# FedNILM: Applying Federated Learning to
NILM Applications at the Edge

Yu Zhang, Guoming Tang, Qianyi Huang, Yi Wang, Xudong Wang, Jiadong Lou Y.
Zhang, G. Tang, Q. Huang and Y. Wang are with the Peng Cheng Laboratory,
Shenzhen, China. X. Wang and J. Lou are with the Chinese University of Hong
Kong, Shenzhen, China.Corresponding author: G. Tang (tanggm@pcl.ac.cn).

###### Abstract

Non-intrusive load monitoring (NILM) helps disaggregate the household’s main
electricity consumption to energy usages of individual appliances, thus
greatly cutting down the cost in fine-grained household load monitoring. To
address the arisen privacy concern in NILM applications, federated learning
(FL) could be leveraged for NILM model training and sharing. When applying the
FL paradigm in real-world NILM applications, however, we are faced with the
challenges of edge resource restriction, edge model personalization and edge
training data scarcity.

In this paper we present FedNILM, a practical FL paradigm for NILM
applications at the edge client. Specifically, FedNILM is designed to deliver
privacy-preserving and personalized NILM services to large-scale edge clients,
by leveraging i) secure data aggregation through federated learning, ii)
efficient cloud model compression via filter pruning and multi-task learning,
and iii) personalized edge model building with unsupervised transfer learning.
Our experiments on real-world energy data show that, FedNILM is able to
achieve personalized energy disaggregation with the state-of-the-art accuracy,
while ensuring privacy preserving at the edge client.

###### Index Terms:

NILM, federated learning, model compression, transfer learning.

##  I Introduction

Non-intrusive load monitoring, also known as energy disaggregation, was first
proposed by Hart in 1992 [1]. It is a single-channel blind source separation
(BSS) problem that aims to decompose the aggregated power readings of a
household into appliance-wise power consumption. One of the major purposes of
NILM is to help reduce household energy consumption efficiently. Evidences
have shown that such feedback of itemized information could encourage
householders to use energy in a more sustainable way, achieving about $15\%$
energy saving [2, 3]. Besides, NILM can be leveraged to evaluate conservation
programs, improve the quality of load forecasting, and provide references for
power grid management [4]. For example, with real-time NILM applications,
utility companies could suggest switching operations on particular appliances
(e.g., air conditioners) for load shifting in peak power hours [3].

Although proposed for decades, the NILM problem has not been addressed
completely, and traditional solutions referring to man-made appliance
signatures have run into bottlenecks [5, 6]. Most recently, it shows that the
deep neural network (DNN) based approaches could greatly improve the
performance of NILM [7, 8, 9, 10], as neural networks are able to
automatically learn appliance features (or _appliance signatures_), either
obvious or latent ones. Various neural network architectures have been
proposed for NILM, including denoising auto-encoder [7], recurrent neural
networks [11], and GAN [12], etc. Among those DNN based methods, the
_Seq2Point_ model [8], a one-dimensional CNN based auto-encoder architecture,
is the current state-of-the-art model for energy disaggregation.

Arguably, the DNN based NILM models largely rely on sufficient and diverse
training data, whereas realistic datasets often exist in the form of isolated
islands. Although there are plenty of meter data in different buildings, it is
almost impossible to transmit or integrate these local user data into a
centralized storage, due to limits in communication bandwidth and legislation
in user privacy and data security. Actually, in the last few years, the
emphasis on data security and user privacy has become a global issue. For
example, in the United States, China, and the European Union, relevant
regulations have been enforced to protect data security and privacy [13, 14],
making it high-risk to gather massive user energy consumption data. In
consequence, it is impractical to train powerful NILM models with existing
paradigms.

###  I-A Motivations and Challenges

To address such problems, federated learning (FL) has recently emerged as a
promising paradigm. In a canonical FL system, user data is kept on client
devices and the NILM model training is realized by i) local model updates with
users’ own data and ii) cloud model fusion with all users’ models. In this
way, client data remains locally and separate clients are integrated to build
a up-to-date cloud model, thus allowing local models to collectively reap the
benefits of cloud model trained from rich data [15]. It seems natural to apply
the FL paradigm to the NILM problem for privacy preserving, which was
theoretically verified in [16] recently. When dealing with real-world NILM
applications at local houses, however, we are still faced with the following
major challenges.

Challenge 1: One critical limitation of adopting the typical FL system to NILM
applications is the constrained resource in local homes (or _edge clients_),
in terms of computation power, communication bandwidth, memory and storage
size, etc. In reality, usually low-end devices (or _edge devices_) appear in
local homes, rather than those powerful CPU/GPU servers. Hence, as the NILM
models usually take large memory and computation overheads, network training
on such resource-constrained edge devices could take prohibitively long time
or even be terminated immediately. Since we cannot perform the model fusion at
cloud without local updates, the resource lack at the edge client literally
hinders the FL paradigm from applying in NILM applications.

Challenge 2: The second one relates to the personalization of NILM models.
Although recent DNN based approaches are promising for NILM, it is not clear
whether existing models are transferable among the diverse edge clients, since
most of the models are trained on public datasets (e.g., REDD [17] and UK-DALE
[18]). In other words, even with satisfied performance on the common training
datasets (or _source domain_), these models may perform badly in a testing
house (or _target domain_), owing to the distribution difference between
training and testing data [19]. Generally speaking, different households
usually have different appliances as well as energy usage patterns, making
those models hardly capture edge clients’ heterogeneity and resulting in poor
scalability in practice [19].

Challenge 3: The third one is on training data scarcity at edge clients. For
NILM applications, obtaining unlabelled testing data (i.e., mains power data)
is quite easy, as current utility smart meters report the whole-home energy
consumption periodically. To acquire labelled training data (i.e., power
consumption data of individual appliances), however, is extremely expensive if
not impossible. Although there are various power sensing devices and we could
equip each appliance with a power sensor, this would incur enormous
installation and maintenance costs, and thus is unscalable across large-scale
households. As current DNN based NILM models are generally trained from large
amounts of labelled data [19], the scarcity of training data further limits
the scalability of NILM systems and applications.

###  I-B Our Ideas and Contributions

To tackle the resource limitation at the edge client (Challenge 1), our idea
is to compress the complex NILM model to a simpler one that the edge device
can afford. More specifically, the cloud NILM model under the FL paradigm
could be pruned before being circulated to edge clients, and thus the edge
devices would be able to implement local computation based on the compressed
global state. Besides, a compressed NILM model is also desirable for cloud-
edge communications. Then, to build personalized NILM models for diverse edge
clients (Challenge 2), particularly under the condition of training data
scarcity (Challenge 3), we propose to leverage the unsupervised transfer
learning. By aligning the feature map distributions between the source and
target domains, the unsupervised transfer learning techniques (e.g., CORAL in
Sec. VI-B) could be incorporated with existing NILM modelling process, and
properly address the domain shift issue for NILM model personalization.

In this work we present FedNILM, a practical FL paradigm for real-world NILM
applications at the edge. FedNILM aims to provide privacy-preserving and
personalized energy disaggregation for the large-scale edge clients, aided by
the cutting-edge DNN model compressing and transfer learning techniques at
resource constrained edge devices. This paper shows that, the adapted FL
framework can be highly preferable to be applied in tackling NILM problems,
which is expected to promote NILM applications and make it practical in real-
world implementations. Our major contributions can be summarized as follows.

  * •

We propose FedNILM, a FL paradigm designed for real-world NILM applications.
Aided by the FL paradigm and with practical considerations, FedNILM is
expected to provide scalable NILM services with the state-of-the-art accuracy
across large-scale households while retaining data privacy for edge clients.

  * •

We adopt cloud model compression for edge adoption in FedNILM. By exploring
how compression techniques influence accuracy and overhead of the NILM model,
we are able to effectively cut down the computation cost at the edge while
retaining satisfied performance.

  * •

We incorporate unsupervised transfer learning with FedNILM for client model
personalization. By introducing the CORAL loss into the state-of-the-art NILM
model, we manage to realize local transfer learning without relying on
labelled training data at the edge client.

  * •

We make extensive evaluations to validate the performance of FedNILM. The
results demonstrate that, aided by the model compression at cloud and model
personalization at edge, FedNILM can provide a comparable performance to the
state-of-the-art without compromising the user privacy.

The rest of the paper is organized as follows. In Sec. II, we give the
background knowledge and related literature. Sec. III briefs the state-of-the-
art NILM model and the federated learning rationale. Then, we present the
design of FedNILM in Sec. IV, including the paradigm overview and workflow.
Two key operations of FedNILM, cloud model compression and edge model
personalization are presented in Sec. V and Sec. VI, respectively. The
experimental evaluations are performed in Sec. VII. The paper is concluded in
Sec. VIII.

##  II Background and Related Work

Recently, there are growing interests on deploying NILM applications and
systems by energy service providers, energy aggregators and distribution
system operators [12]. With the urgent request of privacy preserving, the FL
paradigm has been exploited [16], where the NILM model/service is
deployed/delivered across the edge clients for energy disaggregation. In real-
world implementations, however, some challenges could arise.

###  II-A NILM Model Compression

The first challenge is how to perform DNN-based NILM model inference at edge
clients where only resource constrained devices could be installed. Deploying
compressed NILM models on edge devices might be a promising approach. It has
been demonstrated that model compression may not do harm to the NILM
performance while significantly reducing the computational overhead [20, 21].

There are various algorithms for compressing or pruning neural networks. In
[21], the authors leverage filter pruning [22] and tensor decomposition [23]
methods to compress the convolutional layers, where the former refers to
sparsify the neural network by removing less important parameters and the
later is to perform a low-rank decomposition of the learnt filter matrix.
Surprisingly, experimental results in [24, 21] show that compressing of neural
networks might bring some performance gain due to better generalization in
test cases.

Although the compressed NILM models could be deployed at the edge client for
inference, the training of personal NILM models still needs to be conducted on
the powerful GPU/CPU servers at the cloud end. Thus, one client needs to
upload his/her energy data to the cloud for personal NILM model training, and
the data privacy concern still exists.

###  II-B NILM Model Transfer

The other challenge is on the degradation of the cloud model at the edge
client. When applying the FL paradigm for NILM, the same model trained at the
cloud would be delivered to various client ends. When the distributions
between the client data and the cloud data are different, which is the most
likely case, we would see more or less performance degradation from the NILM
model at the client.

To this end, the transferability issue of NILM models has been preliminarily
explored. In [19], the authors investigate two transfer learning schemes,
i.e., the appliance transfer learning (ATL) and cross domain transfer learning
(CTL). Both ATL and CTL freeze the convolutional layers and merely tune the
fully connected layers during retraining. Recently, apart from the canonical
CNN structure, the authors of [12] develop the TrGAN-NILM, which is based on
the generative adversarial networks (GANs), to automatically extract common
feature representations between source and target domains through minimizing
the statistical distance between different domains.

