# Estimation of optimal control for two-level and three-level quantum systems
with bounded amplitude

Xikun Li  School of Physics and Optoelectronic Engineering, Anhui University,
Hefei, Anhui 230601, China  Max-Planck-Institut für Physik komplexer Systeme,
D-01187 Dresden, Germany  Department of Physics and Astronomy, Aarhus
University, DK-8000 Aarhus C, Denmark

###### Abstract

A systematic scheme is proposed to numerically estimate the quantum speed
limit and temporal shape of optimal control in two-level and three-level
quantum systems with bounded amplitude. For the two-level system, two quantum
state transitions are studied as illustration. Comparisons between numerical
and analytical results are made, and deviations are significantly small. For
the three-level system, two critical time points are determined with high
accuracy, and optimal controls are obtained for different durations. The shape
of optimized control field is simple and does not switch frequently, thus are
easy to implement in experiment. In addition, we compare our method with the
chopped random basis (CRAB), and the performance of our method is much better
than that of CRAB. Our scheme is of importance in estimating the quantum speed
limit and optimal control for cases in which analytical solution is absent.

##  I Introduction

Quantum optimal control (QOC) is crucial to engineer and to manipulate complex
quantum systems in quantum information processing tasks Glaser  _et al._
(2015); D’Alessandro (2021); Krotov (1993); Brif  _et al._ (2010). Operations
in experiments are generally executed very slowly, i.e., adiabatically, to
avoid heating the sample and to guarantee the transition to the target state
with perfect fidelity Gericke  _et al._ (2007). However, the decoherence and
noise in experiments sometimes make such slow operations impossible, thus it
is desirable to achieve a speedup in a fast and robust manner Guéry-Odelin
_et al._ (2019); Chen  _et al._ (2010). This is one important topic of quantum
optimal control theory.

Quantum optimal control theory has been widely applied in various physical
systems such as Bose-Einstein condensate van Frank  _et al._ (2016), NMR
Khaneja  _et al._ (2005), cold atoms in optical lattices Li  _et al._ (2018);
Srivatsa  _et al._ (2021). One of the core problems in QOC theory is to find
the time-optimal control. Time-optimal control problems focus on driving
transitions to the target states in the minimal time which is generally called
the quantum speed limit (QSL) Caneva  _et al._ (2009), and on finding the
temporal shape of the control field. However, analytic solutions are only
available for low-dimensional quantum systems Lloyd and Montangero (2014). For
example, the analytic solutions are obtained for two-level system for energy
minimization D’Alessandro and Dahleh (2001); Boscain  _et al._ (2002) and
time-optimal problems Khaneja  _et al._ (2001); Boscain and Chitour (2005);
Boscain and Mason (2006); Boscain  _et al._ (2014); Hegerfeldt (2013, 2014);
Boozer (2012); Jafarizadeh  _et al._ (2020). In addition, the minimal time and
minimal energy problems in a three-level quantum system with complex controls
are studied with analytical solutions in Ref. Boscain  _et al._ (2002);
Boscain and Chitour (2005).

For multiple-level quantum systems, in general, quantum optimal control relies
on numerical methods, which employ local optimization algorithms, like Krotov
Sklarz and Tannor (2002), GRAPE Khaneja  _et al._ (2005), CRAB Doria  _et al._
(2011), GROUP Sørensen  _et al._ (2018) and GOAT Machnes  _et al._ (2018), as
well as global ones such as differential evolution (DE) and covariance matrix
adaptation evolution strategy (CMA-ES) Li  _et al._ (2018); Zahedinejad  _et
al._ (2014), and reinforcement learning Bukov  _et al._ (2018a). It is worth
noting that in experiments there might be constraints on the control field,
e.g., the amplitude is bounded. In addition, the total duration is also
limited for the time-optimal control problems. In such cases the local
suboptimal traps exist in the quantum control landscape Pechen and Tannor
(2011); Larocca  _et al._ (2018). These local traps in general make the
numerical methods fail to find the global optimal solution, even with the
global optimization algorithms. Moreover, some methods cannot be applied in
the problems where amplitude of control fields are bounded, while optimized
controls found by some methods switch too frequently, or the temporal shape of
control fields is too complicated, such it is hardly possible to implement
them in real experiment.

In this paper we propose a systematic scheme to numerically estimate the QSL
and time-optimal controls. First, as an illustration, we investigate the time-
optimal control problem for quantum state transitions in a two-level quantum
system with bounded amplitude. Analytical results are provided in Refs.
Boscain and Mason (2006). We use the numerical scheme proposed to estimate QSL
and time-optimal controls. In addition, the comparison between the numerical
results and the analytical results is made. We find that deviations from the
analytical results are significantly small. Furthermore, we compare our
results with the results obtained with one state-of-the-art method, CRAB. The
performance of our method is much better than that of CRAB.

Second, we employ this scheme in a three-level quantum system with real
control field which describes an external field without rotating wave
approximation (RWA). This problem is more difficult than the problems with
complex controls studied in Ref. Boscain  _et al._ (2002); Boscain and Chitour
(2005). We estimate the optimal control for different values of total
duration. Two regions of total durations are found in which the optimal
control fields are of different types. QSL and time-optimal control are
estimated. These results are compared with the ones obtained with CRAB.

This paper is organized as follows. We define the control problem and propose
a systematic scheme to estimate QSL in Sec. II. In Sec. III we consider two
processes of quantum state transition in a two-level quantum system, and
demonstrate the quantum control landscapes for certain types of control field.
QSL and time-optimal control fields are estimated numerically, and these
results are compared with the analytical results. In Sec. IV we study the
optimal control problem of quantum state transition in a three-level quantum
system. We conclude in Sec. V.

##  II Model

###  II.1 Hamiltonian and objective function

Here we consider the Hamiltonian with the control field $u(t)$:

$H(t)=H_{0}+u(t)H_{1}$ (1)

where $H_{0}$ is the drift Hamiltonian and $H_{1}$ is the control Hamiltonian
(For simplicity, we consider the case in which there is only one control
Hamiltonian). The bounded control field $u(t)$ is a real function under
constraint $|u(t)|\leq M$. The dynamics of the system is governed by the
controlled Hamiltonian
$\mathrm{d}|\psi(t)\rangle/\mathrm{d}t=-\mathrm{i}H(t)|\psi(t)\rangle$, where
we set $\hbar=1$.

Numerically, for a quantum optimal control problem, we wish to optimize
certain objective function such that the quantum speed limit
$T_{\mathrm{qsl}}$ and optimized control field are obtained. The objective
function is a functional of control field $u(t)$, thus we are able to
investigate the control landscape for certain types of control field. One
common choice of objective function is the fidelity between two quantum
states. In our paper the fidelity is defined as follows:

$\displaystyle F(u(t),T)$
$\displaystyle=|\langle\psi_{t}|\mathcal{T}\exp(-\mathrm{i}\int_{0}^{T}H(t)\mathrm{d}t|\psi_{i}\rangle|^{2}$
(2) $\displaystyle=|\langle\psi_{t}|\psi_{f}\rangle|^{2}.$

where $\mathcal{T}$ is the time-ordering operator, and $|\psi_{t}\rangle$
($|\psi_{i}\rangle$) is the target (initial) state. $T$ is the total duration
of time evolution, and $|\psi_{f}\rangle$ is the final state after the time
evolution. An alternative characterising the similarity between the final
state $|\psi_{f}\rangle$ and the target state$|\psi_{t}\rangle$ is the Bures
distance

$d_{B}=\sqrt{2(1-|\langle\psi_{f}|\psi_{t}\rangle|}=\sqrt{2(1-\sqrt{F})}.$ (3)

Perfect fidelity gives zero Bures distance, and vice versa.

###  II.2 Bang-Off control

Inspired by Ref. Boscain and Mason (2006) we consider the bang-off control as
the control protocol to be optimized. For certain duration $[s,s+\delta s]$,
if the control is restricted to take its maximum (minimum) $u(t)=M$
($u(t)=-M$), it is called a bang control, and is denoted by $P_{\delta s}$
($N_{\delta s}$), here $P$ ($N$) is short for Positive (Negative); Similarly,
if the value of control is zero $u(t)=0$, it is called a off control, and is
denoted by $0_{\delta s}$. Note that in the case of two-level system the off
control is singular control Boscain and Mason (2006). The bang-off control
refers to a finite concatenation of bang and off controls.

We consider two classes of bang-off control. For the first class the duration
for each bang (and off) control is in general different, e.g.,
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ defined in the following

$u(t)=\Bigg{\\{}\begin{matrix}M&0\;\;\leq t<t_{1}\\\ 0&\;\;t_{1}\leq
t<t_{1}+t_{2}\\\ -M&\;\;t_{1}+t_{2}\leq t\leq t_{1}+t_{2}+t_{3},\\\
\end{matrix}$ (4)

where the order of letter sequence is from left to right. For the example
above, the bang-off control is switched from bang ($P$) to off ($0$), then to
bang ($N$), with a switch number being two $N_{s}=2$. The type of such bang-
off controls refers to a way in which bang and off controls are concatenated.
For a given number of switches $N_{s}$, the number of possible types
$N_{type}$ is at most $3\times 2^{N_{s}}$. For certain initial/target quantum
states, $N_{type}$ can be further reduced. The control fields are represented
by the type and vector of durations $\mathbf{t}=[t_{1},t_{2},...]$.

