††thanks: Wooyeong Song and Junghee Ryu contributed equally to this
work††thanks: Wooyeong Song and Junghee Ryu contributed equally to this work

# Average fidelity and fidelity deviation in noisy quantum teleportation

Wooyeong Song  Department of Physics, Hanyang University, Seoul 04763, Korea
Junghee Ryu  Division of National Supercomputing, Korea Institute of Science
and Technology Information, Daejeon 34141, Korea  Kyunghyun Baek  Jeongho Bang
<EMAIL_ADDRESS>School of Computational Sciences, Korea Institute for
Advanced Study, Seoul 02455, Korea

###### Abstract

We analyze the average fidelity (say, $F$) and the fidelity deviation (say,
$D$) in noisy-channel quantum teleportation. Here, $F$ represents how well
teleportation is performed on average and $D$ quantifies whether the
teleportation is performed impartially on the given inputs, that is, the
condition of universality. Our analysis results prove that the achievable
maximum average fidelity ensures zero fidelity deviation, that is, perfect
universality. This structural trait of teleportation is distinct from those of
other limited-fidelity probabilistic quantum operations, for instance,
universal-NOT or quantum cloning. This feature is confirmed again based on a
tighter relationship between $F$ and $D$ in the qubit case. We then consider
another realistic noise model where $F$ decreases and $D$ increases due to
imperfect control. To alleviate such deterioration, we propose a machine-
learning-based algorithm. We demonstrate by means of numerical simulations
that the proposed algorithm can stabilize the system. Notably, the recovery
process consists solely of the maximization of $F$, which reduces the control
time, thus leading to a faster cure cycle.

Quantum teleportation; Quantum machine learning

##  I Introduction

Quantum teleportation makes possible the deterministic transmission of unknown
quantum states from one location to another Bennett et al. (1993). It has been
acknowledged as a fundamental scheme of state transfer. A shared quantum
channel between a sender and a receiver is one of the essential ingredients
for quantum teleportation, and quantum entanglement in the channel is
necessary to ensure that the fidelity is superior to that of classical
communication protocols. Moreover, quantum teleportation provides a useful
framework to study quantum nonlocality Popescu (1994); Gisin (1996); Barrett
(2001); Cavalcanti et al. (2013) and is one of the basic steps in constructing
element gates, for example, single-qubit and CNOT gates, which are used in
continuous-variable (CV) quantum computation Jeong and Kim (2002); Ralph et
al. (2003).

Fidelity $f$ is used to measure the closeness between the input and the
teleported states. One of the methods to quantify the performance of
teleportation based on $f$ without any dependence on the input states involves
averaging $f$ over all the possible inputs. The average fidelity $F$, defined
as a uniform average of $f$, has been widely used as a relevant input-
independent measure of teleportation performance. It is well-known that $F$
can reach $1$ in quantum teleportation, whereas the maximally attainable $F$
is limited in any classical state-transfer scheme without quantum entanglement
Bennett et al. (1993). However, $F$ does not consider universality, which
indicates whether teleportation is performed equally for all of the input
states. For example, a non universal teleportation protocol would be
successful only for a specific set of inputs. This limitation can hinder the
use of a teleportation protocol to implement element gates in CV quantum
computation Jeong and Kim (2002); Ralph et al. (2003). To quantify the
universality condition, a fidelity deviation $D$ defined in terms of the
standard deviation of $f$ was introduced Bang et al. (2012, 2018).

In this study, we analyze the two measures $F$ and $D$ in the context of noisy
qudit—a $d$-level quantum system—teleportation. We show that perfect
universality, $D=0$, is attainable without any dependence on the quantum
channel condition, while the maximum average fidelity $F_{\text{max}}$ is a
function of the degree of entanglement in the quantum channel. We prove that
the condition of $F_{\text{max}}$ is adequate to guarantee perfect
universality. For the case of a qubit, that is, $d=2$, we demonstrate a more
general and tighter relationship between $F$ and $D$ that addresses the
aforementioned behaviors in a clearer manner. Then, we consider a realistic
situation in which operational noises can deteriorate the teleportation
performance, as represented by a decrease in $F$ and an increase in $D$. We
propose a significant machine-learning-based method to alleviate such
deterioration. We numerically demonstrate that the proposed machine-learning-
based method is effective within a certain rate of noise occurrence. Note that
the recovery of $F$ and $D$ can be implemented solely by using the process of
$F\to F_{\text{max}}$. This is because $F_{\text{max}}$ itself suffices the
condition of zero deviation. This feature allows us to reduce the control
and/or learning time, which increases the recovery rate.

##  II Average fidelity and fidelity deviation

We first review the two measures, namely, average fidelity $F$ and fidelity
deviation $D$. Consider a map (or a general quantum operation) ${\cal
T}:\left|\phi\right>\rightarrow\hat{\rho}_{\text{out}}$, where
$\left|\phi\right>$ is an input and $\hat{\rho}_{\text{out}}$ an output. The
fidelity $f$ is defined as Jozsa (1994)

$\displaystyle
f=\mbox{Tr}{\left[\left(\sqrt{\hat{\rho}_{\text{t}}}\hat{\rho}_{\text{out}}\sqrt{\hat{\rho}_{\text{t}}}\right)^{1/2}\right]},$
(1)

where $\hat{\rho}_{\text{t}}$ is a target state that is assumed to be a pure
state $\left|\phi_{t}\right>$. Then, formula (1) can be rewritten as

$\displaystyle
f=\left<\phi_{t}\right|\hat{\rho}_{\text{out}}\left|\phi_{t}\right>.$ (2)

In a well-designed ${\cal T}$, almost all of the input states
$\left|\phi\right>$ are correctly transferred to their corresponding targets
$\left|\phi_{t}\right>$. The average fidelity represents how well a state
transfer is performed. It is defined by averaging $f$ over all possible inputs
$\left|\phi\right>$ as

$\displaystyle F=\int{\text{d}\phi}\,f,$ (3)

where $\text{d}\phi$ is the Haar measure that satisfies $\int\text{d}\phi=1$.
In general, $f$ varies with respect to the input states for a given channel
${\cal T}$. If $f$ is uniform for a task, the task is said to be universal. To
devise a measure of universality, we employ the fidelity deviation $D$ in
terms of the fluctuation of $f$ Bang et al. (2012, 2018):

$\displaystyle D=\left(\int{\text{d}\phi}\,f^{2}-F^{2}\right)^{\frac{1}{2}}.$
(4)

One can show perfect universality, that is, $D=0$ when $f=F$. Otherwise, it is
strictly positive. Additionally,

$\displaystyle D^{2}\leq\int{\text{d}\phi}\,{f}-F^{2}=F(1-F)\leq\frac{1}{4},$
(5)

where the last equality holds when $F=1/2$. Thus, $D$ is bounded as $0\leq
D\leq 1/2$.