Nevertheless, the aforementioned transfer learning techniques are in need of
labelled training data on target domain. In other words, they cannot been
directly applied in the situation where the target domain is unlabeled, which
is quite common in real-world NILM applications. In consequence, for buildings
that are newly added to the NILM service provider portfolio, obtaining
labelled appliance meter data and performing transfer learning are
prohibitively expensive and time consuming.

In this work, we design FedNILM, a practical FL paradigm for real-world NILM
applications. Particularly, FedNILM tackles the computation limitation issue
by incorporating model compression techniques, and addresses the model
personalization problem with unlabelled target domain.

Figure 1: (a) The DNN structure adopted by Seq2Point model. (b) The DNN
structure for MTL-Seq2Point model, i.e., shared convolutional and fully
connected layers for $N$ appliances.

##  III Preliminary

###  III-A NILM Problem Definition

The goal of NILM is to recover the energy consumption of individual appliances
from the mains meter signals. Given the aggregate power readings from $T$ time
periods, we can denote them by $\textbf{y}=(y_{1},y_{2},...,y_{T})$, where
$y_{t}\in R_{+}$. Then, let
$\textbf{x}^{(i)}=(x^{(i)}_{1},x^{(i)}_{2},...,x^{(i)}_{T})$ in which
$x^{(i)}_{t}\in R_{+}$ denotes the power reading of the $i$-th appliance at
time $t$. Therefore, at each time instant $t$, $y_{t}$ is assumed to be the
sum of all $N$ appliances’ power readings. Normally, we are only interested in
the first $N^{\prime}$ appliances used widely and consuming the most energy.
Then, the power consumption of the remaining appliances can be represented as
$\textbf{u}=(u_{1},u_{2},...,u_{T})$ and the aggregate power consumption could
be represented as follows:

$y_{t}=\sum_{i=1}^{N^{\prime}}{x^{(i)}_{t}}+u_{t}+\epsilon_{t}$ (1)

where $\epsilon_{t}$ denotes a Gaussian noise.

###  III-B State-of-the-Art NILM Model

We introduce the state-of-the-art model, i.e., sequence to point (Seq2Point)
[8], recently developed for solving the NILM problem. The Seq2Point learning
model maps a window of the mains signal readings to the midpoint point of the
corresponding window of the target individual appliances. For each time
instant $t$, given a fixed time window with size of $w$, the Seq2Point model
uses the mains power signal sequence
$\textbf{y}_{t:t+w-1}=[y_{t},y_{t+1},\cdots,y_{t+w-1}]$ as the input and the
middle element $x^{(i)}_{t+w/2}$ in power readings of the target appliance $i$
as the output. In other words, instead of estimating the whole power signal
sequence of the target appliance, the Seq2Point model merely predicts the
middle signal element of the appliance in corresponding time window.

Mathematically, for a target appliance $i$, it assumes that there exists a
function $f^{(i)}:\mathbb{R}_{+}^{w}\rightarrow\mathbb{R}_{+}^{1}$, and the
function gives the power estimation of appliance $i$ at time $t+w/2$ by:

$f^{(i)}(\textbf{y}_{t:t+w-1})=x^{(i)}_{t+w/2}$ (2)

Thus, the key task in Seq2Point is to learn the specific form of function
$f^{(i)}$. Once obtained $f^{(i)}$, we are able to estimate the power signal
of the target appliance $i$ with the aggregate (mains) signals, thus achieving
energy disaggregation.

More specifically, to learn the parameters of $f^{(i)}$, Seq2Point employs the
convolutional neural network (CNN) as the training structure, as illustrated
in Fig. 1(a). It was demonstrated that such a DNN structure could inherently
learn the signatures of target appliances and shows superior performance than
other models [7].

###  III-C Federated Learning

At the beginning of FL, the server trains a cloud model based on the original
dataset. When we adopt deep neural networks to learn both the cloud and client
models, the learning objective of the global model could be formulated as:

$\mathop{\arg\min}_{\omega}f_{s}(\omega)=\frac{1}{n}\sum_{i=1}^{n}\mathcal{L}(x_{i},y_{i};\omega)$
(3)

where $\mathcal{L}(x_{i},y_{i};\omega)$ denotes the loss of prediction on
server samples $\\{x_{i},y_{i}\\}_{i=1}^{n}$ made with model parameters
$\omega$, i.e., the weights and bias.

The server then sends the current global state $\omega$ to each of the
clients, thus enabling each client to perform local computation based on the
global state and its local dataset. Technically, the learning objective of
client $u$ can be denoted as:

$\mathop{\arg\min}_{\omega^{u}}f_{u}(\omega^{u})=\frac{1}{n^{u}}\sum_{i=1}^{n^{u}}\mathcal{L}(x_{i}^{u},y_{i}^{u};\omega^{u})$
(4)

where $\\{x_{i}^{u},y_{i}^{u}\\}_{i=1}^{n^{u}}$ are local samples with
${n^{u}}$ denoting their sizes. After the client model $f_{u}$ is trained, the
local computation results, namely local model parameters $\omega^{u}$, would
be uploaded to the server. For parameter transmission between cloud and
client, homomorphic encryption is usually adopted to avoid information leakage
[25].

Then, with enough local updates, the cloud server performs federated
aggregation [15] to align user models and obtain a new global state. Assuming
that there are $K$ clients which are indexed by $k$ and each client locally
updates its gradient to $\omega_{t+1}^{k}$ using its local data. The server
then takes a weighted average of all these local models, which can be
formulated as:

$\omega_{t+1}=\frac{1}{K}\sum_{k=1}^{K}\omega_{t+1}^{k}$ (5)

where $\omega_{t+1}$ denotes the updated global parameters. After adequate
rounds of iterations and the global model has satisfactory generalization
ability, the global model is distributed to the client for local deployment.

##  IV FedNILM Paradigm

Modern DNN based NILM models cannot succeed without access to large amount of
training data at the client side. However, this also causes severe concerns on
user privacy and data security. To tackle these issues, we leverage the
federated learning approach to the NILM problem and design FedNILM in this
section.

###  IV-A Overview

Without loss of generality, we consider one cloud server and multiple served
clients at the edge, as illustrated in Fig. 2.

  * •

At cloud side, the cloud server trains a Seq2Point model with readily-
available open datasets, prunes it into a slim version, and shares it among
the associated edge clients.

  * •

At client side, each takes over the pruned model from cloud, further tailors
it through a local transforming process, and adopts the personalized model for
individual appliance monitoring.

Figure 2: Overview of the FedNILM framework.

###  IV-B Workflow

As shown in Fig. 2, the workflow of FedNILM mainly consists of four steps.

  * •

Step-1: Based on the state-of-the-art NILM model (introduced in Sec. III-B)
and public datasets, the cloud server develops a compressed model with model
pruning techniques. Refer to Sec. V for the details of model pruning. Then,
the compressed NILM model is distributed to all the associated edge clients.

  * •

Step-2: Based on the shared model from the cloud, each edge client further
train a personalized NILM model with their own data at hand. An unsupervised
transfer learning method is applied for the local model personalization. Refer
to Sec. VI for the detailed transfer learning process.

  * •

Step-3: The parameters of personalized models at the edge are encrypted and
uploaded to the cloud server, where the original cloud model is updated
through the federated aggregation process (introduced in Sec. III-C).

  * •

Step-4: The updated cloud model could be distributed to either new edge
clients for personalized model building, or existing edge clients for
continuous model refining with fresh local data.

Generally speaking, the complete FedNILM workflow includes the starting
process (i.e., Step-1 and Step-2 in sequential) and a repeating process (i.e.,
Step-3 and Step-4 in iterative). Next we introduce the two key operations in
the workflow, i.e., cloud model compression in Step-1 and client model
personalization in Step-2, respectively.

##  V Cloud Model Compression

As we have mentioned in Sec. III-B, the Seq2Point model trains a separate
model for each individual appliance. That is to say, to monitor $n$ appliances
for an edge client, we have to train and deploy $n$ appliance-specific models
on the corresponding edge device. This could trigger tremendous resource
demands and hardly be satisfied by the resource constrained edge devices. In
this section, we show how model compression techniques could be leveraged in
FedNILM.

###  V-A MTL-Seq2Point

Multi-task learning (MTL) refers to share representation, usually layers in
neural network, between analogous tasks, and empowers the neural network to
have better generalization ability [26]. Intuitively, leveraging excess
information that comes from auxiliary tasks enables the MTL model to perform
well in main task [27]. By leveraging MTL techniques in the Seq2Point modeling
(we name the variant _MTL-Seq2Point_), we are able to train one single model
for all target appliances for NILM applications. This can significantly reduce
the resource overhead at the edge device.

More specifically, we leverage hard parameter sharing method in Seq2Point
model, where the set of five convolutional layers and one fully connected
layers are commonly shared for all appliances. Then after these shared layers,
this model diverges to several task-specific layers for different appliances.
Refer to Fig. 1(b) for the network structure of MTL-Seq2Point. In contrast
with the original Seq2Point model, MTL-Seq2Point could help save large
computation and memory overheads at the edge devices in our scenario.

###  V-B Pruned MTL-Seq2Point

In addition to the multi-task learning, we also seek other effective
techniques to further compress the NILM model in compatible with the less
powerful edge devices. There has been some progress recently towards this
direction, including weights pruning, filter pruning [22], neuron pruning and
tensor decomposition [23]. As convolutional layers accounting for most of the
computation cost in our NILM model [20, 21], we propose to leverage filter
pruning in convolutional layers to build a “slimmer” MTL-Seq2Point model while
retaining a comparable NILM performance. We name the model _pruned MTL-
Seq2Point_ in this work.

Specifically, in the network structure of MTL-Seq2Point, let $n_{i}$ denote
the number of input channels for the $i$-th convolutional layer and $w_{i}$
the window size of input. Given the kernel size $k$, we have 1D kernel
$\mathcal{K}\in\mathbb{R}^{k\times 1}$ (e.g., 10 $\times$ 1 in the first
layer). Supposing that the number of output channels in the $i$-th
convolutional layer is $n_{i+1}$, the 3D filters matrix
$\mathcal{F}_{i}\in\mathbb{R}^{n_{i+1}\times n_{i}\times k}$ could transform
the input $\mathrm{X_{i}}\in\mathbb{R}^{n_{i}\times w_{i}}$ into the output
$\mathrm{X_{i+1}}\in\mathbb{R}^{n_{i+1}\times w_{i+1}}$. Then, given the
stride size $s$, we have $w_{i+1}=w_{i}-k+s$. Thus, the number of operations
in the $i$-th convolutional layer is $n_{i+1}n_{i}kw_{i+1}$. By removing one
of the $n_{i+1}$ filters in $\mathcal{F}_{i}$, we could eliminate
$n_{i}kw_{i+1}$ operations, as illustrated by Fig. 3. Moreover, pruning a
filter also results in the removal of corresponding feature maps and kernels
of the following layer, thus cutting down another $n_{i+1}kw_{i+2}$
operations. Hence, we can conclude that, by pruning $m$ filters of layer $i$,
we could reduce the computation cost by $m/n_{i+1}$ at both the $i$-th and
$(i+1)$-th layers.

