# Reversing Deep Face Embeddings with Probable Privacy Protection
††thanks: This research work has been partially funded by the German Federal
Ministry of Education and Research and the Hessian Ministry of Higher
Education, Research, Science and the Arts within their joint support of the
National Research Center for Applied Cybersecurity ATHENE and the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No. 860813 - TReSPAsS-ETN.

Dailé Osorio-Roig1, Paul A. Gerlitz2, Christian Rathgeb1, and Christoph Busch1
1- Biometrics and Internet Security Research Group
Hochschule Darmstadt, Germany
<EMAIL_ADDRESS>2- Hochschule
Darmstadt, Germany
<EMAIL_ADDRESS>
###### Abstract

Generally, privacy-enhancing face recognition systems are designed to offer
permanent protection of face embeddings. Recently, so-called soft-biometric
privacy-enhancement approaches have been introduced with the aim of canceling
soft-biometric attributes. These methods limit the amount of soft-biometric
information (gender or skin-colour) that can be inferred from face embeddings.
Previous work has underlined the need for research into rigorous evaluations
and standardised evaluation protocols when assessing privacy protection
capabilities. Motivated by this fact, this paper explores to what extent the
non-invertibility requirement can be met by methods that claim to provide
soft-biometric privacy protection. Additionally, a detailed vulnerability
assessment of state-of-the-art face embedding extractors is analysed in terms
of the transformation complexity used for privacy protection. In this context,
a well-known state-of-the-art face image reconstruction approach has been
evaluated on protected face embeddings to break soft biometric privacy
protection. Experimental results show that biometric privacy-enhanced face
embeddings can be reconstructed with an accuracy of up to approximately 98%,
depending on the complexity of the protection algorithm.

###### Index Terms:

Face recognition, privacy protection, soft-biometrics, irreversibility, attack

##  I Introduction

Face recognition systems are widely used in personal, commercial, and
government sector applications, e.g. border and access control, payments, ID
cards, among others. These techniques mainly focus on deep neural networks
(DNNs) which embed discriminative representations of face images in the latent
space, so-called face embeddings. Recent work has shown that methods based on
de-convolution neural networks can be used to reconstruct face images from
their corresponding embeddings [1, 2, 3]. Moreover, it has been demonstrated
that privacy-sensitive soft-biometric information such as gender, skin-colour,
or age, can be automatically extracted from face embeddings [4]. Privacy is
considered a human right and is subject to regulation. In this context, the
European Union (EU) General Data Protection Regulation 2016/679 (GDPR) [5]
defines biometric information as sensitive data. Therefore, ensuring the
privacy of information stored in face embeddings is an ongoing endeavour.

In the past years, so-called soft-biometric privacy-enhancement approaches [6]
have been explored to achieve protection of soft-biometric attributes in face
embeddings. They can mainly be classified into methods performing on the image
level (e.g. [7]) and representation level (e.g. [8]). The former usually work
by obfuscating soft-biometric attributes of face images (which may greatly
reduce biometric utility), while the latter apply specific transformations on
the face embeddings. On representation level, said transformations are
designed to minimize soft-biometric information (e.g. gender) or distort it
(e.g. simple permutation). However, it should be noted that such approaches do
not depend on user-specific keys or any secret randomness, in contrast to the
well-known biometric template protection (BTP) schemes [9].

The privacy protection capabilities of soft-biometric privacy-enhancement
approaches have mainly been evaluated by machine learning approaches or
interpreted by dimensionality reduction tools. Nevertheless, recently,
vulnerability of these approaches have been shown in [10]. This leads to the
need of investigating further potential attack scenarios [11] that can
facilitate the reconstruction of face images from (privacy-enhanced) face
embeddings. In this context, such a type of reconstruction-based attack can be
estimated with respect to full or partial reversibility [12], where an
adversary retrieves exactly the original face image or a good approximation of
it, respectively. In this work, the irreversibility of face embeddings with
probable protection of soft-biometric attributes is analysed. Experimental
results show that face reconstruction is feasible depending on the
transformation complexity utilised for canceling the soft-biometric
attributes.

The remainder of this paper is organised as follows: Sect. II briefly
introduces the related work. In Sect. III, the irreversibility analysis is
described in detail. Sect. IV presents the experimental setup and the achieved
results, while a summary and concluding remarks are given in Sect. VI.

Figure 1: Conceptual overview of the irreversibility analysis of this work.
Note that unprotected templates used in the upper part (i.e. training process)
are different to the unprotected templates that are protected by PE-MIU (i.e.
bottom part).

