# The IDM and THDMa - current constraints and future prospects

Division of Theoretical Physics, Ruder Boskovic Institute, Zagreb, Croatia
E-mail

###### Abstract:

We discuss two models that extend the Standard Model by additional particles,
leading to additional scalar states, and also providing dark matter
candidates. We briefly review the current constraints and comment on
production cross sections at present and future collider facilities.
RBI-ThPhys-2021-36

##  1 Introduction

We discuss two new physics models that extend the Standard Model (SM) particle
sector by additional scalars and also both provide a dark matter candidate.
The models are confronted with current theoretical and experimental
constraints. From the theoretical side, these include the minimization of the
vacuum as well as the requirement of vacuum stability and positivity. We also
require perturbative unitarity to hold, and perturbativity of the couplings at
the electroweak scale.

Experimental bounds include the agreement with current measurements of the
properties of the 125 GeV resonance discovered by the LHC experiments, as well
as agreement with the null-results from searches for additional particles at
current or past colliders. We also confront the models with bounds from
electroweak precision observables via $S,\,T,\,U$ [1, 2, 3], B-physics
observables
$\left(B\,\rightarrow\,X_{s}\,\gamma,\,B_{s}\,\rightarrow\,\mu^{+}\,\mu^{-},\,\Delta
M_{s}\right)$, as well as agreement with astrophysical observables (relic
density and direct detection bounds). We use a combination of private and
public tools in these analyses, where the latter include HiggsBounds [4],
HiggsSignals [5], 2HDMC [6], SPheno [7], Sarah [8], micrOMEGAs [9, 10], and
MadDM [11]. Experimental numbers are taken from [12, 13] for electroweak
precision observables, [14] for $B_{s}\,\rightarrow\,\mu^{+}\,\mu^{-}$, [15]
for $\Delta M_{s}$ and [16] and [17] for relic density and direct detection,
respectively. Bounds from $B\,\rightarrow\,X_{s}\gamma$ are implemented using
a fit function from [18, 19]. Predictions for production cross sections shown
here have been obtained using Madgraph5 [20].

##  2 Inert Doublet Model

The Inert Doublet Model (IDM) [21, 22, 23] is a two-Higgs-doublet Model (THDM)
that obeys an exact $Z_{2}$ symmetry, under which the SM-like particles are
even, while the novel scalars from the second (dark) doublet are odd. The
exactness of the symmetry induces directly stability of the lightest new
scalar state, providing a dark matter candidate. The model has been widely
discussed in the literature, see e.g. [24, 25, 26, 27]. The dark sector
contains four new particles: $H$, $A$ and $H^{\pm}$. After electroweak
symmetry breaking, the model contains 5 free parameters which we chose to be
$M_{H},M_{A},M_{H^{\pm}},\lambda_{2},\lambda_{345}\,\equiv\,\lambda_{3}+\lambda_{4}+\lambda_{5}$.

In [25] (see also [28]), benchmark points were suggested which cover dark
scalar masses up to 1 TeV and are in agreement with the current constraints
discussed above. We here focus on the investigation of these points at various
current and future colliders, concentrating on pair-production processes

${p\,p\,\rightarrow\,HA,\,HH^{\pm},\,AH^{\pm},H^{+}H^{-}};\;\;\;{\ell^{+}\,\ell^{-}\,\rightarrow\,HA,\,H^{+}H^{-}}$

for various collider options, where $\ell\,\in\,\left\\{e,\,\mu\right\\}$. For
some of these, we also considered VBF-type production processes, where the
above are augmented by 2 additional jets for $pp$ and neutrinos for $\ell\ell$
machines. The different collider specifications considered here are given in
table 1, where more details can be found in [27]. Figure 1 shows the cross-
sections for various collider options and center-of-mass energies, as a
function of the sum of the two pair-produced particles. The dashed lines
signify the minimal cross sections needed for a generation of 1000 events
assuming collider center-of-mass energies and luminosities as specified in
table 1. In summary, we can state that already at the HL-LHC, all considered
processes should be accessible (via the criterium above) up to a mass sum of 2
TeV apart from $AA$ pair-production. The latter process depends on the
parameter
$\bar{\lambda}_{345}\,\equiv\,\lambda_{3}+\lambda_{4}-\lambda_{5}\,=\,\lambda_{345}-2\,\frac{M_{H}^{2}-M_{A}^{2}}{v^{2}}$
which determines the $hAA$ coupling, i.e. also mass-differences between $A$
and $H$ are important. Furthermore, $\lambda_{345}$ is highly constrained by
direct detection [24, 29, 26, 25, 27]. For $AA$ production, therefore, the
mass scale, defined as the sum of produced particle masses, can be up to 600
GeV at the HL-LHC and 2 TeVat the FCC-hh. At lepton colliders, on the other
hand, the VBF channel is needed in order to access this process, where mass
scales are around 600 GeV/ 1.4 TeV/ 2 TeV for CLIC with a center-of-mass
energy of 3 TeV/ a muon-collider with a center-of-mass energy of 10 TeV/ 30
TeV, respectively.

collider | cm energy [TeV] | $\int\mathcal{L}$ | 1000 events [fb]
---|---|---|---
HL-LHC | 13/ 14 | $3\,{\rm ab}^{-1}$ | 0.33
HE-LHC | 27 | $15\,{\rm ab}^{-1}$ | 0.07
FCC-hh | 100 | $20\,{\rm ab}^{-1}$ | 0.05
ee | 3 | $5\,{\rm ab}^{-1}$ | 0.2
$\mu\mu$ | 10 | $10\,{\rm ab}^{-1}$ | 0.1
$\mu\mu$ | 30 | $90\,{\rm ab}^{-1}$ | 0.01
Table 1:  Parameter used for the different collider options discussed in the
text. More details, including references, can be found in [27].

Figure 1:  Predictions for production cross sections for various processes and
collider options. Top left: Predictions for various pair-production cross
sections for a $pp$ collider at 13 TeV, as a function of the mass sum of the
produced particles. Top right: Same for various center-of-mass energies.
Bottom left: VBF-type production of $AA$ and $H^{+}\,H^{-}$ at various center-
of-mass energies for $pp$ colliders. Bottom right: Same for $\mu^{+}\mu^{-}$
colliders. Taken from [27]. The lines correspond to the cross-sections
required to prodce at least 1000 events using the respective design
luminosity.

##  3 THDMa

The THDMa is a type II two-Higgs-doublet model that is extended by an
additional pseudoscalar mixing with the ”standard” pseudoscalar $A$ of the
THDM. In the gauge-eigenbasis, the additional scalar serves as a portal to the
dark sector, with a fermionic dark matter candidate, denoted by $\chi$. This
model has e.g. been discussed in [30, 31, 32, 33, 34, 35, 36].

The model features the following particles in the scalar and dark matter
sector: ${h,\,H,\,H^{\pm},}\,{a,}\,{A,}\,{{\chi}}$. It depends on in total 12
additional parameters

$\displaystyle{v,\,m_{h},\,m_{H},}\,{m_{a},}\,{m_{A},\,m_{H^{\pm}},}\,{m_{\chi}};\;{\cos\left(\beta-\alpha\right),\,\tan\beta,}\,{\sin\theta;\;y_{\chi},}\,{\lambda_{3},}\,{\lambda_{P_{1}},\,\lambda_{P_{2}}},$

where $v$ and either $m_{h}$ or $m_{H}$ are fixed by current measurements in
the electroweak sector. We refer the reader to the above references for more
details.