Figure 3: Pruning a filter results in reduction of computation overhead.

Since not all trained filters are equally important, thus in order to minimize
the performance drop, we choose and prune the less instrumental filters. More
specifically, the relative importance of the filters are measured through
calculating the sum of $\mathcal{L}_{1}$-norm [22] or $\mathcal{L}_{2}$-norm
[28, 29] on convolutional filters. As there are no noticeable differences
between these two criteria in filter selection [22], we leverage the
$\mathcal{L}_{1}$-norm to score the filters and prune $k$% of them in each
layer with the least $\mathcal{L}_{1}$-norm values. By increasing the pruning
percentage of the network iteratively, we can also investigate the variation
of model performance and thus find the optimal pruning percentage for our
model.

Overall, the procedure to obtain the best pruned MTL-Seq2Point model consists
the following four steps: i) train a convergent MTL-Seq2Point model, ii) score
corresponding convolutional filters based on the sum of $\mathcal{L}_{1}$-norm
values, iii) prune the least important filters as per their scores, and iv)
retrain the pruned MTL-Seq2Point model (for a certain iterations) to
compensate for incurred performance degradation.

##  VI Client Model Personalization

When adopting the compressed cloud NILM model at the edge client, however, we
are faced with the domain shift problem, i.e., the difference of distributions
between the cloud and client data. Thus, the common model can perform very
well upon the cloud dataset, while may be greatly degraded at different client
ends. To this end, we leverage the transfer learning technique and build
personalized models for different clients. Particularly, we adopt the
unsupervised transfer learning to work with unlabelled client data, hence
addressing the training data scarcity problem at the edge client.

###  VI-A Transferability Analysis

Transfer learning works under the scenario where observations from the source
domain (denoted by $D_{s}$) have a different distribution with those from the
target domain (denoted by $D_{t}$). In our case, the source domain refers to
the public dataset at cloud, while the target domain is the personal dataset
at each client. There have been interests towards identifying the
transferability of features in each layer of the DNN model, especially in
Computer Vision [30].

For our NILM model, to investigate the transferability of different model
layers, we propose the following three-step procedure.

  1. 1.

Train an initial NILM model (i.e., the pruned MTL-Seq2Point model) on $D_{s}$.

  2. 2.

Fix and fine-tune one of the multiple layers in the pre-trained model, and
randomly initialize the parameters in the rest of layers.

  3. 3.

Retrain the model on $D_{t}$ and compare the prediction performance from the
multiple transferred models.

Note that the NILM model in our scenario is composed of five convolutional
layers and two fully connected layers, as illustrated in Fig. 1. Thus, through
procedurally freezing or tweaking these layers, we have $(2\times 7)$
different transfer learning models, which are then retrained and tested on
target domain data. By wrapping up all the results, we have the following
observations (refer to Sec. VII-C for more details).

  * •

The convolutional layers of the model are good at extracting low-level and
generic features. The transferable load features, such as the ON/OFF switching
points, power level of appliances and typical usage durations, are
insusceptible to the difference between source and target domains.

  * •

The fully connected layers take responsibility for learning high-level
features for specific appliances. Thus, when applying to a new edger client
(target domain), they need to be further fine-tuned. The client model
personalization can then be realized by largely referring to the transferred
features from convolutional layers and substantially fine-tuning the fully
connected layers for specific appliances.

These observations are coincide with those from [19] that, the features in
lower levels of layers are highly transferable as lower layers tend to learn
common and coarse information, whereas the features in higher layers are more
tailored for specific tasks and thus are more personalized. Accordingly, in
our case of transfer learning at the client side, we propose to freeze the
convolutional layers in model transforming (i.e., keep their parameters fixed
in back propagation) and merely update the weights on fully connected layers.

###  VI-B Correlation Alignment

Figure 4: The transfer learning process at the edge client.

Correlation alignment (CORAL) is an instrumental domain adaptation method,
which tackles the domain shift via aligning the feature distributions of
source and target samples [31]. More specifically, deep CORAL aligns the
source and target data distributions through learning a nonlinear
transformation, i.e., a differentiable loss function (CORAL loss), between
source and target layer activations [32]. This nonlinear transformation is
designed to minimize the distance between the second-order statistics
(covariance) of the source and domain layer features.

Given the source domain training data $D_{s}$ (with labels $L_{s}$) and target
input data $D_{t}={u_{i}}$, we aim to align the distribution difference in
layer $I$ which generates the $\sigma$-dimensional deep layer features $x$ of
input $D_{s}$ and $u$ of input $D_{t}$. With $x$ and $u$, we could compute the
covariance matrices $C_{s}$ and $C_{t}$. Therefore, the CORAL loss is defined
as the distance between the second-order statistics, namely covariance, of the
source and target features:

$\mathcal{L}_{CORAL}=\frac{1}{4\sigma^{2}}\|C_{s}-C_{t}\|^{2}_{F}$ (6)

where $\|\cdot\|^{2}_{F}$ denotes the squared matrix Frobenius norm. Let
$\lambda$ denote the trade-off parameter and $\mathcal{L}_{R}$ the regression
loss of the edge model. The loss in the client model training can be designed
as:

$\mathcal{L}_{EDGE}=\mathcal{L}_{R}+\lambda\cdot\mathcal{L}_{CORAL}$ (7)

The above loss serves as a constraint and regulates the distance between
source and target domains during the fine-tuning process. By jointly
optimizing the regression loss and CORAL loss, we can obtain a personalized
client model with both generic signatures pre-trained on the source domain and
specific features working well on the target domain.

Fig. 4 shows the detailed model structure tailored for the unsupervised
transfer learning process. In this way, the local devices are able to reap the
shared benefits of common features pre-trained on a large generic dataset
while retaining the speciality fine-tuned on client data.

##  VII Experiments

In this section, we use real-world energy datasets to evaluate the proposed
FedNILM paradigm, including cloud model compression and client model
personalization, respectively.

###  VII-A Experimental Settings

####  VII-A1 Cloud and Edge Systems

In our experiments, an AMAX server with four Tesla V100 GPUs serves as the
cloud end. A low-end desktop (with Nvidia GTX 960M) and an edge device (Nvidia
Jetson Nano) serve as two different client ends.

####  VII-A2 Datasets

Three benchmark datasets are used to evaluate the performance of FedNILM,
including REFIT [33], UK-DALE [18] and REDD [17]. All contain similar
appliance categories, allowing the evaluation of model transferability. Also,
they have been widely applied in previous NILM researches [7, 8, 19], and thus
enabling performance comparison with the state-of-the-art solutions.

####  VII-A3 Performance Metrics

Mean absolute error (MAE), signal aggregate error (SAE) and F1-score are used
to evaluate the performance of FedNILM, all of which have been leveraged in
prior NILM research [7, 8, 9, 21]. In particular, the former two metrics,
i.e., MAE and SAE, aim to measure the performance of power consumption
estimation, while the F1-score could reflect the performance of appliance
ON/OFF states estimation.

TABLE I: MAE results from cloud model compression on REDD dataset (sequence length 499) Model |  | Washing
---
m.
Fridge |  | Dish
---
w.
| Micro
---
wave
| Time
---
(s)
| Size
---
(MB)
Seq2Point | 20.25 | 25.61 | 14.09 | 13.30 | 9.36 | 367.82
30% pruning | 18.03 | 26.16 | 12.82 | 9.95 | 4.78 | 180.05
60% pruning | 18.30 | 25.75 | 16.91 | 10.65 | 3.06 | 58.80
90% pruning | 20.53 | 44.57 | 48.36 | 13.76 | 2.33 | 3.69
MTL-Seq2Point | 18.74 | 27.04 | 19.40 | 12.37 | 2.69 | 91.97
30% pruning | 18.55 | 27.93 | 21.94 | 12.74 | 1.26 | 45.02
60% pruning | 19.18 | 28.39 | 19.26 | 13.15 | 0.83 | 14.7
90% pruning | 19.39 | 75.21 | 28.86 | 20.60 | 0.68 | 0.93
TABLE II: MAE results from cloud model compression on REDD dataset (sequence length 99) Model |  | Washing
---
m.
Fridge |  | Dish
---
w.
| Micro
---
wave
| Time
---
(s)
| Size
---
(MB)
Seq2Point | 14.97 | 25.80 | 19.35 | 9.16 | 3.13 | 55.32
30% pruning | 14.59 | 23.77 | 11.97 | 9.86 | 2.74 | 27.10
60% pruning | 14.73 | 22.71 | 16.60 | 8.65 | 2.67 | 8.87
90% pruning | 15.46 | 30.70 | 32.23 | 10.52 | 2.55 | 0.58
MTL-Seq2Point | 17.62 | 26.78 | 16.06 | 14.51 | 0.83 | 13.84
30% pruning | 17.26 | 25.40 | 13.05 | 12.65 | 0.75 | 6.79
60% pruning | 18.67 | 28.78 | 10.66 | 15.78 | 0.72 | 2.22
90% pruning | 19.59 | 30.45 | 18.34 | 18.83 | 0.74 | 0.15

(a) MAE

(b) SAE

(c) F1-score

Figure 5: (a) The MAE of transferred models (lower is better). (b) The SAE of
transferred models (lower is better). (c) The F1-score of transferred models
(higher is better). The x-axis indicates till which layer we transfer the
parameters and initialize the rest.

###  VII-B Evaluations on Cloud Model Compression

At the cloud end, we train the Seq2Point model with/without pruning and MTL on
the REDD dataset, and compare the NILM performance, run-time and memory
overhead from different model adoptions. To show the impacts of model
compression, we prune the parameters of Seq2Point and MTL-Seq2Point by 30%,
60% and 90%, respectively. The results are summarized in Table I and Table II,
for input sequence lengths of 499 and 99, respectively.

Based on the experimental results in Table I and Table II, we observe that
both multi-task learning and filter pruning could significantly reduce the
model size along with decent inference time benefits, without drastically
compromising NILM performance. In particular, for the sequence 499, pruning
60% of filters on classical Seq2Point model can save approximately two-thirds
of running time and reduce memory cost by up to 84% while retaining comparable
prediction accuracy. Meanwhile, the multi-task learning structure virtually
takes one fourth of the original computation and memory overhead, as it
simultaneously generates the disaggregation results of four selected
appliances. The combination of filter pruning and multi-task learning
techniques could help save nearly 92% of inference time and 96% of model space
with slight (<10%) performance degradation.

For sequence length 99, the unoptimised model occupies $6\times$ more space
and requires more inference time than the optimised model pruned 60% of
filters with similar performance. Same as we have found with the model of
sequence length 499, for sequence length 99, the multi-task learning structure
helps save nearly three fourths of running time and space, whereas filter
pruning may not have much influence on inference time as the time required for
these three pruned models with different pruning percentages are almost the
same. Note that we merely prune the filters in convolutional layers and in a
forward pass. The majority of computations takes place in fully connected
layers, not convolutional layers, so the reduction of parameters in
convolutional layers may not reduce much operations in inference procedure,
thus explaining this phenomenon.