##  II Related works

Different works have been proposed with the aim of preventing the derivation
of soft-biometric attributes from face embeddings. These approaches have shown
good biometric performance while achieving high privacy protection. They are
usually focused either on the feature representation (embeddings, e.g. [8]) or
on the inference-level (comparison, e.g. [13]). While the former aims at the
soft-biometric minimisation (e.g. [14]) or the feature transformation (e.g.
introduction of noise or simple permutation) without excluding the target
attribute. In the latter concept, transformations are applied and the
biometric comparator is adapted accordingly. In this paper, it is of interest
to explore said approaches that introduce transformations at the feature
level. As mentioned in Sect. I, these transformations do not remove soft-
biometric information and could therefore leak privacy information. In this
context, few works operating at the feature representation have been recently
published. Terhörst et al. [15] proposed a Cosine–Sensitive Noise (CSN)
transformation applied to face embeddings to enhance privacy in terms of
gender and age attributes. To this end, the authors introduced a specific type
of noise over the face representation which hides the soft–biometric
information. Further, Terhörst et al. [8] proposed a Privacy-Enhancing face
recognition approach based on Minimum Information Units (PE-MIU). This method
allows the creation of privacy-enhanced face embeddings by partitioning the
original embedding into smaller parts (called minimum information units).
Then, these blocks are randomly shuffled to obtain a privacy-enhanced
embedding.

The approaches described above have been focused on protecting soft biometric
attributes by transforming the feature space. However, these schemes are
usually assessed in inadequate evaluations that are not based on standardised
protocols [16]. More specifically, it is unknown to what extent such
transformed features (e.g. shuffled features) can be reconstructed to their
original face images. Moreover, in addition to soft-biometric privacy-
enhancement approaches, different privacy-preserving techniques and their
integration with theoretical and practical privacy-preservation mechanisms
[17, 18, 19] have been analysed in the literature in terms of recovery and
estimation of private attributes. In this paper, we consider a practical
privacy-preserving mechanism based on face reconstruction as part of a
protocol for evaluating the privacy leakage of soft biometric privacy-
enhancing approaches.

##  III Reversing protected deep face templates

Fig. 1 shows a conceptual overview depicting the key stages of the analysed
attack scenario. For the process of template inversion, an attacker does not
have full knowledge of the privacy-protection method used (or the involved
random seed), but is in possession of the protected face embeddings (labeled
as “stored template” on the bottom part in Fig. 1). We based our study on a
competitive privacy-protection approach, i.e. PE-MIU [8]. In this context of
the attack, the non-authorised subject is able to prepare and train an
existing DNN-based embedding inversion model to reconstruct face images from
unprotected face embeddings (top part in Fig. 1). The same network is
subsequently used to reconstruct face images from the protected face
embeddings. Note that the protection mechanism of PE-MIU depends on the block
size (i.e. 16,32,64,128) employed. Given an unprotected face embedding, a
random permutation is computed from a previously defined block size. In our
work, an irreversibility analysis is performed by comparing the face images
reconstructed from the PE-MIU-protected face embeddings with the respective
original faces. Finally, the similarity scores computed from the embeddings
extracted from the reconstructed face images and their originals are analysed.

###  III-A Soft-biometric privacy protection

PE-MIU [8] has been selected as the soft-biometric privacy-enhancement
approach since it offers underlying properties that can be easily executed in
a realistic attack scenario or adversarial model. As mentioned in Sect. II,
this method performs a transformation at the feature level preserving entirely
the subject’s identity without excluding the soft-biometric attribute.
Formally, PE-MIU receives a feature embedding of size $S$ containing floating
point-based values, denoted by $V\in\mathbb{R}^{S}$. $V$ is then divided into
$N=S/K$ blocks of size $K$ representing the minimum of information units that
can be transformed into $V$. Further, $N$ blocks of $V$ are randomly exchanged
or shuffled, resulting in a new feature vector $V^{\prime}\in\mathbb{R}^{S}$.
Note that the block size $K$ set to transform $V$ functions as a key that is
previously defined at application or system level. In other words, the method
preserves the same size of blocks and is able to create different transformed
vectors $V^{\prime}$ (i.e. different random permutations). Therefore, given
$K$, the maximum number of permutations in an authentication or enrolment step
would be limited to $N!$.

###  III-B Irreversibility analysis

The irreversibility analysis is depicted in the upper part of Fig. 1. In a
practical context, it is important to point out that the attacker utilises the
previously trained inversion model on features transformed by PE-MIU to
reconstruct face images. Subsequently, the attacker can derive the protected
soft-biometric attributes from reconstructed embeddings. It is worth
mentioning that the main motivation of this work is to estimate the privacy of
PE-MIU empirically taking a closer look at one of the privacy requirements for
biometric template protection schemes [12], i.e. irreversibility. As mentioned
in [11], the reconstruction of biometric samples similar to the original
captured samples from stored protected embeddings is a challenge. However, in
the case of PE-MIU, irreversibility may be improved by the permutation
complexity or number of shuffled blocks. Here, such permutations can lead to
good approximations of the original samples which may reveal soft-biometric
information. In an attack scenario, it should be feasible that non-mated
comparisons for the same attribute (e.g. gender) result in higher similarity
scores according to the broad homogeneity effect [20], and thus to frequent
false matches.

##  IV Experimental Setup

In this section, the implementation details as well as the databases used are
outlined in Sect. IV-A. The main configurations used in the irreversibility
analysis are described in Sect. IV-B.