Before going further, we note that $F=1$ holds iff $f=1$ for all possible
inputs, and therefore, $F=1$ implies perfect universality. However, when the
attainable maximum of $F$ is less than one ($F_{\text{max}}<1$), perfect
universality is not guaranteed at $F_{\text{max}}$ in general. For example, if
$F_{\text{max}}<1$, then $0\leq D\leq\sqrt{F(1-F)}$ [see Eq. (5)]. This is
true for almost all probabilistic tasks, for instance, universal-NOT Bužek et
al. (1999, 2000) and quantum cloning Bužek and Hillery (1996). Therefore, it
is natural to consider the minimization of $D$ independently of the
maximization of $F$.

##  III Noisy quantum teleportation

In this section, we describe noisy quantum teleportation Braunstein et al.
(2000). First, a sender, say Alice, has a $d$-dimensional pure state
$\hat{\rho}_{\phi}=\left|{\phi}\left>\right<{\phi}\right|$ that is to be
transferred to a receiver, say Bob (in general, Alice has no information about
the input state $\hat{\rho}_{\phi}$). A maximally entangled state
$\left|\Psi_{0}\right>=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\left|j\right>\otimes\left|j\right>$
is shared by Alice and Bob. Second, Alice performs a joint measurement by
using a set of maximally entangled bases $\\{\left|\Psi_{\alpha}\right>\\}$
($\alpha=0,1,\ldots,d^{2}-1$) on the composite system of the unknown state and
one of the entangled pairs. The entangled basis is obtained as

$\displaystyle\left|\Psi_{\alpha}\right>=(\hat{U}_{\alpha}\otimes\hat{\openone}_{d})\left|\Psi_{0}\right>,$
(6)

where $\hat{\openone}_{d}$ is the identity of the $d$-dimensional Hilbert
space and $\hat{U}_{\alpha}$ is a unitary conditioned by completeness, that
is,
$\sum_{\alpha=0}^{d^{2}-1}\left|\Psi_{\alpha}\right>\left<\Psi_{\alpha}\right|=\hat{\openone}_{d^{2}}$
Braunstein et al. (2000); Son et al. (2001). The outcome of Alice’s joint
measurement is communicated to Bob through a classical channel. Lastly, Bob
applies a local operation $\hat{V}_{\alpha}$ based on the measurement outcomes
received from Alice. Then, the state $\left|\phi\right>$ is reproduced on
Bob’s side (see Fig. 1).

Sharing the entanglement between Alice and Bob is the crucial step in ensuring
that the better performance is superior to that of any classical protocol
Horodecki et al. (1999). However, under realistic conditions, quantum
entanglement often becomes noisy Carvalho et al. (2004); Burić (2008), and
thus, we need to resolve the noisy case. Herein, we consider the following
($d\times d$)-dimensional noisy entanglement as the quantum channel Werner
(1989):

$\displaystyle\hat{\rho}_{\Psi_{0}}=\gamma\left|\Psi_{0}\right>\left<\Psi_{0}\right|+\frac{1-\gamma}{d^{2}}\hat{\openone}_{d^{2}},$
(7)

which is a statistical mixture of the maximally entangled state
$\left|\Psi_{0}\right>$ and white (symmetric) noise, and $\gamma$ is a
fraction of $\left|\Psi_{0}\right>$.

Figure 1: Schematic of $d$-dimensional quantum state teleportation. The
quantum channel, i.e., shared entangled state $\hat{\rho}_{\Psi_{0}}$, can be
noisy [as in Eq. (7)]. The errors can also arise in $\hat{U}_{\alpha}$ and
$\hat{V}_{\alpha}$ due to the imperfections of control. It can be cured in our
scheme (see Sect. V for details).

.

##  IV Analysis of $F$ and $D$ in noisy quantum teleportation

In this section, we discuss the relationship between the two measures, namely,
the average fidelity $F$ and fidelity deviation $D$, according to the noise
parameter $\gamma$ and dimension $d$.

###  IV.1 Qudit-teleportation

By definition, the value of $F$ ranges between 0 and 1. However, for a given
set of $\gamma$ and $d$, one can restrict the range by using the following
relationship:

######  Result 1.

For a given set of noise parameter $\gamma$ and dimension $d$, $F$ has a
relation such that

$\displaystyle\frac{1}{d}-\frac{\gamma}{d(d+1)}\leq
F\leq\gamma+\frac{(1-\gamma)}{d}.$ (8)

This result is derived from the following extensive analysis of $F$. First, we
write the transmitted state $\hat{\rho}_{\text{out}}$ as

$\displaystyle\hat{\rho}_{\text{out}}=\sum_{\alpha=0}^{d^{2}-1}\left(\hat{\openone}_{d^{2}}\otimes\hat{V}_{\alpha}\right)\left<\Psi^{\text{A}}_{\alpha}\right|\hat{\rho}_{\text{tot}}\left|\Psi^{\text{A}}_{\alpha}\right>\left(\hat{\openone}_{d^{2}}\otimes\hat{V}_{\alpha}^{\dagger}\right),$
(9)

where
$\left|\Psi_{\alpha}^{\text{A}}\right>=\left|\Psi_{\alpha}\right>\otimes\hat{\openone}_{d}$
denotes Alice’s joint measurement,
$\hat{\openone}_{d^{2}}\otimes\hat{V}_{\alpha}$ represents Bob’s local
operation, and
$\hat{\rho}_{\text{tot}}=\hat{\rho}_{\phi}\otimes\hat{\rho}_{\Psi_{0}}$ is the
total initial state of $\hat{\rho}_{\phi}$ that is to be transferred, where
$\hat{\rho}_{\Psi_{0}}$ is the shared entangled state. We can rewrite Eq. (9)
as

