# Safety Verification of Neural Network Control Systems Using Guaranteed
Neural Network Model Reduction

Weiming Xiang and Zhongzhu Shao This research was supported by the National
Science Foundation, under NSF CAREER Award 2143351 and NSF CNS Award
2223035.Weiming Xiang is with School of Computer and Cyber Sciences, Augusta
University, Augusta GA 30912, USA<EMAIL_ADDRESS>Shao is with
School of Electrical Engineering, Southwest Jiaotong University, China.

###### Abstract

This paper aims to enhance the computational efficiency of safety verification
of neural network control systems by developing a guaranteed neural network
model reduction method. First, a concept of model reduction precision is
proposed to describe the guaranteed distance between the outputs of a neural
network and its reduced-size version. A reachability-based algorithm is
proposed to accurately compute the model reduction precision. Then, by
substituting a reduced-size neural network controller into the closed-loop
system, an algorithm to compute the reachable set of the original system is
developed, which is able to support much more computationally efficient safety
verification processes. Finally, the developed methods are applied to a case
study of the Adaptive Cruise Control system with a neural network controller,
which is shown to significantly reduce the computational time of safety
verification and thus validate the effectiveness of the method.

##  I Introduction

Neural networks are currently widely used in various fields, such as image
processing [1], pattern recognition [2], adaptive control [3], unmanned
vehicles [4] and aircraft collision avoidance systems [5], etc., demonstrating
their powerful capabilities in solving complex and challenging problems that
traditional approaches fail to address. As neural networks are further
investigated, the size and complexity of their models continue to increase in
order to improve their performance and accuracy to cope with complex and
difficult tasks and changing environments. However, more complex large-scale
neural network models also imply larger computational resources, such as
larger memory, higher computational power and more energy consumption in
applications [6]. As a result, many neural network model reduction methods
have been developed, such as parameter pruning and sharing, low-rank
factorization, transfer/compact convolution filters, and knowledge
distillation [7]. More results on neural network model reduction can be found
in a recent survey [8].

On the other hand, due to the black-box nature of neural networks, neural
networks are vulnerable in the face of resistance to interference/attacks. It
has been observed that neural networks trained on large amounts of data are
sometimes sensitive to updates and react to even small changes in parameters
in unexpected and incorrect ways [9]. When neural networks are applied as
controllers onto dynamical systems, they will inevitably suffer from safety
problems due to the inevitable disturbances and uncertainties in the control
process, further affecting the stability and safety of the whole closed-loop
system. Therefore, when integrating neural networks into safety-critical
control systems, the safety of the neural network needs to be guaranteed at
all times, i.e., the safety verification of the neural network needs to be
implemented. However, due to the sensitivity of neural networks to
perturbations and the complex structure of neural networks, the verification
of neural networks is extremely difficult. It has been demonstrated that the
verification of simple properties of a small-scale neural network is an
uncertainty polynomial (NP) complete problem [10]. A few results have been
reported in the literature for the formal verification of systems consisting
of neural networks, readers are referred to the recent survey [11].
Specifically, reachability analysis is one of the promising safety
verification tools such as in [12, 13, 14, 15, 16], a simulation-based
approach is proposed that transforms the difficulty of over-approximating the
neural network’s output set into a problem of estimating the neural network’s
maximal sensitivity, which is formulated as a series of convex optimization
problems [13, 14]. Polytope-operation-based approaches were developed in [12,
15, 16] for dealing with a class of neural networks with activation functions
of Rectified Linear Units (ReLU). However, the scalability issue is the major
barrier preventing applying these methods to large-scale neural networks as
well as neural network control systems active in a long period of time which
means a large amount of reachable set computation is required during the time
of interest.

In this paper, we propose a guaranteed model reduction method for neural
network controllers based on the neural network reachability analysis and
apply it to enhance the scalability of the reachability-based safety
verification of closed-loop systems. Firstly, a concept of model reduction
precision is proposed to accurately measure the distance between the outputs
of an original neural network and its reduced-size version, and an approach to
compute the model reduction precision is proposed, which ensures that the
difference between the outputs obtained from two neural networks for a given
input interval, chosen with any identical input, is within the model reduction
precision. This algorithm is then applied to the model reduction of the neural
network control system, enabling computationally efficient verification
processes based on the reduced-size neural network controller. Finally, the
correctness and feasibility of our approach are verified by applying it to the
safety verification through the Adaptive Cruise Control (ACC) case study.

The remainder of the paper is organized as follows: Preliminaries are given in
Section II. The guaranteed model reduction of neural networks is presented in
Section III. The reachable set computation and safety verification algorithm
for the neural network control system are presented in Section IV. The
evaluation on the adaptive cruise control system is given in Section V. The
conclusion is given in Section VI.

##  II Preliminaries

In this paper, we consider a class of continuous-time nonlinear systems in the
form of

$\displaystyle\begin{cases}\mathbf{\dot{x}}(t)=f(\mathbf{x}(t),\mathbf{u}(t))\\\
\mathbf{y}(t)=h(\mathbf{x}(t))\end{cases}$ (1)

where $\mathbf{x}(t)\in\mathbb{R}^{n_{x}}$ is the state vector,
$\mathbf{u}(t)\in\mathbb{R}^{n_{u}}$ is the control input and the
$\mathbf{y}(t)\in\mathbb{R}^{n_{y}}$ is the output vector. In general, the
control input is in the form of

$\displaystyle\mathbf{u}(t)=\gamma(\mathbf{y}(t),\mathbf{r}(t),t)$ (2)

where $\mathbf{r}(t)\in\mathbb{R}^{n_{r}}$ is the reference input for the
controller.

To avoid the difficulties in the controller design when system models are
complex or even unavailable, one effective method is to use input-output data
to train neural networks capable of generating appropriate control input
signals to achieve control objectives. The neural network controller is in the
form of

$\displaystyle\mathbf{u}(t)=\Phi(\mathbf{y}(t),\mathbf{r}(t))$ (3)

where $\Phi$ denotes the neural network mapping output and reference signals
to control input.

In actual applications, the neural network receives input and generates output
in a fraction of the computation time, so the control input generated by the
neural network is generally discrete, generated only at each sampling time
point $t_{k},k\in\mathbb{N}$, and then remains a constant value between two
successive sampling time instants. Therefore, the continuous-time nonlinear
dynamical system with a neural network controller with sampling actions can be
expressed in the following form of

$\displaystyle{\begin{cases}\mathbf{\dot{x}}(t)=f(\mathbf{x}(t),\Phi(\bm{\tau}(t_{k})))\\\
\mathbf{y}(t)=h(\mathbf{x}(t))\end{cases}},\quad t\in[t_{k},t_{k+1})$ (4)

where
$\bm{\tau}(t_{k})=[\mathbf{y}^{\top}(t_{k}),\mathbf{r}^{\top}(t_{k})]^{\top}$.

In this work, we consider feedforward neural networks for controllers in the
form of $\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$ defined by the
following recursive equations in the form of

$\displaystyle\begin{cases}\bm{\eta}_{\ell}=\phi_{\ell}(\mathbf{W}_{\ell}\bm{\eta}_{\ell-1}+\mathbf{b}_{\ell}),~{}\ell=1,\ldots,L\\\
\bm{\eta}_{L}=\Phi(\bm{\eta}_{0})\end{cases}$ (5)

where $\bm{\eta}_{\ell}$ denotes the output of the $\ell$-th layer of the
neural network, and in particular $\bm{\eta}_{0}\in\mathbb{R}^{n_{0}}$ is the
input to the neural network and $\bm{\eta}_{L}\in\mathbb{R}^{n_{L}}$ is the
output produced by the neural network, respectively.
$\mathbf{W}_{\ell}\in\mathbb{R}^{n_{\ell}\times n_{\ell-1}}$ and
$\mathbf{b}_{\ell}\in\mathbb{R}^{n_{\ell}}$ are weight matrices and bias
vectors for the $\ell$-th layer.
$\phi_{\ell}=[\psi_{\ell},\cdots,\psi_{\ell}]$ is the concatenation of
activation functions of the $\ell$-th layer in which
$\psi_{\ell}:\mathbb{R}\to\mathbb{R}$ is the activation function.

In this paper, we aim at reducing the computational cost of safety
verification of neural network control systems in the framework of reachable
set computation.

###### Definition 1

Given a neural network in the form of (5) and an input set $\mathcal{U}$, the
following set

$\displaystyle\mathcal{Y}=\left\\{\bm{\eta}_{L}\in\mathbb{R}^{n_{L}}\mid\bm{\eta}_{L}=\Phi(\bm{\eta}_{0}),~{}\bm{\eta}_{0}\in\mathcal{U}\right\\}$
(6)

is called the output set of neural network (5).

###### Definition 2

A set $\mathcal{Y}_{e}$ is called an output reachable set over-approximation
of neural network (5), if $\mathcal{Y}\subseteq\mathcal{Y}_{e}$ holds, where
$\mathcal{Y}$ is the output reachable set of neural network (5).

###### Definition 3

Given a neural network control system in the form of (1) and (3) with initial
set $\mathcal{X}_{0}$ and input set $\mathcal{V}$, the reachable set at time
$t$ is

$\mathcal{R}(t)=\left\\{\mathbf{x}(t;\mathbf{x}_{0},\mathbf{r}(\cdot))\in\mathbb{R}^{n_{x}}\mid\mathbf{x}_{0}\in\mathcal{X}_{0},~{}\mathbf{r}(t)\in\mathcal{V}\right\\}$
(7)

and the union of $\mathcal{R}(t)$ over $[t_{0},t_{f}]$ defined by

$\mathcal{R}([t_{0},t_{f}])=\bigcup\nolimits_{t\in[t_{0},t_{f}]}\mathcal{R}(t)$
(8)

is the reachable set over time interval $[t_{0},t_{f}]$.

###### Definition 4

A set $\mathcal{R}_{e}(t)$ is an over-approximation of $\mathcal{R}(t)$ at
time $t$ if $\mathcal{R}(t)\subseteq\mathcal{R}_{e}(t)$ holds. Moreover,
$\mathcal{R}_{e}([t_{0},t_{f}])=\bigcup\nolimits_{t\in[t_{0},t_{f}]}\mathcal{R}_{e}(t)$
is an over-approximation of $\mathcal{R}([t_{0},t_{f}])$ over time interval
$[t_{0},t_{f}]$.

###### Definition 5

Safety specification $\mathcal{S}$ formalizes the safety requirements for
state $\mathbf{x}(t)$ of neural network control system (1), and is a predicate
over state $\mathbf{x}(t)$ of neural network control system (1). The neural
network control system (1) is safe over time interval $[t_{0},t_{f}]$ if the
following condition is satisfied:

$\mathcal{R}_{e}([t_{0},t_{f}])\cap\neg\mathcal{S}=\emptyset$ (9)

where $\neg$ is the symbol for logical negation.

As indicated in [12, 13, 14, 15, 16], the computation cost for reachable set
computation heavily relies on the size of neural networks, i.e., numbers of
layers and neurons. In this paper, we aim to reduce the size of the neural
network controller and rigorously compute the model reduction error, i.e.,
guaranteed neural network model reduction, so that the reachable set
computation can be efficiently performed on a significantly reduced-size
neural network and then mapped back to the original neural network to reach
safety verification conclusions.

##  III Guaranteed Neural Network Model Reduction

Given a large-scale neural network $\Phi$, there exist a large number of
neural network model reduction methods as in survey paper [8] to obtain its
reduced-size version $\hat{\Phi}$ as below:

$\displaystyle\begin{cases}\hat{\bm{\eta}}_{\ell}=\hat{\phi}_{\ell}(\hat{\mathbf{W}}_{\ell}\hat{\bm{\eta}}_{\ell-1}+\hat{\mathbf{b}}_{\ell}),~{}\ell=1,\ldots,\hat{L}\\\
\hat{\bm{\eta}}_{L}=\hat{\Phi}(\hat{\bm{\eta}}_{0})\end{cases}$ (10)

To enable guaranteed neural network model reduction, the key is how to
rigorously compute the output difference between the original neural network
$\Phi$ and its reduced-size version $\hat{\Phi}$. Without loss of generality,
the following assumption is given for neural network $\Phi$ and its reduced-
size version $\hat{\Phi}$.

