# Adaptive Few-Shot Learning Algorithm for Rare Sound Event Detection
††thanks: ${*}$ Corresponding Author: Jianzong Wang<EMAIL_ADDRESS>
Chendong Zhao†⋆‡, Jianzong Wang†∗, Leilai Li†, Xiaoyang Qu†, Jing Xiao†
$\ddagger$ Work done during an internship at Ping An Technology. †Ping An
Technology (Shenzhen) Co., Ltd., Shenzhen, China
⋆The Shenzhen International Graduate School, Tsinghua University, China

###### Abstract

Sound event detection is to infer the event by understanding the surrounding
environmental sounds. Due to the scarcity of rare sound events, it becomes
challenging for the well-trained detectors which have learned too much prior
knowledge. Meanwhile, few-shot learning methods promise a good generalization
ability when facing a new limited-data task. Recent approaches have achieved
promising results in this field. However, these approaches treat each support
example independently, ignoring the information of other examples from the
whole task. Because of this, most of previous methods are constrained to
generate a same feature embedding for all test-time tasks, which is not
adaptive to each inputted data. In this work, we propose a novel task-adaptive
module which is easy to plant into any metric-based few-shot learning
frameworks. The module could identify the task-relevant feature dimension.
Incorporating our module improves the performance considerably on two datasets
over baseline methods, especially for the transductive propagation network.
Such as +6.8% for 5-way 1-shot accuracy on ESC-50, and +5.9% on noiseESC-50.
We investigate our approach in the domain-mismatch setting and also achieve
better results than previous methods.

###### Index Terms:

Sound event detection, Few-shot learning, Data augmentation, Deep learning

##  I Introduction

Automatic environmental sound event detection has received increasing
attention in recent years [1, 2]. It deals with audios detecting and
classifying, which leads to multi-form applications in industry. Environmental
sound is naturally different from other audios. It doesn’t exhibit any
stationary patterns like phoneme in speech [3, 4] or rhythm in music. In
contrast, sound event contains very complex temporal structure [5, 6, 7] that
may be continues (e.g. rains), abrupt (e.g. thunder storm) or periodic (e.g.
clock tick). Moreover, speech [8, 9] and music usually distribute on a
relatively fixed frequency bandwidth, but sound event spans a wide frequency
range where different sound’s frequency may distribute in different range that
leads to various patterns, such as the ”Airplane” and ”Dog” sounds’ frequency
differs a lot. The information contained in temporal patterns and frequency
bins of the sound event could be massive [10].

Figure 1: An toy example to illustrate the motivation. (a) defines a 5-way
1-shot task. There’re three features: Tone, Timbre and loudness. Different
color means different value of the feature, same color adds the similarity
score by 1; (b) In the k-shot (k=3) setting, all examples of class c3 share
the same value of Tone even though their Timbre and loudness are different.

To date, many deep-learning (DL) methods greatly improved detection
performance [11, 12, 13, 14, 15, 16]. However, they typically require large
amounts of labeled data, which limits the generalization ability to limited-
data tasks due to the annotation cost. In contrast, children can recognize new
things quickly with proper guidance, even they only see the examples few
times. These motivate the study of Few-shot learning. Meantime, this method
has also been introduced in [17, 18, 19, 20, 21] which concerning rare sound
event detection and achieved promising results. Few-shot model promises
alleviating the problem of data deficit and cold start. It usually follows the
episodic training strategy [17, 22], which considers an $N$-way $K$-shot (e.g
5 way, 1 shot means there’re 5 classes, each contains 1 support example)
classification task in each episode.

In the few-shot learning setting, a model is first trained on labeled data
with base classes. Then, model generalization is evaluated on few-shot tasks,
composed of unlabeled samples from novel classes unseen during training (query
set), assuming only one or a few labeled samples (support set) are given per
novel class. All metric-based few-shot learning frameworks [20, 21, 23, 24,
25] compute similarity between each support (training) example and the query
example independently, resulting in the correlation among support examples
being missed. Figure 1 illustrates our motivation of the task-adaptive module.
The goal of this task is to correctly classify the query example. In Figure 1
(a), we could notice that each support example has different Tone, but may
have same Timbre and loudness with others. During the similarity computation,
the score between support example (c1, c3, c5) and the query example are all
2, making it hard to classify the query example. Moreover, in multi-shot
setting like Figure 1 (b) shows, most of class c3 have the same Tone but
various Timbre and loudness. So it’s obvious that the critical feature in this
task should be Tone thus the label of the query example should be c3. Such an
phenomenon occurs especially when $K$ is low, which motivates us incorporating
the context of all categories in each task to find the relevant features. Our
task-adaptive module aims to integrate all support examples’ information to
value the commonality within per class and the uniqueness among all classes,
thus to find the critical features.

Our contributions are as follows:

(1) We introduce a feature encoder integrating attention mechanism to capture
the temporal&channel context. In addition, a sound event-oriented data
augmentaion strategy is introduced to alleviate the data-shortage problem.

(2) We extend metric-based few-shot learning frameworks with a task-adaptive
module to identify the uniqueness among classes and the commonality within per
class of the whole support set. The output of this module combines both
support set and query set, making the subsequent feature embedding more
effective.

(3) We evaluated our model on two benchmarks. Results verify the superiority
of our model over previous methods. Besides, our model also achieves better
performance in the setting of domain mismatch.

##  II Preliminary and related works

###  II-A Few-shot sound event detection

Recent approaches [13, 17, 19] adopt the prototypical networks [23] and graph
neural networks [26] for few-shot sound event detection. Few-shot sound event
detection aims to correctly classify unseen sounds with a few support labeled
examples to finetune the model.