In conclusion, the general NILM performance decreases with the increasing of
pruning percentage, and the employment of multi-task learning structure also
contribute to the ED performance degradation as all the four appliances use
the same set of signatures for prediction. However, we also observe that in
some circumstances, model pruning might rather give rise to better NILM
results, such as the pruned Seq2Point model performance versus the original
Seq2Point model accuracy on distinguishing the washing machine, which can be
explained by “regularisation effect” as removing the least important weights
from a neural network might lead to better model generalisation. In general,
we choose to build our general cloud model based on Seq2Point model with
multi-task learning and prune 60% of the filters in its convolutional layers.

###  VII-C Evaluations on Client Model Personalization

At the client end, we investigate the transferability of each layer of the
NILM model, determine which layer’s parameters to freeze or fine-tune during
model transfer, and validate the necessity to perform local transfer learning.

####  VII-C1 Layer Transferability of NILM Model

Based on the procedure given in Sec. VI-A, we train a Seq2Point model to
disaggregate the mains signal for a dish washer on REFIT dataset, and then
transfer this model to detect the dish washer in REDD dataset. More
specifically, we gradually transfer the seven layers in Seq2Point model, by
either fixing or fine-tuning them and initializing the rest layers. The
experimental results are shown in Fig. 5, based on which we have the following
observations and empirical findings.

  * •

First, we observe the significantly increasing of both MAE and SAE along with
the decreasing of F1-score, compared with the nearly steady performance from
_conv1_ to _conv2_ , once we begin to share the fully connected layers. This
implies that the convolutional layers virtually learn the generic features
that are common in source and target domains, while the fully connected layers
focus on extracting domain specific signatures. In other words, the number of
transferred convolutional layers generally have little, if any, influence on
the final prediction, while the more the dense layers we transfer the less the
accuracy we get on the target domain.

  * •

Second, we find that the NILM performance of fine-tuning transferred layers is
better than fixing them, with lower MAE and SAE and higher F1-score. This
indicates that fine-tuning the last dense layers could further help us promote
the NILM performance on target domain.

  * •

Finally, we also observe a slight performance improvement as we procedurally
transfer layers from _conv1_ to _conv5_. Note that if we choose to freeze or
fine-tune one particular convolutional layer, the parameters in this
convolutional layer are pre-trained in source domain. Thus when transferring
to target domain, the transferred layers could also leverage the information
pre-trained in source domain, rather than learning from scratch. Arguably, the
extra information from source domain enables the transfer learning model to
generalize better on target domain.

####  VII-C2 Effectiveness of Transfer Learning

To validate the effectiveness of local transfer learning, particularly the
CORAL approach, we conduct experiments on the edge devicxe of Nvidia Jetson
Nano to simulate the procedure of local updates. Basically, we train a pruned
MTL-Seq2Point model with one-month REDD data (for four appliances: washing
machine,fridge, dish washer and microwave), which serves as the cloud model
trained on source domain. Then, to validate the effectiveness of transfer
learning, we leverage the pre-trained cloud model to detect appliances in UK-
DALE dataset. Arguably, the REDD is literally distinct from the UK-DALE, as
the households in former dataset are located in the US while those in latter
dataset are sited in the UK.

For the test without applying transfer learning, we directly employ the pre-
trained cloud model on UK-DALE dataset for energy disaggregation. For the test
with transfer learning, we first retrain the model using the deep CORAL
approach and then leverage the retrained model to detect appliances in UK-DALE
dataset. With results from the above two tests, we can compare their
performances and see whether transfer learning takes effect.

The results are shown in Table III, from which we could observe that: for both
model with sequence length 99 and 499, the model with local transfer learning
outperforms the original one by up to 45% in MAE, 85% in SAE and 40% in
F1-score, respectively. In particular, for MAE, the performance is improved
for all four appliances with transfer learning; for SAE, the accuracy is
improved for three appliances; for F1-score, transfer learning model shows
better performances on dish washer and microwave and comparable performance on
fridge. In general, local transfer learning takes effect on minimizing the
difference in source and target domains, compensating for the performance
degradation due to domain shift.

In our implementations on the edge device, the model training time for each
appliance is around 100 seconds in average, whereas the inference merely takes
less than one second. During each local updating process, once receiving the
cloud model, the edge client could perform transfer learning based on its own
power readings to fine-tune the model parameters in dense layers, and conduct
energy disaggregation with this personalized model promptly. Meanwhile, the
updated local model would be uploaded to the cloud for further federated
averaging, with the goal to collaboratively train more powerful and up-to-date
cloud models.

TABLE III: The effectiveness of transfer learning Appliance | Metric |  | None
---
Transfer
| Transfer
---
Learning
Average Improvement
99 | 499 | 99 | 499
Washing Machine | MAE | 38.83 | 51.59 | 25.43 | 23.54 | 45.84%
SAE | 7.79 | 1.58 | 0.87 | 0.78 | 82.39%
F1-score | 0.47 | 0.30 | 0.25 | 0.31 | -27.27%
Fridge | MAE | 68.78 | 68.75 | 55.67 | 66.95 | 10.84%
SAE | 11.40 | 11.15 | 0.88 | 2.36 | 86.63%
F1-score | 0.58 | 0.58 | 0.58 | 0.58 | 0
Dish Washer | MAE | 13.54 | 14.21 | 11.18 | 10.29 | 22.63%
SAE | 1.17 | 0.76 | 2.21 | 2.49 | -143.52%
F1-score | 0.05 | 0.18 | 0.02 | 0.31 | 43.48%
Microwave | MAE | 41.87 | 31.85 | 25.21 | 23.76 | 33.57%
SAE | 3.01 | 2.17 | 2.08 | 1.97 | 21.81%
F1-score | 0.30 | 0.15 | 0.31 | 0.32 | 40.00%

##  VIII Conclusion

We presented FedNILM, a federated learning paradigm designed for privacy-
preserving and personalized NILM applications at low-end edge devices. FedNILM
realized data privacy-preserving through federated learning, efficient model
compression via filter pruning and multi-task learning, and personalized model
building by unsupervised transfer learning, respectively. The results from the
experiments on real-world energy data demonstrate that, FedNILM can achieve
accurate and personalized energy disaggregation without compromising the user
privacy.

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# Quantum Speed Limit From Tighter Uncertainty Relation

Shrobona Bagchi<EMAIL_ADDRESS>Center for Quantum Information, Korea
Institute of Science and Technology, Seoul, 02792, Korea  Abhay Srivastav
<EMAIL_ADDRESS>Arun Kumar Pati<EMAIL_ADDRESS>Harish-Chandra
Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road,
Jhunsi, Prayagraj 211019, Uttar Pradesh, India

###### Abstract

The quantum speed limit provides a fundamental bound on how fast a quantum
system can evolve between the initial and the final states under any physical
operation. The celebrated Mandelstam-Tamm (MT) bound has been widely studied
for various quantum systems undergoing unitary time evolution. Here, we prove
a new quantum speed limit using the tighter uncertainty relations for pure
quantum systems undergoing arbitrary unitary evolution. We also derive a
tighter uncertainty relation for mixed quantum states and then derive a new
quantum speed limit for mixed quantum states from it such that it reduces to
that of the pure quantum states derived from tighter uncertainty relations. We
show that the MT bound is a special case of the tighter quantum speed limit
derived here. We also show that this bound can be improved when optimized over
many different sets of basis vectors. We illustrate the tighter speed limit
for pure states with examples using random Hamiltonians and show that the new
quantum speed limit outperforms the MT bound.

##  I Introduction

The uncertainty principle and the uncertainty relations are of central
importance in quantum mechanics. The uncertainty relations have helped us to
reveal the behavior of the microscopic world in many different ways. At first
the uncertainty principle was discovered by Heisenberg who heuristically
provided a lower bound on the product of the error and the disturbance for two
canonically conjugate quantum mechanical observables [1]. On the other hand,
the uncertainty relations are capable of capturing the intrinsic restrictions
in preparation of quantum systems, which are termed as the preparation
uncertainty relations [2]. This interpretation was quite fruitful for the
uncertainty relations like position-momentum, angular position-angular
momentum uncertainty relations etc. However, the energy-time uncertainty
relation [3, 4] is different from the above stated uncertainty relations
because time is not treated as an operator in quantum mechanics but as a
classical parameter with no inherent quantum uncertainty in it [5]. The
uncertainty relation for two arbitrary quantum-mechanical observables
formulated by Robertson is essentially a preparation uncertainty relation and
expresses the impossibility of joint sharp preparation of any two incompatible
observables. However, the Robertson uncertainty relation does not completely
express the incompatible nature of two non-commuting observables in terms of
uncertainty quantification. To capture the notion of incompatibility more
efficiently a stronger form of the uncertainty relation based on the sum of
variances was derived in Ref.[6]. In addition, tighter uncertainty relations
and reverse uncertainty relations have been also proved which go beyond the
Robertson-Schrödinger uncertainty relations [7]. The stronger uncertainty
relations and the reverse uncertainty relations have been experimentally
tested using photonic set-ups [8]. However, time, not being a quantum
observable, energy-time uncertainty relation lacked a good interpretation as
such like for those of the other quantum mechanical observables such as
position and momentum.

It was shown by Mandelstam and Tamm that the correct interpretation of the
energy-time uncertainty relation is as a bound on the evolution time of a
quantum system, now known as the MT bound [9]. Subsequently, Margolus and
Levitin derived a new bound on the evolution time based on the expectation
value of the Hamiltonian [10]. The quantum speed limit bounds have since been
studied extensively for closed as well as open system dynamics [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36]. In this paper, we find a tighter form of quantum speed
limit. We show that the new bound provides a tighter expression of quantum
speed limit compared to the MT bound. This bound can also be optimized over
many orthonormal basis vector sets, as in the case of tighter uncertainty
relations. We find various examples that shows the performance of our bound
over the MT bound.

The notion of quantum speed limit is not only of fundamental importance, but
also has practical applications in quantum information and quantum technology.
The quantum speed limit bounds have proven to be very useful in quantifying
the maximal rate of quantum entropy production [39, 40], the maximal rate of
quantum information processing [41], quantum computation [42] and in many
other avenues [43, 44]. Quantum speed limits have also found a wide range of
applications in optimal control theory [45, 46] and quantum thermometry [47].
These explorations motivate us to find better quantum speed limit bounds that
can go beyond the standard quantum speed limit bound such as the MT bound and
find several applications in the aforementioned areas.