We here report on results of a scan that allows all of the above novel
parameters float in specific predefined ranges [36]. In such a scenario, it is
not always straightforward to display bounds from specific constraints in
2-dimensional planes. Two examples where this is possible are given in figure
2. In the first plot, we show bounds in the
$\left(m_{H^{\pm}},\,\tan\beta\right)$ plane from B-physics observables. The
result is similar in a simple THDM, and shows that in general low masses
$m_{H^{\pm}}\lesssim\,800\,{\rm GeV}$ as well as values $\tan\beta\lesssim\,1$
are excluded. The second plot features the relic density as a function of the
mass difference $m_{a}-2\,m_{\chi}$. Here, a behaviour can be observed that is
typical in many models with dark matter candidates: in the region where this
mass difference remains small, relic density annihilates sufficiently to stay
below the observed relic density bound. On the other hand, too large
differences lead to values $\Omega\,h_{c}\,\gtrsim\,0.12$ and therefore are
forbidden from dark matter considerations. Finally, we can investigate which
cross-section values would still be feasible for points that fulfill all
constraints [36] at $e^{+}e^{-}$ colliders. We are specifically interested in
signatures that include missing energy and therefore do not exist in a THDM
without a portal to the dark sector. Due to alignment, processes like
$e^{+}e^{-}\,\rightarrow\,hA,ha$ are suppressed, which makes
$e^{+}e^{-}\,\rightarrow\,HA,Ha$ the most interesting channel featuring novel
signatures. Due to the interplay of B-physics and electroweak constraints,
such points typically have mass scales $\gtrsim\,1\,{\rm TeV}$. We therefore
consider production cross sections for an $e^{+}e^{-}$ collider with a center-
of-mass energy of 3 TeV. The corresponding production cross sections are
displayed in figure 3, where we give predictions for $t\,\bar{t}\,t\,\bar{t}$
and $t\,\bar{t}+\not{E}$ final states using a factorized approach. We see that
there is a non-negligible number of points where the second channel is
dominant. A ”best” point with a large rate for $t\,\bar{t}+\not{E}_{\perp}$
has been presented in [36].

Figure 2:  Left: Bounds on the $\left(m_{H^{\pm}},\,\tan\beta\right)$ plane
from B-physics observables, implemented via the SPheno [7]/ Sarah [8]
interface, and compared to experimental bounds [14, 15]. The contour for low
$\left(m_{H^{\pm},\,\tan\beta}\right)$ values stems from [18, 19]. Right: Dark
matter constraints in the THDMa model. Right: Dark matter relic density as a
function of $m_{a}-2\,m_{\chi}$, with $m_{\chi}$ defining the color coding.
The typical resonance-enhanced relic density annihilation is clearly visible.
Figures taken from [36].

Figure 3:  Production cross sections for $t\bar{t}t\bar{t}$ (x-axis) and
$t\bar{t}+\not{E}$ (y-axis) final state in a factorized approach, for an
$e^{+}e^{-}$ collider with a 3 TeVcenter-of-mass energy. Left: mediated via
$HA$, right: mediated via $HA$ and $Ha$ intermediate states. Color coding
refers to $m_{H}+m_{A}$ (left) and $M_{H}+0.5\times\,\left(m_{A}+m_{a}\right)$
(right). Figures taken from [36].

##  4 Conclusion and Outlook

In this work, we presented two models that extend the particle content of the
SM and also provide at least one dark matter candidate. We have presented
production cross sections for various standard pair-production modes within
these models; for the IDM, we have given an estimate of mass range
reachibility based on a simple counting criterium. A more dedicated
investigation of the corresponding signatures, including background simulation
and cut optimization, is in the line of future work.

## Acknowledgements

This research was supported in parts by the National Science Centre, Poland,
the HARMONIA project under contract UMO-2015/18/M/ST2/00518 (2016-2021), OPUS
project under contract UMO-2017/25/B/ST2/00496 (2018-2021), as well as the
COST action CA16201 - Particleface.

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# Towards a General Deep Feature Extractor for Facial Expression Recognition

Liam Schoneveld  Powder AI Research
Paris, France
<EMAIL_ADDRESS>Alice Othmani  Université Paris-Est, LISSI, UPEC
Vitry sur Seine, France
<EMAIL_ADDRESS>
###### Abstract

The human face conveys a significant amount of information. Through facial
expressions, the face is able to communicate numerous sentiments without the
need for verbalisation. Visual emotion recognition has been extensively
studied. Recently several end-to-end trained deep neural networks have been
proposed for this task. However, such models often lack generalisation ability
across datasets. In this paper, we propose the Deep Facial Expression Vector
ExtractoR (DeepFEVER), a new deep learning-based approach that learns a visual
feature extractor general enough to be applied to any other facial emotion
recognition task or dataset. DeepFEVER outperforms state-of-the-art results on
the AffectNet and Google Facial Expression Comparison datasets. DeepFEVER’s
extracted features also generalise extremely well to other datasets – even
those unseen during training – namely, the Real-World Affective Faces (RAF)
dataset.

###### Index Terms:

Facial Expression Recognition, Visual emotion recognition, Deep learning,
model generalisation, Knowledge distillation

##  I Introduction

Emotional Intelligence in Human-Computer Interaction has attracted increasing
attention from researchers in multidisciplinary fields including psychology,
computer vision, neuroscience, artificial intelligence, and related
disciplines. Humans also tend to interact with computers in a face-to-face
manner. As such, human facial expressions present an important opportunity to
better link humans and computers. Designing interfaces able to understand
human expressions and emotions can thus improve Human-Computer Interaction
(HCI) overall.

Recently, deep neural network-based approaches have become more popular for
facial expression recognition (FER). These approaches have significantly
improved the performance over more traditional approaches to computer vision
[1, 3, 2] due to their capacity to automatically learn both low and high level
descriptors from facial images without manual feature engineering.

Despite considerable advancements in the field, Facial Expression Recognition
(FER) has remained a challenging task. Most existing methods still lack
generalisation ability across datasets acquired under different conditions.
The majority of modern approaches in the emotion recognition literature either
learn or fine-tune an end-to-end network that can only be used for a specific
dataset, and/or use more general features as input to a more basic model.

To overcome this limitation, it is proposed in this paper to learn an
independent feature extractor for images of faces, specialised for facial
expressions, that could be employed for any downstream FER task or dataset.
This approach achieves its generalisation ability by training on multiple
labeled FER datasets, and by employing knowledge distillation (specifically,
_self-distillation_), alongside additional unlabeled data for FER. The
proposed visual facial expression embedding network is described in Section
II. Section III describes the experimental setting and discusses the results
obtained on three FER datasets. Finally, Section IV concludes the paper and
discusses future work.

##  II Proposed Approach

This paper proposes a general and independent deep facial expression feature
extractor called Deep Facial Expression Vector ExtractoR (DeepFEVER).
DeepFEVER is a convolutional neural network (CNN) trained using: (1) two
labeled FER datasets, (2) an additional unlabeled dataset and (3) the
technique known as knowledge distillation [4].

Not only does DeepFEVER outperform state-of-the-art results on both the
datasets it was trained on, it also generalises very well; DeepFEVER achieves
highly competitive results on an additional FER dataset, unseen during
training, without any fine-tuning.

