# Visually quantifying single-qubit quantum memory

Wan-Guan Chang  Department of Physics and Center for Quantum Frontiers of
Research & Technology (QFort), National Cheng Kung University, Tainan 701,
Taiwan  Chia-Yi Ju<EMAIL_ADDRESS>Department of Physics, National
Sun Yat-sen University, Kaohsiung 80424, Taiwan  Center for Theoretical and
Computational Physics, National Sun Yat-sen University, Kaohsiung 80424,
Taiwan  Guang-Yin Chen  Department of Physics, National Chung Hsing
University, Taichung 402, Taiwan  Yueh-Nan Chen<EMAIL_ADDRESS>Department of Physics and Center for Quantum Frontiers of Research &
Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan  Huan-
Yu Ku<EMAIL_ADDRESS>Department of Physics, National Taiwan Normal
University, Taipei 11677, Taiwan

###### Abstract

To store quantum information, quantum memory plays a central intermediate
ingredient in a network. The minimal criterion for a reliable quantum memory
is the maintenance of the entangled state, which can be described by the non-
entanglement-breaking (non-EB) channel. In this work, we show that all single-
qubit quantum memory can be quantified without trusting input state
generation. In other words, we provide a semi-device-independent approach to
quantify all single-qubit quantum memory. More specifically, we apply the
concept of the two-qubit quantum steering ellipsoids to a single-qubit quantum
channel and define the channel ellipsoids. An ellipsoid can be constructed by
visualizing finite output states within the Bloch sphere. Since the Choi-
Jamiołkowski state of a channel can all be reconstructed from geometric data
of the channel ellipsoid, a reliable quantum memory can be detected. Finally,
we visually quantify the single-qubit quantum memory by observing the volume
of the channel ellipsoid.

##  I Introduction

A quantum memory stores input quantum information and, at a later time,
processes the retrieval of the information [1, 2, 3, 4, 5]. To function as a
network, the minimal criteria of a reliable quantum memory is preserving an
entangled resource [6, 7, 8]. In the language of the most general channel
formalism, a reliable quantum memory can be described by a non-entanglement-
breaking (non-EB) channel. Clearly, quantum information tasks based on
entangled resources (i.e., entanglement-based quantum key distribution [9, 10,
11], quantum communication [12, 13, 14, 15], and quantum teleportation [16,
17, 18]) are not functional if there is an entanglement-breaking (EB) quantum
memory in the network.

To verify a functional quantum memory, many independent protocols were
proposed with different assumptions, for instance, the untrusted measurement
apparatus [19, 20, 21, 22], non-local game test [23], and coherence estimation
[24]. Recently, Rosset et al. [25] investigated how to detect quantum memory
with minimal assumptions (see also the continuous-variable version in Ref.
[26]) by expending the measurement-device-independent task into the temporal
domain [27, 28, 29, 30, 31, 32]. Importantly, the minimal assumptions on
verification of quantum memory are two trusted state generation devices. One
of them is an input of the quantum memory, while the other is used to create a
quantum correlation for certifying reliable quantum memory.

In this work, we geometrically quantify all single-qubit quantum memories by
introducing the concept of channel ellipsoids, which are conceptually similar
to quantum steering ellipsoids used to characterize the entanglement between
qubit-qubit systems [33, 34, 35, 36, 37]. To be more specific, we define the
channel ellipsoid of a single-qubit quantum memory as the set of all output
states in the Bloch sphere. Although a channel ellipsoid is defined by all
output states, the geometrical data is accessible with a finite number of
inputs, which is useful for experimental studies on quantum memories. In this
construction, there is no presumption on the input quantum states, in the
sense that one can construct the channel ellipsoids of a single-qubit quantum
memory in a semi-device-independent scenario [38, 39, 40, 41].