###### Assumption 1

The following assumptions hold for neural network $\Phi$ and its reduced-size
version $\hat{\Phi}$:

  1. 1.

The number of inputs of two neural networks are the same, i.e.,
$n_{0}=\hat{n}_{0}$;

  2. 2.

The number of outputs of two neural networks are the same, i.e.,
$n_{L}=n_{\hat{L}}$;

  3. 3.

The number of hidden layers of neural network $\Phi$ is greater than or equal
to the number of hidden layers of neural network $\hat{\Phi}$, i.e.,
$L\geq\hat{L}$.

To characterize the output difference between $\Phi$ and $\hat{\Phi}$, we
define the following metric for model reduction precision.

###### Definition 6

Consider neural network $\Phi$ and its reduced-size version $\hat{\Phi}$ with
one same input set $\mathcal{U}$ and their corresponding output sets
$\mathcal{Y}$ and $\hat{\mathcal{Y}}$, we define the distance between the
outputs of $\Phi$ and $\hat{\Phi}$ with respect to the input set $\mathcal{U}$
by

$\displaystyle\rho(\Phi,\hat{\Phi},\mathcal{U})=\sup\limits_{\bm{\eta}_{0}=\hat{\bm{\eta}}_{0},\bm{\eta}_{0},\hat{\bm{\eta}}_{0}\in\mathcal{U}}\left\|\Phi(\bm{\eta}_{0})-\hat{\Phi}(\hat{\bm{\eta}}_{0})\right\|$
(11)

where $\rho(\Phi,\hat{\Phi},\mathcal{U})$ is called model reduction precision.

In the framework of reachability analysis of neural networks, the following
theorem presents a numerically tractable method to compute model reduction
precision $\rho(\Phi,\hat{\Phi},\mathcal{U})$.

###### Theorem 1

Given neural network $\Phi$ and its reduced-size version $\hat{\Phi}$ with
input set $\mathcal{U}$, the model reduction precision
$\rho(\Phi,\hat{\Phi},\mathcal{U})$ can be computed by

$\displaystyle\rho(\Phi,\hat{\Phi},\mathcal{U})=\sup_{\tilde{\mathbf{\eta}}_{0}\in{\mathcal{U}}}\left\|\tilde{\Phi}(\tilde{\bm{\eta}}_{0})\right\|$
(12)

where neural network $\tilde{\Phi}$ is an augmented neural network of $\Phi$
and $\hat{\Phi}$ defined as follows:

$\displaystyle\begin{cases}\tilde{\bm{\eta}}_{\ell}=\tilde{\phi}_{\ell}(\tilde{\mathbf{W}}_{\ell}\tilde{\bm{\eta}}_{\ell-1}+\tilde{\mathbf{b}}_{\ell}),~{}\ell=1,\ldots,L+1\\\
\tilde{\bm{\eta}}_{L+1}=\tilde{\Phi}(\tilde{\bm{\eta}}_{0})\end{cases}$ (13)

in which input $\tilde{\bm{\eta}}_{0}=\bm{\eta}_{0}=\hat{\bm{\eta}}_{0}$ and

$\displaystyle\tilde{\mathbf{W}}_{\ell}$
$\displaystyle=\begin{cases}\begin{bmatrix}\mathbf{W}_{1}\\\
\hat{\mathbf{W}}_{1}\end{bmatrix},&\ell=1\\\
\begin{bmatrix}\mathbf{W}_{\ell}&\mathbf{0}_{{n_{\ell}}\times{\hat{n}_{\ell-1}}}\\\
\mathbf{0}_{{\hat{n}_{\ell}}\times{n_{\ell-1}}}&\hat{\mathbf{W}}_{\ell}\end{bmatrix},&1<\ell\leq\hat{L}-1\\\
\begin{bmatrix}\mathbf{W}_{\ell}&\mathbf{0}_{{n_{\ell}}\times{\hat{n}_{\hat{L}-1}}}\\\
\mathbf{0}_{{\hat{n}_{\hat{L}-1}}\times{n_{\ell}}}&\mathbf{I}_{\hat{n}_{\hat{L}-1}}\end{bmatrix},&\hat{L}\leq\ell\leq
L-1\\\
\begin{bmatrix}\mathbf{W}_{L}&\mathbf{0}_{{n_{L}}\times{n_{\hat{L}-1}}}\\\
\mathbf{0}_{{n_{\hat{L}}}\times{n_{L-1}}}&\hat{\mathbf{W}}_{\hat{L}}\end{bmatrix},&\ell=L\\\
\begin{bmatrix}\mathbf{I}_{n_{L}}&-\mathbf{I}_{n_{\hat{L}}}\end{bmatrix},&\ell=L+1\end{cases}$
(14)

$\displaystyle\tilde{\mathbf{b}}_{\ell}$
$\displaystyle=\begin{cases}\begin{bmatrix}\mathbf{b}_{\ell}\\\
\hat{\mathbf{b}}_{\ell}\end{bmatrix},&1\leq\ell\leq\hat{L}-1\\\
\begin{bmatrix}\mathbf{b}_{\ell}\\\ \mathbf{0}_{{\hat{n}_{\hat{L}-1}}\times
1}\end{bmatrix},&\hat{L}\leq\ell\leq L-1\\\ \begin{bmatrix}\mathbf{b}_{L}\\\
\hat{\mathbf{b}}_{\hat{L}}\end{bmatrix},&\ell=L\\\
\begin{bmatrix}\mathbf{0}_{{(n_{L}+n_{\hat{L}})}\times
1}\end{bmatrix},&\ell=L+1\end{cases}$ (15)
$\displaystyle\tilde{\phi}_{\ell}(\cdot)$
$\displaystyle=\begin{cases}\begin{bmatrix}\phi_{\ell}(\cdot)\\\
\hat{\phi}_{\ell}(\cdot)\end{bmatrix},&1\leq\ell\leq\hat{L}-1\\\
\begin{bmatrix}\phi_{\ell}(\cdot)\\\
\mathsf{purelin}(\cdot)\end{bmatrix},&\hat{L}\leq\ell\leq L-1\\\ \
\begin{bmatrix}\phi_{L}(\cdot)\\\
\hat{\phi}_{\hat{L}}(\cdot)\end{bmatrix},&\ell=L\\\
\mathsf{purelin}(\cdot),&\ell=L+1\end{cases}$ (16)

where $\mathsf{purelin}(\cdot)$ is linear transfer function, i.e.,
$x=\mathsf{purelin}(x)$.

###### Proof:

Given an input $\bm{\eta}_{0}\in\mathcal{U}$ and
$\tilde{\bm{\eta}}_{0}=\hat{\bm{\eta}}_{0}=\bm{\eta}_{0}$ and considering
layers $1\leq\ell\leq\hat{L}-1$ of $\tilde{\Phi}$, we have

$\displaystyle\tilde{\bm{\eta}}_{\hat{L}-1}=\begin{bmatrix}\phi_{L-1}\circ\cdots\circ\phi_{1}(\mathbf{W}_{1}\bm{\eta}_{0}+\mathbf{b}_{1})\\\
\hat{\phi}_{\hat{L}-1}\circ\cdots\circ\hat{\phi}_{1}(\hat{\mathbf{W}}_{1}\hat{\bm{\eta}}_{0}+\hat{\mathbf{b}}_{1})\end{bmatrix}$
(17)

Specifically, we consider $\ell=1$ such that

$\displaystyle\tilde{\bm{\eta}}_{1}=\begin{bmatrix}\phi_{1}(\mathbf{W}_{1}\bm{\eta}_{0}+\mathbf{b}_{1})\\\
\hat{\phi}_{1}(\hat{\mathbf{W}}_{1}\hat{\bm{\eta}}_{0}+\hat{\mathbf{b}}_{1})\end{bmatrix}=\begin{bmatrix}\bm{\eta}_{1}\\\
\hat{\bm{\eta}}_{1}\end{bmatrix}$ (18)

Moreover, when $1<\ell\leq\hat{L}-1$, it leads to

$\displaystyle\tilde{\mathbf{W}}_{\ell}\tilde{\bm{\eta}}_{\ell-1}+\tilde{\mathbf{b}}_{\ell}=\begin{bmatrix}\mathbf{W}_{\ell}{\bm{\eta}}_{\ell-1}+{\mathbf{b}}_{\ell}\\\
\hat{\mathbf{W}}_{\ell}\hat{\bm{\eta}}_{\ell-1}+\hat{\mathbf{b}}_{\ell}\end{bmatrix},~{}1<\ell\leq\hat{L}-1$
(19)

Staring from $\ell=1$ and recursively using (19) into (17), one can obtain

$\displaystyle\tilde{\bm{\eta}}_{\hat{L}-1}=\begin{bmatrix}\bm{\eta}_{\hat{L}-1}\\\
\hat{\bm{\eta}}_{\hat{L}-1}\end{bmatrix}$ (20)

Furthermore, when $\hat{L}\leq\ell\leq L-1$, one can derive

$\displaystyle\tilde{\mathbf{W}}_{\ell}\tilde{\bm{\eta}}_{\ell-1}+\tilde{\mathbf{b}}_{\ell}$
$\displaystyle=\begin{bmatrix}\mathbf{W}_{\ell}\bm{\eta}_{\ell-1}+\mathbf{b}_{\ell}\\\
\hat{\bm{\eta}}_{\ell-1}\end{bmatrix},~{}\hat{L}\leq\ell\leq L-1$ (21)

and

$\displaystyle\bm{\eta}_{L-1}$
$\displaystyle=\phi_{L-1}\circ\cdots\circ\phi_{\hat{L}}(\mathbf{W}_{\hat{L}}\bm{\eta}_{\hat{L}-1}+\mathbf{b}_{\hat{L}-1})$

which leads to

$\displaystyle\tilde{\bm{\eta}}_{L-1}=$
$\displaystyle\begin{bmatrix}\bm{\eta}_{L-1}\\\
\hat{\bm{\eta}}_{\hat{L}-1}\end{bmatrix}$ (22)

Then, when $\ell=L$, it yields that

$\displaystyle\tilde{\mathbf{W}}_{L}\tilde{\bm{\eta}}_{L-1}+\tilde{\mathbf{b}}_{L}$
$\displaystyle=\begin{bmatrix}\mathbf{W}_{L}\bm{\eta}_{L-1}+\mathbf{b}_{L}\\\
\hat{\mathbf{W}}_{\hat{L}}\hat{\bm{\eta}}_{\hat{L}-1}+\hat{\mathbf{b}}_{\hat{L}}\end{bmatrix}$
(23)

Thus, one can obtain

$\displaystyle\tilde{\bm{\eta}}_{L}=$
$\displaystyle\begin{bmatrix}\bm{\eta}_{L}\\\
\hat{\bm{\eta}}_{\hat{L}}\end{bmatrix}$ (24)