DeepFEVER is trained using knowledge distillation: a two step process whereby
a teacher network is trained on the task of interest, and then a (typically
smaller) student network is trained on predictions made by the teacher.
Specifically in this work, we leverage the benefits of self-distillation,
wherein the student network has the same size as (or at least, is not smaller
than) the teacher network. Self-distillation is used in this work to improve
the performance of the facial expression embedding network, as well as to
provide more flexibility over the final model architecture.

The training procedure for DeepFEVER consists of two phases: the teacher model
training phase and the student model training phase. In the teacher model
training phase, a FaceNet [6] is retrained simultaneously on two different
visual FER datasets (Section II-A). In the student model training phase, we
again train on these two datasets (Section II-B). However, the starting point
for the student network can be any CNN architecture (we opt for a DenseNet
[5]). The student is also trained on an additional unlabeled dataset, whose
training ‘labels’ are provided by the teacher network.

The output of DeepFEVER is a compact vector of dimension $D_{\text{face}}$.
This was set to $D_{\text{face}}=256$ as such generally performed better in
experiments. During training this vector serves as a shared input for various
output heads that each produce predictions for each of the training tasks.
After training, this vector can be extracted from face images to provide
features for any downstream FER task. More details about the teacher and the
student networks follow.

###  II-A The teacher network

The proposed teacher network starts from a pre-trained FaceNet [6]. As FaceNet
is for person re-identification, the teacher network is further trained using
two more datasets to learn specialised features for FER. These datasets are
AffectNet [7] and Google Facial Expression Comparison (FEC) [8]. Further
details about the datasets are provided in Section III-A.

The teacher model’s architecture (Fig. 1) is almost identical to the model
proposed in [8], the only difference is that an additional output head is
added for the AffectNet loss. Thus, following [8], a pre-trained FaceNet111In
this, we used a FaceNet pre-trained on the VGGFace2 dataset, as we found
performance to be slightly improved. The pre-trained FaceNet model
architecture and weights were obtained from
https://github.com/timesler/facenet-pytorch. is taken up until the Inception
4e block. This is followed by a 1x1 convolution and a series of five untrained
DenseNet [5] blocks. After this, another 1x1 convolution followed by global
average pooling reduces this representation to a single $D_{\text{face}}$
dimensional vector. After pooling, two independent linear transformations
serve as output heads. These heads take the $D_{\text{face}}$-dimensional
facial expression representation vector as input and make separate predictions
for the AffectNet and FEC tasks. A 32-dimensional embedding is used for the
FEC triplets task, while an 8-dimensional output head produces class logits
for AffectNet (which has 8 classes). The teacher network training procedure is
detailed in Algorithm 1, while implementation details are provided in Section
III-B.

To improve the regularisation effects of self-distillation through model
ensembling, two teacher networks are trained, and their outputs are
concatenated to serve as distillation targets (see Section II-B for details).
The only difference between the two teachers are the random seeds used for
initialisation, and penultimate layer dimensionalities: one teacher network
was trained with $D_{\text{face}}=256$, the other with $D_{\text{face}}=128$.

Figure 1: The facial expression recognition neural network architecture,
before distillation (i.e. the _teacher network_). Faces are detected and
cropped using MTCNN[10]. The resulting 140x140 RGB images are then fed to an
Inception Resnet V1 until the Inception 4e block. This is followed by a 1x1
convolution layer (1x1 Conv), batch normalisation (BN) and a ReLU activation.
Then, five DenseNet blocks are applied. Finally, another set of 1x1 Conv, BN
and ReLU are applied. The output is then averaged over the spatial dimensions,
resulting in a vector of size $D_{\text{face}}$ (in the figure,
$D_{\text{face}}=128$). Two separate linear layers then give the final model
outputs – a vector for the Google FEC triplets task, and class logits for
AffectNet. The model is trained to minimize both the AffectNet and Google FEC
losses simultaneously. The numbers over each block represent the output shape
of the tensor after applying that block.

for  _iteration in range( $N$)_ do

$(\mathbf{X}_{\text{FEC}},\mathbf{y}_{\text{FEC}})$ $\leftarrow$ batch of
Google FEC triplets and labels

$(\mathbf{X}_{\text{Aff}},\mathbf{y}_{\text{Aff}})$ $\leftarrow$ batch of
AffectNet images and class labels

$\mathbf{e}_{\text{FEC}}\leftarrow f_{\Theta}(\mathbf{X}_{\text{FEC}})$
$\triangleright$ Face embeddings for FEC images

$\mathbf{e}_{\text{Aff}}\leftarrow f_{\Theta}(\mathbf{X}_{\text{Aff}})$
$\triangleright$ Face embeddings for AffectNet images

$\mathbf{v}_{\text{FEC}}\leftarrow g_{\phi}(\mathbf{e}_{\text{FEC}})$
$\triangleright$ Predict vectors for triplet loss

$\mathbf{p}_{\text{Aff}}\leftarrow h_{\theta}(\mathbf{e}_{\text{Aff}})$
$\triangleright$ Predict class probabilities for AffectNet

$L_{\text{FEC}}=\texttt{triplet\\_loss}(\mathbf{v}_{\text{FEC}},\mathbf{y}_{\text{FEC}})$

$L_{\text{Aff}}=\texttt{cross\\_entropy\\_loss}(\mathbf{p}_{\text{Aff}},\mathbf{y}_{\text{Aff}})$

$L=L_{\text{FEC}}+\alpha*L_{\text{Aff}}$ $\triangleright$ Total loss for
training step

Obtain all gradients $\Delta_{\text{all}}=(\frac{\partial
L}{\partial\Theta},\frac{\partial L}{\partial\phi},\frac{\partial
L}{\partial\theta})$

$(\Theta,\phi,\theta)\leftarrow\texttt{SGD}(\Delta_{\text{all}}$)
$\triangleright$ Update feature extractor and output heads’ parameters
simultaneously

end for

Algorithm 1 DeepFEVER: training the teacher network. Given feature extractor
network $f_{\Theta}$, Google FEC output head $g_{\phi}$, AffectNet output head
$h_{\theta}$, number of training steps $N$, AffectNet loss weight $\alpha$.

###  II-B Student network

The student network is a DenseNet201 pre-trained on Imagenet.222We use the
DenseNet implementation and pre-trained Imagenet weights provided in the
torchvision Python package. The student network training procedure is
essentially the same as described in Algorithm 1, except that we additionally
sample batches of unlabeled data from an internal dataset, which we refer to
as PowderFaces (See Section III-A). The sampled batches of face crops from the
Google FEC, AffectNet, and PowderFaces datasets are passed through the two
teacher networks. Each of the two teacher networks produces predictions for
the Google FEC task (32-dimensional) and AffectNet class logits
(8-dimensional). These four vectors (i.e., two vectors from two teacher
networks) are individually L2-normalised. The four normalised vectors are then
concatenated, producing one long vector of dimension 80. A knowledge
distillation loss (we specifically use Relational Knowledge Distillation [9]
for the loss function) is then calculated by comparing the output of a third
output head in the student network to this 80-dimensional target vector. This
knowledge distillation loss is then added to the standard AffectNet and Google
FEC losses, which are calculated as per the teacher network training
procedure. Implementation details for the student network are provided in
Section III-B.

##  III Experiments and Results

###  III-A Datasets

Two datasets are used to train the teacher network:

  * •

AffectNet [7], which consists of around 440,000 in-the-wild face crop images,
each of which is human-annotated into one of eight facial expression
categories (Neutral, Happy, Sad, Surprise, Fear, Disgust, Anger and Contempt).