To quantify all single-qubit quantum memory, we first show that the geometric
data from the channel ellipsoid can reconstruct the quantum channel in the
Choi-Jamiołkowski (CJ) representation. Therefore, all the properties of the
single-qubit quantum memory can be visually characterized by the channel
ellipsoid. Note that this feature does not hold in the quantum steering
ellipsoid, which will be discussed later. In this way, we further give a clear
physical interpretation of the volume of the channel ellipsoid in terms of the
quantification of the quantum memory. Moreover, we can fully distinguish the
unwanted quantum memories by providing the geometrical representations of EB
channels with channel ellipsoids. These representations provide a simple
metric for quantifying a quantum memory in terms of its volume. Finally, we
consider the depolarizing and amplitude damping channels as two concrete
examples and test our results with using the IBM quantum simulator.

The structure of this paper is as the following description. We first briefly
review the definitions of quantum steering ellipsoid and quantum memory in
terms of non-EB channel and their properties in Sec. II. In Sec. III, we will
give the intuition of our main idea for visualizing the channel ellipsoid
enlightened by the quantum steering ellipsoid [34]. Then, in Sec. IV, we will
introduce how to use the volume of the channel ellipsoid to quantify quantum
memory by connecting the volume with the quantum memory robustness. By using
the IBMQ simulation, we explicitly show the boundary of the non-entanglement
breaking channel in Sec. V. Finally, we summarize our results and provide our
outlook in Sec. VI.

Figure 1: Schematic diagram of quantum memory. In this protocol, one can
verify the quantum memory with channel ellipsoid based on semi-device-
independent scenario.

##  II Background knowledge

###  II.1 Quantum steering ellipsoid

We briefly recall the concept of the quantum steering ellipsoid [33, 34, 35],
which is a visualization approach to study the two-qubit entangled state. Any
two-qubit state shared by Alice and Bob can be expressed in the Pauli bases
$\rho^{\rm{AB}}=\openone/4+\sum_{i=1}^{3}A_{i}\sigma_{i}\otimes\openone+\sum_{j=1}^{3}B_{j}\openone\otimes\sigma_{j}+1/4\sum_{i,j=1}^{3}T_{ij}\sigma_{i}\otimes\sigma_{j}$,
where $\sigma_{i}$ denotes the Pauli matrices,
$\mathbf{A}=(A_{1},A_{2},A_{3})^{T}$ [$\mathbf{B}=(B_{1},B_{2},B_{3})^{T}$],
with $T$ standing for transpose, is the Bloch vector of Alice’s (Bob’s)
reduced state, and $\mathbf{T}$ is a $3\times 3$ matrix representing the
correlation of the state. Here, we in particular use the superscript $\rm{AB}$
to emphasize $\rho^{\rm{AB}}$ is a spatially separated state shared by Alice
and Bob. It gives rise to the quantum steering ellipsoid
$\mathcal{E}^{\rm{B}}$ of Bob with center
$\mathbf{C}^{\rm{B}}=\dfrac{\mathbf{B}-\mathbf{T}^{\text{T}}\mathbf{A}}{1-A^{2}}$
where $A^{2}\equiv\mathbf{A}^{\text{T}}\mathbf{A}$. The orientation and
semiaxes lengths $l_{i}=\sqrt{\lambda_{i}}$ can be determined by the
eigenvectors and eigenvalues of the ellipsoid matrix

$Q_{\rm{B}}=\frac{1}{1-A^{2}}(\mathbf{T}^{\text{T}}-\mathbf{B}\mathbf{A}^{\text{T}})\left(\openone+\frac{AA^{\text{T}}}{1-A^{2}}\right)(\mathbf{T}-\mathbf{A}\mathbf{B}^{\text{T}}).$
(1)