###  IV-A Implementation details, databases and metrics

Two well-known face recognition models are utilised in this work, ArcFace [21]
and ElasticFace [22] providing face embeddings of 512-floating point values.
Protected embeddings are computed by the approach presented in Sect. III-A
using available open-source
code111https://github.com/pterhoer/PrivacyPreservingFaceRecognition. For the
embedding inversion, the approach proposed by [23] was selected. Note that
this model was previously trained on unprotected face embeddings extracted
from the FFHQ database [24] as explained in [23]. Subsequently, this trained
model is utilised to reconstruct face images from protected face embeddings
extracted from the LFW database. We consider a single sample per identity in
LFW that obtains the highest quality score computed by the SER-FIQ quality
estimator [25]. Also, the dataset was balanced w.r.t. the gender attribute
resulting in 2,942 unprotected and protected embeddings (identities for each
gender attribute), respectively, as it was done in [10]. Mated and non-mated
comparisons for protected and unprotected embeddings are computed following
the protocol View-2 of the LFW
database222https://gitlab.idiap.ch/bob/bob.db.lfw [26]. Cosine similarity was
used as a biometric comparator. Biometric performance is evaluated according
to the metrics defined in the international standard ISO/IEC19795-1:2021 [27]:
the detection error trade-off curve (DET) comparing the false non-match rate
(FNMR) with the false match rate (FMR) as well as equal error rate (EER).

(a) ArcFace (b) ElasticFace

Figure 2: Evaluation of the biometric performance.

###  IV-B Irreversibility setup

For the irreversibility analysis, different block-sizes $K$ are analysed, i.e.
$K=\\{128,64,32,16\\}$. In the first set of experiments, we explore the
biometric performance, as well as the reversibility success rate (RSR) for
different $K$ values (Sect. V-A). RSR is defined as the ratio of reconstructed
face images from protected embeddings that are accepted by the system at a
pre-defined decision threshold (e.g. at FMR=0.1%). In the experiments,
protected embeddings for a given block-size $K$ are generated (Sect. V-A).
Here, face image reconstruction from different permutations is performed.
Furthermore, gender prediction accuracy from reconstructed face embeddings is
reported. To that end, traditional Support Vector Machines (SVMs) classifiers
are trained on unprotected face embeddings corresponding to the original face
images in LFW using different kernels (Poly, RBF, and Sigmoid). Note that the
hyperparameters of both classifiers were set to basic configurations without
any optimisation and that are used in the training.

In the second set of experiments, we explore the RSR values for different
permutation complexities that can be computed by PE-MIU (Sect. V-B). The
possible number of permutations ($N!$) for each $K$, i.e. $4!$, $8!$, $16!$,
and $32!$, is therefore analysed for the considered face embedding extractors.
To that end, for a fixed $K$, PE-MIU is utilised so that the same permutation
is generated (shuffling the same number of blocks) for all identities. To
analyse the full or partial recovery of original face images, protected deep
face templates are reconstructed for each $K$ by varying the number of all
possible blocks to be shuffled (in this case, $P\in\\{2,\ldots,N$}). Then, the
RSR value for each $K$ and $P$ combination is reported.

TABLE I: Error rates(%) reported on thresholds fixed at the biometric system.

Model Protection EER FMR=0.1 FMR=1.0 FNMR RSR FNMR RSR ArcFace Unprotected
0.30 0.40 100.00 0.27 100.00 16 0.87 1.13 0.11 0.83 2.12 32 0.40 0.40 0.60
0.40 4.66 64 0.37 0.37 4.62 0.37 14.48 128 0.30 0.40 19.22 0.27 34.65
ElasticFace Unprotected 2.17 6.33 99.63 2.93 99.82 16 7.27 15.67 0.08 12.00
1.04 32 4.10 7.53 0.59 5.50 3.80 64 2.63 6.37 5.32 3.63 15.42 128 2.27 6.33
23.94 3.00 39.48

TABLE II: Gender prediction accuracy.

Model Protection SVM Poly RBF Sigmoid ArcFace Unprotected 0.81 $\pm$ 0.02 0.89
$\pm$ 0.01 0.85 $\pm$ 0.02 16 0.50 $\pm$ 0.03 0.50 $\pm$ 0.03 0.49 $\pm$ 0.02
32 0.50 $\pm$ 0.03 0.50 $\pm$ 0.03 0.49 $\pm$ 0.02 64 0.52 $\pm$ 0.02 0.53
$\pm$ 0.03 0.52 $\pm$ 0.03 128 0.55 $\pm$ 0.02 0.57 $\pm$ 0.02 0.56 $\pm$ 0.01
ElasticFace Unprotected 0.85 $\pm$ 0.02 0.88 $\pm$ 0.02 0.84 $\pm$ 0.02 16
0.52 $\pm$ 0.02 0.52 $\pm$ 0.03 0.51 $\pm$ 0.03 32 0.52 $\pm$ 0.02 0.55 $\pm$
0.02 0.54 $\pm$ 0.03 64 0.57 $\pm$ 0.01 0.58 $\pm$ 0.03 0.57 $\pm$ 0.03 128
0.64 $\pm$ 0.03 0.64 $\pm$ 0.02 0.63 $\pm$ 0.03

##  V Experimental Results

###  V-A Face image reconstruction from protected templates

First, the biometric performance between the unprotected face embeddings and
their corresponding protected embeddings for different values of $K$ is shown
in Fig. 2. In this experiment, different random permutations are generated for
each identity by PE-MIU. Note that the biometric performance improves with the
block-size ($K$) for both face recognition systems. However, higher values of
$K$ are assumed to lead to lower privacy protection [8].