  * •

Google Facial Expression Comparison (FEC) [8], which consists of around
700,000 triplets of unique face crop images. Annotations denoting the most
similar pair of face expressions in each triplet are provided. The goal is to
train a model that places the similar pair closer together in a learned
embedding space.

An additional, unlabeled dataset is used to train the student network via
knowledge distillation: the PowderFaces dataset. This dataset was created by
downloading approximately 20,000 short, publicly-available videos from various
online sources such as YouTube. To increase the frequency of faces in the
dataset, specific search terms and topics were used when searching for videos,
such as ‘podcast’, ‘interview’, or ‘monologue’. MTCNN face detection [10] was
then applied to the extracted frames from those videos, producing
approximately 1 million individual face crops.

To evaluate the generalisation ability of DeepFEVER, the Real-World Affective
Faces Database (RAF) [11] is used. RAF consists of 12,271 training face images
and 3,068 validation face images, each annotated into one of seven classes
(Surprise, Fear, Disgust, Happiness, Sadness, Anger and Neutral).

###  III-B Implementation details

Implementation details of the teacher network: The entire network is retrained
(i.e., none of the original FaceNet model’s weights are frozen). The SGD
optimizer is used with a learning rate (LR) of $0.005$ and Nesterov momentum
of $0.9$. No learning rate scheduling is used – rather training was simply
stopped after 13 epochs, based on validation performance 333Here an epoch
refers to a full pass over the FEC triplets dataset. To obtain AffectNet
samples, we simply continually loop over the AffectNet dataset in the
background, reshuffling the dataset at the end of each background loop.. A
batch size of 64 for both the triplets and AffectNet datasets is used. To
bring the AffectNet cross-entropy loss’ magnitude more in line with that of
the triplet loss, it is multiplied by a factor of $\alpha=0.1$ during
training. Dropout of $0.1$ is applied to the $D_{\text{face}}$-dimensional
expression vector before passing it to the output heads.

Implementation details of the student network: For the knowledge distillation
loss, Relational Knowledge Distillation (RKD) (Park et al. [2019]) is used.
The RKD distance-wise loss is multiplied by 25, and the angle-wise loss is
multiplied by 50, to bring their magnitude up to a similar level as per the
AffectNet and Google FEC loss components. The implementation details of the
student network are exactly the same as per the teacher network (Section
II-A), except that it was trained for longer (18 epochs), uses smaller batch
sizes (36 Google FEC triplets, 16 AffectNet images and 16 PowderFaces images
per iteration) and has a penultimate layer dimension of $D_{\text{face}}=256$
(like one of the two teacher networks), with dropout of $0.2$ instead of $0.1$

Implementation details of the linear evaluation on RAF: To evaluate DeepFEVER
on RAF, the $D_{\text{face}}$-dimensional feature vectors are extracted using
the student model for every face in the RAF dataset. A logistic regression
model [17] is then trained (with regularisation parameter $C$ set to $10000$)
on the features extracted from the training set. The accuracy of this model is
then calculated by predicting the validation set labels, using only the
extracted feature vectors as input.

###  III-C Performance of DeepFEVER for visual FER

We evaluate DeepFEVER on the standard evaluation subsets of the two datasets
(AffectNet and Google FEC) used in training, plus a third dataset, unseen
during training (RAF):

  1. 1.

AffectNet: for AffectNet, which requires classifying faces into eight discrete
facial expression classes, a logistic regression model is trained on the
features extracted by the student network for the entire AffectNet training
set.444When training this logistic regression, we re-weight the classes in the
AffectNet training set to have equal representation, as per the validation
set. For eight-class classification, DeepFEVER achieves state-of-the-art
results on the AffectNet validation set, with an accuracy of $61.6\%$ (Table
I). Some papers report seven-class accuracy by excluding images with the
contempt class from the training and validation sets. DeepFEVER also achieves
state-of-the-art results on this seven-class task, with an accuracy of 65.4%.

  2. 2.

Google FEC: following [8], models are evaluated using triplet accuracy on the
Google FEC test set. As shown in Table II, DeepFEVER outperforms state-of-the-
art methods, with an accuracy of $86.5\%$ .

  3. 3.

RAF: A logistic regression trained on DeepFEVER face vectors outperforms all
but one of the previous state-of-the-art results on RAF, including those which
train directly on the RAF dataset (Table III).

TABLE I: Accuracy of DeepFEVER on the AffectNet validation set (7 or 8 classes) compared to existing state-of-the-art methods Methods | 8-class | 7-class
---|---|---
CNNs and BOVW + local SVM [12] | 59.6% |
Pyramid with Super Resolution [15] | 60.7% | 63.8%
PAENet [16] |  | 65.3%
DeepFEVER (Teacher model) | 61.3% | 65.4%
DeepFEVER (Student, no distillation) | 58.8% | 62.6%
DeepFEVER (Distilled student, no PowderFaces) | 61.1% | 65.2%
DeepFEVER (Distilled student) | 61.6% |  65.4%
TABLE II: Triplet accuracy of DeepFEVER on the Google FEC test set compared to the existing state-of-the-art approach Methods | Accuracy
---|---
Fine-tuned FaceNet [8] | 81.8%
DeepFEVER (Teacher model) | 84.5%
DeepFEVER (Student, no distillation) | 85.0%
DeepFEVER (Distilled student, no PowderFaces) | 86.4%
DeepFEVER (Distilled student) | 86.5%
TABLE III: Performances of DeepFEVER on the RAF ‘basic’ test set (7 classes) compared to existing state-of-the-art methods Methods | Accuracy
---|---
IPA2LT [13] | 86.8%
RAN-ResNet18 [14] | 86.9%
PSN [15] |  89.0%
DeepFEVER (Distilled student, no fine-tuning) | 87.4%

###  III-D Ablation study

To experimentally verify the importance of the different components of our
approach, an ablation study is performed (Tables I and II). Training the
student model architecture without the distillation loss component revealed
the importance of distillation: without such, performance dropped
substantially: to 58.8% on AffectNet (eight-class) and 85% on Google FEC.

Similarly, to determine the importance of the unlabeled PowderFaces dataset,
the student model was trained with distillation, but without the additional
distillation targets provided by using this unlabeled data. The results
suggest that the additional unlabeled data is not so important: accuracy on
AffectNet dropped only slightly to 61.1%, and performance on Google FEC
reduced by only 0.1% to 86.4%. Other works successfully applying similar
approaches to self-distillation used an unlabeled dataset on the order of 300
times larger than the labeled dataset [18]. In our case the ratio of unlabeled
data to labeled data is closer to one to one. We leave determining whether
larger volumes of unlabeled data can further improve DeepFEVER’s performance
to future work.

##  IV Conclusion and Future work

This research proposes a new approach to learning a deep facial expression
feature extractor, with high generalisation ability across datasets. The
proposed DeepFEVER network outperforms state-of-the-art approaches on the
datasets it was trained on, and achieves highly competitive results the RAF
dataset without any fine-tuning. In future work, we plan to explore scaling up
the unlabeled dataset for self-distillation and quantisation of the learned
facial embedding space.

## Acknowledgment

This work was supported by Powder, a Paris-based start-up creating the social
network for gamers. https://powder.gg/

## References

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# Fluctuations in Extractable Work and Bounds on the Charging Power of Quantum
Batteries

Shang-Yung Wang<EMAIL_ADDRESS>Department of Physics, Tamkang
University, New Taipei City 25137, Taiwan

###### Abstract

Motivated by a recent disagreement about the claim that fluctuations in the
free energy operator bound the charging power of a quantum battery, we present
a critical analysis of the original derivation. The analysis shows that the
above claim does not hold for both closed- and open-system dynamics. Our
results indicate that the free energy operator is not a consistent quantifying
operator for the work content of a charging quantum battery.