The operational definition of an ellipsoid $\mathcal{E}^{\rm{B}}$ represents
the collection of all steered states generated by Alice’s choices of
measurements. The ellipsoidal surface is constituted when Alice considers all
projectors in her subspace, while the most general quantum measurement,
described by the positive operator-valued measure, generates the state inside
the ellipsoid [34]. It has been shown that any quantum steering ellipsoids
constructed by a separable state
$\rho^{\rm{AB}}=\sum_{i}p(i)\rho^{\rm{A}}(i)\otimes\rho^{\rm{B}}(i)$ is always
inside a tetrahedron which is inside a Bloch sphere. The definition of a
quantum steering ellipsoids is reminiscent of the Einstein–Podolsky–Rosen
steering [42, 43, 44, 45], in which the correlation contains not only Bob’s
steered states but also the probability distribution generated by Alice’s
measurements. Due to the similar operational definitions, several approaches
are proposed to connect quantum steering ellipsoid with
Einstein–Podolsky–Rosen steering; e.g., very recently, it has been shown that
Bob’s steering ellipsoid can be used to predict the quantum steerability of a
quantum state [46, 47, 48]. During a steering experiment, Alice’s measurement
may not be fully characterized, which can be applied to one-sided device-
independent quantum information processing [49, 50, 51, 52, 53].

###  II.2 Quantum memory

Without loss of the generality, the effects of any quantum devices (e.g.,
quantum memory, quantum repeater, and quantum communication line) can be
described by a quantum channel $\Lambda$, which is a complete-positive trace-
preserving (CPTP) map. The properties of a quantum channel can be
characterized by a corresponding CJ state [54, 55], defined as

$\rho^{\Lambda}=\text{id}\otimes\Lambda(\ket{\Psi}\bra{\Psi}),$ (2)

where id is an identity channel and
$\ket{\Psi}=\sum_{i}1/\sqrt{N}\ket{i}\otimes\ket{i}$ is the maximally
entangled state with local dimension $N$. Similar to the superscript in
$\rho^{\rm{AB}}$, we use the superscript $\Lambda$ to emphasize
$\rho^{\Lambda}$ is the CJ state of the channel $\Lambda$. The output of the
channel with an input state $\rho$ can be expressed as

$\Lambda(\rho)=\text{Tr}_{\rm{in}}\left(\openone\otimes\rho^{\text{T}}\rho^{\Lambda}\right),$
(3)

where T denotes the transpose. If a channel is CPTP, the reduced state of
input space for any CJ state must be maximally mixed [56], namely

$\text{Tr}_{\rm in}\left(\rho^{\Lambda}\right)=\openone/d_{\rm in},$ (4)

where $d_{\rm in}$ is the dimension of the input Hilbert space.

A perfect quantum memory should preserve the quantum information for a period
of time without introducing any disturbance. In this sense, the perfect
quantum memory can be described by an identity channel (or unitary channel).
Recently, the minimal criterion of a quantum memory was proposed as the
preservation of the entangled state [25]. In the channel terminology, a
functional quantum memory cannot be EB, namely,

$\Lambda_{\rm{EB}}(\rho)=\sum_{k}\text{Tr}[M_{k}\rho]\rho^{\prime}_{k},$ (5)

where $M_{k}$ denotes the generalized quantum measurement (positive operator-
valued measure), and $\rho^{\prime}_{k}$ is the quantum state conditioned on
the measurement outcome $k$. The set of all EB channels is denoted as
$\mathcal{EB}$. Because the entanglement is destroyed by the measurement, we
can easily observe that the CJ state of an EB channel is a separable state [7,
6] $\rho^{\Lambda_{\rm{EB}}}=\sum_{i}p(i)\rho(i)\otimes\sigma(i)$, where
$\rho(i)$ and $\sigma(i)$ are quantum states sharing with classical
correlation $i$. One can use the memory robustness to measure a quantum memory
$\Lambda$ [23]:

$QM(\Lambda)=\min_{\Lambda^{\prime}}\left\\{t\geq
0\,\left|\,\frac{\Lambda+t\Lambda^{\prime}}{1+t}\right.\in\mathcal{EB}\right\\},$
(6)

where $\Lambda^{\prime}$ can be seen as an arbitrary channel with noisy
parameter $t$ added on the given quantum memory $\Lambda$. This quantity can
be easily computed via a semidefinite program [57] when considering the CJ
representation. Note that the memory robustness is a proper measure under the
resource theory of quantum memory; i.e., it is a nonincreasing function under
any quantum-memory free operation [25], which maps any EB channel to the other
EB channel.