For the second class the duration of each bang/off is the time step-size:
$\delta t=T/N_{T}$ with $T$ being total duration and $N_{T}$ the number of
time slots. Thus the control is usually represented by a control vector
$u(t)\equiv\\{u_{k}|k=1,...,N_{T}\\}$. This class of bang-off control arises
because of a coarse-graining of temporal variable for a numerical
optimization. In general, the simulation is more accurate when $N_{T}$ is
larger. However, the optimization with large $N_{T}$ is time-consuming, and
the dimension of search space is huge. For the problems considered here the
control is allowed to take 3 different values $u_{k}=\\{\pm M,0\\}$, thus the
possible number of control vectors is $3^{N_{T}}$, which is an exponentially
large number. For the quantum optimal control problem of two-level system,
$N_{s}$ is much smaller than $N_{T}$ in general. Thus the optimization of
first class of bang-off controls is much easier than that of second one. In
addition, the optimized controls of second class generally switch more
frequently than the ones of first class, thus are less desirable in real
experiments.

Note that we do not claim that the first class of bang-off control is time-
optimal control protocol for all QOC problems, e.g., time-optimal control of
nonlinear two-level quantum system Chen  _et al._ (2010). Instead, we use the
first class of bang-off control to estimate QSL and to approximate the
temporal shape of time-optimal control. This is due to the fact that the shape
of first class of bang-off control is simple and its universality is in
general better than bang-bang control. In addition, the optimality of first
class of bang-off control has been proved in the two-level Landau-Zener system
Boscain and Mason (2006). Therefore, we mainly focus on the study of first
class of bang-off control.

###  II.3 Optimization algorithm

Optimization algorithms are of great importance for quantum optimal control
problems. Two common choices are gradient-based and gradient-free algorithms.
For the gradient-based algorithm, we choose the quasi-Newton method. For the
gradient-free algorithm, we use stochastic descent (SD) as the optimization
algorithm Bukov  _et al._ (2018a). Stochastic descent starts from a random
configuration and updates control field $u(t)$ only if the candidate increases
(decreases) the fidelity (Bures distance).

###  II.4 Scheme

Specifically, for the first class of bang-off controls, we start from
$N_{s}=0$, and look through all possible types. For each type we optimize
using SD (or quasi-Newton) the vector of durations $\mathbf{t}$ for different
values of total duration $T$, until the perfect fidelity $F=1-\delta$ is
found, where $\delta$ is vanishingly small. We denote $T_{i}^{\mathrm{min}}$
the minimal duration for which the perfect fidelity $F=1-\delta$ is obtained
with number of switches being $N_{s}=i$. If $T_{i}^{\mathrm{min}}\approx
T_{i+1}^{\mathrm{min}}$ and it is zero that one duration of optimized duration
vector $\mathbf{t}$ with $N_{s}=i+1$, then this means the increment of number
of switches does not decrease the minimal duration to reach perfect fidelity.
Hence we can stop optimization and choose the types corresponding perfect
fidelity with optimized $\mathbf{t}$ and $N_{s}=i$ as the estimation of time-
optimal control, and $T_{i}^{\mathrm{min}}$ as the estimation of QSL.

For the second class, we optimize the control vectors using 1-flip SD. Here
1-flip means that the new candidates are generated by randomly choosing one
time-slot and changing the value of control at that time-slot to other values
Bukov  _et al._ (2018a).

##  III Two-level system

Here we consider a two-level quantum system driven by the Hamiltonian
described in the following:

$H(t)=-E\sigma_{z}+u(t)\sigma_{x}$ (5)

where $\sigma_{z}$ is the drift Hamiltonian and $E>0$, and $\sigma_{x}$ is the
controlled Hamiltonian. When the control field is turned off, the lower
(upper) eigenstate is $|0\rangle=[1,0]^{\mathrm{T}}$
($|1\rangle=[0,1]^{\mathrm{T}}$).

One important parameter is defined as: $\alpha\equiv\arctan(M/E)$. The reason
is in the following. For $\alpha<\pi/4$ the optimal control is bang-bang, and
the durations for the first, final and interior bang have to be determined
numerically; for $\alpha>\pi/4$ the optimal control is in general bang-off
control with at most two switches Boscain and Mason (2006). For the latter
case the bang-bang fails to capture time-optimal control, and this is the case
we are interested. Therefore, we choose $E=1$ and $M=4/3$ such that
$\alpha=0.9273>\pi/4$.

We are interested in two transition processes with the bounded controls: (1)
transition from $|0\rangle$ to $|1\rangle$; (2) transition from $|0\rangle$ to
$|\psi_{E}\rangle$, where
$|\psi_{E}\rangle=1/\sqrt{2}(|0\rangle+e^{i9\pi/10}|1\rangle)$. For the first
case the time-optimal control is in fact bang-bang, whereas for the second the
time-optimal control is bang-off type. We wish to estimate the quantum speed
limit, as well as to determine the temporal shape of the time-optimal
controls, of two transitions with the aid of numerical optimization over bang-
off control fields.

###  III.1 Transition from $|0\rangle$ to $|1\rangle$

Figure 1: Quantum control landscape of fidelity $F$ for type
$P_{t_{1}}N_{t_{2}}$ of the first class bang-off controls with initial state
$|0\rangle$ and target state $|1\rangle$. Two time-optimal controls are
indicated by two blue pentagrams with
$[t_{1},t_{2}]=\\{[T_{1},T_{2}],[T_{2},T_{1}]\\}$.

In Ref. Boscain and Mason (2006) it has been proved, for initial state
$|0\rangle$ and target state $|1\rangle$ with $\alpha>\pi/4$, the time-optimal
controls are the first class of bang-off controls with one switch $N_{s}=1$,
i.e.,
$\\{P_{T_{1}}N_{T_{2}},P_{T_{2}}N_{T_{1}},N_{T_{1}}P_{T_{2}},N_{T_{2}}P_{T_{1}}\\}$,
where $T_{1}=(\pi-\arccos{(\cot^{2}{\alpha})})/2\sqrt{E^{2}+M^{2}}=0.6505$ and
$T_{2}=(\pi+\arccos{(\cot^{2}{\alpha})})/2\sqrt{E^{2}+M^{2}}=1.2345$. Thus the
quantum speed limit is
$T_{\mathrm{QSL}}=T_{1}+T_{2}=\pi/\sqrt{E^{2}+M^{2}}=0.6\pi$.

We wish to numerically estimate $T_{\mathrm{QSL}}$ and to obtain the time-
optimal control field using the scheme proposed. We first investigate the
first class with $N_{s}=1$, and show that the minimal duration for perfect
fidelity is equal to $T_{\mathrm{QSL}}$. Then we show the bang-off controls
with $N_{s}=2$ reduce to the ones with $N_{s}=1$ for the time-optimal problem
considered here. At last, we consider the second class of bang-off controls,
and compare these results with the ones obtained using CRAB method.

####  III.1.1 First class with $N_{s}=1$

Figure 2: Bures distance $d_{B}$ versus iterations for different values of
total evolution duration $T$ for type $P_{t_{1}}N_{t_{2}}$: (a)
$T=T_{\mathrm{QSL}}-10^{-2}$; (b) $T=T_{\mathrm{QSL}}-10^{-3}$; (c)
$T=T_{\mathrm{QSL}}-10^{-4}$; (d) $T=T_{\mathrm{QSL}}$; (e)
$T=T_{\mathrm{QSL}}+10^{-4}$. The type of bang-off control is
$P_{t_{1}}N_{t_{2}}$ with initial state $|0\rangle$ and target state
$|1\rangle$, and the optimization algorithm is SD. The number of initial
search points is one hundred, and the number of iteration is ten thousand. (f)
The minimal $d_{B}^{\mathrm{min}}$ obtained after 10000 iterations for
different values of total duration $T$. When $T=T_{\mathrm{QSL}}$, the
sequence of $d_{B}^{\mathrm{min}}$ reaches its minimum.

We start from the first class of bang-off controls with $N_{s}=1$. For the
transition from $|0\rangle$ to $|1\rangle$, the number of possible types is
two, i.e., $\\{P_{t_{1}}N_{t_{2}},N_{t_{1}}P_{t_{2}}\\}$.

In Fig.1 we show the quantum control landscape for type $P_{t_{1}}N_{t_{2}}$.
The control fields are characterized by a two-dimensional vector of durations
$\mathbf{t}=[t_{1},t_{2}]$. A similar landscape is obtained for type
$N_{t_{1}}P_{t_{2}}$. The repetition of landscape over the square domain with
time length $T_{\mathrm{QSL}}$ is because the time evolution of quantum state
is periodic. In Fig.1 no local traps exist, while it is not true for the
second class of bang-off controls. This is one of the advantages of the first
class over the second one.

We now estimate the minimal duration $T_{1}^{\mathrm{min}}$ for which the
perfect fidelity $F=1-\delta$ is obtained. The procedure is in the following:
choose different values of $T$ with $t_{1}+t_{2}=T$; optimize the free
variable $t_{1}$ using SD to maximize (minimize) fidelity $F$ (Bures distance
$d_{B}$). When $T\neq kT_{\mathrm{QSL}},k=1,2,...$ we find that $d_{B}$ does
not converge to zero. In addition, the optimization stops within fewer
iterations when $T$ deviates more from $T_{\mathrm{QSL}}$. When
$T=T_{\mathrm{QSL}}$, however, $d_{B}$ keeps decreasing and the update of
control fields continues for a large number of iterations.