##  I Introduction

Batteries are indispensable to the modern technological world. An immense
amount of effort has been poured into researching the optimal batteries that
have ultralarge capacity, ultracompact size, ultrafast charging, and ultraslow
aging [1, 2]. Recently, the prospect of harvesting energy from the environment
and performing work using quantum machines at the nanoscale has prompted a
rise in the study of quantum batteries [3, 4, 5, 6, 7, 8, 12, 9, 10, 11, 13,
14, 15, 16, 17, 18] (see Ref. [19] for a review). In general, a quantum
battery [3] is a quantum system that can store work extracted from another
quantum system (the energy source system) and release the energy to power
other quantum machines on demand. In light of recent advances in quantum
computing [20, 21], quantum batteries may be able to explore quantum phenomena
to improve charging [10], energy storage [11], and discharging [12]
capabilities.

Like classical batteries, an important figure of merit that characterizes the
performance of quantum batteries is charging power, that is, the rate at which
energy can be stored. The charging power is essentially determined by the
models and charging processes of quantum batteries. Currently, the study of
the charging power of quantum batteries are largely focused on achieving a
quantum advantage of phase coherences and nonlocal correlations to enhance the
charging and discharging processes. Various quantum battery models and
charging processes [4, 5, 6, 7, 8, 12, 9, 10, 11, 13, 14, 15] that utilize
collective operations on many copies of identical quantum batteries to speed
up the charging time have been studied in both theoretical and experimentally
realizable situations. Of particular interest are a practical model with $N$
two-level systems coupled to a single photon mode in a cavity (the
cavity–charger protocol) [6, 13, 14], and a theoretical model with $N$
spin-1/2 battery cells charged collectively by $M$ noninteracting spin-1/2
chargers through a general Heisenberg $XY$ interaction (the spin–charger
protocol) [15]. For the conventional cavity–charger protocol with single-
photon coupling, a collective enhancement in the charging power that scales
like $\sqrt{N}$ [6] and a collective suppression of the energy locked by
correlations relative to the extractable energy that scales as $1/N$ [13] have
been reported for $N\gg 1$. The unconventional cavity–charger protocol with
dominant two-photon coupling shows a more significant collective enhancement
in the charging power that scales like $N$ [14], a clear advantage over the
single-photon coupling case. Moreover, the newly proposed spin–charger
protocol has several advantages over the conventional cavity–charger protocol,
including a high capacity of energy storage, a superior power law in
collective charging, and a full charging for equal number of charger spins and
battery cells [15].

From a thermodynamic perspective, an equally important but less explored
problem is to determine the bound on the charging power of quantum batteries
regardless of the detail of models and charging processes. This bound would be
an analog of the Carnot efficiency that sets the maximum efficiency of a
thermal engine operating between two heat baths at different temperatures. It
is hoped that the bound may shed light on the design of quantum batteries with
ultrahigh charging power.

Initial research along this line has only started recently. In Ref. [16], a
geometric approach is used to derive a bound on the charging power of a
_closed_ quantum battery under a _unitary_ charging process. The bound is
related to the product of the energy fluctuation of the battery and the square
root of the classical Fisher information in the energy space of the battery.
Later, in another study [17], the connection between the charging power of an
_open_ quantum battery and the fluctuation of a “free energy operator” is
studied. It is found that the bound on charging power is proportional to the
fluctuation of the free energy operator for both closed- and open-system
dynamics. The authors thus conclude that fluctuations in the free energy
operator bound the charging power of a quantum battery. In a follow-up study
[18], an “activity operator” akin to the free energy operator is introduced
and a tighter bound proportional to the fluctuation of the activity operator
is derived for open quantum batteries.

Unfortunately, it has been shown [22] that the conclusion of Ref. [17] does
not hold for open-system dynamics. The authors of Ref. [17] acknowledge the
mistakes [23] but, by deriving a modified bound on the charging power, they
reassert “it holds that ‘fluctuations in the free energy operator bound the
charging power of a quantum battery,’ as claimed” . Since closed- and open-
system analyses are physically equivalent approaches to studying the dynamics
of an open quantum battery, the question as to whether the validity of the
conclusion of Ref. [17] holds for both closed- and open-system dynamics
remains unanswered.

In this article, we clarify the situation by critically examining the
derivation in Refs. [17, 23].  In doing so, we find a few mistakes and obtain
the correct bounds on the charging power for both closed- and open-system
dynamics. Our results show that the bounds are _not_ directly proportional to
the fluctuation of the free energy operator; instead there is an _additive_
contribution related to the correlation between the free energy operator and
the operator(s) generating the _nonunitary_ evolution of the battery.
Therefore, the validity of the claim that fluctuations in the free energy
operator bound the charging power of a quantum battery is called into
question.

The rest of this article is organized as follows. In Section II, we introduce
the free energy operator and charging power of a quantum battery defined in
Ref. [17]. The corrected closed- and open-system analyses are presented in
Sections III and IV, respectively. In Section V, we conclude by discussing the
implications of the correct bounds.

##  II Free Energy Operator

The central quantity in the analysis of Refs. [17, 23] is the free energy
operator $\mathcal{F}$. Specifically, for a battery $\mathcal{W}$ with
Hamiltonian $H_{\mathcal{W}}$ and in the state $\rho_{\mathcal{W}}$ it is
defined with respect to a _reference_ heat bath at inverse temperature $\beta$
as

$\mathcal{F}\coloneqq H_{\mathcal{W}}+\beta^{-1}\log\rho_{\mathcal{W}}.$ (1)

Note that $\mathcal{F}$ is a Hermitian operator but does not correspond to a
physical observable [17] because it depends on the state $\rho_{\mathcal{W}}$
of the battery. Physical observables are Hermitian operators but the converse
is not true. A physical observable (e.g., Hamiltonian) obtains its physical
meaning through measurement outcomes in terms of its eigenvalues and
corresponding eigenstates. The density matrix (or state operator) of a quantum
system is a Hermitian operator but not a physical observable. It is defined
through state preparation instead of measurement, and different state
preparations may give rise to the same density matrix. The rationale behind
the definition of $\mathcal{F}$ given in Equation (1) is evident; the
expectation value
$\langle\mathcal{F}\rangle_{\mathcal{W}}=\operatorname{Tr}(\rho_{\mathcal{W}}\mathcal{F})$
of $\mathcal{F}$ in the battery state $\rho_{\mathcal{W}}$ gives the
_nonequilibrium_ free energy
$F(\rho_{\mathcal{W}})=U(\rho_{\mathcal{W}})-\beta^{-1}S(\rho_{\mathcal{W}})$
of the battery in the state $\rho_{\mathcal{W}}$ with respect to a reference
inverse temperature $\beta$. Here,
$U(\rho_{\mathcal{W}})=\operatorname{Tr}(\rho_{\mathcal{W}}H_{\mathcal{W}})$
and
$S(\rho_{\mathcal{W}})=-\operatorname{Tr}(\rho_{\mathcal{W}}\log\rho_{\mathcal{W}})$
are the average energy and von Neumann entropy of the battery in the state
$\rho_{\mathcal{W}}$, respectively. By construction, the expectation value of
$\mathcal{F}$ quantifies the _maximum_ extractable work of the battery in the
presence of heat transfer. That is, the maximum amount of work that can be
extracted on average from a quantum battery in the state $\rho_{\mathcal{W}}$
and in _thermal contact_ with a heat bath at inverse temperature $\beta$ is
given by [24, 25]