##  III Channel ellipsoids

Due to the one-to-one mapping between quantum channel and quantum state [cf.,
Eq. (2)], we can use quantum steering ellipsoids [34] to study the properties
of a single-qubit quantum channel, visually. More specifically, we can span
the CJ state of any single-qubit channel in the Pauli basis, namely

$\rho^{\Lambda}=\openone/4+\sum_{i=1}^{3}A_{i}\sigma_{i}\otimes\openone+\sum_{j=1}^{3}B_{j}\openone\otimes\sigma_{j}+1/4\sum_{i,j=1}^{3}\Theta_{ij}\sigma_{i}\otimes\sigma_{j}.$
(7)

Due to the reduced-state condition of the CJ state [cf., Eq. (4)],
$\mathbf{A}$ must be a null vector. Similar to the case in quantum steering
ellipsoid, $\mathbf{B}=(B_{1},B_{2},B_{3})^{\text{T}}$ represents the Bloch
vector of the output state with input being the maximally mixed state.
Finally, $\mathbf{\Theta}$, a square matrix with components being
$\Theta_{ij}$, is used to describe the correlations between the input and
output states. Thus, the set of all output states can be visualized by the
channel ellipsoid with the center
$\mathbf{C}^{\Lambda}=(B_{1},B_{2},B_{3})^{\text{T}}$. The orientation and
semiaxes lengths $l_{i}=\sqrt{\lambda_{i}}$ of a channel ellipsoid can be
determined by the eigenvectors and eigenvalues $\lambda_{i}$ of the ellipsoid
matrix

$Q^{\Lambda}=\mathbf{\Theta}^{\text{T}}\mathbf{\Theta}.$ (8)

The operational definition of a channel ellipsoid is the collection of all
output states of the channel. Since we only focus on the output states, the
input state generations can be seen as a black box. In this sense, the
construction of a channel ellipsoid is independent of the input quantum
states. In other words, the channel ellipsoids can be constructed with only
trusting measurement apparatus at output space, which leads to the semi-
device-independent scenario.

Furthermore, from the geometric data $(\mathbf{C}^{\Lambda},Q^{\Lambda})$, one
can faithfully reconstruct the CJ state of a channel $\rho^{\Lambda}$ because
$\mathbf{A}$ is a null vector. In this way, we can use the standard approaches
(i.e., Peres–Horodecki criterion [58, 59], and entanglement witness [60]) to
probe the non-EB channels with only the output information. Therefore, all the
single-qubit quantum memory can be certified in a semi-device-independent
scenario. It is worth mentioning that because the CJ state of every EB channel
is separable [8, 7], any corresponding channel ellipsoid can be enclosed by a
tetrahedron [61, 34]. This property is very different from the case of the
quantum steering ellipsoids in which the geometric data of an ellipsoid from
only one side is not sufficient to rebuild the density operator $\rho^{\rm
AB}$ of a two-qubit entangled state [34].

Finally, although the definition of the channel ellipsoid is the set of all
output states, the channel ellipsoid can be constructed with a finite number
of the output states. It is because to obtain a unique solution of an
ellipsoid function, nine inputs (or, equivalently nine output quantum states)
are sufficient, which leads to an experimentally feasible test of a single-
qubit quantum memory in a channel ellipsoid manner (see also Sec. V).