Tab. I reports the RSRs of face images reconstructed from unprotected and
protected embeddings, respectively. As expected, the highest reversibility
chances are achieved for unprotected embeddings (RSRs $\geq 99.63\%$), while
for the protected embeddings, they are in the ranges of 0.08% to 39.48%
depending on $K$. Tab. II confirms the results presented in Tab. I: a low
probability of gender prediction among the different classifiers as a function
of $K$ is perceived.

TABLE III: Reversibility success rates - RSRs (in %) for different
combinations $K$ and $P$ on the two face recognition models at different
security thresholds.

Block size ($\mathbf{K}$) P ArcFace ElasticFace FMR=0.1% FMR=1.0% FMR=0.1%
FMR=1.0% 32 4 89.09 96.64 97.83 99.86 5 84.06 95.45 96.67 99.66 6 78.59 93.51
92.39 99.15 7 70.85 90.59 87.09 98.37 8 58.44 85.39 75.77 95.65 9 43.56 76.18
59.97 89.26 10 32.79 67.01 44.10 79.95 11 20.42 52.97 25.04 63.81 12 9.68
36.29 10.26 43.39 13 3.67 20.93 3.26 22.26 14 1.39 9.85 0.85 7.37 15 0.44 4.35
0.17 2.48 16 0.0 1.39 0.00 0.48 64 2 88.96 96.33 98.03 99.69 3 78.49 93.24
92.05 99.15 4 60.45 87.53 75.33 95.17 5 30.82 65.14 40.27 78.63 6 9.58 35.37
9.0 38.63 7 1.22 10.50 0.78 9.04 8 0.07 1.16 0.00 0.37 128 2 60.58 85.93 75.94
95.62 3 8.83 32.76 8.97 40.13 4 0.03 1.19 0.00 0.37

Figure 3: Examples of face images reconstructed from protected deep face
embeddings for different values $P$ (i.e. $2\leq N\leq 8$) using PE-MIU with
block size of $K=64$. Note that the value zero represents the face
reconstruction from the unprotected embedding, where no blocks in the vector
were swapped. Similarity scores are shown by comparing face images
reconstructed from protected face embeddings with their corresponding face
embeddings extracted from original face images, shown in the left column.

###  V-B Detailed analysis on the permutation complexity

We report in Tab. III the RSR value for reversing face images from face
embeddings in a scenario where the same permutation complexity (i.e. $P$) is
retained for all protected templates. Note that some configurations are
omitted where no change in the observed trend is expected, e.g. $K=16$. We
observe that the RSR decreases as $P$ increases for all values of $K$ across
different security thresholds. In particular, the probability of successfully
inverting a protected face embedding for different values of $K$ is above
85.39% and 95.17% for ArcFace and ElasticFace, respectively, at FMR = 1.0%,
when at least half of the blocks are shuffled. A reduction in RSR down to
58.44% and 75.33% is perceived for both recognition systems for a high safety
threshold, i.e. FMR=0.1%. Note the vulnerability of the ElasticFace face
recognition model in comparison to ArcFace: the highest RSRs yielded by
ElasticFace are about 99% and 97% for FMR=1.0% and FMR=0.1%, respectively,
while ArcFace achieves approximately 96% and 89% for the same thresholds. It
should be also observed that the results attained for lower $P$ values are
directly proportional to the $K$ values, i.e. RSR improves as $P$ and $K$
drop. This observation could be explained due to the minimum information
(entropy) that is managed across the blocks in the shuffling process which
offers “useful” guesses of the floating-point values in the face
reconstruction process. Fig 3 shows examples of faces reconstructed from
different permutations that also reveal gender information. It should be noted
that the analysis in terms of block size and the number of shuffled blocks
provided by the unique design of PE-MIU motivates further works (e.g.
information bottleneck models [17]) to explore which features should be
retained or suppressed to avoid privacy leakage in the considered embedding
space, in particular, in face embeddings of size 512.

###  V-C Discussion on random seeds

It is important to note that the transformations explained in Sect. III-A and
evaluated in Sect. V-A are usually generated from a pseudo-random number
generator stored in e.g. a physical device. Thus, for a specific application,
a transformation (i.e. a permutation) applied to a feature vector would be
computed through a seed produced by a pseudo-random number generator. In this
work, a scenario without seed specification has been evaluated (Sect. V-A).
However, it is worth noting that an adversary or attacker with full knowledge
of the system’s random seed could easily manipulate PE-MIU. In particular, the
attacker could reverse the permutation process by re-shuffling blocks based on
the known seed. Finally, it would be possible to reconstruct face images from
a fully inverted protected template. In this case, RSR values are expected to
be approximately 100%, similar to the reconstruction of face images from
unprotected embeddings presented in Sect V-A. In addition, an attacker could
launch a brute-force attack on the random seed used by the PE-MIU method.
Here, the attacker’s effort has to be taken into account, since a high
permutation complexity (i.e. a large number of shuffled blocks) would have to
be guessed.