In Fig. 2(a)-(e) we show the Bures distance as a function of iteration for
type $P_{t_{1}}N_{t_{2}}$ with five different values of $T$. One hundred
initial search points of $t_{1}$ are randomly generated, and are optimized
using SD. We find that as $T$ approaches toward $T_{\mathrm{QSL}}$, the
convergence rate decreases. Moreover, the minimal Bures distance obtained
after optimization decreases as $T$ approaches toward $T_{\mathrm{QSL}}$, cf.
Fig. 2(f). When $T=T_{\mathrm{QSL}}$, the Bures distance as a function hardly
converge even for very large number of iterations. Therefore, we could
estimate $T_{\mathrm{QSL}}$ by monitoring the convergence of Bures distance as
an alternative method. For instance, in Fig. 2(c)
$T=T_{\mathrm{QSL}}-10^{-4}$, the search point with the minimal $d_{B}$ found
in the last iteration stops decreasing after several hundred iteration. For
$T=T_{\mathrm{QSL}}$, however, the optimization continues after ten thousand
iterations, as shown in Fig. 2(d). Therefore, the minimal duration with one
switch is approximately equal to the quantum speed limit
$T_{1}^{\mathrm{min}}=T_{\mathrm{QSL}}$ with very high precision. In Fig. 2(d)
100 optimized values of $t_{1}$ cluster around $T_{1}$ and $T_{2}$ with
significantly small standard deviation $\sigma\sim\mathcal{O}(10^{-4})$. The
best fidelity using quasi-Newton method is $F=1-\mathcal{O}(10^{-16})$. Same
results are obtained for type $N_{t_{1}}P_{t_{2}}$. Therefore the numerical
results reproduce the analytical solutions with very high accuracy.

Figure 3: Error distributions of $E=1-F$ as a function of standard deviation
$\sigma$ for one time-optimal control $P_{T_{1}}N_{T_{2}}$. Red dots are the
mean, and the vertical blue lines represents the standard deviation of $E$.
(a) Error $E_{m}^{T}$ resulting from the uncertainty of time interval for each
bang control. 10000 random duration vectors $\mathbf{t}$ are drawn from
Gaussian distribution with mean being $[T_{1},T_{2}]$ and standard deviation
$\sigma$. (b) Error $E_{m}^{M}$ resulting from the uncertainty of boundaries
$\pm M$ for each bang control. 10000 random values of boundaries are drawn
from Gaussian distribution with mean being $\pm M$ and standard deviation
$\sigma$.

From the experimental point of view, the errors arise due to the operational
inaccuracy. Therefore, we investigate the robustness of time-optimal bang-off
controls with respect to uncertainty in the duration vector $\mathbf{t}$ and
in the boundary of control field $\pm M$. We sample 10000 duration vectors
$\mathbf{t}$ and calculate the corresponding error distribution $E_{m}^{T}$
for different values of $\sigma$. Similar is done for $E_{m}^{M}$. In Fig. 3
we show $E_{m}^{T}$ and $E_{m}^{T}$ as a function of standard deviation
$\sigma$. The mean of $E$ is roughly a constant multiply the square of
$\sigma$. Same for standard deviation of $E$. In addition, by comparing two
plots in Fig. 3 we observe that the time-optimal control is less robust w.r.t.
the uncertainty of duration vector than that of boundaries.

####  III.1.2 First class with $N_{s}=2$

Figure 4: Quantum control landscape of fidelity $F$ for type
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with initial state $|0\rangle$ and target state
$|1\rangle$, and $t_{1}+t_{2}+t_{3}=T_{\mathrm{QSL}}$. Two time-optimal
controls are indicated by two pentagrams for which
$[t_{1},t_{2}]=\\{[T_{1},0],[T_{2},0]\\}$ The black dots indicate the
optimization process of one search point which climbs towards one of optimal
time vector $[T_{2},0]$ with the global maximum of fidelity $F=1$.

To numerically verify that the time-optimal controls are indeed of type with
$N_{s}=1$ for the case considered in this section, we continue to study the
first class of bang-off controls with $N_{s}=2$. We would show that
$T_{2}^{\mathrm{min}}=T_{1}^{\mathrm{min}}$ with high precision, and one of
three durations is zero, thus controls with $N_{s}=2$ reduce to the ones with
$N_{s}=1$.

Specifically, for the transition from $|0\rangle$ to $|1\rangle$, the number
of possible types is six, i.e.,
$\\{P_{t_{1}}0_{t_{2}}N_{t_{3}},N_{t_{1}}0_{t_{2}}P_{t_{3}},P_{t_{1}}N_{t_{2}}P_{t_{3}},N_{t_{1}}P_{t_{2}}N_{t_{3}},P_{t_{1}}0_{t_{2}}P_{t_{3}},\newline
N_{t_{1}}0_{t_{2}}N_{t_{3}}\\}$. For the first four types, the perfect
fidelity are reached when $T=T_{\mathrm{QSL}}$, and one of the durations is
zero, e.g., $t_{2}=0$ for $P_{t_{1}}0_{t_{2}}N_{t_{3}}$. For the last two
types, the maximal fidelity is $F=0.6399$ when $T=T_{\mathrm{QSL}}$. When
$T<T_{\mathrm{QSL}}$, the perfect fidelity cannot be reached for all six
types. For instance, the maximal fidelity with $T=T_{\mathrm{QSL}}-0.001$ is
$F=1-1.001\times 10^{-6}$ for type $P_{t_{1}}0_{t_{2}}N_{t_{3}}$. Similar
results are obtained for other types. Therefore, we have numerically verified
that $T_{2}^{\mathrm{min}}=T_{1}^{\mathrm{min}}=T_{\mathrm{QSL}}$.

In Fig. 4 we show the quantum control landscape for type
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with $t_{1}+t_{2}+t_{3}=T_{\mathrm{QSL}}$. Two
time-optimal controls are indicated by two pentagrams with
$[t_{1},t_{2}]=\\{[T_{1},0],[T_{2},0]\\}$. The duration for off-control
$u(t)=0$ is zero $t_{2}=0$, thus controls with $N_{s}=2$ reduce to the ones
with $N_{s}=1$. The optimization process of one search point using SD climbs
towards one of the optimal time vector. As search point is climbing towards
the global maximum, smaller step-size and more iterations are required for
better performance.

We optimize the time vector $\mathbf{t}=[t_{1},t_{2}]$ for type
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with $t_{3}=T_{\mathrm{QSL}}-t_{1}-t_{2}$. The
best fidelity obtained using quasi-Newton is $F=1-\mathcal{O}(10^{-16})$, and
the optimized duration vectors deviate from the optimal values, e.g.,
$\mathbf{t}_{\mathrm{Opt}}=[T_{2},0,T_{1}]=[1.2345,\,0,\,0.6505]$ with
vanishingly small value. Similar results are obtained for types
$\\{N_{t_{1}}0_{t_{2}}P_{t_{3}},P_{t_{1}}N_{t_{2}}P_{t_{3}},N_{t_{1}}P_{t_{2}}N_{t_{3}}\\}$.

The increment of switch numbers $N_{s}=1\rightarrow N_{s}=2$ does not result
in the decrement of total duration, but
$T_{2}^{\mathrm{min}}=T_{1}^{\mathrm{min}}=T_{\mathrm{QSL}}$. In addition, we
have shown that one duration of the optimized time vector is approximately
equal to zero, thus the controls with $N_{s}=2$ reduce to the ones with
$N_{s}=1$. Then we can stop studying the controls with larger number of
switches $N_{s}\geq 3$.

In summary we have numerically verified for the transition
$|0\rangle\rightarrow|1\rangle$ the time-optimal controls are
$\\{P_{T_{1}}N_{T_{2}},P_{T_{2}}N_{T_{1}},N_{T_{1}}P_{T_{2}},N_{T_{2}}P_{T_{1}}\\}$,
and the quantum speed limit is $T_{\mathrm{QSL}}$, with little deviations.

####  III.1.3 Second class of bang-off control

Figure 5: The results obtained for transition from $|0\rangle$ to $|1\rangle$
with $T=T_{\mathrm{QSL}}$ using 1-flip SD to optimize the second class of
bang-off controls. The number of time slots is fixed to be $N_{T}=40$. (a) The
optimized control field found with first class of bang-off (red,solid) which
is $P_{T_{1}}N_{T_{2}}$, that with second class (blue, dashed), and with CRAB
(grean, dash-dotted line). (b) Infidelity $1-F$ versus iterations with 1000
search points. (c) Distribution of distance $D_{ij}$ between optimized control
fields.

In this sub-section we consider the second class of bang-off controls as a
comparison. Here we employ the 1-flip SD to optimize the control vectors where
each component is allowed to take one of three values $u_{k}=\\{\pm M,0\\}$.
The number of time slots is fixed to be $N_{T}=40$. 1000 initial control
vectors are randomly generated and are optimized using SD for $10^{4}$
iterations. To further investigate the relations between the optimized control
fields, we calculate the distance between optimized control fields as

$D_{ij}=\frac{1}{N_{T}}||u^{(i)}-u^{(j)}||_{1}$ (6)

where $u^{(i)}$ is the $i$-th optimized control field and $||\cdot||_{1}$ is
the absolute-value norm Larocca  _et al._ (2018).

In addition, we compare the results to one of the state-of-the-art algorithms,
Chopped Random Basis (CRAB). We choose various value of $N_{c}$ which is the
cut-off number of Fourier basis, and optimize the corresponding control field
with $T=T_{\mathrm{QSL}}$. The best fidelity obtained is
$F=1-\mathcal{O}(10^{-15})$ with $N_{c}=5$. The corresponding optimized
control field is shown in Fig. 5(a) by green dash-dotted line.