##  IV Quantifying quantum memory with the volume

As we mentioned in the previous section, we can consider the geometric data
$(\mathbf{C}^{\Lambda},Q^{\Lambda})$ of a channel ellipsoid to test a single-
qubit quantum memory. We now discuss the properties of the volume of the
channel ellipsoid, determined by

$V^{\Lambda}=\frac{4\pi}{3}|l_{1}l_{2}l_{3}|.$ (9)

Later, we show that the volume of the channel ellipsoid provides an upper
bound on the memory robustness. Therefore, the larger volume of the channel
ellipsoid implies better functional quantum memory.

Before we show our main result in this section, we first recall the
concurrence and the negativity of a bipartite quantum state $\rho^{\rm AB}$,
shared by Alice and Bob. The concurrence is used to estimate the number of the
maximally entangled states required to construct the given state $\rho^{\rm
AB}$ by local operations with classical communication [61]. The concurrence
can be computed by

$E_{C}(\rho^{\rm
AB})=\text{max}\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (10)

where $\lambda_{i}$ (in the decreasing order) are the eigenvalues of
$\sqrt{\sqrt{\rho^{\rm AB}}\hat{\rho}^{\rm AB}\sqrt{\rho^{\rm AB}}}$ with
$\hat{\rho}^{\rm
AB}=(\sigma_{2}\otimes\sigma_{2})\rho^{\rm{AB}\dagger}(\sigma_{2}\otimes\sigma_{2})$.
Here, $\dagger$ denotes the conjugate transpose. It has been shown that the
concurrence is a lower bound on the volume of the quantum steering ellipsoids
[35].

On the other hand, the negativity is another entangled measure, based on the
Peres–Horodecki criterion [58, 59]. In short, since the partial transposition
on any qubit-qubit entangled states does not preserve the positivity, there
must exist, at least, one negative eigenvalue for the state after partial
transposition. The negativity is defined as the absolute sum of the negative
eigenvalues of the state after the partial transpose, namely

$E_{N}(\rho^{\rm AB})=\sum_{\lambda_{i}<0}\|\lambda_{i}\|,$ (11)

where $\lambda_{i}$ are the eigenvalues of the partial-transposed state.

Both concurrence and negativity are entanglement measures in the sense that
they are monotonically decreasing functions under any local operations with
classical communications. However, they may not be proper quantum memory
measures because (1) the free operations under the frameworks of resource
theory of entanglement and memory are different, and (2) the mathematical
transformations of a state and a channel under their corresponding free
operations are different. For instance, it has been shown that the negativity
of the CJ state of a quantum channel is the quantum memory measure under the
particular quantum-memory free operation [21]. Note, the memory robustness
cannot be increased under “any” memory free operation.

With the concurrence and negativity, we can now introduce:

######  Result 1.

The volume of the channel ellipsoid is an upper bound on the memory
robustness:

$QM(\Lambda)\leq\left(\frac{3V^{\Lambda}}{4\pi}\right)^{\frac{1}{4}}.$ (12)

_Proof.—_ First, we recall the fact that, for any state $\rho^{\rm AB}$, the
volume of the ellipsoid with respect to Bob is bounded by [35]

$E_{C}(\rho^{\rm AB})\leq\mathcal{R}_{\rm
A}^{-1}\left(\frac{3V^{B}}{4\pi}\right)^{\frac{1}{4}},$ (13)

where $\mathcal{R}_{\rm A}^{-1}=1/\sqrt{1-A^{2}}$ is the Lorentz factor. We
note that if we replace Bob’s ellipsoid with Alice’s one by swapping
$\mathbf{A}\leftrightarrow\mathbf{B}$ and
$\Theta\leftrightarrow\Theta^{\text{T}}$, the above-bounded relation also
holds. From Refs. [62] and [63], we have $E_{C}(\rho^{\rm AB})\geq
E_{N}(\rho^{\rm AB})$ and $E_{N}(\rho^{\rm AB})\geq E_{R}(\rho^{\rm AB})$,
respectively. Here, $E_{R}(\rho^{\rm AB})$ is the entanglement robustness
defined in the same way in Eq. (6), but substituted the channel and EB channel
with a bipartite quantum state and a separable state.

