the sub-Planck structures of these states. This
is demonstrated by comparing the correlative cases of PA and PS in Figs. 4(a)
and 4(b), where $\bar{n}$ and $r$ are kept the same but the number $m$
representing the amount of photons being added (or subtracted) is different.
It is evident that the curves in Fig. 4(b), which correspond to a higher
number of photons added (or subtracted), i.e., $m=11$, are falling more
quickly than the curves in Fig. 4(a), which correspond to a lower number of
photons operated, i.e., $m=5$. Hence, an increase in the number of photons
added or subtracted may cause the corresponding sub-Planck structure to
disappear more quickly in the thermal reservoir.

Next, we investigate how varying the average thermal photon number of the
thermal reservoir affects the decay rate of the sub-Planck structures of the
stated compasslike states. Figs. 4(b) and 4(c) are presenting the cases, where
the values of $m$ and $r$ are kept constant but $\bar{n}$ is varied over each
case. Note that the average thermal photon number has a direct relationship
with the associated temperature of a reservoir [50]; for example, $\bar{n}=0$
corresponds to the zero-temperature thermal reservoir, and an increase in
$\bar{n}$ raises the temperature of the reservoir. Here, we show that the
height of the central sub-Planck structure of both PA and PS cases falls
comparatively faster for higher values of $\bar{n}$; that is, the curve with
$\bar{n}=0.5$ presented in Fig. 4(c) decays faster than its previous case when
$\bar{n}=0$ as shown in Fig. 4(b). This indicates that an increase in the
average thermal photon number of the reservoir increases the decay rate of the
sub-Planck structures related to PA and PS cases, i.e., at a higher
temperature of the thermal reservoir, the sub-Planck structures destroy much
faster.

We now examine how the squeezing parameter associated with these compasslike
states affects the decay rate of the central sub-Planck structures. This is
accomplished by comparing two cases, as presented in Figs. 4(c) and 4(d).
Here, we vary the values of the squeezing parameter $r$, while $m$ and
$\bar{n}$ are kept constant across each case. It shows clearly that the sub-
Planck structures of PA and PS cases decay faster in the specified thermal
reservoir when the squeezing parameter is increased; that is, the curve shown
in Fig. 4(d) with $r=0.8$ is decaying faster than that shown in Fig. 4(c) for
$r=0.5$.

Let us now compare two equivalent PA and PS cases. As illustrated in Fig. 4,
the dashed lines representing the decay rate of the central sub-Planck
structure of the PA case are approaching zero faster than the decay of an
equivalent PS instance. This implies that in the specified thermal reservoir,
the sub-Planck structures associated with the PA cases of the defined
compasslike states vanish faster than their equivalent PS cases.

In summary, we examined the decay rate of the sub-Planck structures associated
with the compasslike states of the present work in the indicated thermal
reservoir. We discovered that increasing the average thermal photon number of
the reservoir, or the number of added or subtracted photons from these
superposed states, as well as the amount of squeezing, may accelerate the
decay of these sub-Planck structures. Furthermore, sub-Planck structures in PA
cases degrade faster in the thermal channel than in PS situations.

###  III.4 Approximating temporal thresholds

Here, we roughly estimate the time at which the height of the central sub-
Planck structure of our compasslike states reaches its minor, i.e.,
$|f(\tau_{d})|\approx 0$, where $\tau_{d}$ is the information of corresponding
times, and for convenience, we denote $\tau^{\text{PA}}_{d}$ for PA and
$\tau^{\text{PS}}_{d}$ for PS situations. We obtain these temporal thresholds
for each of the examples presented in Fig. 4, and let us now discuss these
cases in the following:

First, let us examine the scenario shown in Fig. 4(a) with $m=5$, $\bar{n}=0$,
and $r=0.5$, for which we approximate $\tau^{\text{PA}}_{d}=0.34552$ and
$\tau^{\text{PS}}_{d}=0.35321$. Now, for the case presented in Fig. 4(b) with
$m=11$, $\bar{n}=0$, and $r=0.5$, we have $\tau^{\text{PA}}_{d}=0.32775$ and
$\tau^{\text{PS}}_{d}=0.34657$. This indicates that, in comparison to their
equivalent cases provided in Fig. 4(a) for lower $m$, the values of
$\tau^{\text{PA}}_{d}$ and $\tau^{\text{PS}}_{d}$ presented in Fig. 4(b) for a
greater value of $m$ are decreased. This is another indication that an
increase in the number of added or subtracted photons causes the sub-Planck
structures of the PA and PS instances of the present compasslike states to
decay more quickly. Similarly, for the case represented in Fig. 4(c) with
$m=11$, $\bar{n}=0.5$, and $r=0.5$, we have $\tau^{\text{PA}}_{d}=0.19556$ and
$\tau^{\text{PS}}_{d}=0.20573$, and then for the case shown in Fig. 4(d) with
$m=11$, $\bar{n}=0.5$, and $r=0.8$, we have $\tau^{\text{PA}}_{d}=0.17497$ and
$\tau^{\text{PS}}_{d}=0.20073$. Here, we compare the equivalent situations
shown in Figs. 4(b) and 4(c). In comparison to their equivalents provided in
Fig. 4(b) for smaller $\bar{n}$, this comparison demonstrates that the example
illustrated in Fig. 4(c) with a greater value of $\bar{n}$ has smaller values
of $\tau^{\text{PA}}_{d}$ and $\tau^{\text{PS}}_{d}$. This find another
additional proof that the sub-Planck structures of both PA and PS cases of
these compasslike states degrade more quickly when the average thermal photon
number of the thermal reservoir rises. We finally compare the values
$\tau^{\text{PA}}_{d}$ and $\tau^{\text{PS}}_{d}$ of the cases shown in Fig.
4(d) with their comparable situations shown in Fig. 4(c), and we find that the
values $\tau^{\text{PA}}_{d}$ and $\tau^{\text{PS}}_{d}$ associated with the
lower value of $r$ in Fig. 4(d) are smaller, indicating that the corresponding
sub-Planck structures decay more quickly as the squeezing parameter increases.

The corresponding values of the $\tau^{\text{PA}}_{d}$ and
$\tau^{\text{PS}}_{d}$ cases mentioned in the preceding paragraph are now
compared for two equivalent PA and PS cases of the compasslike states. It is
discovered that, for two identical PA and PS cases,
$\tau^{\text{PA}}_{d}<\tau^{\text{PS}}_{d}$. Again, we verify that, compared
to their counterparts in PS cases, the sub-Planck structures associated with
the PA cases of these compasslike states decay faster in thermal channels.

In summary, in this section, we have provided a numerical verification of all
the results presented in §III.3. In the next section, we provide further
discussion on the physical interpretations and validations of these results.

##  IV Highlights and remarks

In this section, we provide the key findings of the present work and their
extensive physical explanations. Let us first highlight the crucial parts of
our investigations. The Wigner functions of superposed photon-added and
photon-subtracted SVSs may exhibit sub-Planck phase-space structures similar
to the original compass states, and PA and PS cases of these compasslike
states are the main constituents of this work. In particular, here, we
investigated the interaction of these compasslike states with a thermal
reservoir. The sub-Planck structures contained by the Wigner functions of
these states (see Fig. 1) were the main concern of the present analysis, and
we particularly observed the effect of thermal reservoir on these features
(see Figs. 2-4). And it is found that environmental decoherence aroused by the
interaction of these states with the thermal reservoir results in a washout of
the associated sub-Planck structures, and an increase in quantity of the
photons involved in PA and PS, or the squeezing parameter, or the average
thermal photon number contained by the reservoir may lead to the faster decay
of these tiny features. Comparatively, the sub-Planck structures of the PA
case of these compasslike states are found to be more susceptible to
environmental decoherence. However, the sub-Planck structures of these
compasslike states can be preserved in the thermal reservoir for comparatively
higher values of the added or subtracted photons if we set the average thermal
photon number of the reservoir to nearly zero.