$\displaystyle\hat{\rho}_{\text{out}}=\frac{\gamma}{d^{2}}\sum_{\alpha=0}^{d^{2}-1}\hat{X}_{\alpha}\hat{\rho}_{\phi}\hat{X}_{\alpha}^{\dagger}+\frac{1-\gamma}{d}\hat{\openone}_{d},$
(10)

where $\hat{X}_{\alpha}=\hat{V}_{\alpha}\hat{U}_{\alpha}^{\dagger}$. The
fidelity $f$ for a given
$\hat{\rho}_{\phi}=\left|\phi\right>\left<\phi\right|$ is

$\displaystyle
f=\frac{\gamma}{d^{2}}\sum_{\alpha=0}^{d^{2}-1}\xi_{\alpha}+\frac{1-\gamma}{d},$
(11)

where $\xi_{\alpha}$ is the fidelity between $\hat{\rho}_{\phi}$ and
$\hat{X}_{\alpha}\hat{\rho}_{\phi}\hat{X}_{\alpha}^{\dagger}$, and it is
expressed as

$\displaystyle\xi_{\alpha}=\mbox{Tr}{(\hat{X}_{\alpha}\hat{\rho}_{\phi}\hat{X}_{\alpha}^{\dagger}\hat{\rho}_{\phi}})=\left|\left<\phi\right|\hat{X}_{\alpha}\left|\phi\right>\right|^{2}.$
(12)

By using the above descriptions, we evaluate $F$ and $D$. First, we consider
the average fidelity $F$. From Eqs. (3) and (11), we have

$\displaystyle
F=\int\text{d}\phi\,f=\frac{\gamma}{d^{2}}\sum_{\alpha=0}^{d^{2}-1}\int\text{d}\phi\,\xi_{\alpha}+\frac{1-\gamma}{d}.$
(13)

The integral $\int\text{d}\phi\,\xi_{\alpha}$ can be calculated by using a
lemma of identity, called Schur’s lemma Albeverio et al. (2002); Braunstein et
al. (2000):

$\displaystyle\int_{G}\text{d}g\,\left(\hat{U}_{g}^{\dagger}\otimes\hat{U}_{g}^{\dagger}\right)\hat{X}\left(\hat{U}_{g}\otimes\hat{U}_{g}\right)=a\hat{\openone}_{d^{2}}+b\hat{P},$
(14)

where

$\displaystyle
a=\frac{d^{2}\mbox{Tr}{(\hat{X})}-d\mbox{Tr}{(\hat{X}\hat{P})}}{d^{2}(d^{2}-1)},$
$\displaystyle
b=\frac{d^{2}\mbox{Tr}{(\hat{X}\hat{P})}-d\mbox{Tr}{(\hat{X})}}{d^{2}(d^{2}-1)},$
(15)

for any operator $\hat{X}$ in the $d\times d$-dimensional Hilbert space. Here,
$\text{d}g$ denotes the (normalized) Haar measure on the unitary group
$G=U(d)$, satisfying $\int_{G}\text{d}g=1$; $\hat{U}_{g}$ is an irreducible
representation of $g\in G$; and $\hat{P}$ denotes the swap operator defined by
$\hat{P}\left|ij\right>=\left|ji\right>$. By applying Schur’s lemma, we can
compute the integral $\int\text{d}\phi\,\xi_{\alpha}$ such that

$\displaystyle\int\text{d}\phi\,\xi_{\alpha}$ $\displaystyle=$
$\displaystyle\left<\mathbf{0}\mathbf{0}\right|\int\text{d}\phi\,\left(\hat{U}_{\phi}^{\dagger}\otimes\hat{U}_{\phi}^{\dagger}\right)\left(\hat{X}_{\alpha}\otimes\hat{X}_{\alpha}^{\dagger}\right)\left(\hat{U}_{\phi}\otimes\hat{U}_{\phi}\right)\left|\mathbf{0}\mathbf{0}\right>$
(16) $\displaystyle=$
$\displaystyle\frac{1}{d(d+1)}\left(\left|\mbox{Tr}{(\hat{X}_{\alpha})}\right|^{2}+d\right),$

where $\left|\phi\right>=\hat{U}_{\phi}\left|\mathbf{0}\right>$. Then, we
arrive at the final form of $F$ as

$\displaystyle
F=F_{\text{max}}-\frac{\gamma}{d+1}\left(d-\frac{1}{d^{3}}\sum_{\alpha=0}^{d^{2}-1}\left|\mbox{Tr}{(\hat{X}_{\alpha})}\right|^{2}\right),$
(17)

where $F_{\text{max}}=\gamma+\frac{(1-\gamma)}{d}$, which is the maximum value
of $F$ for a given $\gamma$.

We rewrite Eq. (17) in a more useful form as follows:

$\displaystyle F=\frac{dE+1}{d+1},$ (18)

where $E$ is defined as

$\displaystyle
E=\frac{1}{d^{2}}\sum_{\alpha=0}^{d^{2}-1}\left<\Psi_{\alpha}\right|\hat{\rho}_{\Psi_{0}}\left|\Psi_{\alpha}\right>=\frac{\gamma}{d^{4}}\sum_{\alpha=0}^{d^{2}-1}\left|\mbox{Tr}{(\hat{X}_{\alpha})}\right|^{2}+\frac{1-\gamma}{d^{2}}.$
(19)

We find that $E$ is maximized when $\mbox{Tr}{(\hat{X}_{\alpha})}=d$ (or
equivalently, $\hat{X}_{\alpha}=\hat{\openone}_{d}$) for all $\alpha$ and
minimized when $\mbox{Tr}{(\hat{X}_{\alpha})}=0$ for all $\alpha$. Then, the
value of $E$ is bounded by $\gamma$ such that

$\displaystyle\frac{(1-\gamma)}{d^{2}}\leq
E\leq\gamma+\frac{(1-\gamma)}{d^{2}},$ (20)

where the upper bound $\gamma+\frac{(1-\gamma)}{d^{2}}$ is called the
(fullest) entanglement fraction of the channel Albeverio et al. (2002). In
this manner, by using Eq. (18) we can finally prove Result 1. Notably, this
result is consistent with the results obtained in previous studies Horodecki
et al. (1999); Albeverio et al. (2002).