At last, when $\ell=L+1$, the following result can be obtained

$\displaystyle\tilde{\bm{\eta}}_{L+1}=\tilde{\mathbf{W}}_{L+1}\tilde{\bm{\eta}}_{L}+\tilde{\mathbf{b}}_{L+1}=\begin{bmatrix}\mathbf{I}_{n_{L}}&-\mathbf{I}_{n_{\hat{L}}}\end{bmatrix}\begin{bmatrix}\bm{\eta}_{L}\\\
\hat{\bm{\eta}}_{\hat{L}}\end{bmatrix}$

which implies that
$\tilde{\bm{\eta}}_{L+1}=\bm{\eta}_{L}-\hat{\bm{\eta}}_{\hat{L}}$. Therefore,
we can conclude that
$\tilde{\Phi}(\tilde{\bm{\eta}}_{0})=\Phi(\bm{\eta}_{0})-\hat{\Phi}(\hat{\bm{\eta}}_{0})$
as long as $\tilde{\bm{\eta}}_{0}=\hat{\bm{\eta}}_{0}=\hat{\eta}_{0}$ and

$\displaystyle\rho(\Phi,\hat{\Phi},\mathcal{U})=\sup_{\tilde{\mathbf{\eta}}_{0}\in{\mathcal{U}}}\left\|\tilde{\Phi}(\tilde{\bm{\eta}}_{0})\right\|$
(25)

The proof is complete. ∎

###### Remark 1

In the process of augmenting neural networks $\Phi$ and $\hat{\Phi}$ into
$\tilde{\Phi}$, the case of $\ell=1$ in (14)–(16) ensures that augmented
neural network $\tilde{\Phi}$ takes the one same input $\bm{\eta}_{0}$ for the
subsequent calls involving both processes of $\Phi$ and its reduced-size
version $\hat{\Phi}$. Then, for $1<\ell\leq\hat{L}-1$, augmented neural
network $\tilde{\Phi}$ conducts the computation of $\Phi$ and and its reduced-
size version $\hat{\Phi}$ parallelly for the hidden layers of
$1<\ell\leq\hat{L}-1$. When $\hat{L}\leq\ell\leq L-1$, the hidden layers of
reduced-size neural network $\hat{\Phi}$ which has fewer hidden layers are
expanded to match the number of layers of the original neural network $\Phi$
with a larger number of hidden layers, but the expanded layers are forced to
pass the information to subsequent layers without any changes, i.e., the
weight matrices of the expanded hidden layers are identity matrices, and the
bias vectors are zero vectors. This expansion is formalized as in the case of
$\hat{L}\leq\ell\leq L-1$ in (14)–(16). Moreover, as $\ell=L$, this layer is a
combination of output layers of both $\Phi$ and $\hat{\Phi}$ to generate the
same outputs of $\Phi$ and $\hat{\Phi}$. At last, a comparison layer $L+1$ is
added to compute the exact difference between the original neural network
$\Phi$ and its reduced-size version $\hat{\Phi}$.

###### Remark 2

As shown in Theorem 1, the key of computing model reduction precision
$\rho(\Phi,\hat{\Phi},\mathcal{U})$ is to compute the maximal output value of
augmented neural network $\tilde{\Phi}$ with respect to input set
$\mathcal{U}$. This can be efficiently done by neural network reachability
analysis. For instance, as in NNV neural network reachability analysis tool,
the reachable sets are in the form of a family of polyhedral sets [16], and in
the IGNNV tool, the output reachable set is a family of interval sets [14,
13]. With the reachable set $\tilde{\mathcal{Y}}$, the model reduction
precision $\rho(\Phi,\hat{\Phi},\mathcal{U})$ can be easily obtained by
searching for the maximal value of $\left\|\tilde{\bm{\eta}}_{L+1}\right\|$ in
$\tilde{\mathcal{Y}}$, e.g., testing throughout a finite number of vertices in
polyhedral sets.

##  IV Safety Verification of Neural Network Control Systems

In this section, we apply neural network model reduction and model reduction
precision to a neural network control system. By replacing the original neural
network controller with a reduced-size neural network, the computational cost
of safety verification can be significantly reduced. Moreover, the model
reduction precision allows an over-estimation of the difference in behavior
between the original neural network and the reduced-size one. For the
reachability analysis of neural networks, the following result can be
obtained.

###### Proposition 1

Given neural network $\Phi$, its reduced-size version $\hat{\Phi}$ with output
set $\hat{\mathcal{Y}}$, and model reduction precision
$\rho(\Phi,\hat{\Phi},\mathcal{U})$, the output reachable set of original
neural network $\Phi$ satisfies

$\displaystyle\mathcal{Y}\subseteq\hat{\mathcal{Y}}\oplus\mathcal{B}(\mathbf{0}_{n_{L}\times
1},\frac{\rho(\Phi,\hat{\Phi},\mathcal{U})}{2})$ (26)

where $\mathcal{B}(\mathbf{0}_{n\times 1},r)$ denotes a ball centered at
$\mathbf{0}_{n\times 1}$ with a radius of $r$, and $\oplus$ denotes the
Minkowski sum.

###### Proof:

This can be obtained straightforwardly by the definition of model reduction
precision $\rho(\Phi,\hat{\Phi},\mathcal{U})$ which characterizes the maximal
difference between the outputs of $\Phi$ and $\hat{\Phi}$. The proof is
complete. ∎

Input : System dynamics $f$, $h$; Reduced-size neural network $\hat{\Phi}$;
Model reduction precision $\rho(\Phi,\hat{\Phi},\mathcal{U})$; Initial set
$\mathcal{X}_{0}$; Input set $\mathcal{V}$

Output : Reachable set estimation $\mathcal{R}_{e}([t_{0},t_{f}])$.

1 Function  _reachNNCS_

/* Initialization */

2 $k\leftarrow 0$

3 $t_{K+1}\leftarrow t_{f}$

4 $\mathcal{R}_{e}(t_{0})\leftarrow\mathcal{X}_{0}$

/* Iteration for all sampling intervals */

5 while  _$k\leq K$_ do

6
$\mathcal{Y}_{e}(t_{k})\leftarrow\texttt{reachODEy}(h,\mathcal{R}_{e}(t_{k}))$

7 $\mathcal{H}\leftarrow\mathcal{Y}_{e}(t_{k})\times\mathcal{V}$

8
$\hat{\mathcal{U}}_{e}(t_{k})\leftarrow\texttt{reachNN}(\hat{\Phi},\mathcal{H})$

9
$\mathcal{U}_{e}\leftarrow\hat{\mathcal{U}}(t_{k})\oplus\mathcal{B}(\mathbf{0}_{n_{L}\times
1},\frac{\rho(\Phi,\hat{\Phi},\mathcal{U})}{2})$

10
$\mathcal{R}_{e}([t_{k},t_{k+1}])\leftarrow\texttt{reachODEx}(f,\mathcal{U}_{e},\mathcal{R}_{e}(t_{k}))$

11 $k\leftarrow k+1$

12 end while

13 return
_$\mathcal{R}_{e}([t_{0},t_{f}])\leftarrow\bigcup_{k=0,1\ldots,K}\mathcal{R}_{e}([t_{k},t_{k+1}])$_

14

Algorithm 1 Reachable Set computation for Neural Network Control Systems (4)

The reachable set estimation for a sampled-data neural network control system
in the form of (4) generally involves two parts: 1) Output set computation for
neural network controllers denoted by

$\displaystyle\mathcal{Y}=\texttt{reachNN}(\Phi,\mathcal{U})$ (27)

which can be efficiently obtained by neural network reachability tools such as
[13, 16] and (26), and 2) Reachable set computation of system (1). For the
reachable set computation of systems described by ODEs, there exist a variety
of approaches and tools such as those well-developed in [17, 18, 19, 20]. The
following functions are given to denote the reachable set estimation for
sampled data ODE models during $[t_{k},t_{k+1}]$,

$\displaystyle\mathcal{R}_{e}([t_{k},t_{k+1}])$
$\displaystyle=\texttt{reachODEx}(f,\mathcal{U}(t_{k}),\mathcal{R}_{e}(t_{k}))$
(28) $\displaystyle\mathcal{Y}_{e}(t_{k})$
$\displaystyle=\texttt{reachODEy}(h,\mathcal{R}_{e}(t_{k}))$ (29)

where $\mathcal{U}(t_{k})$ is the input set for sampling interval
$[t_{k},t_{k+1}]$. $\mathcal{R}_{e}(t_{k})$ and
$\mathcal{R}_{e}([t_{k},t_{k+1}])$ are the estimated reachable sets for state
$\mathbf{x}(t)$ at sampling instant $t_{k}$ and interval $[t_{k},t_{k+1}]$,
respectively. $\mathcal{Y}_{e}(t_{k})$ is the estimated reachable set for
output $\mathbf{y}(t_{k})$. With the results in Proposition 1, we can use the
reduced-size neural network to compute output set $\hat{\mathcal{U}}(t_{k})$
which is much more computationally efficient due to its smaller size, and
replace the output set of $\mathcal{U}(t_{k})$ by
$\hat{\mathcal{U}}(t_{k})\oplus\mathcal{B}(\mathbf{0}_{n_{L}\times
1},\frac{\rho(\Phi,\hat{\Phi},\mathcal{U})}{2})$ where
$\rho(\Phi,\hat{\Phi},\mathcal{U})$ is normally obtained through a one-time
offline computation. The reachable set computation process is shown in
Algorithm 1.

Based on the estimated reachable set obtained by Algorithm 1, the safety
property can be examined with the existence of intersections between the
estimated reachable set and unsafe region $\neg\mathcal{S}$.

###### Proposition 2

Consider a neural network control system in the form of (4) with a safety
specification $\mathcal{S}$, the system is safe in $[t_{0},t_{f}]$, if
$\mathcal{R}_{e}([t_{0},t_{f}])\cap\neg\mathcal{S}=\emptyset$, where
$\mathcal{R}_{e}([t_{0},t_{f}])$ is an estimated reachable set obtained by
Algorithm 1.

##  V Evaluation on Adaptive Cruise Control Systems

In this section, our approach will be evaluated by the safety verification of
an Adaptive Cruise Control (ACC) system equipped with a neural network
controller as depicted in Fig. 1. The system dynamics is in the form of

$\displaystyle\begin{cases}\dot{x}_{l}(t)=v_{l}(t)\\\
\dot{v}_{l}(t)=\gamma_{l}(t)\\\
\dot{\gamma}_{l}(t)=-2\gamma_{l}(t)+2\alpha_{l}(t)-\mu v^{2}_{l}(t)\\\
\dot{x}_{e}(t)=v_{e}(t)\\\ \dot{v}_{e}(t)=\gamma_{e}(t)\\\
\dot{\gamma}_{e}(t)=-2\gamma_{e}(t)+2\alpha_{e}(t)-\mu
v^{2}_{e}(t)\end{cases}$ (30)

where $x_{l}(x_{e})$, $v_{l}(v_{e})$ and $\gamma_{l}(\gamma_{e})$ are the
position, velocity and actual acceleration of the lead (ego) car,
respectively. $\alpha_{l}(\alpha_{e})$ is the acceleration control input
applied to the lead (ego) car, and $\mu=0.001$ is the friction parameter. The
ACC controller we considered here is a $5\times 20$ feed-forward neural
network with ReLU as its activation functions. The sampling scheme is
considered as a periodic sampling every 0.01 seconds, i.e.,
$t_{k+1}-t_{k}=0.01$ seconds.

Figure 1: Adaptive cruise control system model [13]

The sampled-data neural network controller for the acceleration control of the
ego car is in the form of

$\displaystyle\alpha_{e}(t)=\Phi(v_{set}(t_{k}),t_{gap},v_{e}(t_{k}),d_{rel}(t_{k}),v_{rel}(t_{k}))$
(31)

in which $k\in[t_{k},t_{k+1}]$. The threshold of the safe distance between the
two cars satisfies a function as defined below in the form of

$\displaystyle d_{safe}>d_{thold}=d_{def}+t_{gap}\cdot v_{e}$ (32)

where $d_{safe}$ is the safe distance between the ego car and lead car,
$d_{thold}$ is the threshold of the safe distance, $d_{def}$ is the standstill
default spacing, and $t_{gap}$ is the time gap between the vehicles. The
safety verification scenario we consider is that the lead car decelerates with
$\alpha_{l}=2$ to reduce its speed as an emergency braking occurs. We expect
that the ego car guided by a neural network control system is able to maintain
a safe relative distance to the lead car to avoid the collision.