We now apply the property $\mathbf{A}=0$ of the CJ state on the Lorentz factor
leading $\mathcal{R}_{\rm A}^{-1}=1$. Since the CJ states can be seen as a
subset of the entangled state, we can substitute the entangled state in
$E_{C}(\rho^{\rm AB})$, $E_{N}(\rho^{\rm AB})$, and $E_{R}(\rho^{\rm AB})$
with the CJ state. With the transitive relation, we complete the proof.
$\square$

Despite the fact that it is well-established that all the outputs of the
channel form an ellipsoid [64], the physical meaning of the volume of the
channel ellipsoid was, to some extend, unclear. Our result provides an
explicitly quantitative relationship between the volume and the quantum
memory. More specifically, although the relation between the volume of a
channel ellipsoid and the concurrence of the CJ state of the channel can be
derived [cf., Eq. (13)], the quantum-memory monotonicity of the concurrence of
a CJ state remains unclear. Here, we provide the physical meaning of the
volume of the channel ellipsoid by instead showing the relationship with the
quantum memory robustness, which is a valid quantum memory measure. In this
sense, we call the LHS of Eq. (12) is volume estimated upper bound of memory
robustness. Moreover, since the nature of the CJ states [cf., Eq. (7)] is
equivalent to the canonical form of the qubit-qubit entangled states, all the
results in Ref. [35] are also valid in the channel scenario. For instance, the
maximal volume of channel ellipsoid with the associated center distance
$|\mathbf{C}^{\Lambda}|$ is determined by the channel ellipsoid of the
amplitude damping channel [35].

##  V Implementation with IBM quantum device

In this section, we demonstrate the channel ellipsoids of depolarizing
channels and amplitude damping channels via IBMQ simulator. We use a
depolarizing (amplitude damping) channel to show how to certify (quantify)
quantum memories with the channel ellipsoids by the tetrahedron (volume)
condition.

###  V.1 Depolarizing channel

The depolarizing channel is widely used in quantum information to apply the
quantum correlation certification [65, 66, 67, 68] and measurement mitigation
[69, 70, 71, 72]. The depolarizing channel is a convex combination of the
input quantum state $\rho$ and the maximally mixed state with the mixing
parameter $P$, namely,

$\displaystyle\mathcal{E}_{D}(\rho)=\frac{1-P}{2}\openone+(P)\rho\,.$ (14)

It has been shown that the depolarizing channel is EB when $P\geq\frac{1}{3}$.

Here, we use five qubits to demonstrate the ellipsoid of a depolarizing
channel by the circuit model (see also Fig. 2). We first construct the input
states $\rho$ uniformly on the Bloch sphere by inserting uniform unitary
$U(\theta,\psi)$ on the initial state $\ket{0}$. Since the depolarizing
channel is a convex combination of the input state and the maximally mixed
state, we need to generate a maximally mixed state. Here, we use the property
that the marginal of the maximally entangled state is a maximally mixed state
to finish the construction. Now, we use the Fredkin gate with the control
qubit to be the classical mixture of the $\ket{0}$ and $\ket{1}$. This
classical mixture can be achieved by considering, again, the reduced state of
the pure entangled state. Finally, the control qubit can flip the input state
into the maximally mixed state with probability $1-P$ (see the circuit of the
depolarizing channel in our composer is shown as Fig. 2). We note that the
probability $P$ here in IBMQ composer can be computed by
$P=1-\sin^{2}\frac{\theta}{2}$.