Few-shot learning follows the episodic training paradigm that used in previous
literature [23, 27]. Supposing there’re two non-overlapping datasets of
classes (events) $\bm{C}_{train}$ and $\bm{C}_{test}$, where
$\bm{C}_{train}\cap\bm{C}_{test}=\emptyset$. There are also two procedures:
Meta-training and Meta-testing. In each episode of training, we randomly
sample $N$ classes (a small subset) from $\bm{C}_{train}$ to construct support
set $\mathcal{S}$ and query set $\mathcal{Q}$. A simple $N$-way $K$-shot task
denotes as follow: $\mathcal{S}$ is denoted as
$\mathcal{S}=\\{x_{1}^{1},...,x_{K}^{1},..,x^{N}_{1},..,x^{N}_{K}\\}$, where
$K$ is the number of samples in per class. The query set
$\mathcal{Q}=\\{x_{1}^{*},...,x_{T}^{*}\\}$ contains various examples from the
same $N$ classes. Thus, there are $N*K$ samples in $\mathcal{S}$ and $T$
samples in $\mathcal{Q}$. The support set and query set composed a multi-label
classification task here. During the procedure of meta-testing, $\mathcal{S}$
and $\mathcal{Q}$ sampled from $\bm{C}_{test}$ , and few-shot method is
required to predict on query set (no label) given the support set (with
label).

Figure 2:  Two metric-based few-shot learning methods.

###  II-B Metric-based learning methods

Few-shot learning methods can be divided into three branches: metric-based,
optimization-based and data augmentation-based [28, 29, 30]. In this paper, we
mainly focus on metric-based learning. The metric-based methods could be
categorized into inductive and transductive, and two representative methods
are prototypical network [23] and transductive propagation network [21], as
shown in Figure 2. Given a $N$-way $K$-shot task
$\mathcal{T}=(\mathcal{S},\mathcal{Q})$, and the feature encoder $f_{\varphi}$
decided by its internal parameter $\varphi$.

Prototypical network (ProtoNet): this approach simply integrate nerual network
(NN) baseline into the end-to-end meta-learning framework. It takes the
average of the learned representations of a few examples for each class as
class-wise representations $C$, and then classifies an unlabeled instance by
calculating the Euclidean distance between the input and the class-wise
representations. $x^{s}_{i}\in\mathcal{S}$ and $x^{q}\in\mathcal{Q}$, and
$\mathcal{M}$ is a pair-wise feature distance function. The distance measure
is as follows:

$f_{sim}(x^{s},x^{q};\mathcal{S},\mathcal{Q},\varphi)=\mathcal{M}(\frac{1}{K}\sum_{i=1}^{K}f_{\varphi}(x^{s}_{i}),f_{\varphi}(x^{q}))$
(1)

Transductive propagation network (TPN): this approach utilizes the entire test
set for transductive inference, which is to consider the relationships among
testset and thus predict them as a whole. Transductive inference has shown to
outperform inductive methods [31, 32, 33]. TPN propose to learn to propagate
labels via episodic paradigm. During the propagation, a distance measurement
and example-wise length scale parameter were adopted to obtain a proper
neighborhood graph. After the graph construction, label propagation determines
the labels of the query set. The distance measure is as follows:

$f_{sim}(x_{i},x_{j};\mathcal{S},\mathcal{Q},\varphi,\phi)=exp(-\frac{1}{2}\mathcal{M}(\frac{f(x_{i})}{\sigma_{i}},\frac{f(x_{j})}{\sigma_{j}}))$
(2)

where $\phi$ is the parameters generating example-wise length-scale parameter
($\sigma_{i},\sigma_{j}$). $x_{i},x_{j}\in\mathcal{S}\cup\mathcal{Q}$.

Figure 3: (a). The overall framework of our model, it is composed of three
parts: feature encoder, task-adaptive module and metric-based few-shot
learning network. (b). The temporal&channel attention mechanism. $A_{ch}$ is
the channel attention, $A_{te}$ is the temporal attention. (c). Data
augmentation pipeline for the input log mel-spectrogram.

##  III Approach

The overview of the proposed learning framework is shown in Figure 3. In this
section, we first introduce how to mask and mixup for augmenting the training
data, then use class prototypes to build a classifier and update the
classifier with transductive inference. After that, we discuss how to make use
of the intergreted class prototypes to fine-tune the feature extractor.
Lastly, we summarize the core idea of task-aware learning framework.

###  III-A Masked mixup

To avoid possible overfitting caused by limited trained data, we adopt time
and frequency masking [34] and mix-up as the data augmentation strategy, which
is simple but effective [35]. The strategy adopt multiple time and frequency
masks on input spectrogram to generate multi-masked spectrograms and then
randomly mixes two masked samples, increasing the diversity of samples by the
way. As shown in Figure 3 (c), given a spectrogram $x$ with $T$ frames and $F$
frequency bins. The first step is to use multiple time masking and frequency
masking, generating $M$ masked samples $x^{*}$. To be specific, $Mask_{num}$
assigns $t$ consecutive time frames $[t_{0}:t_{0}+t)$ and $f$ consecutive
frequency bins $[f_{0}:f_{0}+f)$ value to 0, and $num$ denotes the number of
multiple masking. $t$ is chosen from a uniform distribution from 0 to the time
parameter $\tau$, $t_{0}$ is chosen from $[0,T-t)$, $f$ is chosen from a
uniform distribution from 0 to the frequency parameter $\upsilon$, $f_{0}$ is
chosen from $[0,F-f)$. Secondly, the mixup step conducts a convex combination
of two randomly selected $(x_{i}^{*},y_{i})$ and $(x_{j}^{*},y_{j})$ from all
masked samples:

$\displaystyle x^{*}\leftarrow x\odot Mask_{num}$ (3)
$\displaystyle\tilde{x}=\lambda x_{i}^{*}+(1-\lambda)x_{j}^{*}$ (4)
$\displaystyle\tilde{y}=\lambda y_{i}+(1-\lambda)y_{j}$ (5)

where $y_{i}$ and $y_{j}$ are one-hot encoded class labels,
$(\tilde{x},\tilde{y})$ being the new sample. $\lambda\in[0,1]$ is acquired by
sampling from a beta distribution $Beta$($\alpha,\alpha$) with $\alpha$ being
a hyperparameter. Feature encoder: After data augmentation, the samples first
encoded by a ConvNet $f_{\varphi}$ as same as [17], but the CNN layer replaced
with a temporal&channel attention CNN layer [34] as shown in Figure 3 (b). The
architecture of ConvNet contains five blocks and a fully-connected layer, and
the first two block includes a $3\times 3$ convolutional layer, a batch
normalization layer, a ReLU activation layer and a $4\times 4$ max-pooling
layer, and the last block is same as the former except the last $1\times 1$
max-pooling layer.