The present article is organised as follows. In Section II, we give the
background needed for our result that includes the quantum speed limit and the
tighter uncertainty relations for pure quantum states. In section III, we
provide the derivation of tighter uncertainty relation for the case of mixed
quantum states. In section IV and V, we derive the stronger quantum speed
limit for pure and mixed quantum states, respectively, and show that the MT
bound is a special case. In Section VI, we analyse the obtained bound for some
numerical examples using random Hamiltonians chosen from the Gaussian Unitary
ensemble and find that the new bound surpasses the MT bound for the case of
pure quantum states. We also show the better performance of our bound for an
interacting quantum system. Finally, in Section VII, we conclude and point out
future directions.

##  II Background

###  II.1 Standard Quantum Speed Limits

Quantum speed limit (QSL) is a fundamental limitation on the speed of the
evolution of a quantum system imposed by the laws of quantum mechanics.
Historically, Mandelstam and Tamm derived the first expression of the quantum
speed limit time as $\tau_{QSL}=\frac{\pi}{2\Delta H}$, where $\Delta H$ is
the standard deviation of the Hamiltonian $H$ driving the quantum system, and
where the initial and the final states are orthogonal. As a physical
interpretation of their bound, they also argued that $\tau_{QSL}$ quantifies
the life time of quantum states, which has found importance in the foundations
of quantum mechanics as well as in other applications. Their interpretation
was further solidified by Margolus and Levitin, who derived an alternative
expression for $\tau_{QSL}$ in terms of the expectation value of the
Hamiltonian as $\tau_{QSL}=\frac{\pi}{2\langle H\rangle}$. Eventually, it was
also shown that the combined bound,

$\displaystyle\tau_{QSL}=\max\\{\frac{\pi\hbar}{2\Delta
H},\frac{\pi\hbar}{2\langle H\rangle}\\}$ (1)

is tight for the evolution of the system between two orthogonal states [11].
The QSL bound can be generalised for the evolution between two non-orthogonal
states using the Fubini-Study metric on the projective Hilbert space given by

$\displaystyle
ds^{2}=4[\langle\dot{\Psi}(t)|\dot{\Psi}(t)\rangle-(i\langle\Psi(t)|\dot{\Psi}(t)\rangle)^{2}]dt^{2}$
(2)

and the Schrödinger equation for the unitary evolution of a quantum state. The
fact that the total distance travelled by a quantum state in the projective
Hilbert space is always greater than or equal to the shortest distance
connecting the initial and the final points, i.e., the geodesic $s_{0}$
implies

$\displaystyle\tau\geq\tau_{QSL}=\frac{\hbar s_{0}}{2{\Delta{H}}},$ (3)

where $\tau$ is the actual time of evolution and
$s_{0}(t)=2\cos^{-1}|\langle\Psi(t)|\Psi(0)\rangle|$. $\tau_{QSL}$ is the
celebrated MT bound and gives the minimum time required for a quantum system
to evolve between any two states unitarily.

###  II.2 Tighter Uncertainty Relations for pure quantum states

Uncertainty relations hold an important place in the foundations of quantum
mechanics. A quantitative formulation of the Heisenberg uncertainty principle
was given by Robertson. This is also known as the Robertson-Schrödinger
uncertainty relation. For any two generally non-commuting operators A and B,
the Robertson-Schrödinger uncertainty relation for the state of the system
$|\Psi\rangle$ is given by the following inequality:

$\displaystyle\Delta A^{2}\Delta
B^{2}\geq|\frac{1}{2}\langle[A,B]\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-\langle
A\rangle\langle B\rangle|^{2},$ (4)

where the averages and the variances are defined over the state of the system
$|\Psi\rangle$. This relation is a direct consequence of the Cauchy-Schwarz
inequality. However, this uncertainty bound is not optimal. There have been
several attempts to tighten the bound, for some examples see [6, 7]. Here, we
state a tighter bound obtained which can be expressed as

$\displaystyle\Delta A^{2}\Delta
B^{2}\geq\max_{\\{|\psi_{n}\rangle\\}}\frac{1}{2}|\langle[\bar{A},\bar{B}^{\psi}_{n}]_{\Psi}\rangle|^{2}+|\frac{1}{2}\langle\\{\bar{A},\bar{B}^{\psi}_{n}\\}_{\Psi}\rangle|^{2},$
(5)

where $\bar{B}^{\psi}_{n}=|\psi_{n}\rangle\langle\psi_{n}|(B-\langle
B\rangle)$, $\bar{A}=A-\langle A\rangle$ and $\\{|\psi_{n}\rangle\\}$ is the
eigenbasis of any observable other than $A$ and $B$ [7]. This uncertainty
relation was proved to be tighter than Robertson-Schrödinger uncertainty
relation and even outperforms the stronger uncertainty relations by Maccone-
Pati [6] in some cases. We will use this tighter uncertainty relation for
deriving a tighter quantum speed limit bound in the following sections. The
derivation of this type of tighter uncertainty relation for mixed states is
given in the next section.

##  III Tighter Uncertainty Relations for mixed quantum states

######  Theorem 1.

The tighter uncertainty relation for two non-commuting operators A and B for
the mixed quantum state $\rho$ is given by the following inequality

$\displaystyle\Delta A\Delta
B\geq(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|})\geq|\mathrm{Tr}(\bar{A}\rho\bar{B})|,$

where

$\displaystyle\bar{A}=A-\mathrm{Tr}(\rho
A)\mathbb{I},~{}\bar{B}=B-\mathrm{Tr}(\rho
B)\mathbb{I},~{}\bar{B}^{\psi}_{n}=|\psi_{n}\rangle\langle\psi_{n}|\bar{B},$

and $\\{|\psi_{n}\rangle\\}$ form a complete orthonormal basis.

###### Proof.

For proving the tighter uncertainty relation for mixed states, we define the
following quantities:

$\displaystyle f$ $\displaystyle=$
$\displaystyle\bar{A}\rho\bar{A}=\sum_{m,n}\alpha_{m,n}|\psi_{m}\rangle\langle\psi_{n}|,$
$\displaystyle g$ $\displaystyle=$
$\displaystyle\bar{B}\rho\bar{B}=\sum_{i,j}\beta_{i,j}|\psi_{i}\rangle\langle\psi_{j}|,$
(6)

where $\bar{A}$ and $\bar{B}$ are as defined above, and $\alpha_{m,n}$ and
$\beta_{i,j}$ are complex numbers. We will now prove that the operators $f$
and $g$ are Hermitian with non-negative eigenvalues and therefore are positive
operators. We prove it explicitly for $f$ and the same follows for $g$. Let us
define an operator $F$ as follows

$\displaystyle F=\bar{A}\sqrt{\rho},$ (7)

where we have taken the positive square root of $\rho$ without any loss of
generality. Using the above equation then we get

$\displaystyle
FF^{\dagger}=(\bar{A}\sqrt{\rho})(\bar{A}\sqrt{\rho})^{\dagger}=(\bar{A}\sqrt{\rho})(\sqrt{\rho}^{\dagger}\bar{A}^{\dagger}).$
(8)

This implies that

$\displaystyle
FF^{\dagger}=(\bar{A}\sqrt{\rho})(\sqrt{\rho}\bar{A})=\bar{A}\rho\bar{A}=f.$
(9)

where we have used the fact that $\bar{A}$ $\sqrt{\rho}$ are Hermitian
operators. The hermiticity of $\sqrt{\rho}$ can be proved as follows. Let $U$
be the unitary that diagonalises $\rho$ as follows

$\displaystyle\rho=U\rho_{d}U^{\dagger}\Rightarrow\sqrt{\rho}=U\sqrt{\rho_{d}}U^{\dagger},$
(10)

Using the above equation then we see that

$\displaystyle(\sqrt{\rho})^{\dagger}$ $\displaystyle=$
$\displaystyle(U\sqrt{\rho_{d}}U^{\dagger})^{\dagger}=U\sqrt{\rho_{d}}U^{\dagger}$
(11) $\displaystyle=$ $\displaystyle\sqrt{\rho}.$

Therefore, $\sqrt{\rho}$ is a Hermitian operator. Now, using the hermiticity
property of $F$ and Eq.(9) we see that $f$ is also Hermitian as

$\displaystyle
f^{\dagger}=(FF^{\dagger})^{\dagger}=(F^{\dagger})^{\dagger}F^{\dagger}=FF^{\dagger}=f.$
(12)

Now we analyze the eigenvalues of the operator $f=FF^{\dagger}$. For this let
$F$ be diagonalizable in the basis $|k\rangle$ with eigenvalues $F_{k}$, such
that we have the following equation

$\displaystyle F=\sum_{k}F_{k}|k\rangle\langle k|.$ (13)

Then we have the eigenvalues of the operator $f$ from the following equation

$\displaystyle f=FF^{\dagger}=\sum_{i}|F_{i}|^{2}|i\rangle\langle i|.$ (14)

Thus, from the above equation we see that $f$ is diagonalized in the same
basis as $F$ and all is eigenvalues $|F_{i}|^{2}$ are real and positive
semidefinite. Now, we know that if all the eigenvalues of a self adjoint or
Hermitian operator are positive semidefinite, then that operator is positive
semidefinite. Now from the above observations, we note the following
properties about the nature of $\alpha_{mn}$ and $\beta_{ij}$. Note that we
have proved that the operators $f$ and $g$ are positive operators in general.
Thus, we know from the definition of positive operators that the following is
true of $f$ being a positive operator such that $\langle x,fx\rangle>0$ for
all $x\neq 0$. From this definition, we have
$\langle\psi_{n}|f|\psi_{n}\rangle>0$ and this implies that $\alpha_{n,n}>0$.
Similarly, $\beta_{m,m}>0$ as well. Keeping in mind that these properties
hold, we now move on to prove the uncertainty tighter uncertainty relation for
mixed quantum states. From our definitions we get the following

$\displaystyle\Delta A^{2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}(A^{2}\rho)-\mathrm{Tr}(A\rho)^{2}=\mathrm{Tr}(\bar{A}^{2}\rho)=\mathrm{Tr}(f)$
$\displaystyle\Delta B^{2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}(B^{2}\rho)-\mathrm{Tr}(B\rho)^{2}=\mathrm{Tr}(\bar{B}^{2}\rho)=\mathrm{Tr}(g).$

Therefore, we have $\Delta A\Delta B=\sqrt{\mathrm{Tr}(f)\mathrm{Tr}(g)}$.
Using the definition of $f$ and $g$ we have

$\displaystyle\mathrm{Tr}(f)\mathrm{Tr}(g)=\sum_{mn}\alpha_{n,n}\beta_{m,m}.$
(15)

Now we know from the structure of $\\{f,g\\}$ that $\alpha_{n,n}$ and
$\beta_{m,m}$ are real positive numbers in general. Therefore, using the
Cauchy-Schwarz inequality for two real positive vectors
$\\{|\alpha_{1}|,|\alpha_{2}|...,|\alpha_{n}|\\}$ and
$\\{|\beta_{1}|,|\beta_{2}|...,|\beta_{n}|\\}$ we have the following
inequality

$\displaystyle\mathrm{Tr}(f)\mathrm{Tr}(g)=\sum_{mn}\alpha_{n,n}\beta_{m,m}\geq(\sum_{n}|\sqrt{\alpha_{n,n}}\sqrt{\beta_{n,n}}|)^{2}.$