$W_{\mathrm{max}}=\beta^{-1}S(\rho_{\mathcal{W}}\|\tau_{\beta})=F(\rho_{\mathcal{W}})-F(\tau_{\beta}).$
(2)

Here,
$S(\rho\|\sigma)=\operatorname{Tr}(\rho\log\rho)-\operatorname{Tr}(\rho\log\sigma)$
is the quantum relative entropy of the states $\rho$ to $\sigma$,
$\tau_{\beta}=e^{-\beta H_{\mathcal{W}}}/\operatorname{Tr}(e^{-\beta
H_{\mathcal{W}}})$ is the thermal state of the battery at inverse temperature
$\beta$, and $F(\tau_{\beta})=-\beta^{-1}\log\operatorname{Tr}(e^{-\beta
H_{\mathcal{W}}})$ is the equilibrium free energy of the battery at inverse
temperature $\beta$. It is noted that a similar but distinct quantity related
to $W_{\mathrm{max}}$ is the maximum amount of work that can be extracted on
average from a closed driven quantum system under cyclic unitary processes
acting on the system, also known as the ergotropy of the system [26]. An
extraction process is cyclic if the external time-dependent driving field
vanishes at the beginning and at the end of the process. Since unitary
processes preserve entropy, ergotropy corresponds to the maximum extractable
work from a system in the absence of heat transfer.

The authors of Refs. [17, 23] posit that the charging power $P(t)$ of a
quantum battery is defined as

$P(t)\coloneqq\frac{dW_{\mathrm{max}}}{dt}=\frac{d\langle\mathcal{F}\rangle_{\mathcal{W}}}{dt}.$
(3)

We stress that, since there is no consensus on the notion of work in the
quantum regime, the definition of charging power given in Equation (3) is
_not_ without problems. In particular, it follows from Equation (2) that
$W_{\mathrm{max}}$ is a state function in that its value does not depend on
the process that takes the battery to the state $\rho_{\mathcal{W}}$. Thus the
change in $W_{\mathrm{max}}$ is a state function, which implies that the
charging power $P(t)$ defined in Equation (3) is also a state function. This,
however, is in contradiction with the fundamental concept that work and power
are in general process-dependent quantities. We will first focus on examining
the derivation in Refs. [17, 23] and come back to related conceptual issues in
the discussion at the end of this article.

To avoid repetition, in what follows we will skip most of the part of the
derivation in Refs. [17, 23] that does not contain mistakes, and include only
the part that needs to be corrected.

##  III Closed-System Analysis

In the closed-system analysis, the battery is considered to be a subsystem of
a closed quantum system. The closed system $\mathcal{SBAW}$ consists of the
energy source $\mathcal{S}$, bath $\mathcal{B}$, ancilla $\mathcal{A}$, and
battery $\mathcal{W}$, and follows unitary time evolution. The starting point
of our analysis is the charging power $P(t)$ of the battery given by (see
Equation (8) of Ref. [17])

$P(t)=-i\operatorname{Tr}([\rho,\mathcal{F}\otimes\mathds{1}_{\mathcal{SBA}}]V),$
(4)

where $\rho$ is the full state of the closed system $\mathcal{SBAW}$ and $V$
is the interaction Hamiltonian between the battery and the source system
$\mathcal{S}$, bath $\mathcal{B}$, and ancilla $\mathcal{A}$. Following Ref.
[17], we define
$\delta\mathcal{F}=\mathcal{F}-\langle\mathcal{F}\rangle_{\mathcal{W}}$ and
$\delta V=V-\langle V\rangle$, where $\langle V\rangle=\operatorname{Tr}(\rho
V)$. After some algebra, we obtain

$|P(t)|^{2}=|\operatorname{Tr}(\rho[\delta\mathcal{F},\delta V])|^{2},$ (5)

where for notational simplicity we will use the shorthand notation
$\delta\mathcal{F}=\delta\mathcal{F}\otimes\mathds{1}_{\mathcal{SBA}}$ in the
remaining of this section. Note that Equation (5) is an equality instead of an
inequality in Equation (9) of Ref. [17]. It is convenient to rewrite Equation
(5) as

$|P(t)|^{2}=|\operatorname{Tr}(\sqrt{\rho}\,\delta\mathcal{F}\delta
V\sqrt{\rho}-\sqrt{\rho}\,\delta V\delta\mathcal{F}\sqrt{\rho})|^{2},$ (6)

where we have used the fact that $\rho$ is a positive operator. We note that
since $\sqrt{\rho}\,\delta\mathcal{F}\delta V\sqrt{\rho}$ and
$\sqrt{\rho}\,\delta V\delta\mathcal{F}\sqrt{\rho}$ are Hermitian conjugates
of each other, it follows that
$\operatorname{Tr}(\sqrt{\rho}\,\delta\mathcal{F}\delta V\sqrt{\rho})$ and
$\operatorname{Tr}(\sqrt{\rho}\,\delta V\delta\mathcal{F}\sqrt{\rho})$ are
complex conjugates of each other. As a matter of fact, this is the utmost
important point that is missed in the analysis of Ref. [17]. With this point
in mind, we can rewrite Equation (6) as

$|P(t)|^{2}=|\operatorname{Tr}(\sqrt{\rho}\,\delta\mathcal{F}\delta
V\sqrt{\rho})|^{2}+|\operatorname{Tr}(\sqrt{\rho}\,\delta
V\delta\mathcal{F}\sqrt{\rho})|^{2}-2\operatorname{Re}([\operatorname{Tr}(\rho\,\delta\mathcal{F}\delta
V)]^{2}),$ (7)

where Re denotes the real part.

To find the bound on $|P(t)|^{2}$, following Ref. [17], we use the fact that
for a positive operator $A$ and Hermitian operators $B$ and $C$, the Cauchy-
Schwarz inequality implies
$|\operatorname{Tr}(\sqrt{A}BC\sqrt{A})|^{2}\leq|\operatorname{Tr}(AB^{2})|\,|\operatorname{Tr}(AC^{2})|$.
Equation (7) then leads to

$\displaystyle|P(t)|^{2}$ $\displaystyle\leq$ $\displaystyle
2\bigl{(}[\operatorname{Tr}[\rho(\delta\mathcal{F})^{2}]\operatorname{Tr}[\rho(\delta
V)^{2}]-\operatorname{Re}([\operatorname{Tr}(\rho\,\delta\mathcal{F}\delta
V)]^{2})\bigl{)}$ (8) $\displaystyle=$ $\displaystyle
2\bigl{(}\sigma^{2}_{\mathcal{F}}\sigma^{2}_{V}-\operatorname{Re}[\operatorname{Cov}(\mathcal{F},V)^{2}]\bigr{)}.$

Here, $\sigma^{2}_{\mathcal{F}}$ is the variance of $\mathcal{F}$ in the
battery state $\rho_{\mathcal{W}}$, $\sigma^{2}_{V}$ is the variance of $V$ in
the full state $\rho$, and $\operatorname{Cov}(\mathcal{F},V)$ is the
covariance between $\mathcal{F}$ and $V$ in the full state $\rho$.
Specifically, we have