Figure 2: Circuit implementation of the depolarizing channel. The input states
are prepared by scanning uniformly generated unitary operations on the Bloch
sphere. The second and third qubits simulate the maximally mixed state by
applying the Hadamard gate and CNOT gate on the ancilla. Therefore the reduced
state of the second qubit becomes the maximally mixed state. Finally, the
fourth and fifth qubits are used to manipulate the parameter $P$ of state
$\rho^{\prime}=(1-P)\ket{0}\bra{0}+P\ket{1}\bra{1}$ which demonstrates
probability in Eq (14). Finally, the Fredkin gate is applied on the first,
second, and fourth qubits such that the output state can be seen as the input
state after the depolarizing channel.

Since the above circuit required five qubits and multiple CNOT gates, we
implemented this circuit on the IBM quantum simulator. The simulation results
of channel ellipsoids from IBM quantum simulator are presented in Table. 1. We
also present the comparison of the memory robustness and the volume estimated
upper bound in Fig. 3. From the results, one can see that the ellipsoids
shrank when the mixing parameter $P$ decreases (see also Table. 1). One can
also observe that if the channel ellipsoids of the depolarising channels are
in the range of $P\leq\frac{1}{3}$, the channels cannot be used as a
functional quantum memory. This is because the ellipsoids cannot violate the
separability constraint i.g., the ellipsoids are inside the tetrahedrons in
the Bloch sphere.

$\theta=0$ |  $\theta=\frac{\pi}{4}$ |  $\theta=\frac{2\pi}{4}$ |  $\theta=0.609\pi$ |  $\theta=\frac{3\pi}{4}$ |  $\theta=\pi$
---|---|---|---|---|---
|  |  |  |  |
Table 1: Volume of depolarizing channel’s ellipsoid. Here,
$P=1-\sin^{2}\frac{\theta}{2}$ is the mixing parameter. When $\theta$
increases, $P$ and the volumes of channel ellipsoids simultaneously decrease
too. When $\theta=0.609\pi$, which corresponds to $P=0.6679$, there exists the
tetrahedron in the Bloch sphere such that the ellipsoid is inside the
tetrahedron. Figure 3: Memory robustness (red circles and solid curve) and
volume estimated upper bound (blue diamonds and dashed curve) of the
depolarizing channel, characterized by the mixing parameter in Eq. (14). When
the mixing parameter decreases, both the memory robustness and volume
estimated upper bound monotonically decrease too. After $P\geq\frac{2}{3}$,
the channel is EB leading to the vanishing memory robustness.

###  V.2 Amplitude damping channel

The amplitude damping channel is commonly used to characterize the dissipation
of energy with the dissipation rate $\gamma$. The evolution of the input state
after the amplitude damping channel can be described as

$\begin{split}&(a\ket{0}_{\text{in}}+b\ket{1}_{\text{in}})\ket{0}_{\text{anc}}\rightarrow\\\
&(a\ket{0}_{\text{out}}+b\sqrt{1-\gamma}\ket{1}_{\text{out}})\ket{0}_{\text{anc}}+\sqrt{\gamma}\ket{0}_{\text{out}}\ket{1}_{\text{anc}},\end{split}$
(15)

where the subscript ‘in’ and ‘out’ are the input and output of the circuit and
subscript ’anc’ is the ancilla qubit, or equivalently the environment system
(see Fig. 4). If environment goes to $\ket{1}_{\text{anc}}$, the output state
will become $\ket{0}_{\text{out}}$. Otherwise, the output state will be
prepared as the combination of $\ket{0},\ket{1}$. Under this construction, the
Kraus representation of the amplitude damping channel is characterized by

$\displaystyle\mathcal{E}_{AD}(\rho)=E_{0}\rho E_{0}^{\dagger}+E_{1}\rho
E_{1}^{\dagger},$ (16)

where

$\displaystyle E_{0}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&\sqrt{1-\gamma}\end{pmatrix}$ $\displaystyle E_{1}$
$\displaystyle=\begin{pmatrix}0&\sqrt{\gamma}\\\ 0&0\end{pmatrix}.$ (17)

Note that the amplitude damping channel is not an EB channel.