Now, using Eq.(6) we get the following inequality

$\displaystyle\mathrm{Tr}(f)\mathrm{Tr}(g)$ $\displaystyle\geq$
$\displaystyle(\sum_{n}\sqrt{|\langle\psi_{n}|\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|\bar{B}\rho\bar{B}|\psi_{n}\rangle|})^{2}$
$\displaystyle=$
$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|\bar{B}\rho\bar{B}|\psi_{n}\rangle\langle\psi_{n}|)|})^{2}$
$\displaystyle=$
$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n}|)|})^{2},$

where we have defined the operator
$\bar{B}^{\psi}_{n}=|\psi_{n}\rangle\langle\psi_{n}|\bar{B}$. Therefore, we
get the following as the mixed state version of the tighter uncertainty
relation

$\displaystyle\Delta A\Delta
B\geq(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|}).$
(16)

The above equation holds true whichever way we define $\bar{B}^{\psi}_{n}$,
i.e., either as $|\psi_{n}\rangle\langle\psi_{n}|\bar{B}$, or as
$\bar{B}|\psi_{n}\rangle\langle\psi_{n}|$, which is straightforward to deduce
from the above equations. Let us again consider the bound given in Eq.(16)

$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|})$
$\displaystyle=$
$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|\bar{B}\rho\bar{B}|\psi_{n}\rangle\langle\psi_{n}|})$
$\displaystyle=$
$\displaystyle(\sum_{n}\sqrt{|\langle\psi_{n}|\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|\bar{B}\rho\bar{B}|\psi_{n}\rangle|})$
$\displaystyle=$
$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|)||\mathrm{Tr}((\bar{B}\rho\bar{B}|\psi_{n}\rangle\langle\psi_{n}|}).$

Now using Cauchy-Schwarz inequality for complex matrices, we get

$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}|\psi_{n}\rangle\langle\psi_{n}|)||\mathrm{Tr}((\bar{B}\rho\bar{B}|\psi_{n}\rangle\langle\psi_{n}|)}$
(17) $\displaystyle\geq$
$\displaystyle\sum_{n}\sqrt{(|\mathrm{Tr}(\langle\psi_{n}|\bar{A}\sqrt{\rho}\sqrt{\rho}\bar{B}|\psi_{n}\rangle)|^{2})}$
$\displaystyle=$
$\displaystyle(\sum_{n}|\mathrm{Tr}(\langle\psi_{n}|\bar{A}\rho\bar{B}|\psi_{n}\rangle)|)\geq|(\sum_{n}\mathrm{Tr}(\langle\psi_{n}|\bar{A}\rho\bar{B}|\psi_{n}\rangle))|$
$\displaystyle\geq$
$\displaystyle|\mathrm{Tr}(\sum_{n}\bar{A}\rho\bar{B}|\psi_{n}\rangle\langle\psi_{n}|)|=|\mathrm{Tr}(\bar{A}\rho\bar{B})|.$

Therefore, we have proved the following for our tighter uncertainty relation
for mixed quantum states $\rho$.

$\displaystyle\Delta A\Delta
B\geq(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|})\geq|\mathrm{Tr}(\bar{A}\rho\bar{B})|.$
(18)

We know that the term on the right hand side gives us the bound given by the
Robertson-Schrodinger uncertainty relation. As a result therefore we have
shown that the new uncertainty relation derived here for mixed quantum states
outperforms the Robertson-Schrodinger uncertainty relation for mixed quantum
states. We will now show that the above uncertainty relation reduces to the
tighter uncertainty relation for that of the pure states when we have
$\rho=|\Psi\rangle\langle\Psi|$. Using this $\rho$ in Eq.(18) we get

$\displaystyle\Delta A\Delta B$ $\displaystyle\geq$
$\displaystyle(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}|\Psi\rangle\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle\langle\Psi|\bar{B}^{\psi}_{n}|})$
$\displaystyle\geq$
$\displaystyle|\mathrm{Tr}(\bar{A}|\Psi\rangle\langle\Psi|\bar{B})|.$

Simplifying the above equation we get

$\displaystyle\Delta A\Delta
B\geq\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|\geq|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|.$
(19)

This is the tighter uncertainty relation for pure quantum states [7]. Thus, we
have proved that the tighter uncertainty relation for mixed quantum states
reduces to that of the pure quantum states under the right conditions. We can
now optimise Eq.(16) over the set $\\{\ket{\psi_{n}}\\}$

$\displaystyle\Delta A\Delta
B\geq\max_{\\{|\psi_{n}\rangle\\}}(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|}),$
(20)

where the expectation values are defined over the mixed state $\rho$. Thus, we
have proved the mixed state version of the tighter uncertainty relation. Now
we comment on how the same method can be used to derive the tighter quantum
speed limit for mixed quantum states as well. The essential method we have
used here is the Cauchy-Schwarz inequality for two ‘real’ vectors in one of
the steps. We also know that the derivation of the quantum speed limit for the
mixed quantum states by Ulhmann uses the Cauchy-Schwarz inequality in deriving
the main bound. We propose that if we use the Cauchy-Schwarz inequality for
two real vectors in place of the usual Cauchy-Schwarz inequality there, we
will get a tighter version of Ulhmann’s quantum speed limit bound for quantum
mixed states. However, we leave this direction for future research.

##  IV Tighter Quantum Speed Limit for pure quantum states

######  Theorem 2.

The time evolution of a quantum state under a unitary operation generated by a
Hamiltonian is bounded by the following inequality

$\displaystyle\tau\geq\frac{\hbar s_{0}(t)}{2\Delta H}+\frac{2}{\Delta
H}\int^{\tau}_{0}\frac{K(t)}{\sin s_{0}(t)}dt,$ (21)

where we have the following quantities

$\displaystyle s_{0}(t)=2\cos^{-1}|\langle\Psi(0)|\Psi(t)\rangle|,$
$\displaystyle\Delta
H^{2}=\langle\Psi(t)|H^{2}|\Psi(t)\rangle-\langle\Psi(t)|H|\Psi(t)\rangle^{2}~{}\mathrm{and}$
$\displaystyle
K(t)=(\sum_{n}|\langle\Psi(t)|\bar{A}\bar{B}^{\psi}_{n}|\Psi(t)\rangle|-|\langle\Psi(t)|\bar{A}\bar{B}|\Psi(t)\rangle|)\geq
0,$
$\displaystyle\mathrm{where}~{}~{}A=|\Psi(0)\rangle\langle\Psi(0)|~{}\mathrm{and}~{}~{}B=H.$
(22)

###### Proof.

Consider two non-commuting operators $A$ and $B$, the tighter uncertainty
relation then gives

$\displaystyle\Delta A^{2}\Delta B^{2}$
$\displaystyle\geq(\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|)^{2}$
$\displaystyle\Delta A\Delta B$
$\displaystyle\geq\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|,$
(23)

where $\bar{A}=A-\langle A\rangle$ and
$\bar{B}^{\psi}_{n}=|\psi_{n}\rangle\langle\psi_{n}|\bar{B}$, the average
values $\langle A\rangle$ and $\langle B\rangle$ of the Hermitian operators
$A$ and $B$, respectively, being defined with respect to the pure quantum
state $|\Psi\rangle$. Also, we have

$\displaystyle\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|\geq|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|.$
(24)

Now, we add and subtract $|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|$ to the RHS
of Eq.(23)

$\displaystyle\Delta A\Delta
B\geq(\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|-|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|)+|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|.$

We again note that the following equation holds

$|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|^{2}=\frac{1}{4}|\langle\Psi|[A,B]|\Psi\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-\langle
A\rangle\langle B\rangle|^{2}.$

Therefore, we have

$|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|\geq\frac{1}{2}|\langle\Psi|[A,B]|\Psi\rangle|.$

Using the above equation, we have the following uncertainty relation

$\displaystyle\Delta A\Delta
B\geq\frac{1}{2}|\langle\Psi|[A,B]|\Psi\rangle|+K(t),$ (25)

where
$K(t)=(\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|-|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|)\geq
0$ which is time dependent via its dependence on the time evolved quantum
state $|\Psi\rangle=|\Psi(t)\rangle=e^{-iHt}|\Psi(0)\rangle$. We will denote
$K$ for $K(t)$ in short and will use this notation in the coming sections. Let
us now consider the operators $A$ and $B$ as follows

$\displaystyle A=|\Psi(0)\rangle\langle\Psi(0)|~{}\mathrm{and}~{}~{}B=H.$ (26)

For the pure state projector $A=|\Psi(0)\rangle\langle\Psi(0)|$, we have
$\langle
A\rangle=|\langle\Psi(0)|\Psi(t)\rangle|^{2}=\cos^{2}\frac{s_{0}(t)}{2}$,
where $s_{0}(t)$ is called the Bargmann angle. Therefore, the variance of $A$
is given as

$\displaystyle\Delta A^{2}=\langle A^{2}\rangle-\langle
A\rangle^{2}=\frac{1}{4}\sin^{2}s_{0}(t).$ (27)

The range of $s_{0}(t)$ is taken to be from $0$ to $\frac{\pi}{2}$. Using the
equation of motion for the average of $A$, we have

$\displaystyle i\hbar\frac{d}{dt}\langle
A\rangle=\langle\Psi(t)|[A,H]|\Psi(t)]\rangle.$ (28)

Now, using the expectation value of $A$ in terms of the Bargmann angle we have

$\displaystyle|\frac{d\langle A\rangle}{dt}|=\frac{1}{2}\sin
s_{0}(t)\frac{ds_{0}}{dt}.$ (29)

Therefore, putting the values of $\Delta A$ and $\Delta B$ explicitly in
Eq.(25) and using Eq.(29) we get

$\displaystyle\frac{1}{2}\sin s_{0}(t)\Delta H\geq\frac{\hbar}{4}\sin
s_{0}(t)\frac{ds_{0}}{dt}+K(t).$ (30)

Now, integrating the above equation with respect to time we obtain the new
tighter quantum speed limit bound as given by

$\displaystyle\tau\geq\frac{\hbar s_{0}(t)}{2\Delta H}+\frac{2}{\Delta
H}\int^{\tau}_{0}\frac{K(t)}{\sin s_{0}(t)}dt.$ (31)

The first term on the RHS is the MT bound and the second term on the right is
always positive, therefore the above equation gives a quantum speed limit
always tighter than the MT bound. We also expect the above bound to perform
better than the standard quantum speed limit since we have used the tighter
uncertainty relation to derive the new quantum speed limit bound above. Note
that the maximized or optimized speed limit is obtained by optimizing over the
choice of complete basis vectors $\\{|\psi_{n}\rangle\\}$ for $n=1,...,d$ as
follows

$\displaystyle\tau\geq\max_{\\{\ket{\psi_{n}}\\}}\big{[}\frac{\hbar
s_{0}(t)}{2\Delta H}+\frac{2}{\Delta H}\int^{\tau}_{0}\frac{K(t)}{\sin
s_{0}(t)}dt\big{]}.$ (32)

Thus, Eq. (31) and Eq. (32) constitute tighter quantum speed limits for
arbitrary unitary evolutions of pure quantum states. The standard QSL such as
the MT bound follows as a special case of the new bound.