In Fig. 5(a) the optimized control field found with second class is indicated
by the blue dashed line, while one of the time-optimal control fields
$P_{T_{1}}N_{T_{2}}$ is marked by the red solid line. We observe that the
temporal shape of the optimized control field found by 1-flip is similar to
that of time-optimal control $P_{T_{1}}N_{T_{2}}$, but the number of switch is
three. Both the optimized control fields obtained with second class of bang-
off and that with CRAB approximate reasonably the time-optimal control field.

In Fig. 5(b) we plot the infidelity $1-F$ as a function of optimization
iteration. 1-flip SD stops updating after about 1000 iterations, and the
maximal fidelity obtained is $F=1-1.35\times 10^{-5}$. Presumably, the
performance of second class of bang-off controls would be improved by $k$-flip
SD ($k\geq 2$), yet the number of iterations required is much larger Bukov
_et al._ (2018a). In Fig. 5(c) distances between optimized control fields
$D_{ij}$ form a multimodal distribution with four peaks. This yields the fact
that the optimized control fields cluster around four time-optimal controls
$\\{P_{T_{1}}N_{T_{2}},P_{T_{2}}N_{T_{1}},N_{T_{1}}P_{T_{2}},N_{T_{2}}P_{T_{1}}\\}$
with a standard deviation much larger than that obtained with first class of
bang-off controls.

Therefore, for the time-optimal problem of quantum transition from $|0\rangle$
to $|1\rangle$, the performance of first class of bang-off control is much
better than that of second class optimized using 1-flip SD, concerning the
maximal fidelity obtained, the optimized control fields found and the their
distribution.

###  III.2 Transition from $|0\rangle$ to $|\psi_{E}\rangle$

In this section we consider the transition from $|0\rangle$ to
$|\psi_{E}\rangle$ with $\alpha>\pi/4$. The time-optimal controls are of first
class bang-off controls with two switches $N_{s}=2$:
$\\{P_{\tau_{1}}0_{\tau_{2}}N_{\tau_{3}},P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}\\}$
Boscain and Mason (2006). Three durations are: $\tau_{1}=T_{1}=0.6505$ is the
duration for which the time-evolving quantum state reaches the equator of
Bloch sphere for the first time under the bang control $u(t)=M$. The quantum
state after this process is
$|A^{+}\rangle=1/\sqrt{2}(|0\rangle+e^{i(\pi+\beta)}|1\rangle)$ (up to a
global phase) with $\beta=\arccos{E/M}=0.7227$;
$\tau_{2}=(\beta-\pi/10)/2E=0.2043$ is the duration for which the quantum
state reaches
$|\psi_{E}^{\prime}\rangle=1/\sqrt{2}(|0\rangle+e^{i11\pi/10}|1\rangle)$ along
the equator starting from $|A^{+}\rangle$ with the off control $u(t)=0$; and
$\tau_{3}=0.2978$ is the duration for which the quantum state reaches target
state $|\psi_{E}\rangle$ from $|\psi_{E}^{\prime}\rangle$ with the control
fields $u(t)=\pm M$. The quantum speed limit is
$\tau_{\mathrm{QSL}}=\tau_{1}+\tau_{2}+\tau_{3}=1.1525$. Here we use a
different notation $\tau_{\mathrm{QSL}}$ for QSL to distinguish it from the
one in Sec. III.1.

####  III.2.1 First class with $N_{s}=1$

For the transition considered in this section, the possible types of first
class are four:
$\\{P_{t_{1}}0_{t_{2}},P_{t_{1}}N_{t_{2}},N_{t_{1}}0_{t_{2}},N_{t_{1}}P_{t_{2}}\\}$.
When $T=\tau_{\mathrm{QSL}}$, the maximal fidelity of four types are:
$F=[0.9997,1-5.0937\times 10^{-6},0.4999,0.9873]$. Therefore, the time-optimal
controls cannot be the ones with $N_{s}=1$ because
$T_{1}^{\mathrm{min}}>\tau_{\mathrm{QSL}}$. It is worth noting that although
$T_{1}^{\mathrm{min}}>\tau_{\mathrm{QSL}}$, the perfect fidelity can be
obtained using $N_{s}=1$ controls with a larger duration, e.g.,
$P_{\tau_{1}}0_{\lambda}$ with $\lambda=\beta+\pi/10=0.5184$ and the total
duration $T=\tau_{1}+\lambda=1.1689$ which is a little larger than
$\tau_{\mathrm{QSL}}=1.1525$. Because of the operational inaccuracy in
experiment, the larger $N_{s}$ is, the larger the error is. Therefore, the
controls with $N_{s}=1$ is acceptable, provided the perfect fidelity can be
obtained and the total duration which does not exceed $\tau_{\mathrm{QSL}}$
much is tolerable.

####  III.2.2 First class with $N_{s}=2$

Figure 6: Quantum control landscape of fidelity $F$ for type
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with initial state $|0\rangle$ and target state
$|\psi_{E}\rangle$, and $t_{1}+t_{2}+t_{3}=\tau_{\mathrm{QSL}}$. The time-
optimal control is indicated by a blue pentagram with
$\mathbf{\tau}_{\mathrm{Opt}}=[\tau_{1},\tau_{2}]$. The black dots indicate
the optimization process of one search point which climbs towards one of
optimal time vector $\mathbf{t}_{\mathrm{Opt}}$ with the global maximum of
fidelity $F=1$.  Figure 7: (a) Quantum control landscape of log-Bures distance
$\log_{10}d_{B}$ for type $P_{t_{1}}N_{t_{2}}0_{t_{3}}$ with
$t_{1}+t_{2}+t_{3}=\tau_{\mathrm{QSL}}$.Two local minima are shown with blue
color. (b) $\log_{10}d_{B}$ as a function of optimization iteration with 100
initial search points. 100 search points saturated towards two branches with
values of $\log_{10}d_{B}$ being $[-1.7916,-2.6825]$ after 10000 iterations.

We could estimate $T_{2}^{\mathrm{min}}$ using the same scenario in Sec.
III.1. An alternative method is to calculate the fidelity for time vectors in
the fine-grained $3D$ temporal space. Find the locations of perfect fidelity
and choose the shortest duration as the estimate of $T_{2}^{\mathrm{min}}$.

In Fig. 6 we show the quantum control landscape of fidelity for
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with $t_{1}+t_{2}+t_{3}=\tau_{\mathrm{QSL}}$.
The perfect fidelity is indicated by a blue pentagram with the optimal time
vector
$\mathbf{t}_{\mathrm{Opt}}=[\tau_{1},\tau_{2},\tau_{3}]=[0.6505,0.2043,0.2978]$.
By investigating the landscape with durations $T<\tau_{\mathrm{QSL}}$, perfect
fidelity cannot be reached. It is interesting to notice that there is only one
maximum, which is also the global maximum, in the landscape of fidelity.
Therefore, all search points should converge towards the optimal time vector,
and indeed this is true in our numerical optimization process. 100 search
points are optimized using SD, and the maximal fidelity obtained after
$10^{6}$ iterations is $F=1-2.220\times 10^{-16}$. The optimized time vector
is $\mathbf{t}=[0.6504,0.2046,0.2976]$. Therefore we numerically verified
$T_{2}^{\mathrm{min}}=\tau_{\mathrm{QSL}}$ with high accuracy.

One of the benefits using SD optimizing over first class of bang-off controls
is that we could estimate the number of local extremes and their values, as
well as the location of corresponding controls. For instance, in Fig. 7(a) we
show the quantum control landscape of log-Bures distance $\log_{10}(d_{B})$
for type $P_{t_{1}}N_{t_{2}}0_{t_{3}}$ with
$t_{1}+t_{2}+t_{3}=\tau_{\mathrm{QSL}}$. Note that perfect fidelity cannot be
reached for type $P_{t_{1}}N_{t_{2}}0_{t_{3}}$ with total duration being
$\tau_{\mathrm{QSL}}$. Two local minima exist in the landscape, and are
estimated by optimizing 100 search points which are randomly sampled in the
beginning of optimization. It is clearly shown in Fig. 7(b) two branches of
search points saturate towards different values of $\log_{10}(d_{B})$. It is
worth noting that since the two local minima are far from each other, given
that the step-size of optimization is small, it is hardly possible for search
points to jump from one local trap to another. If the distance between two
local minima is small, the search points might jump from one to another.
Similar results are obtained for other types.

Therefore, we have estimated that $T_{2}^{\mathrm{min}}=\tau_{\mathrm{QSL}}$
and the perfect fidelity is reached for control fields
$\\{P_{\tau_{1}}0_{\tau_{2}}N_{\tau_{3}},P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}\\}$.

####  III.2.3 First class with $N_{s}=3$

Figure 8: Bures distance $d_{B}$ versus iterations for type
$P_{t_{1}}0_{t_{2}}N_{t_{3}}0_{t_{4}}$with the total evolution duration
$T=\tau_{\mathrm{QSL}}$. 200 initial search points are optimized using SD with
$10^{5}$ iterations.

To further verify the time-optimal controls are of types with $N_{s}=2$, we
continue to study the first class bang-off controls with $N_{s}=3$. For
illustration, we consider the type $P_{t_{1}}0_{t_{2}}N_{t_{3}}0_{t_{4}}$. We
would demonstrate that $T_{3}^{\mathrm{min}}=T_{2}^{\mathrm{min}}$, and one of
the durations is zero $t_{4}=0$ .