###  III-B Task-adaptive module

This module contains three parts and leverages the $f_{\varphi}(.)$ as input,
and outputs the task-adapted feature embedding, which will be passed to
subsequent metric-based learning network.

Task-adaptive-extractor for commonality among classes: Task-adaptive-extractor
(TAE) aims to find the commonality among all instances within a class. Denote
the output shape from feature encoder $f_{\varphi}$ as $(N\times
K,m_{1},w_{1},h_{1})$, where $m_{1},w_{1},h_{1}$ indicate the number of
channel and spatial size respectively. TAE is defined as follows:

$f_{\varphi}(\mathcal{S}):(N\times K,m_{1},w_{1},h_{1})\stackrel{{\scriptstyle
TAE}}{{\longrightarrow}}o:(N,m_{2},w_{2},h_{2})$ (6)

where $m_{2},w_{2},h_{2}$ denote the output number of channel and spatial
size. In this part, we first utilize a simple CNN layer to perform the
dimension reduction. Then the samples in each class are averaged to a final
output $o$. The purpose of TAE is to extract the commonality among a category.
Specifically, for 1-shot setting, there is no average operation. The purpose
of TAE is to eliminate the differences among instances and extract the
commonalities in the same category.

We intend to make use of $o$ denoting the mean of the learned representation
of the support samples on the support set to represent the learned
representation of query samples. $N$ denotes the number of query samples. In
our experiments, $sim$ stands for cosine similarity. We do not use $M$ for the
reason that query sample is randomly chosen.

Projector for characteristics among classes: The goal of the second component
namely projector is to find the characteristics of various classes. Projector
takes the output of TAE as input and produce a mask for the support and query
set by observing all the support classes at the same time.

$o:(N,m_{2},w_{2},h_{2})\stackrel{{\scriptstyle
reshape}}{{\longrightarrow}}\hat{o}\stackrel{{\scriptstyle
CNN}}{{\longrightarrow}}p:(1,m_{3},w_{3},h_{3})$ (7)

During the projector process, firstly, we reshape the
$o:(N,m_{2},w_{2},h_{2})$ into $\hat{o}:(1,N\times m_{2},w_{2},h_{2})$, then a
small CNN is applied to $\hat{o}$, producing the mask
$p:(1,m_{3},w_{3},h_{3})$. Finally, a softmax is also applied to the dimension
$m3$. For making the output of projector $p$ influence the feature encoder
output $f_{\varphi}(.)$, we need to match the shape between $p$ and
$f_{\varphi}(.)$. This could be achieved as follows:

$f_{\varphi}(.)\stackrel{{\scriptstyle
Reshaper}}{{\longrightarrow}}r(.):(N\times K,m_{3},w_{3},h_{3})$ (8)

where Reshaper means a light-weight CNN network, and $r(.)$ is regarded as the
Reshaper network.

This idea is inspired by [36], which aims to maximize the mutual information
between the query features and their label predictions for a few-shot task at
inference. It means that the model has seen these unlabeled data before making
final prediction.

Portable to metric-based backbone: The task-adaptive module is portable, which
could be easily integrated with any metric-based few-shot learning methods. In
this paper, we investigate two classical metric-based methods: prototypical
network (inductive) and transductive propagation network (transductive) in Sec
II-B, both do not consider the whole support set at the same time. For support
set, mask $p$ directly onto the embedding. For the query set, $\odot$ stands
for broadcasting the value of $p$ along the sample dimension ($NK$) in
$\mathcal{Q}$. So the distance measurement could be modified as follows.
Specifically, $\theta$ is a model parameter in TPN.

$f_{sim}(\mathcal{S},\mathcal{Q},\varphi,\theta)=\mathcal{M}(p\odot
r(f_{\varphi}(\mathcal{S})),p\odot r(f_{\varphi}(\mathcal{Q})))$ (9)

Our loss function is based on the cross entropy following most of state-of-
the-art methods:

$\mathcal{L}=\frac{exp(\sum_{j}^{K}f_{sim}(x_{j},x_{q}))}{\sum_{i=1}^{N}exp(\sum_{j}^{K}f_{sim}(x_{j}^{i},x_{q}))}$
(10)

where $x_{j}^{i}$ , $x_{q}$ denoting support and query examples respectively.

##  IV Experiments

###  IV-A Experimental setting

Data preparation: We utlize ESC-50 and noiseESC-50 datasets in this work,
which is from DCASE2021 task 5 [37]. The ESC-50 dataset contains 2,000
5-seconds audio clips that belonged to 50 classes, each having 40 examples.
The sound categories cover sounds of human behaviors (sneezing, crying,
laughing), animals (bird, dog, sheep), machine sounds (train, clock-alarm,
airplane). Our work also follows [17] to evaluate the performance under noise
condition called noiseESC-50 that selected from audio recordings of 15
different acoustic scenes. So, the performance on ESC-50 and noiseESC-50 would
reflect the generalization ability of the model in real-world applications.
What’s more, although ESC-50 is relatively smaller than AudioSet, which
suffers from the class imbalance problem, this also is the reason for our
choice. To directly compare our model with other baselines, we follow the
setting of [22] as same as [17]. Two datasets are divided into 35 classes for
training, 10 classes for test and other classes for validation. All audio
clips are down-sampled from 44.1kHz to 16kHz. 128-bin log mel-spectrogram of
raw audio is extracted as the input.

TABLE I: The result of sound detection (in %) on ESC-50 and noiseESC-50. All
baselines reported here are directly reprint the experimental results from the
literature [17]. TA means the task-adaptive module. DA means the data
augmentation method.