##  VI Summary

In this work, it was empirically demonstrated that face embeddings protected
by soft-biometric privacy-enhancement schemes may be successfully
reconstructed, depending on certain parameters of the privacy-enhancement
approach (e.g. $K$ defined by PE-MIU and a permutation complexity $P$).
Experimental evaluations conducted on an open-source privacy-enhancement
method and a freely available database, i.e. LFW, showed that the
reversibility success rates (RSRs) are low in a scenario where the attacker
does not have any knowledge about employed random seeds. A detailed analysis
of the transformation complexity used by the protection mechanism showed that
RSR values can increase up to 95.62%. In experiments, face image were
reconstructed that achieved high similarity scores with respect to their
corresponding original face images for different state-of-the-art face
recognition models. Generally, high RSRs were obtained when transformations
exhibit low complexity or an attacker knows or is able to guess a random seed.
Future work will be focused on a deeper analysis that considers the
reversibility and reconstruction of protected face embeddings as part of a
joint training process across different soft-biometric privacy-enhancement
approaches.

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# Lower-vs-Higher Order Non-classicality of Photon-added Bell-type Entangled
Coherent States

Deepak<EMAIL_ADDRESS>Arpita Chatterjee<EMAIL_ADDRESS>Department of Mathematics, J. C. Bose University of Science and Technology,
YMCA, Faridabad 121006, India

###### Abstract

We compare the lower and higher order non-classicality of a class of the
photon-added Bell-type entangled coherent states (PBECS) got from Bell-type
entangled coherent states using creation operators. We obtained lower and
higher order criteria namely Mandel’s $Q_{m}^{l}$, antibunching $d_{h}^{l-1}$,
Subpoissioning photon statistics $D_{h}(l-1)$ and Squeezing $S(l)$ for the
states obtained. Further we observe that first three criteria does not gives
non-classicality for any state and higher order criteria gives very high
positive values for all values of parameters. Also the fourth or last
criterion $S(l)$ gives non-classicality for lower order as well as higher
order.

###### keywords:

Creation and annihilation operators, Two mode coherent state, Entanglement
etc.

††journal: Conference proceedings

##  1 INTRODUCTION

Since Einstein et al. [1], who created a two-particle state that was highly
entangled both in position and momentum space, entanglement has played a
significant role in quantum information processing and quantum computing. The
presence of a significant class of entangled states violates Bell-type
inequalities [2], which suggests that no local theory can explain their
existence. Entanglement in continuous variables has been rapidly developed
from both a theoretical and an experimental standpoint in the recent few
years, promising to be more compact and efficient in both coding and
processing quantum information [3]. A two-mode continuous-variable state is
commonly referred to as an entangled coherent state (ECS) [4]. A variety of
studies of ECSs have revealed their quantum nonlocality [5]. Quantum
teleportation [6], quantum computation [7], entanglement purification [8],
quantum error corrections [9], and other jobs have been discovered to benefit
from the ECS. Furthermore, various theoretical techniques for generating ECS
in cavity fields have been developed [10].

A class of nonclassical states formed by stimulating specific states, on the
other hand, has gotten a lot of attention. Agarwal and Tara initially proposed
photon-added coherent states (PACS) [11] or excited coherent states [12],
which are the outcome of successive elementary one-photon excitations of a
coherent state. In ref. [13], the nonclassical quasi-probability distribution
and amplitude of the PACS are addressed in detail. Using parametric down-
conversion, the Zavatta [14] and Kalamidas [15] groups created a single PACS
and a two PACS, respectively. Many authors have already presented photon-added
compressed states [16], photon-added even and (or) odd coherent states [17],
modified photon-added coherent states [18], photon-added thermal states [19],
and so on.

We now introduce different class of quantum states obtained in section 2.
Again in section 3 we discuss the non-classicalities of different states and
in the last section 4 we gives the conclusion.

##  2 Quantum States of Interest

In this section we consider the four different states from entanglement of
coherent states as discribed below;

$\displaystyle\ket{\psi}_{1}=N_{1}(\ket{\gamma,\gamma}+\ket{-\gamma,-\gamma})$
$\displaystyle\ket{\psi}_{2}=N_{2}(\ket{\gamma,\gamma}-\ket{-\gamma,-\gamma})$
(1)
$\displaystyle\ket{\psi}_{3}=N_{3}(\ket{\gamma,-\gamma}+\ket{-\gamma,\gamma})$
$\displaystyle\ket{\psi}_{4}=N_{4}(\ket{\gamma,-\gamma}-\ket{-\gamma,\gamma})$

Where $\gamma$ is the coherent state amplitude and $N_{i}$’s are the
normalization constants for $i=1,2,3,4$ and given by [20]