$\begin{gathered}\sigma^{2}_{\mathcal{F}}=\langle\mathcal{F}^{2}\rangle_{\mathcal{W}}-\langle\mathcal{F}\rangle^{2}_{\mathcal{W}},\quad\sigma^{2}_{V}=\langle
V^{2}\rangle-\langle V\rangle^{2},\\\
\operatorname{Cov}(\mathcal{F},V)=\langle(\mathcal{F}\otimes\mathds{1}_{\mathcal{SBA}})V\rangle-\langle\mathcal{F}\rangle_{\mathcal{W}}\langle
V\rangle.\end{gathered}$ (9)

Moreover, the inequality
$\sigma^{2}_{\mathcal{F}}\sigma^{2}_{V}\geq|\operatorname{Cov}(\mathcal{F},V)|^{2}$
implies
$\sigma^{2}_{\mathcal{F}}\sigma^{2}_{V}-\operatorname{Re}[\operatorname{Cov}(\mathcal{F},V)^{2}]\geq
0$ as it should be. Equation (8) is a slightly corrected expression for
Equations (9) and (12) of Ref. [17]. As a result, the charging power of the
battery is bounded _not only_ by the fluctuation of the free energy operator
_but also_ by the covariance between the free energy operator and the
interaction Hamiltonian, which generates the nonunitary evolution of the
battery. Therefore, the conclusion of Ref. [17] that fluctuations in the free
energy operator bound the charging power of a quantum battery does not hold
for closed-system dynamics.

Finally, we consider the case in which the battery state is an instantaneous
eigenstate of the free energy operator. Suppose
$\rho_{\mathcal{W}}=|j\rangle\langle j|$ and
$\mathcal{F}|j\rangle=f_{j}|j\rangle$ with $f_{j}$ being the real eigenvalue;
we obtain $\sigma^{2}_{\mathcal{F}}=\operatorname{Cov}(\mathcal{F},V)=0$,
which implies $P(t)=0$. However, even though the total system is initially in
a product state with the battery in an eigenstate of $\mathcal{F}$, the
interaction $V$ will make the battery entangled with the other subsystems,
giving rise to a mixed battery state. It is conceivable that there exist
entangled full states $\rho$ and mixed battery states
$\rho_{\mathcal{W}}=\operatorname{Tr}_{\mathcal{SBA}}(\rho)$ with nonzero
$\sigma_{\mathcal{F}}$ and $\operatorname{Cov}(\mathcal{F},V)$ but $P(t)=0$.
We therefore stress that under the assumption of a general charging process
with $\sigma_{V}\neq 0$, the battery state
$\rho_{\mathcal{W}}=|j\rangle\langle j|$ is only a _sufficient_ condition for
the battery to have a vanishing charging power, as opposed to a _sufficient
and necessary_ condition in the original invalid analysis of Ref. [17].

##  IV Open-System Analysis

In the open-system analysis the battery is treated as an open quantum system
per se. The original analysis [17] is not valid [22] and an improved one is
presented in Ref. [23]. Following Ref. [23], we express the state of the
battery $\rho_{\mathcal{W}}$ in its instantaneous eigenbasis
$\\{|\alpha\rangle\\}$ as
$\rho_{\mathcal{W}}=\sum_{\alpha}p_{\alpha}|\alpha\rangle\langle\alpha|$,
where $p_{\alpha}\geq 0$ are the eigenvalues. The charging power $P(t)$ of the
battery is given by (see the first two equalities in Equation (2) of Ref.
[23])

$P(t)=\operatorname{Tr}\Bigl{(}\frac{d\rho_{\mathcal{W}}}{dt}\mathcal{F}\Bigr{)}=\sum_{\alpha,\beta}\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle},$
(10)

where
$\delta\mathcal{F}_{\alpha\beta}=\langle\alpha|\delta\mathcal{F}|\beta\rangle$.
It is convenient to rewrite Equation (10) as

$P(t)=\sum_{\begin{subarray}{c}\alpha,\beta\\\
p_{\alpha}+p_{\beta}>0\end{subarray}}\frac{\sqrt{p_{\alpha}+p_{\beta}}}{\sqrt{p_{\alpha}+p_{\beta}}}\,\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle}+\sum_{\begin{subarray}{c}\alpha,\beta\\\
p_{\alpha}+p_{\beta}=0\end{subarray}}\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle},$
(11)

where we have separated the terms in the summation into those with
$p_{\alpha}+p_{\beta}>0$ and $p_{\alpha}+p_{\beta}=0$. We stress that since
$\rho_{\mathcal{W}}$ is in general _not_ of full rank, there exist eigenstates
$|\alpha\rangle$ of $\rho_{\mathcal{W}}$ with $p_{\alpha}=0$. Hence the
additional factor of
$\sqrt{p_{\alpha}+p_{\beta}}/\sqrt{p_{\alpha}+p_{\beta}}=1$ in the first
summation is valid if and only if $p_{\alpha}+p_{\beta}>0$. As a matter of
fact, this is the utmost important point that is missed in the derivation of
Ref. [23].

To find the bound on $|P(t)|$, we first use the triangle inequality to obtain

$|P(t)|\leq\Biggl{|}\sideset{}{{}^{\prime}}{\sum}_{\alpha,\beta}\frac{\sqrt{p_{\alpha}+p_{\beta}}}{\sqrt{p_{\alpha}+p_{\beta}}}\,\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle}\Biggr{|}+\Biggl{|}\sideset{}{{}^{\prime\prime}}{\sum}_{\alpha,\beta}\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle}\Biggr{|},$
(12)

where we have introduced the shorthand notation
$\sideset{}{{}^{\prime}}{\textstyle\sum_{\alpha,\beta}}=\sum_{\alpha,\beta:p_{\alpha}+p_{\beta}>0}$
and
$\sideset{}{{}^{\prime\prime}}{\textstyle\sum_{\alpha,\beta}}=\sum_{\alpha,\beta:p_{\alpha}+p_{\beta}=0}$.
Applying the Cauchy–Schwarz inequality to the first term of the bound (12), we
then obtain

$\displaystyle|P(t)|$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{\alpha,\beta}(p_{\alpha}+p_{\beta})\,|\delta\mathcal{F}_{\alpha\beta}|^{2}}\sqrt{2\sideset{}{{}^{\prime}}{\sum}_{\alpha,\beta}\frac{|\langle\beta|\frac{d\rho_{\mathcal{W}}}{dt}|\alpha\rangle|^{2}}{p_{\alpha}+p_{\beta}}}+\Biggl{|}\sideset{}{{}^{\prime\prime}}{\sum}_{\alpha,\beta}\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle}\Biggr{|}$
(13) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}\sum_{\alpha,\beta}(p_{\alpha}+p_{\beta})\,|\delta\mathcal{F}_{\alpha\beta}|^{2}}\sqrt{2\sideset{}{{}^{\prime}}{\sum}_{\alpha,\beta}\frac{|\langle\beta|\frac{d\rho_{\mathcal{W}}}{dt}|\alpha\rangle|^{2}}{p_{\alpha}+p_{\beta}}}+\Biggl{|}\sideset{}{{}^{\prime\prime}}{\sum}_{\alpha,\beta}\delta\mathcal{F}_{\alpha\beta}\Bigl{\langle}\beta\Bigl{|}\frac{d\rho_{\mathcal{W}}}{dt}\Bigr{|}\alpha\Bigr{\rangle}\Biggr{|},$

where in the last equality we have used the identity
$\sum_{\alpha,\beta}(p_{\alpha}+p_{\beta})|\delta\mathcal{F}_{\alpha\beta}|^{2}=\sideset{}{{}^{\prime}}{\textstyle\sum_{\alpha,\beta}}(p_{\alpha}+p_{\beta})|\delta\mathcal{F}_{\alpha\beta}|^{2}$.
The first factor in the first term of the bound (13) is the fluctuation
$\sigma_{\mathcal{F}}$ of $\mathcal{F}$ in the battery state
$\rho_{\mathcal{W}}$. The second factor is a _finite_ quantity, as opposed to
the one obtained in Ref. [23] that can become _divergent_. It is the square
root of the quantum Fisher information $I_{Q}(t)$ of the state
$\rho_{\mathcal{W}}$ with $t$ being the parameter [27]. Specifically, we have