Model |  ESC-50 |  noiseESC-50
---|---|---
5-way acc | 10-way acc | 5-way acc | 10-way acc
1-shot | 5-shot | 1-shot | 5-shot | 1-shot | 5-shot | 1-shot | 5-shot
MatchingNet [22] |  53.7% |  67.0% |  34.5% |  47.9% |  51.0% |  61.5% |  31.7% |  43.0%
RelationNet [24] | 60.0% | 70.3% | 41.7% | 52.0% | 56.2% | 74.5% | 39.2% | 52.5%
SimilarityEmbeddingNet [38] | 61.0% | 78.1% | 45.2% | 65.7% | 63.2% | 78.5% | 44.2% | 62.0%
ProtoNet [23] | 67.9% | 83.0% | 46.2% | 74.2% | 66.2% | 83.0% | 46.5% | 72.2%
ProtoNet + AS [17] | 74.0% | 87.7% | 55.0% | 76.5% | 69.7% | 85.7% | 51.5% | 73.5%
TPN [21] | 74.2% | 86.9% | 55.2% | 76.7% | 72.7% | 86.1% | 52.7% | 72.9%
ProtoNet+DA | 70.5% | 83.3% | 48.9% | 74.7% | 69.8% | 83.3% | 47.7% | 72.1%
TA+ProtoNet | 70.2% | 84.0% | 48.6% | 74.8% | 69.5% | 83.5% | 50.1% | 72.4%
TA+ProtoNet+DA | 70.8% | 84.6% | 50.9% | 75.3% | 71.5% | 84.8% | 50.2% | 72.3%
TA+ProtoNet+DA (temporal&channel) | 71.6% | 85.2% | 51.5% | 75.7% | 72.1% | 85.2% | 51.3% | 72.9%
TPN+DA | 77.3% | 86.5% | 60.1% | 75.7% | 77.1% | 86.6% | 55.9% | 73.1%
TA+TPN | 76.5% | 87.1% | 57.8% |  76.9% | 76.3% | 86.3% | 57.3% | 73.2%
TA+TPN+DA | 80.2% | 86.4% | 62.3% | 76.1% | 77.9% | 86.8% | 59.3% | 76.1%
TA+TPN+DA (temporal&channel) | 81.0% |  87.2% |  63.6% | 76.8% | 78.6% |  87.1% |  60.5% |  76.7%

TABLE II: Architecture of ConvBlock layer of TAE and Encoder Architecture.

ConvBlock Architecture | Encoder Architecture
---|---
Sub-Layer 1 | Conv2D | Layer 1 | ConvBlock 128
Sub-Layer 2 | BatchNorm | Layer 2 | ConvBlock 128
Sub-Layer 3 | ReLU | Layer 3 | ConvBlock 128
Sub-Layer 4 | MaxPool2D((2,2)) | Layer 4 | Flatten

Implentation detail: We list the model arcgitecture in Table II. The librosa
[39] is used for feature extraction. During the episodic training, for each
task, we only perform the mixup on the query set $\mathcal{Q}$. Empirically,
$\tau=24$, $\upsilon=36$ and $num=2$ are used for time and frequency masking,
and $\alpha=0.2$ is used for masked mixup. We conduct Adam optimization with
stochastic gradient descent algorithm in all experiments with an initial
learning rate of 0.0001. The model is trained by using back-propagation with
cross entropy loss function as we mentioned before. The network is trained for
a max of 60 epochs and a decaying factor 0.01 is set for the learning rate to
avoid overfitting.

Data augmentation: We utilize SpecAug, a cheap yet effective data augmentation
method for spectrograms, has been introduced [31]. SpecAug randomly sets time-
frequency regions to zero within an input log-Mel spectrogram with $D$ (here
64) number of frequency bins and $T$ frames. Time modification is applied by
masking $y_{t}$ times $x_{t}$ consecutive time frames, where $x_{t}$ is chosen
to be uniformly distributed between [ $t_{0}$, $t_{0}$ \+ $x_{t}$ ] and
$t_{0}$ is uniformly distributed. Frequency modification is applied by masking
$y_{f}$ times consecutive frequency bins[ $f_{0}$, $f_{0}$ \+ $x_{f}$ ), where
randomly chosen from a uniform distribution in the range and $f_{0}$ is
uniformly chosen. For all experiments we set $y_{t}$ = 2, $y_{f}$ = 2.

###  IV-B Performance on ESC-50 and noiseESC-50

Experimental results are shown in Table I. Our model outperforms the previous
model on two datasets. As shown in Table I, the absolute improvement of our
best model (TA+TPN+DA (temporal&channel)) over published SOTA (TPN) is +6.8%
in 5-way 1-shot, +8.4% in 10-way 1-shot on ESC-50. On noiseESC-50, +5.9% in
5-way 1-shot and +7.8% in 10-way 1-shot. At the same time, on ESC-50, we also
notice that the performance gains slightly improvement than the SOTA, +0.1% in
10-way 5-shot, 0.3% in 5-way 5-shot on ESC-50, +3.8% in 10-way 5-shot, +1.0%
in 5-way 5-shot on noiseESC-50. Obviously, our model gains more improvement in
1-shot setting. Two metric-based methods can be continuously improved by
integrating TA. Specifically, the experiment results of TPN on two datasets is
produced by ourself. All results are averaged over 1000 Meta-train $\&$ Meta-
test episodes.

From the statistics of Table I, data augmentation and temporal&channel
attention mechanism are also contributive. The data augmentation strategy
could continuously improve the effect of various few-shot models. The
enhancement is more obvious of TA+TPN, 2.7% for 5-way 1-shot and 2.0% for
10-way 1-shot. The improvement brought by data augmentation and novel
attention mechanism illustrate that the performance of baseline methods is
severely underestimated.

###  IV-C Analysis of experimental results

First, the temporal&channel attention can significantly improve the sound’s
representation, thus promote model’s performances. It is acknowledged that
temporal&channel attention and data augmentation strategy could reduce the
intra-class variation [40]. With regarding the noise, the performance on
ESC-50 is inferior to noiseESC-50. In addition, another significant
observation is that 5-shot is less significantly improved than 1-shot. For
example, in 5-way of ESC-50, the performance of our model over published
state-of-the-art is 0.3% for 5-shot but 6.8% for 1-shot. Moreover, the margin
of the various model decreases with the increasing of shots is because more
labeled data are used. The superiority of task-adaptive module and other
modules will be decreased when more labeled data are available. In this paper,
We also make detail experiments (5-way k-shot $k\in\\{1,2,...,10\\}$) of
various model, and the results are presented in Figure 4 and Figure 5, which
verify the viewpoints above. This gives a qualitative comparison of the result
of the prototypical network with and without attentional similarity for 3
query examples from the noiseESC-50 test set. Those picked by the model
without attentional similarity (marked in cyan) do not share the same class as
the queries; they are picked possibly because both the query and the picked
one have a long silence. In contrast, the model with attentional similarity
finds correct matches (marked in red).