Next, we consider the fidelity deviation $D$. Moreover, because the range of
$F$ is limited, we derive a condition of $D$ for a given $\gamma$ and $d$ as
follows.

######  Result 2.

For a given set of noise parameter $\gamma$ and dimension $d$, the fidelity
deviation $D$ has a condition such that

$\displaystyle 0\leq D\leq\gamma\,\overline{\Delta},$ (21)

where
$\overline{\Delta}=\frac{1}{d^{2}}\sum_{\alpha=0}^{d^{2}-1}\Delta_{\alpha}$,
and

$\displaystyle\Delta_{\alpha}^{2}=\int\text{d}\phi\,\xi_{\alpha}^{2}-\left(\int\text{d}\phi\,\xi_{\alpha}\right)^{2},$
(22)

which can be regarded as the fidelity deviation of $\xi_{\alpha}$.

To prove this result, we express $D$ as follows by using Eqs. (11) and (13)

$\displaystyle
D=\left(\int\text{d}\phi\,f^{2}-F^{2}\right)^{\frac{1}{2}}=\frac{\gamma}{d^{2}}\left(\sum_{\alpha,\beta=0}^{d^{2}-1}C_{\alpha\beta}\right)^{\frac{1}{2}},$
(23)

where $C_{\alpha\beta}$ are elements of covariance matrix $\mathbf{C}$ given
by

$\displaystyle
C_{\alpha\beta}=\int\text{d}\phi\,\xi_{\alpha}\xi_{\beta}-\int\text{d}\phi\,\xi_{\alpha}\int\text{d}\phi\,\xi_{\beta}.$
(24)

Note that $\mathbf{C}$ is symmetric, that is,
$C_{\alpha\beta}=C_{\beta\alpha}$, and its diagonal elements
$C_{\alpha\alpha}$ are equal to $\Delta_{\alpha}^{2}$ in Eq. (22).
Furthermore, each element of $\mathbf{C}$ is bounded as

$\displaystyle\left|C_{\alpha\beta}\right|\leq\sqrt{\Delta_{\alpha}^{2}\Delta_{\beta}^{2}},$
(25)

which is known as the variance-covariance inequality Bhatia and Davis (2000).
Then, by using Eq. (25), we obtain Result 2. The perfect universality $D=0$
can be achieved when $\xi_{\alpha}$ is constant for all $\alpha$.

Based on the above results, we discuss the conditions of the sets
$\\{\hat{U}_{\alpha}\\}$ and $\\{\hat{V}_{\alpha}\\}$ from the viewpoint of
achieving the maximum average fidelity and perfect universality. First, we
consider the maximization of $F$. According to Eq. (17), $F_{\text{max}}$ is
achieved when

$\displaystyle\hat{X}_{\alpha}=\hat{\openone}_{d},~{}\text{or
equivalently,}~{}\hat{U}_{\alpha}=\hat{V}_{\alpha},~{}(\forall\alpha).$ (26)

That implies that the maximization of $E$ straightforwardly leads to
$F_{\text{max}}$. Therefore, our main result is as follows:

######  Result 3.

The condition of Eq. (26) naturally suffices perfect universality, that is,

$\displaystyle\hat{X}_{\alpha}=\hat{\openone}_{d}~{}\rightarrow~{}\text{the
perfect universality}~{}D=0.$ (27)

The proof is simple. By using Eq. (12), we rewrite condition (26) as

$\displaystyle\xi_{\alpha}=1~{}(\forall\alpha).$ (28)

Then, $\mathbf{C}$ becomes a zero (or null) matrix because $C_{\alpha\beta}=0$
for $\xi_{\alpha}=const$ ($\forall\alpha$), thus leading to $D=0$, see Eq.
(23). However, the opposite is not always true, that is, perfect universality
does not guarantee the maximum fidelity. We emphasize that Result 3 is a non-
common trait in limited maximum fidelity tasks, for instance, universal-NOT or
cloning Bang et al. (2012).

###  IV.2 Qubit-teleportation

In the case of a qubit, that is, $d=2$, we can derive a tighter relationship
between $F$ and $D$. To this end, we first write the input state
$\hat{\rho}_{\phi}$ in terms of the Bloch representation as

$\displaystyle\hat{\rho}_{\phi}=\frac{1}{2}\left(\hat{\openone}_{2}+\boldsymbol{\phi}^{\text{T}}\boldsymbol{\sigma}\right),$
(29)

where $\boldsymbol{\phi}=(\phi_{x},\phi_{y},\phi_{z})^{\text{T}}$ is a Bloch
vector of unit norm (i.e., $|\boldsymbol{\phi}|=1$) in three-dimensional real
vector space $\mathbb{R}^{3}$, and
$\boldsymbol{\sigma}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})^{\text{T}}$
is a vector operator whose components $\hat{\sigma}_{j}$ ($j=x,y,z$) are Pauli
operators. Then, we can express $\xi_{\alpha}$ in Eq. (12) as

$\displaystyle\xi_{\alpha}=\frac{1}{2}\left(1+\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi}\right),$
(30)

where $\mathbf{R}_{\alpha}$ is a $3\times 3$ rotation matrix in
$\mathbb{R}^{3}$, whose elements $[\boldsymbol{R}_{\alpha}]_{jk}$ are given as