The safety specification parameter we consider in the simulation is
$t_{gap}=1.4$ seconds and $d_{def}=10$. The time horizon that we want to
verify is 3 seconds, i.e., 300 sampling intervals, after the emergency braking
comes into play. The initial sets are $x_{l}(0)\in[94,96]$,
$v_{l}(0)\in[30,30.2]$, $\gamma_{l}(0)=0$, $x_{e}(0)\in[10,11]$,
$v_{e}(0)\in[30,30.2]$, and $\gamma_{e}(0)=0$.

As mentioned above, the size of the hidden layer of the original neural
network controller is $5\times 20$, i.e., 5 layers with 20 neurons in each
layer, Through neural network model reduction, we replace the original neural
network controller with a reduced-size neural network of hidden layer size
$2\times 5$, i.e., 2 layers with 5 neurons in each layer, combined with a
model reduction precision $\rho=1.0234$. For the continuous-time nonlinear
dynamics, we use CORA [18] to do the reachability analysis for the time
interval between two sampling instants, and IGNNV in [13] is used for neural
network reachability analysis.

Figure 2: Reachable set of ACC systems. The green (blue) pipe indicates the output reachable set of the dynamic system with the original (reduced-size) neural network. It is obvious that the blue pipe completely wraps around the green pipe. The yellow (red) pipe denotes the safe distance threshold region for the dynamical system with the original (reduced-size) neural network. Figure 3: Reachable set estimation for relative distance and the relative velocity between lead and ego cars. The green (blue) pipe indicates the reachable set of the dynamical system with the original (reduced-size) neural network. The yellow (red) pipe denotes the safe distance threshold and the relative velocity between lead and ego cars for the dynamical system with the original (reduced-size) neural network. TABLE I: Comparison of ACC reachable set calculation times Dynamical Systems | Computational Time
---|---
ACC with original neural network |  $12.355$ s
ACC with reduced-size neural network |  $1.099$ s

The output reachable set of the ACC system for the relative distance between
the lead car and the ego car over time can be shown in Figs. 2 and 3. Notably,
the computation time has been significantly reduced from 12.355 seconds to
1.099 seconds when using a reduced-size neural network as shown in Fig. 2 and
Table I. The system is safe when the output reachable set of relative
distances does not intersect with the safe distance threshold region.

In summary, the simulations show that the closed-loop system with the reduced-
size neural network can be used for safety verification of the original system
as long as the model reduction precision can be provided. The reduced-size
neural network can significantly reduce the computational time of the entire
process of solving the neural network control system for the output reachable
set.

##  VI Conclusions

This paper investigates the problem of simplifying the safety verification of
neural network control systems, proposes a concept of model reduction
precision that characterizes the minimum upper bound on the outputs between a
neural network and its reduced-size one, and proposes an algorithm to
calculate the model reduction precision. By using a reduced-size neural
network as the neural network controller and introducing the model reduction
precision in the computation of the output reachable. Combined with the
calculation of reachable sets for dynamical systems, we give the reachable set
computation algorithm based on model reduction of neural network control
systems. In this way, we can obtain the over-approximated output reachable set
of the original neural network control system with less computation time and
enable a simplification of safety verification processes. The developed
results are applied to the ACC system to verify its effectiveness and
feasibility.

## References

  * [1] G. Litjens, T. Kooi, B. E. Bejnordi, A. A. A. Setio, F. Ciompi, M. Ghafoorian, J. A. van der Laak, B. van Ginneken, and C. I. Sánchez, “A survey on deep learning in medical image analysis,” _Medical Image Analysis_ , vol. 42, pp. 60–88, 2017.
  * [2] J. Schmidhuber, “Deep learning in neural networks: An overview,” _Neural Networks_ , vol. 61, pp. 85–117, 2015.
  * [3] K. Hunt, D. Sbarbaro, R. Żbikowski, and P. Gawthrop, “Neural networks for control systems—a survey,” _Automatica_ , vol. 28, no. 6, pp. 1083–1112, 1992.
  * [4] M. Bojarski, D. Del Testa, D. Dworakowski, B. Firner, B. Flepp, P. Goyal, L. D. Jackel, M. Monfort, U. Muller, J. Zhang _et al._ , “End to end learning for self-driving cars,” _arXiv preprint arXiv:1604.07316_ , 2016.
  * [5] K. D. Julian, J. Lopez, J. S. Brush, M. P. Owen, and M. J. Kochenderfer, “Policy compression for aircraft collision avoidance systems,” in _2016 IEEE/AIAA 35th Digital Avionics Systems Conference (DASC)_ , 2016, pp. 1–10.
  * [6] S. Wiedemann, H. Kirchhoffer, S. Matlage, P. Haase, A. Marban, T. Marinč, D. Neumann, T. Nguyen, H. Schwarz, T. Wiegand, D. Marpe, and W. Samek, “Deepcabac: A universal compression algorithm for deep neural networks,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 14, no. 4, pp. 700–714, 2020.
  * [7] Y. Zhang, W. Ding, and C. Liu, “Summary of convolutional neural network compression technology,” in _2019 IEEE International Conference on Unmanned Systems (ICUS)_ , 2019, pp. 480–483.
  * [8] L. Deng, G. Li, S. Han, L. Shi, and Y. Xie, “Model compression and hardware acceleration for neural networks: A comprehensive survey,” _Proceedings of the IEEE_ , vol. 108, no. 4, pp. 485–532, 2020.
  * [9] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus, “Intriguing properties of neural networks,” in _International Conference on Learning Representations_ , 2014.
  * [10] G. Katz, C. Barrett, D. L. Dill, K. Julian, and M. J. Kochenderfer, “Reluplex: An efficient smt solver for verifying deep neural networks,” in _International Conference on Computer Aided Verification_. Springer, 2017, pp. 97–117.
  * [11] H.-D. Tran, W. Xiang, and T. T. Johnson, “Verification approaches for learning-enabled autonomous cyber–physical systems,” _IEEE Design & Test_, vol. 39, no. 1, pp. 24–34, 2022.
  * [12] W. Xiang, H.-D. Tran, and T. T. Johnson, “Reachable set computation and safety verification for neural networks with relu activations,” _arXiv preprint arXiv:1712.08163_ , 2017.
  * [13] W. Xiang, H.-D. Tran, X. Yang, and T. T. Johnson, “Reachable set estimation for neural network control systems: A simulation-guided approach,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 32, no. 5, pp. 1821–1830, 2021.
  * [14] W. Xiang, H.-D. Tran, and T. T. Johnson, “Output reachable set estimation and verification for multilayer neural networks,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 29, no. 11, pp. 5777–5783, 2018.
  * [15] H.-D. Tran, D. M. Lopez, P. Musau, X. Yang, L. V. Nguyen, W. Xiang, and T. T. Johnson, “Star-based reachability analysis of deep neural networks,” in _International Symposium on Formal Methods_. Springer, 2019, pp. 670–686.
  * [16] H.-D. Tran, X. Yang, D. Manzanas Lopez, P. Musau, L. V. Nguyen, W. Xiang, S. Bak, and T. T. Johnson, “Nnv: the neural network verification tool for deep neural networks and learning-enabled cyber-physical systems,” in _International Conference on Computer Aided Verification_. Springer, 2020, pp. 3–17.
  * [17] G. Frehse, C. Le Guernic, A. Donzé, S. Cotton, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang, and O. Maler, “SpaceEx: Scalable verification of hybrid systems,” in _International Conference on Computer Aided Verification_. Springer, 2011, pp. 379–395.
  * [18] M. Althoff, “An introduction to CORA 2015,” in _Workshop on Applied Verification for Continuous and Hybrid Systems_ , 2015.
  * [19] X. Chen, E. Ábrahám, and S. Sankaranarayanan, “Flow*: An analyzer for non-linear hybrid systems,” in _International Conference on Computer Aided Verification_. Springer, 2013, pp. 258–263.
  * [20] S. Bak and P. S. Duggirala, “HyLAA: A tool for computing simulation-equivalent reachability for linear systems,” in _Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control_. ACM, 2017, pp. 173–178.


# Hyperfine-interaction limits polarization entanglement of photons from
semiconductor quantum dots

Christian Schimpf  Institute of Semiconductor and Solid State Physics,
Johannes Kepler Universtiy Linz, 4040 Linz, Austria<EMAIL_ADDRESS>Francesco Basso Basset  Department of Physics, Sapienza University of Rome,
00185 Rome, Italy  Maximilian Aigner  Institute of Semiconductor and Solid
State Physics, Johannes Kepler Universtiy Linz, 4040 Linz, Austria  Wolfgang
Attenender  Institute of Semiconductor and Solid State Physics, Johannes
Kepler Universtiy Linz, 4040 Linz, Austria  Laia Gínes  Department of Physics,
Stockholm University, 106 91 Stockholm, Sweden  Gabriel Undeutsch  Institute
of Semiconductor and Solid State Physics, Johannes Kepler Universtiy Linz,
4040 Linz, Austria  Marcus Reindl  Institute of Semiconductor and Solid State
Physics, Johannes Kepler Universtiy Linz, 4040 Linz, Austria  Daniel Huber
Institute of Semiconductor and Solid State Physics, Johannes Kepler Universtiy
Linz, 4040 Linz, Austria  Dorian Gangloff  Department of Engineering Science,
University of Oxford, Oxford, OX1 3PJ, United Kingdom  Evgeny A. Chekhovich
Department of Physics and Astronomy, University of Sheffield, United Kingdom
Christian Schneider  Institute of Physics, Carl von Ossietzky Universität
Oldenburg, 26111 Oldenburg, Germany  Sven Höfling  Technische Physik and
Wilhelm Conrad Röntgen Research Center for Complex Material Systems,
Universität Würzburg, 97074 Würzburg, Germany  Ana Predojević  Department of
Physics, Stockholm University, 106 91 Stockholm, Sweden  Rinaldo Trotta
Department of Physics, Sapienza University of Rome, 00185 Rome, Italy  Armando
Rastelli  Institute of Semiconductor and Solid State Physics, Johannes Kepler
Universtiy Linz, 4040 Linz, Austria

###### Abstract

Excitons in quantum dots are excellent sources of polarization-entangled
photon pairs, but a quantitative understanding of their interaction with the
nuclear spin bath is still missing. Here we investigate the role of hyperfine
energy shifts using experimentally accessible parameters and derive an upper
limit to the achievable entanglement fidelity. Our results are consistent with
all available literature, indicate that spin-noise is often the dominant
process limiting the entanglement in InGaAs quantum dots, and suggest routes
to alleviate its effect.

††preprint: APS/123-QED

Semiconductor quantum dots (QDs) have emerged as outstanding sources of single
photons [1, 2] and photon pairs [3, 4], allowing for high brightness and
ultra-low multi-photon emission probability [5, 6] for potential applications
in quantum communication and information processing. Most of the proofs of
principle demonstrations in these areas have relied on In(Ga)As QDs obtained
via the Stranski-Krastanow (SK) epitaxial growth [7] on GaAs substrates. A key
advantage of QDs over other solid-state quantum emitters is their capability
of emitting polarization-entangled photon pairs using the biexciton-exciton
(XX-X) decay cascade [8, 9, 10]. However, in spite of many careful
investigations on SK and other In-containing QDs [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19], entanglement fidelity values beyond 0.9 could so far only be
achieved by resorting to lossy time-filtering techniques [20, 21, 22]. In
contrast, fidelities values of up to 0.98 [23] have been reported for GaAs QDs
obtained via local Al-droplet etching (LDE) on AlGaAs [24, 25, 26]. The
physical origin of this large discrepancy is still unclear and solving this
puzzle would enhance the understanding of the physics of QDs in general and
guide the improvement of the QD performance as sources of entangled photons.