As this sample shows, the TPN model is indeed capable of sound localization,
specifically for events with a duration information, such as “gun shot”.
However, it seems to struggle with longer events, such as “dog bark,” at which
it exhibits a peaking behavior, chunking the event into small pieces. On the
contrary, TA can predict and localize both short and long events for this
sample. Specifically, TA + TPN was unable to notice the short event, yet
predicted its presence. We believe that this is due to the low time-resolution
could skip over the presence of short events.

Figure 4: 5-way performance on ESC-50 with various training/test shots Figure
5: 5-way performance on noiseESC-50 with various training/test shots TABLE
III: The result of few-shot sound detection in domain mismatch. The AUC (Area
Under Curve) is used for evaluation.

Model | 1-shot | 5-shot
---|---|---
Music | Animal | Music | Animal
ProtoNet[23] |  0.712 |  0.644 |  0.824 |  0.729
ProtoNet+AS[17] | 0.736 | 0.677 | 0.839 | 0.750
TPN[21] | 0.747 | 0.685 | 0.843 | 0.748
ProtoNet+DA | 0.752 | 0.680 | 0.836 | 0.739
TA+ProtoNet | 0.759 | 0.691 | 0.851 | 0.763
TA+ProtoNet+DA | 0.762 | 0.694 | 0.854 | 0.761
TA+ProtoNet+DA(temporal&channel) | 0.779 | 0.705 | 0.857 | 0.768
TPN+DA | 0.755 | 0.693 | 0.848 | 0.751
TA+TPN | 0.762 | 0.691 | 0.854 | 0.753
TA+TPN + DA | 0.768 | 0.696 | 0.850 | 0.754
TA+TPN+DA(temporal&channel) | 0.779 | 0.705 | 0.857 | 0.761

###  IV-D Analysis of domain mismatch

During this subsection, we follow the setting created by [41]. While the
current evaluation focus on recognizing novel class with limited training
examples, these novel classes are sampled from the same domain. So, we follow
the experiments from [41], such a out-of-domain testing could display the
ability of few-shot learning methods to generalize [42, 43]. Following the
previous setup, the selected AudioSet [44] with 99 events for meta-train,
meta-validation with 21 events and meta-test with 21 events. In addition, the
pre-processing are same as the previous description about the setup of [17].
The number of ways is set to 5, and we rerun the experiments: ProtoNet,
ProtoNet+AS and TPN. For the domain mismatch setting, we experiments ”Music”
and ”Animals” domain as same as [41]. The events and associated audios from
the two domains are removed from train set.

Results on ProtoNet are slightly better than those on TPN, which is mainly
because in Audioset music sounds are common and some patterns exist in other
sounds as well such as bell ringing. On the other hand animal sounds such as
“dog growling” rarely resemble other sounds. The gain of meta-learning
approaches over supervised baselines are diminished due to domain mismatch.
This implies the potential overfitting issue in meta-learning. Meta-learning
models learn to utilize the correlation between classes. However, if all
training classes come from one domain, the model tends to overfit to that
domain and performance on new domain would drop.

##  V Conclusion

In this paper, we proposed a task-adaptive module for few-shot sound event
detection. The module contains a task-adaptive extractor (TAE) and a
projector. By considering all support examples at same time, TAE could extract
the inner-class commonality and the projector could find cross-class
characteristic features. Besides, the data augmentation strategy and the novel
attention mechanism could further improve model’s performance. We demonstrated
that it significantly improved accuracy on two benchmarks (ESC-50 and
noiseESC-50), achieving state-of-the-art performance. In addition, we compared
with several baselines under the experimental domain mismatch setting, our
model could also gain improvement over baselines.

In future work we plan to further analyze why adaptive training of current
state-of-the-art models does not yield substantial improvements in the multi-
shots setting. We would also like to devise a method that will perform better
in fast adaptation and unsupervised adaptation. Finally, we plan to evaluate
our method on more challenge tasks such as domain adaptation or language
adaptation of ASR models.

## Acknowledgment

This paper is supported by the Key Research and Development Program of
Guangdong Province under grant No.2021B0101400003. Corresponding author is
Jianzong Wang from Ping An Technology (Shenzhen) Co., Ltd (jzwang@188.com).

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2023

[1]M. E. Tunalioglu

1]Department of Physics Engineering, Ankara University, Dögol St., Ankara,
06100, Ankara, Türkiye

# On the geometric phases in entangled states

<EMAIL_ADDRESS>H. O. Cildiroglu  A. U. Yilmazer  [

###### Abstract

Correlation relations for the spin measurements on a pair of entangled
particles scattered by the two separate arms of interferometers in hybrid
setups of different types are investigated. Concurrence, entanglement of
formation, quantum fidelity, Bures distance are used to clarify how the
geometric phase affects the initial bipartite state. This affect causes a
quantum interference due to the movement of charged particles in regions where
electromagnetic fields are not present. We shown that in some cases the
geometric phase information is carried over to the final bipartite entangled
state.

###### keywords:

Geometric phase, CHSH inequality, Concurrence, Entanglement of formation,
Fidelity, Bures distance

###### pacs:

[

###### pacs:

[

Physics and Astronomy Classification]03.65.Ta, 03.65.Ud, 03.65.Vf

Mathematics Subject Classification]81P40, 81P42, 81Q70

##  1 Introduction

In classical and quantum physics, the geometric phase is a unifying and
innovative concept, and it appears as an additional phase factor due to the
geometric and topological properties of the Hamiltonian parameter space for
various driven systems. The geometric phase was first studied by Ehrenberg
Ehrenberg_1949 , Kato doi:10.1143/JPSJ.5.435 , Pancharatnam
pancharatnam1956generalized , Longuet-Higgins et al. longuet1958studies ,
Aharonov and Bohm aharonov1959significance  and later generalized by Berry in
1984 berry1984quantal . Geometric and topological phases play a considerable
role in physics and a vast literature exist today covering many plausible
applications in different fields aharonov1984topological ; PhysRevA.47.3424 ;
PhysRevLett.72.5 ; PhysRevLett.83.2486 ; sponar2010geometric ; sponar2010new ;
lepoutre2012he ; Werner2012 ; gillot2013measurement ; cohen2019geometric . On
the other hand, entanglement is another striking aspect of quantum mechanics
and its dates back to the article published by Einstein et.al. einstein1935can
and its deep meaning was exhibited later by Bell bell1964einstein .