$\displaystyle[\boldsymbol{R}_{\alpha}]_{jk}=\frac{1}{2}\mbox{Tr}{(\hat{X}_{\alpha}\hat{\sigma}_{j}\hat{X}_{\alpha}^{\dagger}\hat{\sigma}_{k})}~{}\text{for}~{}j,k=x,y,z.$
(31)

The rotation angles $\vartheta_{\alpha}$ and axes $\mathbf{n}_{\alpha}$ of
$\mathbf{R}_{\alpha}$ are found in the general expression of a single-qubit
unitary as

$\displaystyle\hat{X}_{\alpha}=e^{-i\frac{\vartheta_{\alpha}}{2}\mathbf{n}_{\alpha}^{\text{T}}\boldsymbol{\sigma}}=\cos{\frac{\vartheta_{\alpha}}{2}}\,\hat{\openone}_{2}-i\sin{\frac{\vartheta_{\alpha}}{2}}\mathbf{n}_{\alpha}^{\text{T}}\boldsymbol{\sigma}.$
(32)

Then, by using Eqs. (13) and (30), the $F$ corresponding to $d=2$ can be
expressed as

$\displaystyle F=\frac{\gamma}{4}\sum_{\alpha=0}^{3}\frac{1}{2}\left[1+\int
d\boldsymbol{\phi}\left(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi}\right)\right]+\frac{1-\gamma}{2},$
(33)

where $\text{d}\boldsymbol{\phi}$ is the Haar measure over the surface of the
Bloch sphere, and it is normalized as $\int\text{d}\boldsymbol{\phi}=1$. Here,
we again employ Schur’s lemma to calculate the integral
$\int\text{d}\boldsymbol{\phi}(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi})$:

$\displaystyle\int_{G}\text{d}g\,\mathbf{O}_{g}\mathbf{X}\mathbf{O}_{g}^{\text{T}}=\frac{1}{r}\mbox{Tr}{(\mathbf{X})}\,\mathbf{I}_{r},$
(34)

where $\mathbf{I}_{r}$ is the identity matrix in $\mathbb{R}^{r}$,
$\mathbf{O}_{g}$ is an irreducible orthogonal representation of an element
$g\in G$, and $\text{d}g$ is the measure, normalized as $\int_{g}\text{d}g=1$.
By applying this lemma to $O(3)$ of three-dimensional rotations, we can
calculate the integral in Eq. (33) as

$\displaystyle\int\text{d}\boldsymbol{\phi}(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi})=\frac{1}{3}\mbox{Tr}{(\mathbf{R}_{\alpha})},$
(35)

and we immediately obtain the following form of $F$:

$\displaystyle
F=\frac{1}{2}+\frac{\gamma}{24}\sum_{\alpha=0}^{3}\mbox{Tr}{(\mathbf{R}_{\alpha})}.$
(36)

Next, we consider $D$. First, we express $D$ as

$\displaystyle
D=\frac{\gamma}{4}\left(\sum_{\alpha=0}^{3}\Delta_{\alpha}^{2}+\sum_{\alpha\neq\beta}C_{\alpha\beta}\right)^{\frac{1}{2}}.$
(37)

Subsequently, we (re)calculate the inequality of Eq. (25) for $d=2$, and a
tighter lower bound can be found as follows (for more details, see appendix B
in Ref. Bang et al. (2012)):

$\displaystyle-\frac{1}{2}\Delta_{\alpha}\Delta_{\beta}\leq
C_{\alpha\beta}\leq\Delta_{\alpha}\Delta_{\beta}~{}(\alpha\neq\beta),$ (38)

where we obtain the lower bound when the two rotation axes
$\mathbf{n}_{\alpha}$ and $\mathbf{n}_{\beta}$ are orthogonal to each other,
that is, $\mathbf{n}_{\alpha}^{\text{T}}\mathbf{n}_{\beta}=0$, and the upper
bound when $\mathbf{n}_{\alpha}$ and $\mathbf{n}_{\beta}$ are parallel or
antiparallel, that is, $\mathbf{n}_{\alpha}^{\text{T}}\mathbf{n}_{\beta}=\pm
1$. Then, the upper bound of $D$ is [from Eq. (38)]

$\displaystyle
D\leq\frac{\gamma}{4}\sqrt{\sum_{\alpha=0}^{3}\Delta_{\alpha}^{2}+\sum_{\alpha\neq\beta}\Delta_{\alpha}\Delta_{\beta}}=\gamma\overline{\Delta}_{d=2},$
(39)

where $\overline{\Delta}_{d=2}=\frac{1}{4}\sum_{\alpha=0}^{3}\Delta_{\alpha}$.
Here, $\Delta_{\alpha}$ are given in the Bloch form, such that

$\displaystyle\Delta_{\alpha}=\frac{1}{2}\left[\int\text{d}\boldsymbol{\phi}\,\left(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi}\right)^{2}-\frac{1}{9}\mbox{Tr}{(\mathbf{R}_{\alpha})}^{2}\right]^{\frac{1}{2}}.$
(40)

Now, we calculate
$\int\text{d}\boldsymbol{\phi}\,(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi})^{2}$
by using the product form of Schur’s lemma:

$\displaystyle\int\text{d}g\left(\mathbf{O}_{g}^{\text{T}}\otimes\mathbf{O}_{g}^{\text{T}}\right)\mathbf{X}\left(\mathbf{O}_{g}\otimes\mathbf{O}_{g}\right)=a\mathbf{I}_{d^{2}}+b\mathbf{D}+c\mathbf{P},$
(41)

where

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{(r+1)\mbox{Tr}{(\mathbf{X})}-\mbox{Tr}{(\mathbf{X}\mathbf{D})}-\mbox{Tr}{(\mathbf{X}\mathbf{P})}}{r(r-1)(r+2)},$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{-\mbox{Tr}{(\mathbf{X})}+(r+1)\mbox{Tr}{(\mathbf{X}\mathbf{D})}-\mbox{Tr}{(\mathbf{X}\mathbf{P})}}{r(r-1)(r+2)},$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{-\mbox{Tr}{(\mathbf{X})}-\mbox{Tr}{(\mathbf{X}\mathbf{D})}+(r+1)\mbox{Tr}{(\mathbf{X}\mathbf{P})}}{r(r-1)(r+2)}.$