######  Proposition 1.

K(t) is positive semidefinite.

###### Proof.

The expression for $K(t)$ is given as follows:

$\displaystyle
K(t)=(\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|-|\langle\Psi|\bar{A}\bar{B}|\Psi\rangle|),$
(33)

where we have $\bar{A}=A-\langle A\rangle$, $\bar{B}=B-\langle B\rangle$ and
$\bar{B}^{\psi}_{n}=|\psi_{n}\rangle\langle\psi_{n}|\bar{B}$. Using the fact
that the sum of the absolute values of complex numbers is greater than or
equal to the absolute values of the sum of the complex numbers we have

$\displaystyle(\sum_{n}|\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|)$
$\displaystyle\geq|(\sum_{n}\langle\Psi|\bar{A}\bar{B}^{\psi}_{n}|\Psi\rangle|$
$\displaystyle=|(\langle\Psi|\bar{A}\sum_{n}(\bar{B}^{\psi}_{n})|\Psi\rangle|$
$\displaystyle=(\langle\Psi|\bar{A}\sum_{n}(|\psi_{n}\rangle\langle\psi_{n}|\bar{B})|\Psi\rangle|$
$\displaystyle=(\langle\Psi|\bar{A}\bar{B})|\Psi\rangle|,$ (34)

where we have used the completeness relation
$\sum_{n}|\psi_{n}\rangle\langle\psi_{n}|=I$. Therefore, using the above
inequality, we get $K(t)\geq 0$.

##  V Tighter quantum speed limit for mixed quantum states

######  Theorem 3.

Tighter quantum speed limit for the case of mixed state $\rho$ is given by the
following inequality

$\displaystyle\tau\geq\Big{[}\hbar\big{(}\frac{\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})})-\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})})}{\Delta
H}\big{)}+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt\Big{]},$
$\displaystyle\mathrm{and~{}the~{}optimized~{}version~{}is~{}given~{}as}$
$\displaystyle\tau\geq\max_{\\{\ket{\psi_{n}}\\}}\Big{[}\hbar\big{(}\frac{\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})})-\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})})}{\Delta
H}\big{)}+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt\Big{]},$
$\displaystyle\mathrm{where~{}we~{}have~{}the~{}following~{}definitions}$
$\displaystyle
s_{0}(t)=2\arccos{\sqrt{\frac{\mathrm{Tr}(\rho_{0}\rho_{t})}{\mathrm{Tr}(\rho_{0}^{2})}}},~{}\Delta
H^{2}=\mathrm{Tr}(\rho H^{2})-(\mathrm{Tr}(\rho
H))^{2},~{}K(t)=(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{\rho}_{0}\rho_{t}\bar{\rho}_{0}\bar{H}^{\psi}_{n}\rho_{t}\bar{H}^{\psi}_{n})|})-|\mathrm{Tr}(\bar{\rho}_{0}\rho_{t}\bar{H})|.$
(35)

###### Proof.

Consider two non-commuting operators $A$ and $B$, the tighter uncertainty
relation for a mixed state $\rho$ using Eq.(16) is given by

$\displaystyle\Delta A\Delta
B\geq(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|}).$
(36)

Now adding and subtracting $|\mathrm{Tr}(\bar{A}\rho\bar{B})|$ to the RHS of
the above equation, we get

$\displaystyle\Delta A\Delta B\geq$
$\displaystyle\Big{[}(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\psi}_{n}\rho\bar{B}^{\psi}_{n})|})-|\mathrm{Tr}(\bar{A}\rho\bar{B})|\Big{]}$
$\displaystyle+|\mathrm{Tr}(\bar{A}\rho\bar{B})|.$ (37)

We will now analyze the term $|\mathrm{Tr}(\bar{A}\rho\bar{B})|$. For this we
use a more convenient notation as
$|\mathrm{Tr}(\bar{A}\rho\bar{B})|=|\langle\bar{A}\bar{B}\rangle|$. We note
that the following equation holds for all mixed quantum states and where the
expectation values denoted by the angled brackets are with respect to the
mixed quantum state $\rho$, i.e.,

$\displaystyle|\langle\bar{A}\bar{B}\rangle|^{2}=\frac{1}{4}|\langle[A,B]\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-2\langle
A\rangle\langle B\rangle|^{2}.$

Since both the terms on the R.H.S are positive, we have

$\displaystyle|\langle\bar{A}\bar{B}\rangle|\geq\frac{1}{2}|\langle[A,B]\rangle|.$
(38)

Using Eq.(37) and the inequality from the above equation we get

$\displaystyle\Delta A\Delta B\geq\frac{1}{2}|\langle[A,B]\rangle|+K(t),$ (39)

where
$K(t)=\Big{[}(\sum_{n}\sqrt{|\mathrm{Tr}(\bar{A}\rho\bar{A}\bar{B}^{\Psi}_{n}\rho\bar{B}^{\Psi}_{n})|})-|\mathrm{Tr}(\bar{A}\rho\bar{B})|\Big{]}$
is positive semidefinite using Eq.(18). Let us now take the operators $A$ and
$B$ as follows $A=\rho(0)$ and $B=H$, and
$\rho=\rho(t)=e^{-iHt}\rho(0)e^{iHt}$. The variance of the operator $A$ is
then given by

$\displaystyle\Delta A^{2}$
$\displaystyle=\mathrm{Tr}(\rho(0)^{2}\rho(t))-(\mathrm{Tr}(\rho(0)\rho(t)))^{2}$
$\displaystyle=\mathrm{Tr}(\rho_{0}^{2}\rho_{t})-(\mathrm{Tr}(\rho_{0}\rho_{t}))^{2},$
(40)

where we have used the notation $\rho(0)=\rho_{0}$ and $\rho(t)=\rho_{t}$. We
can now take the following parametrization

$\displaystyle\langle
A\rangle=\mathrm{Tr}(\rho(0)\rho(t))=\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2}.$
(41)

Now, using the equation of motion for the average of $A$, we get

$\displaystyle|\hbar\frac{d}{dt}\langle A\rangle|=|\langle[A,H]\rangle|,$

where the averages are all with respect to the mixed quantum state $\rho$ and
$A$ has no explicit time dependence. Using Eq.(41), we get

$\displaystyle|\frac{d\langle
A\rangle}{dt}|=\mathrm{Tr}(\rho_{0}^{2})\frac{\sin
s_{0}(t)}{2}\frac{ds_{0}}{dt}$ (42)

Therefore, putting the values of $A$ and $B$ explicitly in the above derived
equations we get

$\displaystyle\Delta A\Delta H\geq\mathrm{Tr}(\rho_{0}^{2})\frac{\hbar\sin
s_{0}(t)}{2}\frac{ds_{0}}{dt}+K(t).$ (43)

Now let us analyze the structure of $\Delta A^{2}$ as follows

$\displaystyle\Delta
A^{2}=\mathrm{Tr}(\rho_{0}^{2}\rho_{t})-(\mathrm{Tr}(\rho_{0}\rho_{t}))^{2}.$
(44)

Let $\\{\ket{k}\\}$ be the eigenbasis of $\rho_{0}$ then we have

$\displaystyle\rho_{0}=\sum_{k}\lambda_{k}|k\rangle\langle
k|~{}\mathrm{and}~{}\rho_{0}^{2}=\sum_{k}\lambda_{k}^{2}|k\rangle\langle k|.$
(45)

Using the above equation then we have the following quantities

$\displaystyle\mathrm{Tr}(\rho_{0}\rho_{t})$
$\displaystyle=\sum_{k}\lambda_{k}\langle k|\rho_{t}|k\rangle~{}\mathrm{and}$
$\displaystyle\mathrm{Tr}(\rho_{0}^{2}\rho_{t})$
$\displaystyle=\sum_{k}\lambda_{k}^{2}\langle k|\rho_{t}|k\rangle.$ (46)

Since, we know that $0\leq\lambda_{k}^{2}\leq\lambda_{k}\leq 1~{}\forall~{}k$
and also $\langle k|\rho_{t}|k\rangle\geq 0~{}\forall~{}k$ because $\rho_{t}$
is a positive operator. Therefore, we get

$\displaystyle\mathrm{Tr}(\rho_{0}\rho_{t})\geq\mathrm{Tr}(\rho_{0}^{2}\rho_{t}).$
(47)

Adding $-(\mathrm{Tr}(\rho_{0}\rho_{t}))^{2}$ on both side of the above
equation we get

$\displaystyle\mathrm{Tr}(\rho_{0}\rho_{t})-(\mathrm{Tr}(\rho_{0}\rho_{t}))^{2}$
$\displaystyle\geq\mathrm{Tr}(\rho_{0}^{2}\rho_{t})-(\mathrm{Tr}(\rho_{0}\rho_{t}))^{2}$
$\displaystyle=\Delta A^{2}.$ (48)

Now, using Eq.(41) we get

$\displaystyle\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2}(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})\geq\Delta
A^{2}$ (49)

Taking square root on both sides and multiplying by $\Delta H$ we get

$\displaystyle\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}\Delta
H\geq\Delta A\Delta H.$ (50)

The above inequality using Eq.(43) becomes

$\displaystyle\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}\Delta
H$ $\displaystyle\geq\mathrm{Tr}(\rho_{0}^{2})\frac{\hbar\sin
s_{0}(t)}{2}\frac{ds_{0}}{dt}+K(t).$ (51)

Now, integrating the above equation with respect to t and $s_{0}(t)$ we obtain
the new quantum speed limit bound for mixed quantum states as follows

$\displaystyle\tau\geq\frac{\hbar\sqrt{\mathrm{Tr}(\rho_{0}^{2})}}{2\Delta
H}\int_{s_{0}(0)}^{s_{0}(\tau)}\frac{\sin
s_{0}(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}ds_{0}$
$\displaystyle+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt.$

The first term on the right hand side can be integrated in the analytical
form, so that we get the following relation

$\displaystyle\tau$
$\displaystyle\geq\frac{\hbar\sqrt{\mathrm{Tr}(\rho_{0}^{2})}}{2\Delta
H}\Big{[}-2\frac{\sin^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\cos\frac{s_{0}(t)}{2})}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}}\Big{]}_{s_{0}(0)}^{s_{0}(\tau)}$
$\displaystyle+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt.$

Now, we know that $\cos\frac{s_{0}(0)}{2}=1$ and
$\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\cos\frac{s_{0}(\tau)}{2}=\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})}$.
Thus, putting these value in the above equation and simplifying, we get

$\displaystyle\tau$ $\displaystyle\geq\frac{\hbar}{\Delta
H}\Big{[}\sin^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})})-\sin^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})})\Big{]}$
$\displaystyle+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt.$