The study of topological phases for entangled states has gained notable
interest recently and it has been considered as a useful resource to analyze
the quantum information processing. From this perspective, in this work,
quantum correlations are briefly discussed for entangled particles in hybrid
Aharonov-Bohm (AB), Aharonov-Casher (AC), He-McKellar-Wilkens (HMW), Dual
Aharonov-Bohm (DAB) type Einstein, Podolsky and Rosen (EPR) setups. In this
study, Clauser-Horne-Shimony-Holt (CHSH) inequality clauser1969proposed  for
certain choices of spin measurements is discussed in a hybrid AC-EPR setup and
then various entanglement measures such as concurrence ($\mathcal{C}$),
entanglement of formation ($EoF$) are examined bennett1996mixed ;
wootters1998entanglement ; WERNER2006233 ; 10.5555/2011706.2011707  in
connection with the quantum states acquiring geometric and topological phases.
At the same time, in order to better understand the possible role of geometric
phase as an indicator of the entanglement concepts such as quantum fidelity
($\mathcal{F}$) and Bures distance ($\mathcal{D}_{B}$) are utilized.

##  2 An Examplary Model: Entangled State in a Hybrid AC-EPR Setup

In order to investigate the quantum spin correlations related to the Aharonov-
Casher effect performed with an entangled spin pairs a hybrid AC-EPR
experimental setup with two electric charge lines ($\lambda_{1},\lambda_{2}$)
provides a useful and practical configuration (see Fig. 1). Consider a pair of
entangled spin-$\frac{1}{2}$ neutral particles produced in a spin singlet
state by the source at the center:

$|\psi(t)\rangle=\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}\otimes|\\!\downarrow\rangle_{R}-|\\!\downarrow\rangle_{L}\otimes|\\!\uparrow\rangle_{R}].$
(1)

Thus we are considering a gedanken experiment with two magnetic dipoles
(neutrons) from a single source, one to the right, one to the left, and are
under the influence of two electric charge lines.

Figure 1: A hybrid AC-EPR setup. To investigate the result of the measurement
of the spin components of the particles going left and right, one can deﬁne
projection operators along arbitrary vectors $\alpha$, $\alpha^{\prime}$,
$\beta$ and $\beta^{\prime}$. Along these vectors the expectation values of
joint measurements necessary for the CHSH inequality are calculated.

Our goal is to discuss the possible results of the spin measurements in
various directions and to find the correlation functions associated with the
joint probabilities; and furthermore we will investigate the CHSH violation,
calculate the concurrence ($\mathcal{C}$), entanglement of formation ($EoF$)
and quantum fidelity ($\mathcal{F}$), Bures distance ($\mathcal{D}_{B}$) step
by step.

The initial singlet state in (1) will gain an AC phase as follows,

$|\psi(t^{\prime})\rangle=\frac{1}{\sqrt{2}}[e^{-i\mu\lambda_{1}}|\\!\uparrow\rangle_{L}\otimes
e^{i\mu\lambda_{2}}|\\!\downarrow\rangle_{R}\\\
-e^{i\mu\lambda_{1}}|\\!\downarrow\rangle_{L}\otimes
e^{-i\mu\lambda_{2}}|\\!\uparrow\rangle_{R}].$ (2)

This equation can be written more simply,

$|\psi(t^{\prime})\rangle=\frac{1}{\sqrt{2}}[e^{-i\mu(\lambda_{1}-\lambda_{2})}|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}\\\
-e^{i\mu(\lambda_{1}-\lambda_{2})}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}].$
(3)

If one defines $\lambda_{1}-\lambda_{2}=\lambda_{E}$, then above equation
becomes,

$|\psi(t^{\prime})\rangle=\frac{1}{\sqrt{2}}e^{-i\mu\lambda_{E}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-e^{2i\mu\lambda_{E}}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}].$
(4)

The total phase factor can obviously be removed as it will not be included in
the expectation value calculation, and the final spin wave function will be
the following,

$|\psi(t^{\prime})\rangle=\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-e^{2i\mu\lambda_{E}}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}].$
(5)

The quantum correlations can be examined by measuring the spin components of
the particles moving to the right and left, along the directions as in the
usual Bell tests. The correlation function ($S$) for the CHSH inequality
associated with the angles ($\alpha,\beta,\alpha^{\prime},\beta^{\prime}$) for
the spin measurements shown in Figure 1 can be found to be as:

$\begin{split}S(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=|E(\alpha,\beta)-E(\alpha,\beta^{\prime})|+|E(\alpha^{\prime},\beta)+E(\alpha^{\prime},\beta^{\prime})|\\\
=|-\cos\alpha\cos\beta-\sin\alpha\sin\beta\cos(2\mu\lambda_{E})\\\
+\cos\alpha\cos\alpha^{\prime}+\sin\alpha\sin\alpha^{\prime}\cos(2\mu\lambda_{E})|\\\
+|-\cos\alpha^{\prime}\cos\beta^{\prime}-\sin\alpha^{\prime}\sin\beta^{\prime}\cos(2\mu\lambda_{E})\\\
+\cos\beta\cos\beta^{\prime}+\sin\beta\sin\beta^{\prime}\cos(2\mu\lambda_{E})|\leq
2.\end{split}$ (6)

Here $E\boldsymbol{(\alpha,\beta)}$’s are expectation values of the joint spin
measurements defined as,
$E\boldsymbol{(\alpha,\beta)}=|\langle\psi(t^{\prime})[P^{l}_{+}\boldsymbol{(\alpha)}-P^{l}_{-}\boldsymbol{(\alpha)]}\otimes[P^{r}_{+}\boldsymbol{(\beta)}-P^{r}_{-}\boldsymbol{(\beta)]}|\psi(t^{\prime})\rangle|$
and $P^{l}_{\pm}\boldsymbol{(\alpha)}$ being the spin projection operators
along the angles $\boldsymbol{{\pm}(\alpha)}$, etc. CILDIROGLU2021127753 .