Here, $\mathbf{P}$ is a swap matrix
$\mathbf{P}\,(\mathbf{x}_{i}\otimes\mathbf{x}_{j})=\mathbf{x}_{j}\otimes\mathbf{x}_{i}$,
or equivalently,
$\mathbf{P}=\sum_{i,j=0}^{r-1}\left(\mathbf{x}_{j}\otimes\mathbf{x}_{i}\right)\left(\mathbf{x}_{i}\otimes\mathbf{x}_{j}\right)^{\text{T}}$,
and
$\mathbf{D}=\left(\sum_{i=0}^{r-1}\mathbf{x}_{i}\otimes\mathbf{x}_{i}\right)\left(\sum_{j=0}^{r-1}\mathbf{x}_{j}\otimes\mathbf{x}_{j}\right)^{\text{T}}$
where $\\{\mathbf{x}_{i}\\}$ is an orthonormal basis set in $\mathbb{R}^{r}$.
Then, by using this lemma, we can rewrite
$\int\text{d}\boldsymbol{\phi}\,(\boldsymbol{\phi}^{\text{T}}\mathbf{R}_{\alpha}\boldsymbol{\phi})^{2}$
as

$\displaystyle\mathbf{x}_{00}^{\text{T}}\int\text{d}\boldsymbol{\phi}\Big{(}\mathbf{O}_{\boldsymbol{\phi}}^{\text{T}}\otimes\mathbf{O}_{\boldsymbol{\phi}}^{\text{T}}\Big{)}\left(\mathbf{R}_{\alpha}\otimes\mathbf{R}_{\alpha}\right)\Big{(}\mathbf{O}_{\boldsymbol{\phi}}\otimes\mathbf{O}_{\boldsymbol{\phi}}\Big{)}\mathbf{x}_{00},$
(42)

where $\mathbf{x}_{00}=\mathbf{x}_{0}\otimes\mathbf{x}_{0}$ and
$\boldsymbol{\phi}=\mathbf{O}_{\boldsymbol{\phi}}\mathbf{x}_{0}$. Eq. (40) is
then calculated and we have

$\displaystyle\Delta_{\alpha}=\frac{1}{2\sqrt{5}}\left(1-\frac{1}{3}\mbox{Tr}{(\mathbf{R}_{\alpha})}\right),$
(43)

where we used the following properties:

$\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha}\otimes\mathbf{R}_{\alpha})}$
$\displaystyle=$ $\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha})}^{2},$
$\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha}\otimes\mathbf{R}_{\alpha}\,\mathbf{D})}$
$\displaystyle=$
$\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha}\mathbf{R}_{\alpha}^{\text{T}})}=\mbox{Tr}{(\mathbf{I}_{3})}=3,$
$\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha}\otimes\mathbf{R}_{\alpha}\,\mathbf{P})}$
$\displaystyle=$ $\displaystyle\mbox{Tr}{(\mathbf{R}_{\alpha}^{2})}.$ (44)

Consequently, we can derive a tighter relationship between $F$ and $D$ as
follows:

$\displaystyle D\leq\frac{1}{\sqrt{5}}\left(F_{\text{max}}-F\right),$ (45)

where $F_{\text{max}}=\frac{1+\gamma}{2}$. This confirms Result 3, that is,
$D=0$ iff $F=F_{\text{max}}$. Here, we can prove that the maximum value of
$D$, that is, the worst case of universality, implies the minimum average
fidelity, $F_{\text{min}}=\frac{1}{2}-\frac{\gamma}{6}$.

##  V Machine-learning-based stabilization of control in teleportation

Result 3 implies the following: [T.1] The maximization of $F$ naturally
includes the minimization of $D$ in noisy teleportation. This is indeed a
structural trait of teleportation. We conjecture that quantum teleportation
degraded by operational noises can be effectively cured and further provide a
benefit of utilizing the aforementioned trait [T.1] can be achieved. To
investigate this, we assume the operational noises in the control of Alice’s
joint measurements (i.e., $\\{\hat{U}_{\alpha}\\}$) and Bob’s local operations
(i.e., $\\{\hat{V}_{\alpha}\\}$), which deteriorate $F$ and $D$. We propose a
machine-learning-based algorithm to stabilize the teleportation system against
noise. For qubit teleportation (i.e., $d=2$), we demonstrate by means of
numerical simulations that our machine-learning-based algorithm can cure the
deterioration of both $F$ and $D$ solely by maximizing $F$ (see Fig. 1).

###  V.1 Effects of operational noise

In general, a unitary operation in the $d$-dimensional Hilbert space can be
parameterized as follows:

$\displaystyle\hat{U}(\mathbf{p})=\exp{(-i\,\mathbf{p}^{\text{T}}\mathbf{G})},$
(46)

where $\mathbf{p}=(p_{1},p_{2},\ldots,p_{d^{2}-1})^{\text{T}}$ is a
$(d^{2}-1)$-dimensional real vector and
$\mathbf{G}=(\hat{g}_{1},\hat{g}_{2},\ldots,\hat{g}_{d^{2}-1})^{\text{T}}$ is
a vector operator whose components are SU($d$) group generators $\hat{g}_{j}$
($j=1,2,\ldots d^{2}-1$) Hioe and Eberly (1981); Son et al. (2004); Bang et
al. (2008). Here, the components of $\mathbf{p}$ correspond to a set of
control parameters in a real experiment, such as multiport beam splitters and
phase shifters in linear optical systems Reck et al. (1994) or the pulse
sequences of solid system qubits Kim et al. (2000). The operational noises of
$\\{\hat{U}_{\alpha}\\}$ and $\\{\hat{V}_{\alpha}\\}$ can be formulated as
follows: the control parameters of $\hat{U}_{\alpha}$ and $\hat{V}_{\alpha}$
fluctuates such that

$\displaystyle\mathbf{p}\rightarrow\mathbf{p}+\eta\,\boldsymbol{\epsilon},$
(47)

where
$\boldsymbol{\epsilon}=(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{d^{2}-1})^{\text{T}}$
is a vector of the stochastic errors $\epsilon_{j}\in[-\pi,\pi]$. The factor
$\eta\in[0,1]$ is casted to represent the degree of immaturity in control.