$\displaystyle
N_{1}^{-2}=N_{3}^{-2}=2m!n![L_{m}(-|\gamma|^{2})L_{n}(-|\gamma|^{2})+e^{-4|\gamma|^{2}}L_{m}(|\gamma|^{2})L_{n}(|\gamma|^{2})]$
$\displaystyle
N_{2}^{-2}=N_{4}^{-2}=2m!n![L_{m}(-|\gamma|^{2})L_{n}(-|\gamma|^{2})-e^{-4|\gamma|^{2}}L_{m}(|\gamma|^{2})L_{n}(|\gamma|^{2})]$
(2)

Also the expectation values of operator $a^{\dagger p}a^{q}$ with respect to
all the four states are given by;

$\bra{\alpha,\beta}a^{p+m}b^{p+n}a^{\dagger q+m}b^{\dagger
q+n}\ket{\gamma,\delta}=\left\\{\begin{array}[]{ll}(p+m)!(p+n)!(\alpha\beta)^{*q-p}L_{p+m}^{q-p}(-\alpha^{*}\gamma)L_{p+n}^{q-p}(-\beta^{*}\delta)&\mbox{if
}p\leq q\\\
(q+m)!(q+n)!(\gamma\delta)^{p-q}L_{q+m}^{p-q}(-\alpha^{*}\gamma)L_{q+n}^{p-q}(-\beta^{*}\delta)&\mbox{if
}q\leq p\end{array}\right.$ (3)

Case-I, $p\leq q$

$\langle a^{p}a^{\dagger
q}\rangle_{1,2}=N_{1,2}^{2}(p+m)!(p+n)!z^{*2(q-p)}2[L_{p+m}^{q-p}(-|z|^{2})L_{p+n}^{q-p}(-|z|^{2})\pm
L_{p+m}^{q-p}(|z|^{2})L_{p+n}^{q-p}(|z|^{2})]$ (4)

Case-II, $p\geq q$

$\langle a^{p}a^{\dagger
q}\rangle_{1,2}=N_{1,2}^{2}(q+m)!(q+n)!z^{2(p-q)}2[L_{q+m}^{p-q}(-|z|^{2})L_{q+n}^{p-q}(-|z|^{2})\pm
L_{q+m}^{p-q}(|z|^{2})L_{q+n}^{p-q}(|z|^{2})]$ (5)

Case-I, $p\leq q$

$\langle a^{p}a^{\dagger
q}\rangle_{3,4}=N_{3,4}^{2}(p+m)!(p+n)!z^{*2(q-p)}2(-1)^{q-p}[L_{p+m}^{q-p}(-|z|^{2})L_{p+n}^{q-p}(-|z|^{2})\pm
L_{p+m}^{q-p}(|z|^{2})L_{p+n}^{q-p}(|z|^{2})]$ (6)

Case-II, $p\geq q$

$\langle a^{p}a^{\dagger
q}\rangle_{3,4}=N_{3,4}^{2}(q+m)!(q+n)!z^{2(p-q)}2(-1)^{q-p}[L_{q+m}^{p-q}(-|z|^{2})L_{q+n}^{p-q}(-|z|^{2})\pm
L_{q+m}^{p-q}(|z|^{2})L_{q+n}^{p-q}(|z|^{2})]$ (7)

Now it is clear from equations (4) to (7) that if $p=q$ then states
$\ket{\psi}_{3}$ and $\ket{\psi}_{4}$ are exactly same as states
$\ket{\psi}_{1}$ and $\ket{\psi}_{2}$ respectively.

##  3 Non-Classicality Criteria

In this section we find out the most important and useful criteria as (i)
Mandel’s $Q$ parameter(mqp), (ii) Higher and lower order antibunching
$d_{h}^{l-1}$ (hloa), (iii) higher and lower order sub-poisoning photon
statistics $D_{h}(l-1)$(hlsps) and (iv) higher and lower order squeezing
$S(l)$(hls).

###  3.1 Mandel’s $Q$ Parameter

The Mandel’s parameter $Q_{M}$ [21] illustrates the nonclassicality of a
quantum state through its photon number distribution. The introductory
definition of $Q_{M}$ can be generalized to an arbitrary order $l$ as [22]

$\displaystyle Q_{M}^{(l)}$ $\displaystyle=$
$\displaystyle\frac{\braket{(\Delta{\mathcal{N}})^{l}}}{\braket{a^{\dagger}a}}-1,$
(8)

where $\Delta{\mathcal{N}}\,=\,a^{\dagger}a-\braket{a^{\dagger}a}$ is the
dispersion in the number operator $\mathcal{N}=a^{\dagger}a$. Using the
identity [22]

$\displaystyle\braket{(\Delta{\mathcal{N}})^{l}}=\sum_{k=0}^{l}{l\choose
k}(-1)^{k}\langle(a^{\dagger}a)^{l-k}\rangle{\langle a^{\dagger}a\rangle}^{k}$

and [23]

$(a^{\dagger}a)^{r}=\sum_{n=0}^{r}S_{r}^{(n)}a^{\dagger n}a^{n},$

where $S_{r}^{(n)}$ is the Stirling number of second kind [24]