To demonstrate the amplitude damping channel in the circuit model, we use a
similar approach to the case in the depolarizing channel to construct the
input state. After the initialization, we apply a control-$Ry$ gate which
rotates the target qubit from $\ket{0}$ to an angle $\theta$ along the y-axis.
With this construction, the dissipation rate satisfies
$\gamma=\sin^{2}(\theta)$. Then, a CNOT gate is implemented to finish the
amplitude damping channel (see Fig. 4).

Figure 4: Circuit implementation of the amplitude damping channel. The input
here is also the prepared state by scanning uniformly the Bloch sphere. The
rest of the construction is a typical amplitude damping channel by applying
the control $R_{y}$ gate to simulate the energy dissipation.

The demonstration results by IBM quantum simulation with
$P\in\\{0,\frac{\pi}{4},...,\pi\\}$ are shown in Table. 2. From the data
points, one can compute the volume of the ellipsoid (see also Fig. 5). As
expected, both the memory robustness and volume estimated upper bound
monotonically decrease as the dissipation rate increases. Note that both the
memory robustness and the volume estimate upper bound are non-zero because the
amplitude damping channels are not EB channels. Our result also shows that the
amplitude damping channel provides the maximal volume of the channel
ellipsoids with the associated center distance $|\mathbf{C}^{\Lambda}|$ [35].

$\theta=0$ |  $\theta=\frac{\pi}{4}$ |  $\theta=\frac{2\pi}{4}$ |  $\theta=\frac{3\pi}{4}$ |  $\theta=\pi$
---|---|---|---|---
|  |  |  |
Table 2: Volume of amplitude damping channel’s ellipsoid. Here, we rotate the
target qubit along the y-axis by angles $\\{0,\pi/4,\pi/2,3\pi/4,\pi\\}$. When
$\theta$ increases, the volumes of channel ellipsoids simultaneously decrease.
Figure 5: Memory robustness (red circles and solid curve) and volume estimated
upper bound (blue diamonds and dashed curve) of the amplitude damping channel,
characterized by the dissipation rate in Eq. (17). When the dissipation rate
increases, both the memory robustness and volume estimated upper bound
monotonically decrease too. Since the amplitude damping channel is not EB
unless $\gamma=1$, both the memory robustness and the volume estimate upper
bound are always non-zero.

##  VI Discussion

In this work, we investigate the relationship between channel ellipsoids and
quantum memories. We show that all functional single-qubit quantum memory can
not only be certified with finite geometrical data of the ellipsoid but also
be visually quantified with the volume of the ellipsoid. Therefore, we provide
a clear operational definition of the volume of the ellipsoid. We also
simulate our results with the memories described by depolarizing and
amplitudedamping channels.

This work also raises many interesting questions. As we have shown that for a
qubit channel, it is sufficient to construct the ellipsoid with a finite
amount of output geometrical data. Can this property generalize to a high-
dimensional channel? In other words, can we construct the Choi state of a
high-dimensional channel with finite output states? In this work, we only show
that the volume of a channel ellipsoid is an upper bound of the memory
robustness. Can we use the properties of an ellipsoid to define a quantum
memory monotone?

## Acknowledgments

H.-Y. K. is supported by the Ministry of Science and Technology, Taiwan,
(Grants No. MOST 112-2112-M-003-020-MY3), and Higher Education Sprout Project
of National Taiwan Normal University (NTNU) and the Ministry of Education
(MOE) in Taiwan. This work is supported partially by the National Center for
Theoretical Sciences and Ministry of Science and Technology, Taiwan, Grants
No. MOST 110-2811- M-006-546, and the Army Research Office (under Grant No.
W911NF-19-1-0081). C.Y.J. is partially supported by the National Science and
Technology Council (NSTC) through Grant No. NSTC 112-2112-M-110-013-MY3.
G.Y.C. is partially supported by the NCTS and NSTC through Grant No.
112-2112-M-005-006 and 112-2123-M-006 -001.

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