By investigating the control landscape, we verify that the time vector
corresponding the perfect fidelity with minimal duration is
$\mathbf{t}_{\mathrm{Opt}}=[\tau_{1},\tau_{2},\tau_{3},0]$ (data not shown).
This is further confirmed by the optimization. Two hundred initial time
vectors $t=[t_{1},t_{2},t_{3},t_{4}]$ are optimized using SD with
$t_{1}+t_{2}+t_{3}+t_{4}=\tau_{\mathrm{QSL}}$ for $10^{5}$ iterations. In Fig.
8 we show Bures distance $d_{B}$ as a function of iteration. It is interesting
to notice from Fig. 8 that there are one local minimum and one global minimum
of landscape, and they are not far from each other. Therefore, some of the
search points are able to jump outside the local minimum and move towards the
global one.

The best fidelity obtained using quasi-Newton is $F=1-\mathcal{O}(10^{-16})$,
and the optimized time vector is $\mathbf{t}=[0.6505,0.2043,0.2977,4.064\times
10^{-4}]$ with one duration being approximately equal to zero $t_{4}\approx
0$. Similar results are obtained for type
$\\{P_{t_{1}}0_{t_{2}}N_{t_{3}}P_{t_{4}},P_{t_{1}}0_{t_{2}}P_{t_{3}}0_{t_{4}},P_{t_{1}}0_{t_{2}}P_{t_{3}}N_{t_{4}}\\}$,
while the perfect fidelity cannot be reached for other types with the total
duration being $\tau_{\mathrm{QSL}}$. Therefore, we have numerically checked
that $T_{3}^{\mathrm{min}}=T_{2}^{\mathrm{min}}=\tau_{\mathrm{QSL}}$ with high
accuracy, hence the optimal controls with $N_{s}=3$ reduce to the ones with
$N_{s}=2$.

Considering that the optimal controls with $N_{s}=3$ reduce to the ones with
$N_{s}=2$ and $T_{3}^{\mathrm{min}}=T_{2}^{\mathrm{min}}$, there is no need to
study the first class of controls with larger switch numbers $N_{s}\geq 4$.

In summary we have numerically verified for the transition considered in this
section the time-optimal controls are
$\\{P_{\tau_{1}}0_{\tau_{2}}N_{\tau_{3}},P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}\\}$,
and the quantum speed limit is $\tau_{\mathrm{QSL}}$, with high precision.

####  III.2.4 Second class of bang-off control

Figure 9: The results obtained for transition from $|0\rangle$ to
$|\psi_{E}\rangle$ using 1-flip SD optimizing the second class of bang-off
controls with total duration being $T=\tau_{\mathrm{QSL}}$. The number of time
slots is fixed to be $N_{T}=20$. (a)The optimized control field found with
first class of bang-off (red,solid) which is
$P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}$, that with second class (blue, dashed),
and with CRAB (grean, dash-dotted line). (b) Infidelity $1-F$ versus
iterations with 1000 search points. (c) Distribution of distance $D_{ij}$
between optimized control fields.

Same as Sec. III.1 we investigate the performance of the second class of bang-
off controls using the 1-flip SD. 1000 initial control vectors are randomly
generated and are optimized for $10^{4}$ iterations with the number of time
slots being $N_{T}=20$. Distance between optimized control fields $D_{ij}$ are
calculated.

In Fig. 9 the results of optimization are shown with total duration
$T=\tau_{\mathrm{QSL}}$. In Fig. 9(a) one of the time-optimal control fields
$P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}$ is marked by the red solid line, and
the optimized control field with The temporal shape of the optimized control
field found by 1-flip SD is again similar to that of time-optimal control
$P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}}$, except for certain duration the
optimized control takes the value $u(t)=-M$, rather than the expected value
$u(t)=0$. However, the best fidelity found by second class of bang-off is
$F=1-2.77\times 10^{-6}$. In addition, we optimize the CRAB control field, the
best fidelity obtained is $F=1-\mathcal{O}(10^{-13})$ with $N_{c}=2$. The
corresponding control field is indicated by green dash-dotted line,

In Fig. 9(b) we plot the infidelity $1-F$ as a function of optimization
iteration. 1-flip SD stops updating after about 100 iterations. In Fig. 9(c)
distances between optimized control fields $D_{ij}$ form a bimodal
distribution, which indicates the optimized control fields cluster around two
time-optimal controls
$\\{P_{\tau_{1}}0_{\tau_{2}}P_{\tau_{3}},P_{\tau_{1}}0_{\tau_{2}}N_{\tau_{3}}\\}$.

Concerning the best fidelity obtained, the performance of first class of bang-
off is much better than CRAB and second class, while second class of bang-off
performs worst.

##  IV Three-level system

Figure 10: Fidelity $F$ versus total duration $T$ in three-level system
obtained with the first class of bang-off control field. Two critical time
points are $T_{c}$ and $T_{\mathrm{QSL}}$. For $T\leq T_{c}$, the optimal
control is $P_{T}$ (and $N_{T}$); for $T>T_{c}$ the optimal control is of type
$N_{s}=4$.

To test the performance of the systematic scheme proposed, we apply it in the
three-level quantum system. Specifically, we consider the Hamiltonian of a
three-level quantum system described in the following:

$H(h(t))=-EH_{0}+\mu_{1}H_{1}+\mu_{2}u(t)H_{2},$ (7)

where the drift Hamiltonian $H_{0}$ and $H_{1}$, and the control Hamiltonian
$H_{2}$ are

$\centering{\begin{matrix}H_{0}=\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{bmatrix},H_{1}=\begin{bmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{bmatrix},H_{2}=\begin{bmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{bmatrix},\end{matrix}}\@add@centering$ (8)

respectively. $H_{1}$ couples state $|1\rangle$ and $|2\rangle$, and $H_{2}$
couples state $|2\rangle$ and $|3\rangle$, where
$|1\rangle=[1,0,0]^{\mathrm{T}}$, etc.

Without loss of generality, we set $E_{0}=1$, the coupling strengths
$\mu_{1}=1$, and $\mu_{2}=2$. The control field is constrained, i.e., $-1\leq
u(t)\leq 1$. We wish to drive the system from the initial state $|1\rangle$ to
the target state $|3\rangle$.

It is worthy noting that

$\langle 3|H^{n}(h)|1\rangle=-\langle 3|H^{n}(-h)|1\rangle,$ (9)

where $n$ is integer. Therefore, it is straightforward to show that there is a
mirror symmetry $F(h(t),T)=F(-h(t),T)$:

$\displaystyle F(h(t),T)$ $\displaystyle=$ $\displaystyle|\langle
3|\;\mathcal{T}\exp(-\mathrm{i}\int_{0}^{T}H[h(t)]\mathrm{d}t\;|1\rangle|^{2}$
(10) $\displaystyle=$ $\displaystyle|\langle
3|\;\mathcal{T}\exp(-\mathrm{i}\int_{0}^{T}H[-h(t)]\mathrm{d}t\;|1\rangle|^{2}$
$\displaystyle=$ $\displaystyle F(-h(t),T).$

This symmetry can help reduce by half the number of types to be calculated.

In Fig. 10 we show the best fidelity obtained as a function of the total
duration $T$ by optimizing the first class of bang-off controls. Two critical
time points are shown, i.e., $T_{c}$ and $T_{\mathrm{QSL}}$. For different
total duration $T$, the corresponding optimal control is in general different.
For $T\in[0,T_{c}]$ the optimal control is of type $N_{s}=0$, and
$T\in(T_{c},T_{\mathrm{QSL}}]$ the optimal control is of type $N_{s}=4$.

###  IV.1 Optimal control for $T\in[0,T_{c}]$

To determine the optimal control for given $T$, we optimize over all types of
the first class of bang-off control. It is found that for $T\leq 0.73157$, the
best fidelity obtained with $N_{s}\geq 1$ is equal to that obtained with
$N_{s}=0$, while for $T>0.73157$ the former is larger than the latter; cf.
Fig. 11. For $N_{s}=0$ the best fidelity is obtained with both $P_{T}$ and
$N_{T}$, which is implied by Eq. (9). In addition, for $T\leq 0.73157$ the
optimal control fields obtained with $N_{s}\geq 1$ are either $P_{T}$ or
$N_{T}$.

Figure 11: $F_{1}-F_{0}$ versus total duration $T$. $F_{1}$ is the best
fidelity obtained with $N_{s}=1$, and $F_{0}$ with $N_{s}=0$. $F_{1}-F_{0}=0$
when $T\leq T_{c}$, and $F_{1}-F_{0}>0$ when $T>T_{c}$. $T_{c}=0.73517$ is
indicated by an arrow.  Figure 12: Infidelity $1-F$ versus total duration $T$
obtained with $N_{s}=4$ bang-off control, which is the estimation of optimal
control for $T\geq T_{c}$. The $y$-axis is in $\log$-scale.

In Ref. Bukov  _et al._ (2018b) the region with $T\in[0,T_{c}]$ is called
overconstrained since the landscape in this region is convex such that the
search for local minimum is easy. By observing the behavior of correlator and
other functions, the value of $T_{c}$ is roughly estimated Bukov  _et al._
(2018b). By using our method, it is much easier to determine the value of
$T_{c}$ with very high accuracy. For the three-level quantum system considered
here, we compare the best fidelity obtained with $N_{s}=1$ and that with
$N_{s}=0$, the value of $T_{c}$ is located by checking the duration $T$ with
which $F_{1}-F_{0}$ becomes nonzero. See Fig. 11 for more information.