Several potential dephasing- and mixing-channels have been identified that
could limit the polarization entanglement in QDs. Besides the electron-hole
exchange interaction, which causes the well-known fine-structure-splitting
(FSS) between the bright X levels [27] in QDs with in-plane asymmetry,
possible limitations from spin-scattering [28], phonon-induced dephasing [29,
30], excitation-laser induced AC-Stark shifts [31], and their combination [32]
have been considered. In addition, nuclear spin noise in the form of
stochastic hyperfine shifts, known as Overhauser fluctuations [33, 34, 35]
have been proposed as the probable dominant dephasing mechanism in In(Ga)As
QDs, motivated by the high nuclear spin (9/2) of In [36, 37, 38] compared to
Ga and As (3/2). However, no sufficiently accurate statements about the weight
of nuclear spin noise have been made so far.

In this work we investigate the polarization entanglement of photon pairs
generated by state-of-the-art InGaAs QDs [39, 40] with radiative lifetimes
$T_{1}<$500\text{\,}\mathrm{p}\mathrm{s}$$ and an FSS ranging from $\lesssim
0.4$ to $\sim$2.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
We provide a quantitative estimate of the upper limit of the degree of
entanglement achievable with such QDs due to nuclear spin noise and show that
this limit is fully consistent with our measurements as well as former
experimental results [16, 28, 41, 15, 4], including those resorting to time-
filtering [20, 21, 19, 22, 17] and Purcell enhanced emission [4]. Our
numerical results are based on direct measurements of the nuclear hyperfine-
noise on the ground-state electron in InGaAs QDs, which reveal an
inhomogeneous electron spin coherence time
$T_{2}^{*}\simeq$1.7\text{\,}\mathrm{n}\mathrm{s}$$ [34, 42, 43]. We further
show that the consistently higher degree of entanglement found in LDE GaAs QDs
[23] can be tracked back to the lower $T_{1}$
($\sim$$230\text{\,}\mathrm{p}\mathrm{s}$) and the simultaneously higher
measured electron spin $T_{2}^{*}$ ($\sim$$2.6\text{\,}\mathrm{n}\mathrm{s}$
[44]).

Figure 1:  (a) Spectrum of an InGaAs QD (QD1) under two-photon-excitation
(TPE). The sketch on the left depicts the biexciton-exciton (XX-X) level
scheme. TPE is performed by tuning the energy $E_{\text{L}}$ of a pulsed laser
to half of the XX energy. Here, the XX level lies
$E_{\text{B}}=$2.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ lower than two
times the X energy (2X). The X states are split by the constant fine structure
splitting $S$ and – in the presence of randomly oriented nuclear spins – also
by the inhomogeneous Overhauser shift (OS) with a normally distributed
amplitude with a standard deviation $\sigma$. (b) Rabi oscillation of the
emitted XX signal with varying average laser power $P_{\text{E}}$. (c)
Lifetime measurements of the XX and X emission. The depicted values are
obtained by a convolved fit with the instruments response function (IRF).

The energy levels relevant for generating entangled photon pairs from QDs are
sketched in Fig. 1(a). The level structure of the neutral exciton states (X)
is primarily defined by the electron-hole exchange-interaction [27]. An
anisotropy of this interaction in the $x-y$ plane, perpendicular to the growth
direction $z$, leads to a fine structure splitting (FSS) between the “bright”
(optically dipole-allowed) excitons. In addition to the FSS, the interaction
between the exciton and the nuclear spins leads to a hyperfine shift depending
on the X spin configuration and nuclear spins´ orientations. The biexciton
state (XX), as a singlet state, is neither affected by FSS nor by the
hyperfine shifts. In the absence of stabilizing Knight-fields [34] or
externally applied magnetic fields [27, 45] the nuclear spins change their
orientation and magnitude randomly at timescales in the order of
$\tau_{\text{S}}\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$
[33, 35]. Consequently, the Overhauser shift (OS) changes many times during
entanglement measurements, which take place on typical timescales of seconds,
so the OS acts as an inhomogeneous broadening for the X level, following a
Gaussian distribution. The standard deviation $\sigma$ of the OS amplitudes
can be identified via their connection to the inhomogeneous spin coherence
time $T_{2}^{*}$ of the X. The hole’s hyperfine contribution is typically at
least 10 times smaller than the electron’s [46], so that the electron spin
$T_{2}^{*}$ constitutes a good estimate of the X coherence time within about
$10\text{\,}\mathrm{\char 37\relax}$. The expression for $\sigma$ then reads
as [33, 42]:

$\sigma\simeq\frac{\hbar}{T_{2}^{*}}=\sqrt{\frac{\sum_{n}x_{n}\,A_{n}\,I_{n}(I_{n}+1)}{N}},$
(1)

with $x_{n}$ the fraction of the nuclear species $n$, $A_{n}$ the hyperfine
coupling constant, $I_{n}$ the nuclear spin and $N$ the number of nuclei in
contact with the electron wavefunction. The electron $T_{2}^{*}$ can be
determined by measuring its free induction decay via Ramsey interferometry
[47, 43]. For InGaAs QDs, values of around
$T_{2}^{*}\simeq$1.7\text{\,}\mathrm{n}\mathrm{s}$$ are typically observed
[48, 34, 42, 43], which results in
$\sigma\simeq$0.39\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
From these values, a time-averaged loss of X coherence of approximately
$1-\text{exp}[-(T_{1}/T_{2}^{*})^{2}]=$6\text{\,}\mathrm{\char 37\relax}$$
[47, 44], and thus a non-negligible degradation of the polarization
entanglement of the XX-X photon pair, can be expected. In comparison, for LDE
GaAs QDs [25], with $T_{1}\simeq$230\text{\,}\mathrm{p}\mathrm{s}$$ and
$T_{2}^{*}\simeq$2.6\text{\,}\mathrm{n}\mathrm{s}$$ [44], the loss amounts to
$<$1\text{\,}\mathrm{\char 37\relax}$$, providing already a clear hint on the
origin of the discrepancy between the fidelity values observed for InGaAs and
GaAs QDs.

We now investigate the generation of entangled photon pairs in individual
InGaAs QDs at a temperature of $5\text{\,}\mathrm{K}$. The QDs were obtained
via the Stranski-Krastanow growth mode and the partial capping and annealing
method, resulting in $\sim$$2\text{\,}\mathrm{n}\mathrm{m}$-high QDs with an
average In fraction of $x_{\text{In}}=0.45(5)$ [40]. The employed material can
be considered of state-of-the-art quality, as it originates from the same
wafer used in Ref. [39]. To limit the generation of free carriers in the QD
surroundings, we create the XX state via coherent two-photon excitation (TPE)
[49]. The TPE process is sketched in the inset of Fig. 1(a), where a pulsed
laser with a pulse duration of $7\text{\,}\mathrm{p}\mathrm{s}$ is tuned to
$E_{\text{L}}$, corresponding to half of the XX energy. To maximize chances of
observing entanglement fidelities beyond state-of-the-art, we focus on QDs
selected out of a much larger ensemble because of their intrinsically small
value $S$ of the FSS, see inset of Fig. 1(a), which we obtain from
polarization-resolved photoluminescence (PL) spectra with an accuracy of about
$0.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$. Figure 1(a)
depicts the TPE-PL spectrum from an InGaAs QD (QD1), with
$S=$1.3(3)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ and an
energy difference between the XX and the X photons of
$E_{\text{B}}=$2.4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$
($1.6\text{\,}\mathrm{n}\mathrm{m}$), while Fig. 1(b) shows the corresponding
excitation dynamics under varying average laser power $P_{\textrm{E}}$. At the
$\pi$-pulse condition, seen from the first Rabi-flop, the XX inversion
efficiency is determined to be $\eta_{\text{P}}=0.70(2)$ (see supplementary
for details). The radiative lifetimes for the XX-to-X and X-to-ground-state
decays, obtained via a convolved fit of the traces in Fig. 1(c) with the
instrument response function (IRF), are
$T_{1,\text{XX}}=$150(4)\text{\,}\mathrm{p}\mathrm{s}$$ and
$T_{1,\text{X}}=$346(8)\text{\,}\mathrm{p}\mathrm{s}$$, respectively. The
$T_{1,\text{X}}$ value observed for this QD and others in the same sample is
shorter than typical values of $\sim$1 ns reported for InGaAs QDs and favors
the observation of highly entangled photons.

Figure 2:  (a) Sketch of the experimental setup used to determine the
entanglement fidelity of the QD photon source. A 50:50 non-polarizing
beamsplitter (BS) is placed directly after the source. The exchangeable
waveplate before and the polarizers (P) after the BS set the measurement basis
of XX and X. The emitted photons are spectrally filtered (F) before impinging
on avalanche photodiodes (APD), between which correlation measurements are
performed. (b) HBT measurements performed on emitted XX (red) and X (blue)
photons under pulsed, resonant TPE for QD2. The resulting $g^{(2)}(0)$ values,
obtained by using a coincidence window of $3\text{\,}\mathrm{n}\mathrm{s}$,
are indicated in each panel. (c) Cross-correlation histograms for the six
2-qubit measurement bases used in the fidelity estimation for QD1p, with a
resulting fidelity of 0.89(1). (d) Measured fidelity values for different QDs
(squares) plotted against the respective FSS values. The red dots show the
expected fidelity when only accounting for the FSS and the lifetime, the blue
area depicts the range of expected values when including fluctuating
Overhauser shifts (OS) according to the values of the electron $T_{2}^{*}$
reported in the literature.

The fidelity of a XX-X two-photon state $\rho$ to the $\ket{\phi^{+}}$ Bell
state is defined as $f:=\braket{\phi^{+}}{\rho}{\phi^{+}}$. We estimate $f$
via the setup depicted in Fig. 2(a), in which the emitted state can be
projected into arbitrary 2-qubit states by rotating the waveplate(s) in front
of the beam-splitter (BS). When removing the polarizers, the same setup can be
used to perform Hanbury Brown and Twiss measurements to obtain auto-
correlation histograms, as shown for QD2 in Fig. 2(b). Using a coincidence
window of $3\text{\,}\mathrm{n}\mathrm{s}$, the $g^{(2)}(0)$ values for XX and
X are found to be $g^{(2)}_{\text{XX}}(0)=0.009(3)$ and
$g^{(2)}_{\text{X}}(0)=0.002(1)$, respectively. These values indicate a
negligible multi-pair emission probability, comparable with former reports on
similar QDs [15]. The fidelity estimation is performed by inserting the
polarizers and projecting the XX-X two-photon state into three bases,
resulting in six cross-correlation histograms (Fig. 2(c) for QD1p, where “p”
stands for “processed on a piezo-actuator”, see supplementary for a full list
of measured QDs), from which a fidelity of $f=0.89(1)$ can be calculated [37].
The fidelity measurement is repeated for seven different QDs on the same
sample. The resulting $f$ values are plotted (squares) as a function of $S$ in
Fig. 2(d). In absence of dephasing mechanisms we would expect a monotonic
decrease of the time-averaged $f$ with increasing $S$ beginning with $f\simeq
1$ at $S=0$, as a finite FSS induces a time-dependent evolution of the
entangled state [28]. However, we find that none of the measured fidelities
exceed a value of 0.9 and that reducing $S$ below
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$ (corresponding to
about the natural linewidth of the X transitions of
$\delta_{\text{F}}\approx$1.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$)
leads to no significant increase in $f$. These findings, which are in line
with previous reports, clearly indicate the presence of dephasing processes.