$S_{r}^{(n)}=\frac{1}{n!}\sum_{j=0}^{n}(-1)^{n-j}{n\choose j}j^{r},$

the higher-order Mandel parameter $Q_{M}^{(l)}$ can be evaluated explicitly
upto order $l$. The negativity of $Q_{M}^{(2)}$ signifies the negativity of
the conventional Mandlel’s $Q_{M}$. All expectations in (8) have been
calculated with help of (4) to (7). Also the function $Q_{M}^{(l)}$ as a
function of $\gamma$ in fig. 1

Figure 1: Mandel’s $Q$ as a function of $\gamma$ for lower order($l=2$) and
higher order($l=3$) for $\ket{\psi}_{1}$ state (a) $m=1,\,n=3$, (c)
$m=2,\,n=6$ and for $\ket{\psi}_{2}$ state (b) $m=1,\,n=3$, (d) $m=2,\,n=6$

Also as stated in section 2 the behaviour of $Q_{M}^{(l)}$ for states
$\ket{\psi}_{3}$ and $\ket{\psi}_{4}$ is same as states $\ket{\psi}_{1}$ and
$\ket{\psi}_{2}$ respectively. Also from the fig. 1 it can be notified that
with increase in $m$ and $n$ the values of parameter increase for both the
states $\ket{\psi}_{1}$ and $\ket{\psi}_{2}$ and value of higher order
parameter is more positive than lower order and due to positive value of
$Q_{M}^{(l)}$ for different parameters we can’t classified all the four states
as they are classical or not. Further values of $Q_{M}^{(l)}$ is less for
state $\ket{\psi}_{1}$ than $\ket{\psi}_{2}$.

###  3.2 Higher and Lower-order antibunching

Different well-known criteria for detecting higher-order nonclassicality can
be expressed in compact forms for the class of states described in (2). In
this subsection, we focus on higher and lower-order antibunching. The concept
of hloa, by using the theory of majorization, was introduced by Lee [25].
Later it was modified by Pathak and Gracia [26] to provide a clear physical
meaning and a more simple expression. The $(l-1)$-th order antibunching is
observed in a quantum state if it satisfies the following condition:

$d(l-1)=\langle a^{\dagger l}a^{l}\rangle-{\langle
a^{\dagger}a\rangle}^{l}\,\,<\,\,0$ (9)

Since the negativity of $d(l-1)$ indicates that the probability of photons
coming bunched is less compared to that of coming independently, therefore the
nonclassicality feature (9) typifies how suitable the state $\ket{\psi}$ is as
a single photon resource. Also the hloa parameter can be plotted in fig. 2

Figure 2: hloa $d_{h}^{(l-1)}$ as a function of $\gamma$ for lower
order($l=2$) and higher order($l=3$) for $\ket{\psi}_{1}$ state (a)
$m=1,\,n=3$, (c) $m=2,\,n=6$ and for $\ket{\psi}_{2}$ state (b) $m=1,\,n=3$,
(d) $m=2,\,n=6$

Also as stated in section 2 the behaviour of $d_{h}^{(l-1)}$ for states
$\ket{\psi}_{3}$ and $\ket{\psi}_{4}$ is same as states $\ket{\psi}_{1}$ and
$\ket{\psi}_{2}$ respectively. Also from the fig. 2 it can be notified that
with increase in $m$ and $n$ the values of parameter increase for both the
states $\ket{\psi}_{1}$ and $\ket{\psi}_{2}$ and value of higher order
parameter is more positive than lower order and due to positive value of
$d_{h}^{(l-1)}$ for different parameters we can’t classified all the four
states as they are classical or not. Further values of $d_{h}^{(l-1)}$ is less
for state $\ket{\psi}_{1}$ than $\ket{\psi}_{2}$.

###  3.3 Higher and Lower-order sub-Poissonian photon statistics

Higher-order sub-Poissonian photon statistics is an important feature that
affirms the existence of higher-order nonclassicality of a radiation field.
The lower-order antibunching and sub-Poissonian photon statistics are closely
connected as the presence of later ensures the possibility of observing the
first one. But recently these two phenomena are proved to be independent of
each other [27, 28]. It is also reported that the higher-order antibunching
and sub-Poissonian photon statistics can exist irrespective of whether their
lower-order counterparts exist or not [29].

The generalized criteria for observing the $(l-1)$-th order sub-Poissonian
photon statistics (for which
$\langle(\Delta\mathcal{N})^{l}\rangle<\langle(\Delta\mathcal{N})^{l}\rangle_{\ket{\mathrm{Poissonian}}}$
is given by [30]

$\mathcal{D}_{h}(l-1)=\sum_{e=0}^{l}\sum_{f=1}^{e}S_{2}(e,f)^{l}C_{e}(-1)^{e}d(f-1){\langle
a^{\dagger}a\rangle}^{l-e}\,\,<\,\,0$ (10)

where $S_{2}(e,f)=\sum_{r=0}^{f}{{}^{f}C_{r}}(-1)^{r}r^{e}$ is the Stirling
number of second kind, ${}^{l}C_{e}$ is the usual binomial coefficient. The
analytic expression of hlsps for the superposed state can be obtained by
substituting (2) in (10). Also the obtained parameter is plotted in fig. 3