###  IV.2 Optimal control for $T\in(T_{c},T_{\mathrm{QSL}}]$

As $T$ increases and is larger than $T_{c}$, the optimal control is not
$P_{T}$ or $N_{T}$ anymore. For $T>T_{c}$, we optimize over the first class of
bang-off control with different value of $N_{s}$. It is found that the best
fidelity obtained with $N_{s}=4$ is larger than those obtained with $N_{s}\leq
3$, and is equal to those obtained with $N_{s}\geq 5$. In addition, the
optimal control fields obtained with $N_{s}\geq 5$ are reduced to the ones
with $N_{s}=4$.

Figure 13: One of the optimal control fields $h(t)$ obtained by the first
class of bang-off control (red, solid)
$P_{t_{1}}0_{t_{2}}N_{t_{3}}P_{t_{4}}N_{t_{5}}$ and by CRAB (blue, dashed),
with the total duration being the estimated quantum speed limit
$T=2.850\approx T_{\mathrm{QSL}}$.

We conjecture that for $T\in(T_{c},T_{\mathrm{QSL}}]$ the optimal control
field is bang-off control with $N_{s}=4$. The reasons are as follows. In Ref.
Bukov  _et al._ (2018b) for total duration in the overconstrained region and
glassy region, the optimal control is of bang-off type, which is obtained by
averaging one thousand optimized bang-bang controls. This is further verified
in Ref. Li (2022) by employing the systematic method proposed in this paper.
The number of switches of the bang-off control is small, i.e., $N_{s}=1$
($N_{s}=2$) for overconstrained (glassy) region. However, for total duration
in the symmetry-broken region, the optimal control field obtained is
continuous, rather than bang-off control which is piece-wise constant Bukov
_et al._ (2018b). This is also reflected in Ref. Li (2022) through the fact
that the number of switch of the optimized bang-off control in this region is
$N_{s}=9$. The bang-off control with such a large number of switch serves as a
reasonable approximation to the continuous control. In such case, $u(t)$ takes
values between the maximal and minimal value, and is in general referred to as
’singular’ Boscain and Mason (2006); Brady  _et al._ (2021). Note that in the
case of two-level quantum system considered here, the singular control is in
fact off control Boscain and Mason (2006). In general, however, this
conclusion does not hold in multiple-level quantum system Brady  _et al._
(2021). Given the fact that for $T\in(T_{c},T_{\mathrm{QSL}}]$ the optimized
bang-off control is of type with $N_{s}=4$, which is small, and the fact that
the optimized control field obtained with CRAB is very similar to the bang-off
control (see Fig. 13), we conjecture that the optimal control in this region
is bang-off control with $N_{s}=4$.

From the numerical results we observed that there are eight types of optimal
control fields in this region, i.e., $P0NPN$, $PNPNP$, $PN0PN$, $0NPNP$, and
other four types are the negative of the former four types, see Eq. (10). In
Fig. 12 we show the infidelity $1-F$, obtained with $N_{s}=4$ optimal control
field, as a function of the total duration. A vertical asymptote of infidelity
is present around $T\approx 2.84$. Furthermore, the unit fidelity
$F=1-\mathcal{O}(10^{-16})$ is obtained with $T=2.850$. Therefore, we estimate
the the quantum speed limit to be $T_{\mathrm{QSL}}\approx 2.850$.

In Fig. 13 we show one of the time-optimal controls obtained numerically with
type $P_{t_{1}}0_{t_{2}}N_{t_{3}}P_{t_{4}}N_{t_{5}}$. As a comparison, we
employ the CRAB method to optimize the control field for $T=2.850$ with
various values of $N_{c}$. The best fidelity obtained is $F=1-3.996\times
10^{-6}$ with $N_{c}=3$. It is worth noting that the optimized control field
obtained with CRAB approximates the bang-bang control, and is very similar to
the one obtained with bang-off control. However, the infidelity obtained with
CRAB is ten orders of magnitude greater than the one obtained with bang-off
control. Therefore, the performance of our method of bang-off is much better
than that of CRAB.

##  V Conclusions

We have proposed a systematic scheme for numerically estimating the quantum
speed limit and approximating the temporal shape of optimal control with
bounded amplitude. It has been studied as examples the optimal control
problems of quantum state transition processes in two-level and three-level
quantum system.

For the two-level quantum system, the numerical results are compared to the
analytic solutions. We found that the deviations from analytic solutions are
significantly small. We have also investigated the robustness of first class
of bang-off controls. In addition, the performance of first class is better
than that of second class and that of CRAB, concerning the fidelity obtained
and the optimized control field found.

For the three-level quantum system, we study the control problem with real
control field with bounded amplitude. Optimal controls for different regions
of total duration are obtained. QSL and time-optimal control are estimated. We
found that although the optimized control field obtained using CRAB is very
similar to the one obtained using our method, the infidelity obtained in the
former is much greater than the latter with the total duration being QSL.

Since the temporal shape of the first class of bang-off control is simple, it
is easy to implement such control fields in experiment. Our scheme might shed
light on quantum optimal control problems in which the analytical solution is
absent.

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# On the boundedness of certain generalized Hilbert operators in $\ell^{p}$

Nikolaos Athanasiou American College of Thessaloniki, V. Sevenidi 17, Pylaia
55535, Thessaloniki, Greece. E-mail address<EMAIL_ADDRESS>
(May 2022)

###### Abstract

The Hilbert matrix $\mathcal{H}_{n,m}=(n+m+1)^{-1}$ has been extensively
studied in previous literature. In this paper we look at generalized Hilbert
operators arising from measures on the interval $[0,1]$, such that the Hilbert
matrix is obtained by the Lebesgue measure. We provide a necessary and
sufficient condition for these operators to be bounded in $\ell^{p}$ and
calculate their norm.

##  1 Introduction

###  1.1 Motivation and history

The study of the classical Hilbert operator has its origins on the classical
inequality of the same name.

######  Theorem 1.1.

(Hilbert’s inequality) Suppose that $(a_{m})_{m=1}^{\infty}$ and
$(b_{n})_{n=1}^{\infty}$ are sequences of non-negative terms and let $p,\hskip
1.42262ptq>1$ be Hölder conjugates, i.e. such that
$\frac{1}{p}+\frac{1}{q}=1$. Assume that $(a_{m})\in\ell^{p}$ and
$(b_{n})\in\ell^{q}$. Then there holds

$\sum_{m=1}^{\infty}\hskip 1.42262pt\sum_{n=1}^{\infty}\frac{a_{m}\hskip
1.42262ptb_{n}}{m+n}\leq\frac{\pi}{\text{sin}\left(\frac{\pi}{p}\right)}\left(\sum_{m=1}^{\infty}a_{m}^{p}\right)^{\frac{1}{p}}\left(\sum_{n=1}^{\infty}b_{n}^{q}\right)^{\frac{1}{q}}.$

It was Hilbert who first proved the above for the special case $p=q=2$ in one
of his lectures on integral equations. His proof was published by Weyl (1908).
The first proof for general $p,\hskip 1.42262ptq>1$ was given by Hardy and
Riesz [5]. Perhaps the best reference for the proof is [4]. A sharp version of
Hilbert’s inequality is the following: For $(a_{m})_{m=0}^{\infty}$ and
$(b_{n})_{n=0}^{\infty}$ both sequences of non-negative numbers such that
$(a_{m})\in\ell^{p}$ and $(b_{n})\in\ell^{q}$, there holds

$\sum_{m=0}^{\infty}\hskip 1.42262pt\sum_{n=0}^{\infty}\frac{a_{m}\hskip
1.42262ptb_{n}}{m+n+1}\leq\frac{\pi}{\text{sin}\left(\frac{\pi}{p}\right)}\left(\sum_{m=0}^{\infty}a_{m}^{p}\right)^{\frac{1}{p}}\left(\sum_{n=0}^{\infty}b_{n}^{q}\right)^{\frac{1}{q}}.$
(1.1)

The Hilbert operator and its associated Hilbert matrix are now introduced as
follows: Consider the operator $H$ on $\ell^{p}$ given by

$H:(a_{n})_{n=0}^{\infty}\mapsto\left(\sum_{m=0}^{\infty}\frac{a_{m}}{m+n+1}\right)_{n=0}^{\infty}.$

The fact that this is a bounded operator on $\ell^{p}$ for $p>1$ follows from
Hilbert’s inequality as follows.

We recall that the norm of $(a_{n})\in\ell^{p}$ is
$\lVert(a_{n})\rVert_{\ell^{p}}=\left(\sum_{n=0}^{\infty}a_{n}^{p}\right)^{\frac{1}{p}}$.Moreover
there must hold

$\lVert(a_{n})\rVert_{\ell^{p}}=\sup_{h\in(\ell^{p})^{*},\hskip
1.42262pt\hskip 1.42262pt\lVert h\rVert\leq 1}h((a_{n})),$

where $h$ is any element of the dual space $\ell^{q}$ with norm $1$. Therefore

$\lVert(a_{n})\rVert_{\ell^{p}}=\sup_{(b_{n})\in\ell^{q},\hskip
1.42262pt\hskip 1.42262pt\rVert(b_{n})\rVert_{\ell^{q}}\leq
1}\bigg{\lvert}\sum_{n=0}^{\infty}a_{n}b_{n}\bigg{\rvert}.$

By fixing an arbitrary number $k$ and taking the supremum over all sequences
$(b_{n})\in\ell^{q}$ such that $\lVert(b_{n})\rVert_{\ell^{q}}=k$, we arrive
at the desired result. The associated matrix

$H=\begin{pmatrix}1&\frac{1}{2}&\frac{1}{3}&\dots\\\
\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\dots\\\
\frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\dots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

acting on column vectors whose entries are terms of a sequence in $\ell^{p}$
is another way to look at the operator (we shall not be making a notational
distinction between $H$ and its associated matrix). Further historical notes
and properties of the Hilbert matrix can be found in [1].