Considering the relevant properties of the studied QDs (X lifetime of about
$400\text{\,}\mathrm{p}\mathrm{s}$, corresponding to a natural linewidth
$\delta_{\text{F}}$ of about
$1.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$, which is much
larger than the minimum value of $S$) and the excitation dynamics (TPE with
pulse lengths of $7\text{\,}\mathrm{p}\mathrm{s}$, much shorter than the XX
lifetime of $200\text{\,}\mathrm{p}\mathrm{s}$), we can safely conclude that
excitation-induced dephasing [31] and re-excitation cannot explain the
observed reduction in $f$. Both effects combined would account for less than
0.01 drop of fidelity. The measurement temperature of $5\text{\,}\mathrm{K}$
is sufficiently low to exclude significant phonon-induced dephasing [30, 29,
32], as this would be incompatible with the negligible change of $f$ we
observed when varying the temperature from $5\text{\,}\mathrm{K}$ to
$20\text{\,}\mathrm{K}$ (see supplementary). We therefore ascribe the limited
maximum value of entanglement to the interplay of finite FSS and nuclear spin
noise, as supported by the model we derive in the following.

The typical bright-dark exciton splitting of
$\sim$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ [27, 34]
is much larger than the $S$ and $\sigma$ of the QDs considered here (in the
order of µeV). Therefore we can describe the X dynamics in the 2-dimensional
bright exciton subspace in good approximation. In the basis given by
X${}_{H/V}=\\{(\ket{+1}\pm\ket{-1})/\sqrt{2}\\}$, where $\ket{\pm 1}$ are the
eigenstates of the total electron-hole angular momentum along the
$z$-direction, the Hamiltonian is then given by [27]:

$H=\begin{pmatrix}\delta_{1}&\delta_{2}\\\
\delta_{2}^{*}&-\delta_{1},\end{pmatrix}$ (2)

where $\delta_{1}=S/2$ and $\delta_{2}=ih_{z}$, with $h_{z}$ the OS due to the
finite nuclear spin polarization in the QD. In the frozen spin approximation
(when $\tau_{\text{S}}\gg T_{1}$) [33, 34] $H$ can be treated as time-
independent and we can construct the entangled state of the two photons
emitted in the XX-X radiative decay cascade for a fixed $h_{z}$ as

$\ket{\psi_{h_{z}}}(t)=\frac{1}{\sqrt{2}}\left(\ket{j^{*}}\otimes\ket{j}+e^{-\frac{i}{\hbar}\,\Delta_{E}\,t}\ket{l^{*}}\otimes\ket{l}\right),$
(3)

with $t$ the emission time of the X photon relative to that of the XX photon.
The orthonormal eigenstates $\ket{j}$ and $\ket{l}$ of Eq. (2) are defined in
the basis of the horizontal $\ket{H}$ and vertical $\ket{V}$ polarization and
have an energy difference $\Delta_{E}=2\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}$.
Due to conservation of angular momentum, the XX eigenstates are necessarily
the complex conjugates (∗) of the X eigenstates (see supplementary for
details). The time evolution of the two-photon density matrix
$\rho_{h_{z}}(t)=\ket{\psi_{h_{z}}}\bra{\psi_{h_{z}}}$ is then given by

$i\,\hbar\,\frac{\text{d}}{\text{d}t}\rho_{h_{z}}(t)=\left[\mathbb{I}^{(2)}\otimes{H},\rho_{h_{z}}(t)\right],$
(4)

with $\mathbb{I}^{(2)}$ the $2\times 2$ identity matrix. The differential
equation Eq. (4) can readily be solved numerically. The time-averaged density
matrix $\braket{\rho}_{t}$ is calculate by drawing $h_{z}$ multiple times from
a normal distribution with a mean value of zero and a standard deviation of
$\sigma$. The value of $S$ remains constant. By indicating with $k$ the
fraction of events with no more than one photon pair per excitation cycle, we
account for the multi-photon emission probability $1-k$ as in [6]:

$1-k\approx\frac{g^{(2)}_{\text{XX}}(0)+g^{(2)}_{\text{X}}(0)}{2}\,\eta_{\text{P}}.$
(5)

By measuring the quantities in the above equation for each QD, we add a mixing
channel to $\braket{\rho}_{t}$ so that

$\tilde{\rho}=k\,\braket{\rho}_{t}+\frac{1-k}{4}\,\mathbb{I}^{(4)},$ (6)

with $\mathbb{I}^{(4)}$ the $4\times 4$ identity matrix. In the supplementary
we provide further details about the calculation of $\tilde{\rho}$ and how to
derive the fidelity

$f\approx\frac{1}{4}\left(1+k+\frac{2k}{1+4\,T_{1}^{2}\left(S^{2}+\sigma^{2}\right)/\hbar^{2}}\right).$
(7)

We note that expressions similar to Eq. (7) were previously derived, in which
phenomenological spin-scattering and dephasing times were included [28].
However, the physical origin of such effects has not been sufficiently
clarified and no clear connection to measured quantities have been drawn so
far, while the stochastic energy shift $\sigma$ in our model is a direct
consequence of the random OS routinely measured in experiments on electron
spins confined in QDs [33, 48, 34, 47, 43].

The red dots in Fig. 2(d), which depict the fidelity calculated from Eq. (7)
with $\sigma=0$ and using the measured $S$, $T_{1}$ and $k$ for each
individual dot, confirm that in the absence of other mechanisms $f$ should
reach values exceeding 0.98. The blue points correspond instead to the
estimation for
$\sigma\simeq$0.39\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$
($T_{2}^{*}\simeq$1.7\text{\,}\mathrm{n}\mathrm{s}$$ [48, 43]) and the filled
blue area depicts the range of $f$ between the shortest
($T_{2}^{*}=$1\text{\,}\mathrm{n}\mathrm{s}$$ [34]) and longest
($T_{2}^{*}=$3.2\text{\,}\mathrm{n}\mathrm{s}$$ [47]) coherence times in
In(Ga)As QDs reported in the literature. We see that no data point lies above
the expectations from the model including the realistic effect of OS. Most
importantly, our model reproduces the plateau of $f$ values for
$S\lesssim\sigma$ and allows us to ascribe it to the dominant role played by
the Overhauser shifts while, for $S\gg\sigma$ the effect of the FSS dominates.

Figure 3:  (a) Illustration of the piezoelectric device used to tune the FSS
via three-axial strain-fields acting on QD1p, individually controlled via
three input voltages $\left(V_{p1},V_{p2},V_{p3}\right)$ (b) Measured fidelity
values of QD1p, for which the FSS was tuned to different values by the piezo-
actuator. The red line shows the expected fidelity for $\sigma=0$ (no spin
noise), the blue area depicts the expected fidelity range when including OS.
(c) Real- and imaginary part of the 2-qubit density matrix, measured by state
tomography on QD1p ($T_{1}=$430(4)\text{\,}\mathrm{p}\mathrm{s}$$,
$S=$0.4(2)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ and
$k=0.990(5)$) with 16 measurement bases, calculated using the maximum-
likelihood approach with a resulting fidelity of $0.85(1)$, a purity of
$0.74(2)$ and a concurrence of $0.69(2)$. (d) Modelled density matrix for
QD1p, assuming
$\sigma=$0.41\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$
($T_{2}^{*}=$1.6\text{\,}\mathrm{n}\mathrm{s}$$), with a resulting fidelity of
0.89, a purity of 0.81 and a concurrence of 0.79.

Since the spin-properties of InGaAs QDs can vary strongly from QD to QD due to
atomistic disorder related to alloying and consequent inhomogenous strain and
piezoelectric fields, it is desirable to measure the FSS-dependent fidelity
for one single QD (QD1p). Such measurements have been formerly conducted on
InGaAs QDs using magnetic, electric, and/or strain fields [28, 50, 16, 41, 51]
but only under non-resonant excitation, which may introduce additional
entanglement-degrading effects. To eliminate this uncertainty we use TPE and
tune $S$ using a microprocessed piezoelectric strain-tuning actuator [23], as
illustrated in Fig. 3(a). The actuator allows the in-plane stress tensor in
the QD structure to be manipulated by independently expanding or compressing
three pairs of opposing “legs” through the application of different voltages
$\mathbf{V}=\left(V_{p1},V_{p2},V_{p3}\right)$ between the bottom contacts and
the common ground-contact on top. The green squares in Fig. 3(b) show the
measured values of $f$ for different $S$, adjusted by varying $\mathbf{V}$.
The red line shows the expected valued of $f$ according to Eq. (7) for
$\sigma=0$. The blue dashed line, just like in the previous measurements,
depicts the case for
$\sigma=$0.39\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$
($T_{2}^{*}=$1.7\text{\,}\mathrm{n}\mathrm{s}$$ [48, 43]) and the filled blue
area depicts the range of $f$ for the range of reported $T_{2}^{*}$ values.
Again, all $f$ values lie well within the range predicted by Eq. (6), when
considering the measured $T_{1}$, $k$ and varied $S$ of QD1p.

To confirm the reliability of our fidelity measurements, which may be affected
by polarization-altering effects in the used setup (Fig. 2(a)), we performed a
state tomography with 16 measurement bases on QD1p at the lowest value of
$S=$0.4(2)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$,
obtaining the density matrix $\rho_{\text{T}}$ shown in Fig. 3(c) via the
maximum-likelihood estimator [52]. The fidelity calculated from
$\rho_{\text{T}}$ is $f=0.85(1)$, the purity and the concurrence $C$ [53],
which are independent from a unitary rotation of the state, are calculated as
0.74(2) and 0.69(2), respectively. The slightly lower $f$ compared to the
0.89(1) obtained by the fidelity estimation could stem from the additional
wave-plates required for the state tomography, which are known to induce
slight state-mixing by the inhomogeneities in the birefringent material. For
comparison, the density matrix from Eq. (6) is shown in Fig. 3(d) using the
measured values of $S$, $T_{1}$, and $k$. The value for the electron spin
$T_{2}^{*}$ was estimated to $1.6\text{\,}\mathrm{n}\mathrm{s}$
($\sigma=$0.41\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$) to
match the $f=0.89$ measured using the fidelity estimation setup.

Before comparing the results of our model with former experimental results we
emphasize that our model provides an upper limit for the degree of
entanglement which may be observed experimentally, given the measured electron
spin $T_{2}^{*}$. Other entanglement-degrading effects may still be present.
As an example, the concurrence of 0.79 derived from the density matrix in Fig.
3(d) is higher than the measured 0.69(2) and some entries of the
experimentally reconstructed density matrix are not reproduced by the model
(e.g. the $\braket{HH}{\rho}{HV}$ component). In addition, one data point in
Fig. 2, with $f\simeq 0.64$ for an $S$ of only
$0.4(2)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$ deviates from
the trend obtained from the others and was found to correspond to a QD showing
a strong degree of linear polarization of the X signal, see supplementary.
This brings us to the conclusion that effects like heavy-hole–light-hole
mixing [54, 55, 56] or strain-activated quadrupolar double-spin-flips [57, 34,
58] could act as non-negligible mixing channels in InGaAs QDs (and possibly
also in other material systems). In fact, state mixing could explain the
discrepancy between the expected and the measured concurrence for the QD1p in
Fig 3, while the fidelity matches better. The reason is that the concurrence
drops significantly faster compared to the fidelity in the presence of a
mixing channel than for pure dephasing channels (see supplementary for
details).

Figure 4:  (a) Measured maximum $f$ from InGaAs QDs with $S\approx 0$ from the
indicated references, compared against the values expected from Eq. (4),
taking into account the reported $T_{1}$. The leftmost data point refers to
QD1p. (b) Measured concurrence from the references indicated on the x axis,
The measurements are compared against the values calculated by numerically
propagating Eq. (4) from 0 up to the used coincidence window (indicated in the
graph in ps), taking into account the reported values for $T_{1}$ and $S$. The
leftmost datapoint refers to QD1p at its lowest value of $S$.