Now we provide a brief physical interpretation of these findings by comparing
them with the previous results, and let us start with the well-known coherent-
state superpositions that are comparable to the states of the present work.
For example, in the case of macroscopic superposition of coherent states,
particularly cat states [77], higher values of the separation between two
coherent states cause to enhance the non-classical phase-space attributes of
such states [78, 77]. It has been found that the deformation of a cat state in
a thermal reservoir depends on the separation between the component states;
that is, higher values of the macroscopic parameter may enhance the fragility
of such states against environmental decoherence [50]. Moreover, the
macroscopic parameter of a cat state is directly correlated with its average
photon number and the non-classical nature [78, 77]; that is, an increase in
this parameter directly increases these two quantities. This means that a cat
state with a greater macroscopic parameter holds a greater non-classical
nature and a higher average photon number, hence being more susceptible to
external decoherence [50]. This concept directly applies to our quantum
states, which we discuss in the following.

Now consider the compasslike states of the present work. These two states
differ significantly from cat states in the way that now the number of added
or subtracted photons plays the role of the macroscopic parameter; that is,
when photons are added or removed from SVSs, the non-classical phase-space
features and average photon number of these states are boosted in a manner
similar to cat states [65, 66]. Interestingly, for the same number of photons
added (or subtracted) to SVSs, the resulting quantum states hold different
non-classical properties and average photon number; that is, photon-added
cases are always richer in these two quantities [65, 66]. This difference
between these two states also has an impact on their phase-space structures;
that is, the sub-Planck structures of the photon-added case are smaller than
their equivalent in the photon-subtracted case, and hence the photon-added
case appeared to be more useful for metrological applications [18]. Holding
these differences, the photon-added and photon-subtracted SVSs of the present
work also behave differently in the thermal reservoir; that is, the photon-
added case is highly fragile against decoherence as compared to the photon-
subtracted case.

Finally, we remark that our findings may be useful for researchers interested
in the generation of such states in cavities, and perhaps our observations may
also help in the development of a technique to protect such states in noisy
channels. Protecting quantum states from the environment has always been an
important topic [54, 55]. How to preserve the specified quantum states of this
study in a noisy environment is an open question that can be addressed in
future research.

## Acknowledgement

This work is supported by the Natural Science Foundation of Jiangsu Province
(Grant No. BK20231320) and the National Natural Science Foundation of China
(Grant No. 12174157). NA acknowledges Ms. Chuchi for her helpful feedback on
the work. MA acknowledges support from the Khalifa University of Science and
Technology under Award No. FSU-2023-014.

## Footnote

To eliminate the $\mathrm{e}^{\gamma st}$ terms from our complicated
exponentials, we use the following sum series:

$\displaystyle\exp(\gamma_{1}s+\gamma_{2}t+\gamma_{3}st)=\sum_{l=0}^{\infty}\frac{\gamma_{3}^{l}}{l!}\frac{\partial^{2l}}{\partial\gamma_{1}^{l}\partial\gamma_{2}^{l}}\left[\exp\left(\gamma_{1}s+\gamma_{2}t\right)\right].$
(67)

Notice the generating function of the Hermite polynomial

$H_{n}(x)=\big{.}\frac{\partial^{n}}{\partial
s^{n}}\exp\left(2xs-s^{2}\right)\big{|}_{s=0}$ (68)

and its recursive relation

$\frac{\mathrm{d}^{l}}{\mathrm{d}x^{l}}H_{n}(x)=\frac{2^{l}n!}{(n-l)!}H_{n-l}(x).$
(69)

The integral formula

$\displaystyle\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta\exp{\bigg{[}A|\beta|^{2}+B\beta+C\beta^{*}+D\beta^{2}+E\beta^{*2}}\bigg{]}$
(70)
$\displaystyle=\frac{\pi}{\sqrt{A^{2}-4DE}}\exp{\bigg{[}\frac{-ABC+B^{2}E+C^{2}D}{A^{2}-4DE}}\bigg{]}.$
(71)

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