Notice that each entry in $H$ can be written as

$H_{n,\hskip 1.42262ptk}=\int_{0}^{1}g_{n,k}(t)\hskip 1.42262pt\text{d}t,$

where $g_{n,k}(t):=\binom{n+k}{k}\hskip 1.42262pt(1-t)^{n}\hskip
1.42262ptt^{k}$. Motivated by this integral representation, in the current
paper we shall be looking at a class of generalized Hilbert operators, arising
from replacing $\text{d}t$ with finite, positive Borel measures. The
definition is as follows.

######  Definition 1.2.

Let $\mu$ be a finite, positive Borel measure on $[0,1]$. We define the
generalized Hilbert matrix $\mathcal{C}^{\mu}$ by

$\mathcal{C}^{\mu}_{n,k}:=\binom{n+k}{k}\hskip
1.42262pt\int_{0}^{1}(1-t)^{n}\hskip 1.42262ptt^{k}\hskip
1.42262pt\text{d}\mu(t),$ (1.2)

for $n,k$ integers ranging from 0 to $\infty$.

The generalized Hilbert matrix can be viewed as an operator on $\ell^{p}$
given by $\ell^{p}\ni(c_{n})_{n=0}^{\infty}\mapsto(d_{n})_{n=0}^{\infty}$,
with

$d_{n}:=\sum_{k=0}^{\infty}\mathcal{C}^{\mu}_{n,k}c_{k}.$ (1.3)

Let us, finally, note that the corresponding question to the one addressed in
this paper for the case of generalized Césaro operators was tackled in [3] and
throughout this work, our general approach significantly borrows from the
ideas presented in it.

###  1.2 Preliminaries

In this subsection we will be introducing a series of lemmata which will
become relevant later on.

######  Lemma 1.3.

There holds

$\sum_{n=0}^{\infty}\binom{n+m}{m}\hskip 1.42262pts^{n}=(1-s)^{-m-1},$

for all $s\in(0,1).$

###### Proof.

That the left-hand side converges (absolutely) is a direct consequence of the
ratio test for series. One can use induction on $m$. For $m=0$, the result
holds. Assume the result holds for $m=k$. Then we have

$(1-s)^{-m-2}=\frac{1}{1-s}(1-s)^{-m-1}=\left(1+s+s^{2}+\dots\right)\hskip
1.42262pt\sum_{n=0}^{\infty}\binom{n+m}{m}s^{n}.$

Since both series above converge absolutely, their product is given by the
series

$\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\binom{m+l}{m}\right)\hskip
1.42262pts^{n},$

which is convergent by Mertens’ theorem on the Cauchy product of series.
Therefore, all one has to show is that

$\sum_{l=0}^{n}\binom{m+l}{m}=\binom{m+n+1}{m+1},$

which itself can be shown by a straightforward induction, using the fact that

$\binom{m+n+1}{m+1}+\binom{m+n+1}{m}=\binom{m+n+2}{m+1}.$

∎

######  Lemma 1.4.

There holds

$\frac{1-t}{1-t\hskip 1.42262pte^{-x}}\geq e^{-\frac{t}{1-t}\hskip
1.42262ptx},$

for $x$ non-negative and $t\in[0,1)$.

###### Proof.

One can consider, for a fixed $t$, the function

$f_{t}(x)=(1-t\hskip 1.42262pte^{-x})e^{-\frac{t}{1-t}\hskip 1.42262ptx}.$

Differentiating with respect to $x$ we see that the only critical point is at
$x=0$, whence the result follows. ∎

######  Lemma 1.5.

There holds

$\frac{1}{\Gamma(\omega)}\int_{0}^{\infty}\text{e}^{-a\hskip
1.42262ptx}x^{w-1}\hskip 1.42262pt\text{d}x=a^{-w},$

for any $a>0$ and $w>0$.

###### Proof.

As long as $a$ is positive, the integral is well-defined and convergent.
Recall that

$\Gamma(\omega)=\int_{0}^{\infty}e^{-x}x^{w-1}\hskip 1.42262pt\text{d}x.$

Setting $u=a\hskip 1.42262ptx$, we have $\text{d}x=\frac{\text{d}u}{a}$ and
the integral becomes

$\int_{0}^{\infty}e^{-u}u^{w-1}a^{-w}\hskip 1.42262pt\text{d}u=a^{-w}\hskip
1.42262pt\Gamma(\omega).$

The result follows. ∎

##  2 Boundedness of $\mathcal{C}^{\mu}$ on $\ell^{p}$

In the current section we shall prove two theorems regarding the boundedness
of $\mathcal{C}^{\mu}$ as an operator on $\ell^{p}$ spaces. The first concerns
the case where there are no Dirac masses at $0$ or $1$, while the latter
allows for the existence of such masses at the endpoints of $[0,1]$.

######  Theorem 2.1.

Fix $1\leq p\leq\infty$. If $\mu$ has no Dirac masses at $0$ or $1$, then the
matrix $\mathcal{C}^{\mu}$ is bounded as an operator from $\ell^{p}$ to itself
if and only if

$\mathcal{N}^{\mu}_{p}:=\int_{0}^{1}\frac{1}{(1-t)^{1-\frac{1}{p}}\hskip
1.42262ptt^{\frac{1}{p}}}\hskip 1.42262pt\text{d}\mu(t)<\infty.$

Moreover, if that is the case, the operator norm equals
$\mathcal{N}^{\mu}_{p}$ (here we understand that when $p=\infty$, the integral
becomes $\int_{0}^{1}\frac{1}{1-t}\hskip 1.42262pt\text{d}\mu(t)$).

###### Proof.

Suppose that $\int_{0}^{1}\frac{1}{(1-t)^{\frac{p-1}{p}}\hskip
1.42262ptt^{\frac{1}{p}}}\hskip 1.42262pt\text{d}\mu(t)$ is finite and there
are no Dirac masses at $0$ or $1$. We will, in the first instance, show that
$\mathcal{C}^{\mu}$ is bounded from $\ell^{p}$ to $\ell^{p}$ for $1\leq
p<\infty$ and we will calculate its norm. We will then treat the case
$p=\infty$.

We begin by defining, for a given sequence
$(a_{n})_{n=0}^{\infty}\in\ell^{p}$, the function $e_{n}:[0,1]\to\mathbb{R}$
given by

$e_{n}(t)=\sum_{m=0}^{\infty}\binom{n+m}{m}t^{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262pta_{m}.$ (2.1)

We claim that $e_{n}$ is well-defined on $[0,1]$ and that there holds, for
$t\in(0,1)$,

$\sum_{n=0}^{\infty}e_{n}^{p}\leq\frac{1}{(1-t)^{p-1}\hskip
1.42262ptt}\sum_{n=0}^{\infty}a_{n}^{p}.$ (2.2)

To check this, we first look at the case $p>1$. Here, Hölder’s inequality
gives

$e_{n}(t)\leq\left(\sum_{m=0}^{\infty}\underbrace{{\binom{n+m}{m}}^{q}\hskip
1.42262ptt^{q\hskip 1.42262ptm}\hskip 1.42262pt(1-t)^{q\hskip
1.42262ptn}}_{:=\hskip
1.42262ptB_{m}}\right)^{\frac{1}{q}}\left(\sum_{m=0}^{\infty}a_{m}^{p}\right)^{\frac{1}{p}},$
(2.3)

where $q=\frac{p}{p-1}$ is the Hölder conjugate of $p$, i.e.
$\frac{1}{q}+\frac{1}{p}=1$. To check that the first term on the right-hand
side in (2.3) is convergent, the ratio test for series gives

$\frac{B_{m+1}}{B_{m}}=\left(\frac{\binom{n+m+1}{m+1}\hskip
1.42262ptt^{m+1}\hskip 1.42262pt(1-t)^{n}}{\binom{n+m}{m}\hskip
1.42262ptt^{m}\hskip
1.42262pt(1-t)^{n}}\right)^{q}=\left(\frac{m+n+1}{m+1}\hskip
1.42262ptt\right)^{q}\to t^{q},$ (2.4)

as $m\to\infty$. Hence, for $t\in[0,1)$, we have

$\lim_{m\to\infty}\frac{B_{m+1}}{B_{m}}=t^{q}<t<1$

and therefore the series is (absolutely) convergent. Finally, for $t=1$, the
partial sums $\sum_{m=0}^{K}B_{m}=0$ and hence the series equals zero. This
shows that for all $t\in[0,1]$, $e_{n}(t)$ is well-defined.

For $p=1$, it is enough to show that, for a fixed $t\in[0,1]$ and for any
fixed $n$, the sequence

$c_{m}(n;t):=\binom{n+m}{m}\hskip 1.42262ptt^{m}(1-t)^{n}$

is in $\ell^{\infty}$. Consider a biased coin with probability of heads $t$
and probability of tails $1-t$. The probability of landing $m$ heads in
$(n+m)$ throws for this coin is precisely $c_{m}(n;t)$. Hence each term is
bounded by 1 and the result follows. This proves that $e_{n}(t)$ is well-
defined when $(a_{n})\in\ell^{1}$.