It is easily seen that the CHSH inequality is violated by quantum mechanics,
if the appropriate angles are chosen for the $S$ function. At this point, one
notes that Bell-type inequalities have a generalized upper violation limit
called Tsirelson limit $(|\langle E\rangle|_{QMcorrelations}\leq 2\sqrt{2})$
cirel1980quantum , and for appropriate choices of relevant angles below, one
obtains:

$S(0,\frac{\pi}{4},\frac{3\pi}{4},\frac{\pi}{2})=\sqrt{2}+\sqrt{2}|\cos(2\mu\lambda_{E})|.$
(7)

This result reveals the clear dependence of the CHSH correlation function $S$
on the geometric AC phase $2\mu\lambda_{E}$.

##  3 Related Entanglement Measures

Now in order to discuss the entanglement measures in connection with the Bell-
type correlations for the entangled states we continue to examine the above
mentioned hybrid AC-EPR setup with two electric charge lines. Firstly, let us
calculate concurrence. One needs to write a more suitable version of the wave
function, namely $|\widetilde{\psi}(t^{\prime})\rangle$, according to the
definition given in wootters1998entanglement ,

$|\widetilde{\psi}(t^{\prime})\rangle=\sigma_{y}\otimes\sigma_{y}|\psi^{*}\rangle=\frac{1}{\sqrt{2}}[(|\\!\downarrow\uparrow\rangle-e^{-2i\mu\lambda_{E}}|\\!\uparrow\downarrow\rangle)].$
(8)

Pauli spin matrices here will spin flip the complex conjugate of the wave
function, thus the concurrence becomes,

$\mathcal{C}(|\psi(t^{\prime})\rangle)=\left|\langle\psi(t^{\prime})|\widetilde{\psi}(t^{\prime})\rangle\right|=\\\
\frac{1}{2}\left|(\langle\uparrow\downarrow\\!|-e^{-2i\mu\lambda_{E}}\langle\downarrow\uparrow\\!|)(|\\!\downarrow\uparrow\rangle-e^{-2i\mu\lambda_{E}}|\\!\uparrow\downarrow\rangle)\right|.$
(9)

If we arrange the above result, and performs simple inner products the
concurrence is found as

$\mathcal{C}(|\psi(t^{\prime})\rangle)=\left|-e^{-2i\mu\lambda_{E}}\right|=1.$
(10)

Thus, the entanglement of formation, $EoF$, is defined via the concurrence
value and the function $h(x)=-xlog_{2}x-(1-x)log_{2}(1-x)$ (see ref.
wootters1998entanglement ). If we use the above result in the $EoF$
calculation (10) we get

$EoF=h\left(\frac{1+\sqrt{1-\mathcal{C}^{2}(\rho)}}{2}\right)=h\left(\frac{1}{2}\right)=1.$
(11)

In the case when both states $(|\psi(t)\rangle$ and
$|\psi(t^{\prime})\rangle)$ are pure states then quantum fidelity can be
calculated as an overlap of states,

$\mathcal{F}=|\langle\psi(t)|\psi(t^{\prime})|=\\\
\frac{1}{2}\big{|}\langle\uparrow\downarrow\\!|\\!\uparrow\downarrow\rangle-e^{2i\mu\lambda_{E}}\langle\uparrow\downarrow\\!|\\!\downarrow\uparrow\rangle\\\
-\langle\downarrow\uparrow\\!|\\!\uparrow\downarrow\rangle+e^{2i\mu\lambda_{E}}\langle\downarrow\uparrow\\!|\\!\downarrow\uparrow\rangle\big{|}.$
(12)

Thus quantum fidelity for a process of hybrid AC-EPR scattering with spin
entangled initial state turns out to be

$\mathcal{F}=\frac{1}{2}\big{|}1+e^{2i\mu\lambda_{E}}\big{|}=|\cos(\mu\lambda_{E})|.$
(13)

Furthermore, the Bures distance is given as
$\mathcal{D}_{B}=\sqrt{2(1-\mathcal{F})}$ (see ref.yuan2017quantum ), so
$\mathcal{D}_{B}$ between initial and final entangled states in a hybrid AC-
EPR setup can be found as

$\mathcal{D}_{B}(AC)=\sqrt{2-2|\cos(\mu\lambda_{E})|}.$ (14)

The state in (1) is a pure entangled state of two qubits, so by Gisin’s
theorem GISIN1991201 , it violates the CHSH inequality and our result in (7)
confirms that prediction. Also by the equations (10), (11) and (13); one
observes that the geometric phase has no effects on the concurrence and
entanglement of formation, but on the contrary it has an explicit appearance
in the quantum fidelity expression and the latter ranges between
$0\leq\mathcal{F}\leq 1$. Also $|\psi(t)\rangle$ and
$|\psi(t^{\prime})\rangle$ which are connected via an AC phase as in (5) and
the non-zero Bures distance between them is given by the equation (14).

In this context, AB, HMW, DAB type experimental hybrid setups can be designed
similar to the hybrid AC-EPR setup and analogous calculations can be repeated
for them. Thus, the summarized results, including a generalized version of the
geometric Berry phase, can be obtained as written in Table 1.

Table 1: Summary of Aharonov-Bohm (AB), Aharonov-Casher (AC), He-McKellar-
Wilkens (HMW), Berry and Dual Aharonov-Bohm (DAB) phases gained by entangled
states and the values of Concurrence ($\mathcal{C}$), Entanglement of
Formation (EoF), Quantum Fidelity ($\mathcal{F}$), Bures Distance
($\mathcal{D}_{B}$). Here $\mu$ is magnetic dipole, $\lambda_{E}$ is electric
charge density, $d$ is electric dipole, $\lambda_{B}$ is magnetic charge
density, $\gamma$ is Berry phase and $g$ is magnetic charge.