$I_{Q}(t)=2\sideset{}{{}^{\prime}}{\sum}_{\alpha,\beta}\frac{|\langle\beta|\frac{d\rho_{\mathcal{W}}}{dt}|\alpha\rangle|^{2}}{p_{\alpha}+p_{\beta}}.$
(14)

It is important to note that in the statement that follows Equation (88) of
Ref. [27], it is stressed that if the sum extends over all $\alpha$ and
$\beta$, including those with vanishing $p_{\alpha}$ and $p_{\beta}$, then the
quantum Fisher information becomes ill-defined and one has to find an
alternative way to define it [28, 29]. Equation (13) is a slightly corrected
expression for Equations (2) and (4) of Ref. [23]. In the weak-coupling and
Markovian limit, the explicit expressions for $I_{Q}(t)$ and the second term
of the bound (13) can be found using the Lindblad equation for
$\rho_{\mathcal{W}}$ given by Equation (14) of Ref. [17]. For the purpose of
our discussion, it however suffices to observe that the second term of the
bound is related to the correlations between $\mathcal{F}$ and the Lindblad
operators $L_{j}$ in the _kernel_ of the battery state $\rho_{\mathcal{W}}$.
As a result, the charging power of the battery is bounded _not only_ by the
fluctuation of the free energy operator _but also_ by the correlation between
the free energy operator and the Lindblad operators, which generate the
nonunitary evolution of the battery. Therefore, the conclusion of Ref. [23]
that fluctuations in the free energy operator bound the charging power of a
quantum battery does not hold for open-system dynamics.

To illustrate the last point, we consider the case in which the battery state
is an instantaneous eigenstate of the free energy operator. Write
$\delta\mathcal{F}=\sum_{m}w_{m}|m\rangle\langle m|$ with $w_{m}$ the real
eigenvalues and $|m\rangle$ the corresponding instantaneous eigenstates, and
suppose $\rho_{\mathcal{W}}=|n\rangle\langle n|$ with $|n\rangle$ an
instantaneous eigenstate of $\mathcal{F}$. For
$\rho_{\mathcal{W}}=|n\rangle\langle n|$ the condition
$\langle\delta\mathcal{F}\rangle_{\mathcal{W}}=0$ implies $w_{n}=0$; thus we
obtain $\sigma_{\mathcal{F}}=0$ and

$|P(t)|\leq\sum_{j}\gamma_{j}\sum_{\begin{subarray}{c}m\\\ m\neq
n\end{subarray}}|w_{m}|\,|\langle m|L_{j}|n\rangle|^{2},$ (15)

where $\gamma_{j}>0$ are relaxation rates. The bound (15) evidently agrees
with Equation (8) of Ref. [22], which can be written in our notation as

$P(t_{0})=\sum_{j}\gamma_{j}\sum_{\begin{subarray}{c}m\\\ m\neq
n\end{subarray}}w_{m}\,|\langle m|L_{j}|n\rangle|^{2}\neq 0.$ (16)

Contrary to the result of Ref. [23], for open-system dynamics the contribution
to the nonzero charging power of a battery in an eigenstate of $\mathcal{F}$
comes from the correlations between $\mathcal{F}$ and $L_{j}$ in the kernel of
$\rho_{\mathcal{W}}=|n\rangle\langle n|$ instead of the divergence of an _ill-
defined_ quantum Fisher information of the state.

##  V Discussion

We have shown that the claim that fluctuations in the free energy operator
bound the charging power of a quantum battery does not hold for both closed-
and open-system dynamics. In particular, a battery in an eigenstate of
$\mathcal{F}$ is only a sufficient condition for the battery to have a
vanishing charging power in the closed-system analysis, but the same battery
state has a nonzero charging power in the open-system analysis. Since the
interaction Hamiltonian in the closed-system analysis can always be engineered
to give rise to the Lindblad equation in the open-system analysis, this
inconsistency between closed- and open-system dynamics calls for a deeper
examination of the free energy operator $\mathcal{F}$ introduced in Ref. [17]
to quantify the work content of a charging quantum battery.

As stressed in Section II, the definition of the charging power $P(t)$ in
Equation (3) is not problem-free. First, recall that $W_{\mathrm{max}}$ is a
state function corresponding to the maximum extractable work of the battery
when it is in thermal contact with a heat bath at inverse temperature $\beta$.
However, the battery cannot be in contact with a heat bath; otherwise part of
the energy transfer will be heat rather than work. As a matter of fact, the
authors of Refs. [17, 23] never address the issue of why the bounds (8) and
(13) on the charging power of a battery would depend on the arbitrary
parameter $\beta$ that is introduced by hand solely for the purpose of
constructing the operator $\mathcal{F}$. Second, we note that the operator
$\mathcal{F}$ has logarithmic singularities and is not bounded in the kernel
of $\rho_{\mathcal{W}}$ [30]. While one may choose to restrict $\mathcal{F}$
(and hence $H_{\mathcal{W}}$) to the support of $\rho_{\mathcal{W}}$, it is
likely to give rise to inconsistent battery dynamics because $H_{\mathcal{W}}$
and the Lindblad equation are defined on the _full_ Hilbert space of the
battery. Hence, despite that _by construction_ the expectation values of
$\mathcal{F}$ yield the formal expression for $W_{\mathrm{max}}$ in Equation
(2), the operator $\mathcal{F}$ and its fluctuations may not have physical
meaning per se. Finally, the second law of thermodynamics dictates that the
maximum extractable work $W_{\mathrm{max}}$ is achieved if and only if the
extraction process is thermodynamically reversible (quasistatic and
nondissipative) [25, 24]. Thus to have a nonzero charging power the charging
efficiency cannot be maximum [31], that is, $P(t)$ in general cannot be
$dW_{\mathrm{max}}/dt$, which vanishes identically in the quasistatic limit.
Moreover, there is a _separation of time scales_ in the battery charging
problem: the infinitely long quasistatic time scale associated with
$W_{\mathrm{max}}$ and the much shorter relaxation time scale associated with
$P(t)$. Defining $P(t)=dW_{\mathrm{max}}/dt$ sends the relaxation time scale
to infinity and erroneously equates the two separated time scales. A
consistent treatment of the problem is to consider $P(t)=dW/dt$, where $W$ is
the work stored in the battery on the relaxation time scale. In general, we
have $W<W_{\mathrm{max}}$ because of a tradeoff between the charging
efficiency and the charging power for finite-time processes [32].

In conclusion, our results indicate that the free energy operator introduced
in Ref. [17] does not consistently quantify the work content of a charging
quantum battery at the relaxation time scale of the battery. The question as
to whether there is a consistent quantifying operator is certainly an
important problem that warrants further investigation.

###### Acknowledgements.

This work was supported in part by the Ministry of Science and Technology of
Taiwan under Grant No. 110-2112-M-032-010.

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