In Fig. 4(a) we show a compilation of $f$ values for InGaAs QDs with
$S\leq$0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ from
representative work [16, 28, 41, 15], with $T_{1}$ ranging from
$400\text{\,}\mathrm{p}\mathrm{s}$ to $1\text{\,}\mathrm{n}\mathrm{s}$ and one
QD in a Purcell-enhancing photonic structure with with
$T_{1}=$127(1)\text{\,}\mathrm{p}\mathrm{s}$$ and
$S\leq$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ [4] and
compare them against the range of $f$ predicted by Eq. (7). We see that the
values consistently lie inside the range expected from the reported
$T_{2}^{*}$ as used in Figs. 2 and 3, indicating that OS are probably the
dominant source of dephasing in those measurements. Finally, Fig. 4(b) shows
the values of $C$ from the literature, measured for other In-containing QDs
with finite $S$ when relying on time-filtering [20, 21, 19, 22, 17]. The
emission wavelengths range from $870\text{\,}\mathrm{n}\mathrm{m}$ to
$1550\text{\,}\mathrm{n}\mathrm{m}$. The data points show the maximum achieved
$C$ when including only photons within the coincidence windows stated in the
graph. The values are compared against the range calculated numerically from
Eq. (4) again using the $T_{2}^{*}$ available only for InGaAs QDs, taking into
account the coincidence window, $T_{1}$ and $S$ (see supplementary for
details). We see that all measured values lie well below the maximum expected
for the used coincidence window. The value of $C$ for QD1p (Fig. 3), increases
from 0.69(2) to 0.87(9) when decreasing the coincidence window from
$3\text{\,}\mathrm{n}\mathrm{s}$ to $256\text{\,}\mathrm{p}\mathrm{s}$.
Propagating Eq. (4) up to the detector resolution of
$350\text{\,}\mathrm{p}\mathrm{s}$, however, results in $C\approx 0.98$, again
supporting the presence of additional entanglement-degrading processes
requiring further attention.

In summary, we have studied the behaviour of the polarization entanglement
fidelity on several InGaAs quantum dots (QDs) with different fine structure
splittings (FSS) and on a single strain-tuned QD on a sample bonded onto a
piezoelectric actuator. For the latter, full state tomography was performed
for the lowest achieved FSS to quantify the purity and the concurrence of the
entangled state. The results are in good agreement with a theoretical model,
which predicts an upper limit for the observable fidelity using the measured
FSS of the exciton (X), its radiative lifetime, the multi-photon-pair emission
probability, and the inhomogeneous electron coherence time $T_{2}^{*}$ as an
input. From the model we conclude that slowly varying random Overhauser shifts
(OS), prominent in QDs with finite nuclear spin, lead to a pronounced
dephasing of the bright X states over time, which limits the time-averaged
polarization entanglement in InGaAs QDs. By comparing the model predictions
with our data and values reported in the literature, we find that OS often
constitute the dominant degradation mechanism. This result is consistent with
the higher fidelity values of about 0.98 observed in LDE GaAs QDs, mainly due
to the lower X lifetimes of about $230\text{\,}\mathrm{p}\mathrm{s}$ [23].
Additionally, the electron spin $T_{2}^{*}$ in LDE GaAs QDs was found to be
higher ($\approx$2.6\text{\,}\mathrm{n}\mathrm{s}$$) [44], as expected from
Eq. (1) when considering the lack of In and the approximately doubled number
of nuclei $N$ in contact with the electron [44], compared to InGaAs QDs [43].
Our results are also compatible with previous claims of dephasing-free In-
containg QD sources of polarization-entangled photon pairs [20, 21] in the
limit of low detector time-jitter $<$50\text{\,}\mathrm{p}\mathrm{s}$$. In
fact, the slowly varying OS, with correlation times above
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, lead to coherent phase
rotations over the time $t$ elapsed between the XX and X recombination. This
may lead to the observation of almost optimal entanglement for $t<<T_{2}^{*}$
and entanglement degradation for $t$ comparable to $T_{2}^{*}$ due to
averaging of the phase evolution occurring at different frequencies and
directions. To improve the performance of InGaAs QDs as sources of entangled
photons, the amplitude of the hyperfine fluctuations can be decreased by
lowering the In-contribution or by developing QD structures in which the
number of nuclei in contact with the electron wavefunction is increased and/or
the X lifetime is decreased [59]. Alternatively, the Overhauser fluctuations
could be mitigated by optically pre-cooling the nuclear spin bath below the
thermal limit [47, 43]. These measures could boost the polarization
entanglement of InGaAs QDs, making them compatible with state-of-the-art
quantum communication protocols and demanding tasks in quantum computation.

The authors thank S. Moser for his assistance with the optical setup and U.
Kainz for the help in device processing. This work has been supported by the
Austrian Academy of Sciences (ÖAW), the QuantERA HYPER-U-P-S project, which
received funding from the QuantERA ERA-NET Cofund in Quantum Technologies
implemented within the European Union’s Horizon 2020 Programme and the
Austrian Science Fund (FWF) I 3762, the European Union’s Horizon 2020 research
and innovation program under Grant Agreements No. 899814 (Qurope) and No.
871130 (Ascent+), the QuantERA II Programme that has received funding from the
European Union’s Horizon 2020 research and innovation programme under Grant
Agreement No 101017733 via the project QD-E-QKD and the FFG Grant No. 891366,
the Linz Institute of Technology (LIT), the LIT Secure and Correct Systems Lab
funded by the State of Upper Austria. We further acknowledge funding by the
DFG within the project PR1749/1-1. A.P would like to acknowledge the support
of the Swedish research council. L. G. was supported by the Knut&Alice
Wallenberg Foundation [through the Wallenberg Centre for Quantum Technology
(WACQT)]. D.G. was supported by a University Research Fellowship from the
Royal Society. E.A.C. was supported by the Royal Society and through an EPSRC
grant EP/V048333/1.