We investigate the effects of operational noises on qubit teleportation.
First, we assume that the protocol is already set with the optimal condition
as in Eq. (26), and in this case, operational noise occurs continuously in
both $\\{\hat{U}_{\alpha}\\}$ and $\\{\hat{V}_{\alpha}\\}$. To understand the
effects of operational noise, we perform Monte-Carlo simulations with
increasing $\eta$. The results are shown in Fig. 2, where each data point is
created by averaging $10^{4}$ simulations. $F$ decreases from $1$ to
$F_{\text{err}}$, and $D$ increases from $0$ to $D_{\text{err}}$. The fullest
deterioration of $F$, that is, $F_{\text{err}}\simeq 0.5102$ is approximately
$98\%$ to $F_{\text{rand}}=\frac{1}{2}$, whereas $D_{\text{err}}$ increases to
$\simeq 0.0431$, or by approximately $39\%$, to
$\frac{1}{2}D_{\text{err}}^{\text{max}}\simeq\frac{1}{4\sqrt{5}}$ even in the
worst case of $F_{\text{err}}\simeq\frac{1}{2}$. This implies that the
deterioration of $F$ is more conspicuous than that of $D$. Here,
$F_{\text{rand}}$ is the average fidelity of the purely random protocol and
$\frac{1}{2}D_{\text{err}}^{\text{max}}$ represents half (mean) of the maximum
fidelity deviation at $F_{\text{err}}$, that is,
$\frac{1}{2}D_{\text{err}}^{\text{max}}=\frac{1}{2\sqrt{5}}(F_{\text{max}}-F_{\text{err}})$
(see the dashed line in Fig. 2).

Figure 2: Deteriorations of $F$ (left) and $D$ (right) with respect to $\eta$.
Each data point is created by averaging $10^{4}$ simulations. The error bar
denotes the standard deviation. The teleportation system breaks down due to
operational noise, as indicated by $F=1\rightarrow F_{\text{err}}<1$ and
$D=0\rightarrow D_{\text{err}}>0$. The dashed lines are created based on
simulations of the random protocol.

###  V.2 Machine-learning-based algorithm for stabilization of control

To cure the unstable control, we propose a machine-learning-based algorithm
built on the so-called differential evolution concept Bang et al. (2012),
where the control parameters of $\\{\hat{U}_{\alpha}\\}$ and
$\\{\hat{V}_{\alpha}\\}$ are allowed to evolve during the process. The
algorithm runs as follows: First, $N_{\text{pop}}$ sets of the control
parameter vectors are prepared as the candidate solutions
$\\{\mathbf{p}^{(u)}_{\alpha,i},\mathbf{p}^{(v)}_{\alpha,i}\\}$, where
$\mathbf{p}^{(u)}_{\alpha,i}$ and $\mathbf{p}^{(v)}_{\alpha,i}$ are
respectively the control parameter vectors of the candidate operations
$\hat{U}_{\alpha,i}$ and $\hat{V}_{\alpha,i}$ ($i=1,2,\ldots,N_{\text{pop}}$).
Thus, we have $2d^{2}N_{\text{pop}}$ parameter vectors. Then, the prepared
sets of the candidate solutions are allowed to evolve through the following
steps: (1) We generate $2N_{\text{pop}}$ mutant vectors
$\boldsymbol{\nu}^{(u,v)}_{\alpha,i}$ for $\hat{U}_{\alpha,i}$ and
$\hat{V}_{\alpha,i}$ according to

$\displaystyle\boldsymbol{\nu}^{(u,v)}_{\alpha,i}=\mathbf{p}^{(u,v)}_{\alpha,a}+W\left(\mathbf{p}^{(u,v)}_{\alpha,b}-\mathbf{p}^{(u,v)}_{\alpha,c}\right),$
(48)

where $\mathbf{p}^{(u,v)}_{\alpha,a}$, $\mathbf{p}^{(u,v)}_{\alpha,b}$, and
$\mathbf{p}^{(u,v)}_{\alpha,c}$ are randomly selected for
$a,b,c\in\\{1,2,\ldots,N_{\text{pop}}\\}$. These vectors are selected to be
different from each other. The free parameter $W$, also called a differential
weight, is a real and constant number. (2) Thereafter, all
$2d^{2}N_{\text{pop}}$ parameter vectors,

$\displaystyle\mathbf{p}^{(u)}_{\alpha,i}=\begin{pmatrix}p^{(u)}_{1}\\\
p^{(u)}_{2}\\\ \vdots\\\
p^{(u)}_{d^{2}-1},\end{pmatrix}_{\alpha,i},~{}\mathbf{p}^{(v)}_{\alpha,i}=\begin{pmatrix}p^{(v)}_{1}\\\
p^{(v)}_{2}\\\ \vdots\\\ p^{(v)}_{d^{2}-1},\end{pmatrix}_{\alpha,i}$ (49)

are reformed to trial vectors,

$\displaystyle\boldsymbol{\tau}^{(u)}_{\alpha,i}=\begin{pmatrix}\tau^{(u)}_{1}\\\
\tau^{(u)}_{2}\\\ \vdots\\\
\tau^{(u)}_{d^{2}-1},\end{pmatrix}_{\alpha,i},~{}\boldsymbol{\tau}^{(v)}_{\alpha,i}=\begin{pmatrix}\tau^{(v)}_{1}\\\
\tau^{(v)}_{2}\\\ \vdots\\\ \tau^{(v)}_{d^{2}-1},\end{pmatrix}_{\alpha,i}$
(50)

by the rule: For each $j=1,2,\ldots,d^{2}-1$,

$\displaystyle\left\\{\begin{array}[]{ll}\tau^{(u,v)}_{j}\leftarrow
p^{(u,v)}_{j}&~{}~{}\text{if}~{}R_{j}>C_{r},\\\
\tau^{(u,v)}_{j}\leftarrow\nu^{(u,v)}_{j}&~{}~{}\text{otherwise},\\\
\end{array}\right.$ (53)