Phase |  | $\mathbf{|}{\psi(t)\rangle^{1,3}}$
---
| $\mathbf{|}{\psi(t^{\prime})\rangle^{2}}$
---
$\mathcal{C}$ | EoF | $\mathcal{F}$ | $\mathcal{D}_{B}$
AB |  $\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$ |  | $\frac{1}{\sqrt{2}}e^{-i\phi_{B}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$
---
1 | 1 | 1 | 0
AC |  | $\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-e^{2i\mu\lambda_{E}}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$
---
1 | 1 | $|\cos(\mu\lambda_{E})|$ | $\sqrt{2-2|\cos(\mu\lambda_{E})|}$
HMW |  | $\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-e^{2id\lambda_{B}}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$
---
1 | 1 | $|\cos(d\lambda_{B})|$ | $\sqrt{2-2|\cos(d\lambda_{B})|}$
Berry |  | $\frac{1}{\sqrt{2}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-e^{2i\gamma}|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$
---
1 | 1 | $|\cos(\gamma)|$ | $\sqrt{2-2|\cos(\gamma)|}$
DAB |  | $\frac{1}{\sqrt{2}}e^{-ig\phi_{E}}[|\\!\uparrow\rangle_{L}|\\!\downarrow\rangle_{R}-|\\!\downarrow\rangle_{L}|\\!\uparrow\rangle_{R}]$
---
1 | 1 | 1 | 0

11footnotetext: Initial State Vector at $t$22footnotetext: Final State Vector
at $t^{\prime}$

33footnotetext: The entangled state at time $t$ for all five cases

Quantum computers are analog machines, which are expected to become a part of
our lives in the near future. Although the codes to be run in the quantum
computer are applied on various quantum gates, they are actually represented
by real variables. Applying a quantum gate to a qubit means subjecting it to a
wave from an indiscriminate wave generator. All properties of this wave, such
as amplitude and frequency, are real-valued variables. Errors may occur from
time to time in these variables. These errors are different from those caused
by the interaction of the qubit with the environment. Even if a required
quantum gate is prepared and applied perfectly, however exactly the desired
quantum state may not be achieved.

In fact, this is where quantum fidelity comes into play. Fidelity is a link
between regular digital needs and quantum hardware. A quantum state will be
less entangled the closer it is to the set of separable states, or more
entangled the farther it is liang2019quantum . With fidelity calculations, one
finds a way to track how well real circuits produced by a quantum computer fit
the original state. Mathematically, it provides guidance on the degree between
quantum states irtf-qirg-principles-10 , and there are several benefits for
making such a comparison. As a result fidelity has become one of the most
widely used quantities to measure the degree of similarity between quantum
states.

Figure 2: Plots of (a) Fidelity and (b) Bures distance for AC-EPR and HMW-EPR
Hybrid Setups, respectively. The white areas are the regions where the
absolute value of the fidelity expression given for AC
($|\cos(\mu\lambda_{E})|$) and HMW setups ($|\cos(d\lambda_{B})|$) in Table
LABEL:Table1 attains unity, and in the dark areas are this absolute value
becomes zero. Similarly, as to the Bures distance expression given for AC
($\sqrt{2-2|\cos(\mu\lambda_{E})|}$) and HMW
($\sqrt{2-2|\cos(d\lambda_{B})|}$), the region where the absolute value
reaches unity is the dark areas, and conversely the white areas are the
regions where the absolute value is zero. As a result, if the fidelity value
is $1$, the Bures distance value is $0$ as expected. This tells us that the
geometric phase information is preserved (observed) for the parameter values
of the white areas but not preserved (seen) in the dark areas. In this plot
exemplary $\mu$, $d$, $\lambda_{E}$ and $\lambda_{B}$ are the relevant
physical variables having certain values within some exemplary intervals
chosen as ($\mu,d=[0,4]$ and $\lambda_{E},\lambda_{B}=[-4,4]$) and each
product $\mu\lambda_{E}$ and $d\lambda_{B}$ is an angle. It is to be noted
that; since there is no geometric phase information in AB and DAB type hybrid
experimental setups and neither in the Berry case, no figures are needed for
them.

Besides, Bures distance describes the infinitesimal distance between the
density matrices describing the quantum states, and this approach is a
generalization of the Fisher distance. If it is limited to pure states only,
it gives similar results as the Fubini–Study distance
doi:10.1080/09500349414552171 ; HUBNER1992239 ; bures1969extension ;
HELSTROM1967101 ; facchi2010classical . These distances can be used as a
function of distance, metric. The Fisher metric is a Riemann metric that can
be defined on a smooth statistical manifold. It is generally used to calculate
the difference in information between measurements.

In this regard, statistical distances such as the Bures distance are
determined by the size of the statistical fluctuations that occur in the
measurements made to distinguish between the initial state and the evolved
state. These statistical fluctuations raise the interesting possibility that
quantum mechanics may be partially responsible for the Hilbert-space
structure, so Bures distance is thought to be a link between statistics and
geometry. By counting the number of intermediate states, the distance between
states can be found wootters1981statistical .

##  4 Conclusion

Summary Table 1 and Figure 2 shows us that, AB and DAB phase information does
not appear in the results of the measurements in the experimental setups to be
made with entangled particles. However, the situation is different for AC, HMW
and Berry phases. For these processes, geometric phase information appears in
the joint spin measurements. Thus, it seems possible in the future to code an
information via the geometric phase relevant in the process. It is understood
that investigation of the entanglement properties by the geometric phase will
provide a better understanding of the nature of entanglement and its role in
quantum technologies sandhya2012geometric .

Nevertheless, quantum logic gate operations can also be implemented with
geometric phase. When a quantum system is in a cyclic evolution, it acquires a
geometric phase determined by the path in which the system moves. Geometric
phases are useful for combatting errors in quantum gate operations and can
help quantum error correction codes reach below the error threshold. There are
many studies examining how geometric phases can be used to better analyze
entangled systems and to process the quantum information Vedral2003 ;
Sjoqvist2015 ; Thomas2016 .

Since the geometric phase is not dependent on time and energy like the dynamic
phase, but is connected to the closed path, it is not affected by changes such
as noise distortions. Therefore, it can also be used for quantum logic gate
applications Song2017 ; ji-li-na . Obviously one of the main questions here is
the entanglement content of the measured state, so after a physical process
providing a geometric phase the different measures of entanglement are
calculated for this purpose.

Otherwise, as is well known interferometers of different types, such as
Mach–Zehnder, Hanbury Brown and Twiss, etc. are effectively used to study
counter intuitive predictions of quantum mechanics doi:10.1119/1.12387 ;
Grangier_1986 ; SILVERMAN-BOOK . In such works, one observes that the
probability calculations based on the correlations between the number of
particle counts at detectors have interesting similarities with the
probabilities via spin measurements of the particles in certain directions as
used in the present work.

## Acknowledgments

We thank Abdullah Vercin for helpful comments.

## Declarations

The authors did not receive support from any organization for the submitted
work. The authors have no relevant financial or non-financial interests to
disclose.

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