As before, $K(t)$ is always greater than or equal to zero in all cases. Other
than that, the maximized version follows in the same way as in the case of
mixed states without any further need of extra steps. Let us now check the
bound for the case of pure quantum states as follows

$\displaystyle\rho_{0}=|\Psi(0)\rangle\langle\Psi(0)|,~{}\rho_{t}=|\Psi(t)\rangle\langle\Psi(t)|.$
(52)

In this case, our bound becomes the following

$\displaystyle\tau\geq\frac{\hbar}{\Delta
H}\big{(}\sin^{-1}(1)-\sin^{-1}(\cos\frac{s_{0}(\tau)}{2})\big{)}+\frac{2}{\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\sin s_{0}(t)}dt.$

Using $\sin^{-1}(1)=\frac{\pi}{2}$ and the following inverse trigonometric
identity

$\displaystyle\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2},~{}\forall~{}x~{}\in[-1,1],$
(53)

we get the following

$\displaystyle\tau\geq\Big{[}\hbar\big{(}\frac{\cos^{-1}(\cos\frac{s_{0}(t)}{2})}{\Delta
H}\big{)}\Big{]}+\frac{2}{\Delta H}\int_{0}^{\tau}\frac{K(t)}{\sin
s_{0}(t)}dt.$

Therefore, this gives us the following quantum speed limit for pure quantum
states

$\displaystyle\tau\geq\frac{\hbar s_{0}(t)}{2\Delta H}+\frac{2}{\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\sin s_{0}(t)}dt.$

Thus, the quantum speed limit bound for mixed quantum states reduces to that
of the pure quantum states derived from the tighter uncertainty relation for
pure quantum states in the appropriate limit. Note that in a more conventional
notation using the trigonometric identity of inverses of $\sin$ and $\cos$
functions, we can write the bound as follows

$\displaystyle\tau$
$\displaystyle\geq\Big{[}\hbar\big{(}\frac{\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})})-\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})})}{\Delta
H}\big{)}$ $\displaystyle+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt\Big{]}.$

The optimized version can be expressed as

$\displaystyle\tau$
$\displaystyle\geq\max_{\\{\ket{\psi_{n}}\\}}\Big{[}\hbar\big{(}\frac{\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}\rho_{t})})-\cos^{-1}(\sqrt{\mathrm{Tr}(\rho_{0}^{2})})}{\Delta
H}\big{)}$ $\displaystyle+\frac{1}{\sqrt{\mathrm{Tr}(\rho_{0}^{2})}\Delta
H}\int_{0}^{\tau}\frac{K(t)}{\cos\frac{s_{0}(t)}{2}\sqrt{(1-\mathrm{Tr}(\rho_{0}^{2})\cos^{2}\frac{s_{0}(t)}{2})}}dt\Big{]}.$

Therefore, the quantum speed limit for mixed quantum states derived from the
tighter uncertainty relation for mixed quantum states is another important
result derived in the paper. The performance of this bound depends on the
value of the second integration and it cannot be said a priori in a
straightforward way in which regions this will give better bounds than the
other existing bounds in the literature. As a result, we leave this direction
for future research. Next, we demonstrate the better performance of our bounds
over the MT bound with some examples in the case of pure quantum states.

##  VI Examples

In this section, we illustrate the tighter QSL for few examples where we see
dramatic improvement over the standard QSL such as the MT bound. In the first
example, we discuss the tighter QSL for quantum sytsem whose dynamics is
governed by random Hamiltonians. In the second example, we discuss the tighter
QSL for interacting systems of spins.

###  VI.1 Tighter QSL with random Hamiltonians

In this subsection, we calculate and compare the tighter quantum speed limit
bound with that of the MT bound using random Hamiltonians from the Gaussian
Unitary ensemble (GUE). Random Hamiltonians drawn from one of the random
matrix ensembles such as the GUE can be applicable to a large class of
physically important models in quantum physics, quantum information and
computation where long-range interactions are important. Recently, the random
Hamiltonian setup as in the models of random quantum circuits have been used
for analyzing features of quantum entanglement [48], universal properties of
the out-of-time correlation function [49, 50, 51, 52], quantum entanglement
tsunami [53], unitary design [54] and also measurement induced phase
transitions [55]. As a result, due to such physical applicability we study the
performance of the tighter quantum speed limit bound for the random
Hamiltonians.

Here, we state how we draw the random Hamiltonians and its mathematical
properties. A random Hamiltonian is a $D\mathrm{x}D$ Hermitian operator $H$
drawn from a Gaussian Unitary Ensemble (GUE), described by the following
probability distribution function $P(H)=Ce^{-\frac{D}{2}\mathrm{Tr}(H^{2})}$,
where $C$ is the normalization constant and the elements of $H$ are drawn from
the Gaussian probability distribution. In this way $H$ is also Hermitian. A
random Hamiltonian dynamics is a unitary time-evolution generated by a fixed
(time-independent) GUE Hamiltonian. The matrices of this form can be drawn
from mathematica in a straightforward way using routine procedure.

For the example considered here, we take the Hilbert space of dimension 3. The
initial state is taken as
$|\Psi(0)\rangle=\sqrt{0.1}|0\rangle+\sqrt{0.2}|1\rangle+\sqrt{0.7}|2\rangle$.
The random eigenbasis is taken as the set of eigenvectors of a random
Hermitian operator obtained in the same way as the Hamiltonian, i.e., from the
Gaussian Unitary Ensemble. In Figure 1, we plot $\Delta=\tau_{tqsl}-\tau_{MT}$
vs the actual time $t$ of evolution, for 3 different random Hamiltonians. It
is clear that $\Delta$ is always positive, showing that the tighter quantum
speed limit bound always outperforms the MT bound. At $t=0$, all the values of
$\Delta$ are zero because all the random Hamiltonians start with being
identity at $t=0$. All the Hamiltonians taken here are time independent. As a
result, we are able to show that with these Hamiltonians our bound outperforms
the MT bound.

Figure 1: The difference $\Delta$ between the tighter quantum speed limit
$\tau_{tqsl}$ and the MT bound $\tau_{MT}$, obtained for 3 different random
Hamiltonians obtained from a Gaussian Unitary Ensemble. As expected from the
theory, the tighter quantum speed limit bound outperforms the MT bound always
for these random Hamiltonians. The same holds for many other random
Hamiltonian obtained in the same way from the Gaussian Unitary Ensemble. The
initial state is taken as
$\sqrt{0.1}|0\rangle+\sqrt{0.2}|1\rangle+\sqrt{0.7}|2\rangle$ in the three
dimensional Hilbert space.

###  VI.2 Interacting quantum systems of spins

In this subsection, we work out another example to illustrate our bound. We
choose the basis vectors to be from any Hermitian operator chosen from a GUE.
We consider a chain of $M$ spins that evolve under the Hamiltonian as follows
$H=\sum_{i=1}^{M}H_{i}+H_{int}$,

Figure 2: The difference $\Delta$ between the tighter quantum speed limit
$\tau_{tqsl}$ and the MT bound $\tau_{MT}$ obtained for the example in section
VI (B).

where we have $H_{i}=\hbar\omega_{0}(1-\sigma_{x}^{i})$ [15, 16]. Here,
$H_{i}$ is the local Hamiltonian that evolves the individual spin systems
independent of each other, whereas $H_{int}$ acts on subsystems jointly. Here,
we assume that the interaction takes place in each of the $Q$ number of blocks
present in the spin chain. Each block consists of $K$ number of spins. The $K$
spins in the $j^{th}$ block interact through the Hamiltonian $H_{j}$. We,
therefore, have the following [16]

$\displaystyle
H_{j}=\hbar\omega(1-S_{j}),~{}~{}H_{int}=\hbar\omega\sum_{j=1}^{Q}(1-S_{j}),$
$\displaystyle\mathrm{where}~{}~{}S_{j}=\sigma_{x}^{i_{1_{j}}}\otimes\sigma_{x}^{i_{2_{j}}}\otimes....\otimes\sigma_{x}^{i_{k_{j}}},j=1,2,...,Q.$
(54)

Under this Hamiltonian, starting from a completely product state
$|\Psi(0)\rangle$, the time evolved quantum state is of the following form

$\displaystyle|\Psi(t)\rangle=$ $\displaystyle
C\bigotimes_{i=1}^{M}(\cos\omega_{0}t+i\sigma_{x}^{i}\sin\omega_{0}t)$
$\displaystyle\Pi_{j=1}^{Q}(\cos\omega t+iS_{j}\sin\omega t)|\Psi(0)\rangle,$

where $C=e^{-i(\omega+\omega_{0})t}$. For our case, we take two qubit system
and the initial state as the product state $|0\rangle|0\rangle$, where
$|0\rangle$ is the eigenstate of the operator $\sigma_{z}$. Also, we take the
simplest case of a single block. We take this state as the initial state and
evolve it under the Hamiltonian as stated above. The random eigenbasis is
again taken as the set of eigenvectors of a random Hermitian operator obtained
from the Gaussian Unitary Ensemble. In Figure 2, we plot
$\Delta=\tau_{tqsl}-\tau_{MT}$ vs the actual time $t$ of evolution. The figure
clearly shows that our bound performs better than the MT bound.

##  VII Conclusions and Future Directions

In this work, we have derived a tighter quantum speed limit. We first derived
the mixed state generalization of the tighter uncertainty relation of pure
quantum states. Using the tighter uncertainty relations for the pure and mixed
quantum states, we have derived the tighter quantum speed limit bounds for
pure and mixed quantum states, respectively. We have shown that the new bound
performs better than the MT bound. Also, the tighter quantum speed limit bound
has been shown to coincide with that of the pure quantum states in the
appropriate limit. Hereafter, we have shown numerically using random
Hamiltonians obtained from Gaussian Unitary Ensemble that our bound performs
better than the MT speed limit bound. Since, we have shown that our bound is
always better than the MT bound in all cases, therefore, all the cases where
the MT bound performs better than the Margolus-Levitin bound, our new bound
also performs better than the Margolus- Levitin bound in those cases. It
remains a subject of future investigation to compare our bound with the
Margolus-Levitin bound where MT bound is weaker than the Margolus-Levitin
bound. In future, it will be interesting to see how much the speed limit
bounds can be improved by optimization over different choice of sets of
orthonormal bases and which basis set will be the optimal one in deriving the
optimal quantum speed limit bound. Also, one could generalize our tighter
quantum speed limit bound to the case of mixed quantum states for open systems
dynamics.

We believe that the tighter quantum speed limit derived here will have
important applications in quantum computation, quantum information and quantum
control.

## ACKNOWLEDGEMENTS

S. B. acknowledges support from the National Research Foundation of Korea
(2020M3E4A1079939, 2022M3K4A1094774) and the KIST institutional program
(2E31531). A.S. acknowledges the support from the INFOSYS scholarship. A. K.
P. acknowledges the support from the QUEST Grant Q-117 and J C Bose grant from
the Department of Science and Technology, India.

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