Figure 3: hloa $D_{h}(l-1)$ as a function of $\gamma$ for lower order($l=2$)
and higher order($l=3$) for $\ket{\psi}_{1}$ state (a) $m=1,\,n=3$, (c)
$m=2,\,n=6$ and for $\ket{\psi}_{2}$ state (b) $m=1,\,n=3$, (d) $m=2,\,n=6$

Also as stated in section 2 the behaviour of $d_{h}^{(l-1)}$ for states
$\ket{\psi}_{3}$ and $\ket{\psi}_{4}$ is same as states $\ket{\psi}_{1}$ and
$\ket{\psi}_{2}$ respectively. Also from the fig. 3 it can be notified that
with increase in $m$ and $n$ the values of parameter increase for both the
states $\ket{\psi}_{1}$ and $\ket{\psi}_{2}$ and value of higher order
parameter is more positive than lower order and due to positive value of
$D_{h}^{(}l-1)$ for different parameters we can’t classified all the four
states as they are classical or not. Further values of $D_{h}^{(}l-1)$ is less
for state $\ket{\psi}_{1}$ than $\ket{\psi}_{2}$.

###  3.4 Higher and Lower-order squeezing

Coherent state, being the minimum uncertainty state, the product of the
fluctuations in two field quadratures becomes minimum and the fluctuations in
each quadrature become equal. For lower-order squeezing, the variance in one
of the field quadrature (defined by a linear combination of annihilation and
creation operators) reduces below the coherent state limit at the cost of
enhanced fluctuation in the other quadrature. The idea of higher-order
squeezing is originated by the pioneering work of Hong and Mandel [31].
According to them, the $l$-th order higher-order squeezing ($l>2$) is obtained
while the $l$-th order moment of a field quadrature operator is less than the
corresponding coherent state value. Hong-Mandel’s criteria for higher-order
squeezing can be described by the following inequality

$S(l)=\frac{\langle(\Delta
X)^{l}\rangle-{\left(\frac{1}{2}\right)}_{\left(\frac{l}{2}\right)}}{{\left(\frac{1}{2}\right)}_{\left(\frac{l}{2}\right)}}\,\,<\,\,0,$
(11)

where $(x)_{l}$ is the conventional Pochhammer symbol and the quadrature
variable is defined as $X=\frac{1}{\sqrt{2}}(a+a^{\dagger})$. The inequality
in (11) can also be rewritten as

$\langle(\Delta
X)^{l}\rangle\,\,<\,\,{\left(\frac{1}{2}\right)}_{\left(\frac{l}{2}\right)}=\frac{1}{2^{\frac{l}{2}}}(l-1)!!,$
(12)

with

$\begin{array}[]{rcl}\langle(\Delta
X)^{l}\rangle&=&\sum_{r=0}^{l}\sum_{i=0}^{\frac{r}{2}}\sum_{k=0}^{r-2i}(-1)^{r}\frac{1}{2^{\frac{1}{2}}}(2i-1)!^{2i}\\\
&&C_{k}^{l}C_{r}^{r}C_{2i}\langle a^{\dagger}+a\rangle^{l-r}\langle a^{\dagger
k}a^{r-2i-k}\rangle,\end{array}$ (13)

where $l$ is an even number and

$\displaystyle n!!=\left\\{\begin{array}[]{lll}&n(n-2)(n-4)\ldots 4.2&\mbox{if
$n$ is even},\\\ \\\ &n(n-2)(n-4)\ldots 3.1&\mbox{if $n$ is
odd},\end{array}\right.$

The analytic expression for the Hong-Mandel type HOS can be obtained by using
(2) in (11)-(13) and it is plotted in fig 4.

Figure 4: hloa $D_{h}(l-1)$ as a function of $\gamma$ for lower order($l=2$)
and higher order($l=3$) with $m=1,\,n=3$ for (a)$\ket{\psi}_{1}$,
(b)$\ket{\psi}_{2}$, (c)$\ket{\psi}_{3}$, (d)$\ket{\psi}_{1}4$ states and
$m=2,\,n=6$ (a)$\ket{\psi}_{1}$, (b)$\ket{\psi}_{2}$, (c)$\ket{\psi}_{3}$,
(d)$\ket{\psi}_{4}$ states

From the fig. 4 it can be notified that with increase in $m$ and $n$ the
values of parameter increase for three states except $\ket{\psi}_{3}$ in both
positive and negative directions and value of higher order parameter is more
positive(negative) than lower order and due to negative value of $S(l)$ for
different parameters we can classified the three states except
$\ket{\psi}_{3}$ as non-classical states. Further values of $S(l)$ for state
$\ket{\psi}_{3}$ is positive for higher order but negative for lower order so
we can conclude that higher or criterion $S(l)$ gives the better result for
three states except $\ket{\psi}_{3}$ while lower order for state
$\ket{\psi}_{3}$.

##  4 Conclusion

From the above discussion we can conclude that all the four states stated
above are non-classical. First three criteria does not gives any result
regarding the non-classical while in fourth that is in squeezing parameter
three states except $\ket{\psi}_{3}$ non-classicality is identified by higher
order($l=4$) parameter while for $\ket{\psi}_{3}$ state non-classicality is
identified by lower order($l=2$) criteria.

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