## References

  * Senellart  _et al._ [2017] P. Senellart, G. Solomon, and A. White, High-performance semiconductor quantum-dot single-photon sources, Nat. Nanotechnol. 12, 1026 (2017).
  * Tomm  _et al._ [2021] N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. Löbl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, and R. J. Warburton, A bright and fast source of coherent single photons, Nature Nanotechnology 2021 16:4 16, 399 (2021).
  * Liu  _et al._ [2019] J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. da Silva, Y. Yu, J. Iles-Smith, K. Srinivasan, A. Rastelli, J. Li, and X. Wang, A solid-state source of strongly entangled photon pairs with high brightness and indistinguishability, Nat. Nanotechnol. 14, 586 (2019).
  * Wang  _et al._ [2019] H. Wang, H. Hu, T. H. Chung, J. Qin, X. Yang, J. P. Li, R. Z. Liu, H. S. Zhong, Y. M. He, X. Ding, Y. H. Deng, Q. Dai, Y. H. Huo, S. Höfling, C. Y. Lu, and J. W. Pan, On-Demand Semiconductor Source of Entangled Photons Which Simultaneously Has High Fidelity, Efficiency, and Indistinguishability, Phys. Rev. Lett. 122, 113602 (2019).
  * Schweickert  _et al._ [2018] L. Schweickert, K. D. Jöns, K. D. Zeuner, S. F. Covre Da Silva, H. Huang, T. Lettner, M. Reindl, J. Zichi, R. Trotta, A. Rastelli, and V. Zwiller, On-demand generation of background-free single photons from a solid-state source, Appl. Phys. Lett. 112, 093106 (2018).
  * Neuwirth  _et al._ [2022] J. Neuwirth, F. B. Basset, M. B. Rota, J.-G. Hartel, M. Sartison, S. F. C. da Silva, K. D. Jöns, A. Rastelli, and R. Trotta, A multipair-free source of entangled photons in the solid state, Phys. Rev. B - Condens. Matter Mater. Phys. 106, L241402 (2022).
  * Vural  _et al._ [2020] H. Vural, S. L. Portalupi, and P. Michler, Perspective of self-assembled InGaAs quantum-dots for multi-source quantum implementations, Appl. Phys. Lett. 117, 030501 (2020).
  * Benson  _et al._ [2000] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, Regulated and entangled photons from a single quantum dot, Phys. Rev. Lett. 84, 2513 (2000).
  * Akopian  _et al._ [2006] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, Entangled photon pairs from semiconductor quantum dots, Phys. Rev. Lett. 96, 130501 (2006).
  * Young  _et al._ [2006] R. J. Young, R. M. Stevenson, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, Improved fidelity of triggered entangled photons from single quantum dots, New J. Phys. 8, 29 (2006).
  * Hafenbrak  _et al._ [2007] R. Hafenbrak, S. M. Ulrich, P. Michler, L. Wang, A. Rastelli, and O. G. Schmidt, Triggered polarization-entangled photon pairs from a single quantum dot up to 30 K, New J. Phys. 9, 315 (2007).
  * Salter  _et al._ [2010] C. L. Salter, R. M. Stevenson, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, An entangled-light-emitting diode, Nature 465, 594 (2010).
  * Dousse  _et al._ [2010] A. Dousse, J. Suffczyński, A. Beveratos, O. Krebs, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, Ultrabright source of entangled photon pairs, Nature 466, 217 (2010).
  * Juska  _et al._ [2013] G. Juska, V. Dimastrodonato, L. O. Mereni, A. Gocalinska, and E. Pelucchi, Towards quantum-dot arrays of entangled photon emitters, Nature Photonics 7, 527 (2013).
  * Müller  _et al._ [2014] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl, and P. Michler, On-demand generation of indistinguishable polarization-entangled photon pairs, Nat. Photonics 8, 224 (2014).
  * Trotta  _et al._ [2014] R. Trotta, J. S. Wildmann, E. Zallo, O. G. Schmidt, and A. Rastelli, Highly entangled photons from hybrid piezoelectric-semiconductor quantum dot devices, Nano Lett. 14, 3439 (2014).
  * Olbrich  _et al._ [2017] F. Olbrich, J. Höschele, M. Müller, J. Kettler, S. Luca Portalupi, M. Paul, M. Jetter, and P. Michler, Polarization-entangled photons from an InGaAs-based quantum dot emitting in the telecom C-band, Appl. Phys. Lett. 111, 133106 (2017).
  * Müller  _et al._ [2018] T. Müller, J. Skiba-Szymanska, A. B. Krysa, J. Huwer, M. Felle, M. Anderson, R. M. Stevenson, J. Heffernan, D. A. Ritchie, and A. J. Shields, A quantum light-emitting diode for the standard telecom window around 1,550 nm, Nat. Commun. 9, 1 (2018).
  * Zeuner  _et al._ [2021] K. D. Zeuner, K. D. Jöns, L. Schweickert, C. Reuterskiöld Hedlund, C. Nuñez Lobato, T. Lettner, K. Wang, S. Gyger, E. Schöll, S. Steinhauer, M. Hammar, and V. Zwiller, On-Demand Generation of Entangled Photon Pairs in the Telecom C-Band with InAs Quantum Dots, ACS Photonics 8, 2337 (2021).
  * Winik  _et al._ [2017] R. Winik, D. Cogan, Y. Don, I. Schwartz, L. Gantz, E. R. Schmidgall, N. Livneh, R. Rapaport, E. Buks, and D. Gershoni, On-demand source of maximally entangled photon pairs using the biexciton-exciton radiative cascade, Phys. Rev. B 95, 235435 (2017).
  * Fognini  _et al._ [2019] A. Fognini, A. Ahmadi, M. Zeeshan, J. T. Fokkens, S. J. Gibson, N. Sherlekar, S. J. Daley, D. Dalacu, P. J. Poole, K. D. Jöns, V. Zwiller, and M. E. Reimer, Dephasing Free Photon Entanglement with a Quantum Dot, ACS Photonics 6, 1656 (2019).
  * Anderson  _et al._ [2020] M. Anderson, T. Müller, J. Skiba-Szymanska, A. B. Krysa, J. Huwer, R. M. Stevenson, J. Heffernan, D. A. Ritchie, and A. J. Shields, Gigahertz-Clocked Teleportation of Time-Bin Qubits with a Quantum Dot in the Telecommunication C Band, Phys. Rev. Appl. 13, 054052 (2020).
  * Huber  _et al._ [2018] D. Huber, M. Reindl, S. F. Covre da Silva, C. Schimpf, J. Martín-Sánchez, H. Huang, G. Piredda, J. Edlinger, A. Rastelli, and R. Trotta, Strain-Tunable GaAs Quantum Dot: A Nearly Dephasing-Free Source of Entangled Photon Pairs on Demand, Phys. Rev. Lett. 121, 033902 (2018).
  * Gurioli  _et al._ [2019] M. Gurioli, Z. Wang, A. Rastelli, T. Kuroda, and S. Sanguinetti, Droplet epitaxy of semiconductor nanostructures for quantum photonic devices, Nat. Mater. 18, 799 (2019).
  * da Silva  _et al._ [2021] S. F. C. da Silva, G. Undeutsch, B. Lehner, S. Manna, T. M. Krieger, M. Reindl, C. Schimpf, R. Trotta, and A. Rastelli, GaAs quantum dots grown by droplet etching epitaxy as quantum light sources, Appl. Phys. Lett. 119, 120502 (2021).
  * Zhai  _et al._ [2022] L. Zhai, G. N. Nguyen, C. Spinnler, J. Ritzmann, M. C. Löbl, A. D. Wieck, A. Ludwig, A. Javadi, and R. J. Warburton, Quantum interference of identical photons from remote GaAs quantum dots, Nature Nanotechnology , 1 (2022), arXiv:2106.03871 .
  * Bayer  _et al._ [2002] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots, Phys. Rev. B 65, 195315 (2002).
  * Hudson  _et al._ [2007] A. J. Hudson, R. M. Stevenson, A. J. Bennett, R. J. Young, C. A. Nicoll, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, Coherence of an entangled exciton-photon state, Phys. Rev. Lett. 99, 266802 (2007).
  * Hohenester  _et al._ [2007] U. Hohenester, G. Pfanner, and M. Seliger, Phonon-assisted decoherence in the production of polarization-entangled photons in a single semiconductor quantum dot, Phys. Rev. Lett. 99, 047402 (2007).
  * Reigue  _et al._ [2017] A. Reigue, J. Iles-Smith, F. Lux, L. Monniello, M. Bernard, F. Margaillan, A. Lemaitre, A. Martinez, D. P. McCutcheon, J. Mørk, R. Hostein, and V. Voliotis, Probing Electron-Phonon Interaction through Two-Photon Interference in Resonantly Driven Semiconductor Quantum Dots, Phys. Rev. Lett. 118, 233602 (2017).
  * Seidelmann  _et al._ [2022] T. Seidelmann, C. Schimpf, T. K. Bracht, M. Cosacchi, A. Vagov, A. Rastelli, D. E. Reiter, and V. M. Axt, Two-Photon Excitation Sets Fundamental Limit to Entangled Photon Pair Generation from Quantum Emitters, Phys. Rev. Lett. 129, 193604 (2022).
  * Seidelmann  _et al._ [2023] T. Seidelmann, T. K. Bracht, B. U. Lehner, C. Schimpf, M. Cosacchi, M. Cygorek, A. Vagov, A. Rastelli, D. E. Reiter, and V. M. Axt, Two-photon excitation with finite pulses unlocks pure dephasing-induced degradation of entangled photons emitted by quantum dots (2023), arXiv.2301.10820 .
  * Merkulov  _et al._ [2002] I. A. Merkulov, A. L. Efros, and M. Rosen, Electron spin relaxation by nuclei in semiconductor quantum dots, Phys. Rev. B - Condens. Matter Mater. Phys. 65, 205309 (2002).
  * Urbaszek  _et al._ [2013] B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin, P. Maletinsky, A. Högele, and A. Imamoglu, Nuclear spin physics in quantum dots: An optical investigation, Rev. Mod. Phys. 85, 79 (2013).
  * Kuhlmann  _et al._ [2013] A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. J. Warburton, Charge noise and spin noise in a semiconductor quantum device, Nat. Phys. 9, 570 (2013).
  * Kuroda  _et al._ [2013] T. Kuroda, T. Mano, N. Ha, H. Nakajima, H. Kumano, B. Urbaszek, M. Jo, M. Abbarchi, Y. Sakuma, K. Sakoda, I. Suemune, X. Marie, and T. Amand, Symmetric quantum dots as efficient sources of highly entangled photons: Violation of Bell’s inequality without spectral and temporal filtering, Physical Review B 88, 041306 (2013).
  * Huber  _et al._ [2017] D. Huber, M. Reindl, Y. Huo, H. Huang, J. S. Wildmann, O. G. Schmidt, A. Rastelli, and R. Trotta, Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots, Nat. Commun. 8, 15506 (2017).
  * Keil  _et al._ [2017] R. Keil, M. Zopf, Y. Chen, B. Höfer, J. Zhang, F. Ding, and O. G. Schmidt, Solid-state ensemble of highly entangled photon sources at rubidium atomic transitions, Nat. Commun. 8, 15501 (2017).
  * He  _et al._ [2013] Y. M. He, Y. He, Y. J. Wei, D. Wu, M. Atatüre, C. Schneider, S. Höfling, M. Kamp, C. Y. Lu, and J. W. Pan, On-demand semiconductor single-photon source with near-unity indistinguishability, Nat. Nanotechnol. 8, 213 (2013).
  * Braun  _et al._ [2016] T. Braun, S. Betzold, N. Lundt, M. Kamp, S. Höfling, and C. Schneider, Impact of ex situ rapid thermal annealing on magneto-optical properties and oscillator strength of In(Ga)As quantum dots, Phys. Rev. B 93, 155307 (2016).
  * Zhang  _et al._ [2015] J. Zhang, J. S. Wildmann, F. Ding, R. Trotta, Y. Huo, E. Zallo, D. Huber, A. Rastelli, and O. G. Schmidt, High yield and ultrafast sources of electrically triggered entangled-photon pairs based on strain-tunable quantum dots, Nature communications 6, 1 (2015).
  * Stockill  _et al._ [2016] R. Stockill, C. Le Gall, C. Matthiesen, L. Huthmacher, E. Clarke, M. Hugues, and M. Atatüre, Quantum dot spin coherence governed by a strained nuclear environment, Nat. Commun. 7, 1 (2016).
  * Gangloff  _et al._ [2019] D. A. Gangloff, G. Éthier-Majcher, C. Lang, E. V. Denning, J. H. Bodey, D. M. Jackson, E. Clarke, M. Hugues, C. Le Gall, and M. Atatüre, Quantum interface of an electron and a nuclear ensemble, Science (80-. ). 364, 62 (2019).
  * Zaporski  _et al._ [2023] L. Zaporski, N. Shofer, J. H. Bodey, S. Manna, G. Gillard, M. H. Appel, C. Schimpf, S. F. Covre da Silva, J. Jarman, G. Delamare, G. Park, U. Haeusler, E. A. Chekhovich, A. Rastelli, D. A. Gangloff, M. Atatüre, and C. Le Gall, Ideal refocusing of an optically active spin qubit under strong hyperfine interactions, Nat. Nanotechnol. 10.1038/s41565-022-01282-2 (2023).
  * Welander  _et al._ [2014] E. Welander, J. Hildmann, and G. Burkard, Influence of Hyperfine Interaction on the Entanglement of Photons Generated by Biexciton Recombination, arXiv e-prints , arXiv:1409.6521 (2014), arXiv:1409.6521 [cond-mat.mes-hall] .
  * Eble  _et al._ [2009] B. Eble, C. Testelin, P. Desfonds, F. Bernardot, A. Balocchi, T. Amand, A. Miard, A. Lemaître, X. Marie, and M. Chamarro, Hole-nuclear spin interaction in quantum dots, Phys. Rev. Lett. 102, 146601 (2009).
  * Éthier-Majcher  _et al._ [2017] G. Éthier-Majcher, D. Gangloff, R. Stockill, E. Clarke, M. Hugues, C. Le Gall, and M. Atatüre, Improving a Solid-State Qubit through an Engineered Mesoscopic Environment, Phys. Rev. Lett. 119, 130503 (2017).
  * Press  _et al._ [2010] D. Press, K. De Greve, P. L. McMahon, T. D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Höfling, A. Forchel, and Y. Yamamoto, Ultrafast optical spin echo in a single quantum dot, Nat. Photonics 4, 367 (2010).
  * Stufler  _et al._ [2006] S. Stufler, P. Machnikowski, P. Ester, M. Bichler, V. M. Axt, T. Kuhn, and A. Zrenner, Two-photon Rabi oscillations in a single Inx Ga1-x As GaAs quantum dot, Phys. Rev. B - Condens. Matter Mater. Phys. 73, 125304 (2006).
  * Bennett  _et al._ [2010] A. Bennett, y. M. Pooley, n. R. Stevenson, M. Ward, R. Patel, A. B. de La Giroday, N. Sköld, I. Farrer, C. Nicoll, D. Ritchie, _et al._ , Electric-field-induced coherent coupling of the exciton states in a single quantum dot, Nature Physics 6, 947 (2010).
  * Lettner  _et al._ [2021] T. Lettner, S. Gyger, K. D. Zeuner, L. Schweickert, S. Steinhauer, C. Reuterskiöld Hedlund, S. Stroj, A. Rastelli, M. Hammar, R. Trotta, K. D. Jöns, and V. Zwiller, Strain-Controlled Quantum Dot Fine Structure for Entangled Photon Generation at 1550 nm, Nano Lett. 21, 10501 (2021).
  * James  _et al._ [2001] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Phys. Rev. A 64, 052312 (2001).
  * Wootters [1998] W. K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).
  * Belhadj  _et al._ [2010] T. Belhadj, T. Amand, A. Kunold, C. M. Simon, T. Kuroda, M. Abbarchi, T. Mano, K. Sakoda, S. Kunz, X. Marie, and B. Urbaszek, Impact of heavy hole-light hole coupling on optical selection rules in GaAs quantum dots, Appl. Phys. Lett. 97, 051111 (2010).
  * Tonin  _et al._ [2012] C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, Polarization properties of excitonic qubits in single self-assembled quantum dots, Physical Review B 85, 155303 (2012).
  * Plumhof  _et al._ [2013] J. D. Plumhof, R. Trotta, V. Křápek, E. Zallo, P. Atkinson, S. Kumar, A. Rastelli, and O. G. Schmidt, Tuning of the valence band mixing of excitons confined in gaas/algaas quantum dots via piezoelectric-induced anisotropic strain, Phys. Rev. B 87, 075311 (2013).
  * Bulutay [2012] C. Bulutay, Quadrupolar spectra of nuclear spins in strained In xGa 1-xAs quantum dots, Phys. Rev. B - Condens. Matter Mater. Phys. 85, 1 (2012).
  * Denning  _et al._ [2019] E. V. Denning, D. A. Gangloff, M. Atatüre, J. Mørk, and C. Le Gall, Collective Quantum Memory Activated by a Driven Central Spin, Phys. Rev. Lett. 123, 140502 (2019).
  * Huber  _et al._ [2019] D. Huber, B. U. Lehner, D. Csontosová, M. Reindl, S. Schuler, S. F. Covre Da Silva, P. Klenovský, and A. Rastelli, Single-particle-picture breakdown in laterally weakly confining GaAs quantum dots, Phys. Rev. B 100, 235425 (2019).