where $R_{j}\in[0,1]$ is a randomly generated number, and the crossover rate
$C_{r}$ is another free parameter ranging between $0$ and $1$. Note that these
free parameters $W$ and $C_{r}$ are set to achieve the best learning
efficiency. (3) Finally, the control parameter vectors are evaluated by using
the fitness criteria, that is, how well do the given parameters fit to the
protocol. More specifically,
$\\{\boldsymbol{\tau}^{(u)}_{\alpha,i},\boldsymbol{\tau}^{(v)}_{\alpha,i}\\}$
are taken if they yield a higher level of fitness; if not,
$\\{\mathbf{p}^{(u)}_{\alpha,i},\mathbf{p}^{(v)}_{\alpha,i}\\}$ are retained.
In our algorithm, we extract the best fitness among $N_{\text{pop}}$ and
retain the corresponding parameters
$\\{\mathbf{p}_{\alpha,\text{best}},\mathbf{p}_{\alpha,\text{best}}\\}$. Steps
(1)-(3) are then repeated.

Figure 3: Remediation of fully broken teleportation. The graphs of $F$ (left)
and $D$ (right) are plotted based on the results of the numerical simulations.
Here, we consider three cases: $\gamma=\gamma_{\text{C}}=\frac{1}{3}$,
$\gamma=\gamma_{\text{BV}}=\frac{1}{\sqrt{2}}$, and $\gamma=1$. Each data
point is created based on $10^{4}$ repeating simulations.

We investigate numerically whether even a fully broken teleportation system
can be cured using the proposed machine-learning-based algorithm. The
numerical simulation is performed for $d=2$. Here, we take
$N_{\text{pop}}=100$, and the free parameters of our algorithm are selected
such that: $W=0.5$ and $C_{r}=0.1$. To use the structural trait [T.1] of
teleportation, we define fitness in terms of $F$; in other words, there is no
minimization of $D$. Note that in general, fitness should be defined as a
function of $F$ and $D$ (for example, see Ref. Bang et al. (2012)). Such a
setting is indeed beneficial, as described later. In Fig. 3, we present the
results in the form of graphs of $F$ and $D$ for three cases:
$\gamma=\gamma_{\text{C}}=\frac{1}{3}$,
$\gamma=\gamma_{\text{BV}}=\frac{1}{\sqrt{2}}$, and $\gamma=1$. Here,
$\gamma_{\text{C}}$ denotes the condition of the separability of the channel
and $\gamma_{\text{BV}}$ is the critical value that the entanglement of the
channel allows the violation of CHSH inequality. Each data point is created
based on $10^{4}$ repeated simulations. The results indicates that the broken
teleportation system can be recovered; $F$ approaches $F_{\text{max}}$, and
$D$ decreases to zero. Specifically, we obtain ($F\simeq 0.6640$, $D\simeq
0.0002$) for $\gamma=\gamma_{\text{C}}$, ($F\simeq 0.8478$, $D\simeq 0.0005$)
for $\gamma=\gamma_{\text{BV}}$, and ($F\simeq 0.9915$, $D\simeq 0.0009$) for
$\gamma=1$.

Figure 4: Real-time remediation of control fluctuations. The results of the
numerical simulations are given as graphs of $F$ (left) and $D$ (right). Each
data point is created based on $10^{4}$ simulations. It is assumed that
fluctuation occurs after every $10$ or $50$ iterations of our algorithm. For
more convincing analysis, we consider the worst case, namely, $\eta=1$. It is
observed that the teleportation can be cured continuously.

We further investigate through numerical simulations whether the system can be
stabilized, by assuming that the fluctuation occurs abruptly after intervals
of some iterations of our algorithm. Note that such a model is realistic Viola
et al. (1999); Khaneja01 et al. (2001). Here, we consider the scenarios in
which the controls fluctuate after every $10$ and $50$ iterations. For more
faithful and confident analysis, we consider the scenario with the worst
deterioration, that is, $\eta=1$. Fitness is defined solely by $F$. In Fig. 4,
we present our simulation results. $F$ and $D$ deteriorate to their fullest
extents, but the system is cured continuously.

##  VI Summary and remarks

We analyzed the average fidelity and fidelity deviation for noisy
teleportation. We proved that teleportation can be zero fidelity deviations
(or equivalently, the perfect universality) independently of the quantum
channel condition, while the achievable maximum average fidelity is limited by
the fraction of entanglement in the channel. Based on these analyses, we
derived Result 3: the maximum average fidelity ensures perfect universality in
quantum teleportation. For the case of $d=2$, we derived a tighter
relationship between the two measures. Taking into account other realistic
noises, namely, the fluctuations in system control, we proposed a machine-
learning-based algorithm to stabilize teleportation. We demonstrated by means
of numerical simulations that even the fullest deteriorations can be cured. It
is remarkable that the process of fidelity maximization guarantees the
minimization of fidelity deviation without additional processes. The
aforementioned trait (coming from Result 3) is indeed beneficial to reduce the
algorithm time and realize faster system remediation; in fact, if the
minimization of fidelity deviation was considered in the algorithm, we may not
have obtained a cure cycle (i.e., sufficient time for iterations to cure
abrupt fluctuations). Such a gain is expected to be more conspicuous in
large-$d$ teleportation.

## Acknowledgments

JB thanks to M. Wieśniak and T. Vértesi for discussions. WS and JB thank to
the financial support of the National Research Foundation (NRF) of Korea
Grants (no. 2019R1A2C2005504 and no. NRF-2019M3E4A1079666), funded by the MSIP
(Ministry of Science, ICT and Future Planning), Korea government. JR
acknowledges the National Research Foundation of Korea (NRF) Grants no.
NRF-2020M3E4A1079792). JB and KB was supported by KIAS Individual Grants (no.
CG061003 and no. CG074701), respectively. JB and KB was supported by the NRF
Grant funded by the Korea government(MSIT) (no. 2020M3E4A1079939).

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