By Hölder’s inequality again, we have

$\begin{split}e_{n}^{p}(t)&\leq\left(\sum_{m=0}^{\infty}\binom{n+m}{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262ptt^{m}\hskip
1.42262pta_{m}^{p}\right)\left(\sum_{m=0}^{\infty}\binom{n+m}{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262ptt^{m}\right)^{p-1}\\\
&\leq\frac{1}{(1-t)^{p-1}}\sum_{m=0}^{\infty}\binom{n+m}{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262ptt^{m}\hskip 1.42262pta_{m}^{p}.\end{split}$
(2.5)

Summing over $n$, we get (for $0<t<1$),

$\begin{split}\sum_{n=0}^{\infty}e_{n}^{p}&\leq\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{(1-t)^{p-1}}\hskip
1.42262pt\binom{n+m}{m}(1-t)^{n}\hskip 1.42262ptt^{m}\hskip
1.42262pta_{m}^{p}\\\ &\leq\sum_{m=0}^{\infty}\frac{1}{(1-t)^{p-1}}\hskip
1.42262ptt^{m}\hskip 1.42262pta_{m}^{p}\hskip
1.42262pt\sum_{n=0}^{\infty}\binom{n+m}{m}\hskip 1.42262pt(1-t)^{n}\\\
&\leq\sum_{m=0}^{\infty}\frac{1}{(1-t)^{p-1}\hskip
1.42262ptt}a_{m}^{p},\end{split}$ (2.6)

where we have used Lemma 1.3. We now have

$\begin{split}b_{n}&=\sum_{m=0}^{\infty}\int_{0}^{1}\binom{n+m}{m}\hskip
1.42262ptt^{m}\hskip 1.42262pt(1-t)^{n}\hskip 1.42262pta_{m}\hskip
1.42262pt\text{d}\mu(t)\\\
&=\int_{0}^{1}\sum_{m=0}^{\infty}\binom{n+m}{m}\hskip 1.42262ptt^{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262pta_{m}\hskip 1.42262pt\text{d}\mu(t)\hskip
14.22636pt\text{(by the D.C.T.)}\\\ &=\int_{0}^{1}e_{n}(t)\hskip
1.42262pt\text{d}\mu(t).\end{split}$

By Minkowski’s inequality, we have

$\left(\sum_{n=0}^{\infty}b_{n}^{p}\right)^{\frac{1}{p}}\leq\int_{0}^{1}\left(\sum_{n=0}^{\infty}e_{n}^{p}\right)^{\frac{1}{p}}\hskip
1.42262pt\text{d}\mu(t)\leq\int_{0}^{1}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip
1.42262pt\left(\sum_{n=0}^{\infty}a_{n}^{p}\right)^{\frac{1}{p}}\hskip
1.42262pt\text{d}\mu(t),$ (2.7)

or equivalently

$\lVert(b_{n})\rVert_{\ell^{p}}\leq\left(\int_{0}^{1}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip
1.42262pt\text{d}\mu(t)\right)\lVert(a_{n})\rVert_{\ell^{p}}.$ (2.8)

To show that $\mathcal{N}_{p}^{\mu}$ is the optimal constant (and hence the
norm of the corresponding operator, when it is finite) we work as follows. Fix
any $\eta>0$ small. There exists a $\delta>0$ such that

$\int_{\delta}^{1-\delta}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip 1.42262pt\text{d}\mu(t)>(1-\eta)\hskip
1.42262pt\mathcal{N}_{p}^{\mu}.$ (2.9)

That such a $\delta$ exists follows from the monotone convergence theorem,
when applied for example to the functions
$f_{n}(t)=\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\mathds{1}_{[\frac{1}{n},\frac{n-1}{n}]}$. We
choose $c,\hskip 1.42262ptN,\hskip 1.42262pt\varepsilon$ (in this order) such
that

  * •

There holds $\left(1+\frac{1}{c}\right)^{-\frac{2}{p}}>1-\eta$.

  * •

There holds $N\geq\lceil\frac{c}{\delta}\rceil$, so that

$\int_{\frac{c}{n}}^{1-\frac{c}{n}}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip
1.42262pt\text{d}\mu(t)>\int_{\delta}^{1-\delta}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip 1.42262pt\text{d}\mu(t)>(1-\eta)\hskip
1.42262pt\mathcal{N}_{p}^{\mu},$

for $n\geq N$.

  * •

There holds $0<\varepsilon<\frac{1}{p}$. Moreover, denoting
$w=\frac{1}{p}+\varepsilon$, we must choose $\varepsilon$ so that the
following two conditions also hold:

$\int_{\delta}^{1-\delta}t^{-w}\hskip 1.42262pt(1-t)^{w-1}\hskip
1.42262pt\text{d}\mu(t)>(1-\eta)\hskip
1.42262pt\int_{\delta}^{1-\delta}\frac{1}{t^{\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{p-1}{p}}}\hskip 1.42262pt\text{d}\mu(t)$ (2.10)

and such that

$\sum_{n=N}^{\infty}(n+1)^{-p\hskip 1.42262ptw}>(1-\eta)\hskip
1.42262pt\sum_{n=0}^{\infty}\hskip 1.42262pt(n+1)^{-p\hskip 1.42262ptw}.$
(2.11)

The latter condition can be met for $\varepsilon$ sufficiently small, since
the $p-$series diverges for $p=1$ and converges for $p>1$.

Finally, we pick the sequence $a_{n}=(n+1)^{-w}$. Using Lemma 1.5, we have

$a_{n}=\frac{1}{\Gamma(\omega)}\hskip
1.42262pt\int_{0}^{\infty}\mathrm{e}^{-(n+1)\hskip 1.42262ptx}\hskip
1.42262ptx^{w-1}\hskip 1.42262pt\text{d}x$ (2.12)

Using this and (2.1), we can get the following estimate for $e_{n}(t)$.

$\begin{split}e_{n}(t)&=\frac{1}{\Gamma(\omega)}\hskip
1.42262pt\int_{0}^{\infty}\mathrm{e}^{-x}\hskip 1.42262ptx^{w-1}\hskip
1.42262pt\sum_{m=0}^{\infty}\binom{n+m}{m}\hskip 1.42262ptt^{m}\hskip
1.42262pt(1-t)^{n}\hskip 1.42262pt\mathrm{e}^{-m\hskip 1.42262ptx}\text{d}x\\\
&=\frac{(1-t)^{n}}{\Gamma(\omega)}\hskip
1.42262pt\int_{0}^{\infty}\mathrm{e}^{-x}\hskip 1.42262ptx^{w-1}\hskip
1.42262pt(1-t\hskip 1.42262pt\mathrm{e}^{-x})^{-n-1}\hskip
1.42262pt\text{d}x\\\ &=\frac{1}{1-t}\hskip
1.42262pt\frac{1}{\Gamma(\omega)}\hskip
1.42262pt\int_{0}^{\infty}\mathrm{e}^{-x}\hskip 1.42262ptx^{w-1}\hskip
1.42262pt\left(\frac{1-t}{1-t\hskip
1.42262pt\mathrm{e}^{-x}}\right)^{n+1}\hskip 1.42262pt\text{d}x\\\
&\geq\frac{1}{1-t}\hskip 1.42262pt\frac{1}{\Gamma(\omega)}\hskip
1.42262pt\int_{0}^{\infty}\mathrm{e}^{-x}\hskip 1.42262ptx^{w-1}\hskip
1.42262pt\hskip 1.42262pt\mathrm{e}^{-(n+1)\frac{t}{1-t}\hskip
1.42262ptx}\hskip 1.42262pt\text{d}x\\\ &=\frac{1}{1-t}\left(\frac{1+n\hskip
1.42262ptt}{1-t}\right)^{-w},\end{split}$ (2.13)

where the penultimate step was obtained by Lemma 1.4 and the last step from
Lemma 1.5. When $\frac{c}{n}\leq t\leq 1-\frac{c}{n}$, we therefore have

$\begin{split}e_{n}(t)&\geq\hskip 1.42262pt(1-t)^{w-1}\hskip
1.42262ptt^{-w}\hskip 1.42262pt\left(1+\frac{1}{c}\right)^{-w}\hskip
1.42262pt(n+1)^{-w}\\\ &\geq(1-t)^{w-1}\hskip 1.42262ptt^{-w}\hskip
1.42262pt(1-\eta)\hskip 1.42262pta_{n}.\end{split}$ (2.14)

Using (2.14), we have

$\begin{split}b_{n}=\int_{0}^{1}e_{n}(t)\hskip
1.42262pt\text{d}\mu(t)&\geq\int_{\frac{c}{n}}^{1-\frac{c}{n}}e_{n}(t)\hskip
1.42262pt\text{d}\mu(t)\\\ &\geq(1-\eta)\hskip 1.42262pta_{n}\hskip
1.42262pt\int_{\frac{c}{n}}^{1-\frac{c}{n}}t^{-w}\hskip
1.42262pt(1-t)^{w-1}\hskip 1.42262pt\text{d}\mu(t)\\\ &\geq(1-\eta)\hskip
1.42262pta_{n}\hskip 1.42262pt\int_{\delta}^{1-\delta}t^{-w}\hskip
1.42262pt(1-t)^{w-1}\hskip 1.42262pt\text{d}\mu(t)\\\ &\geq(1-\eta)^{2}\hskip
1.42262pta_{n}\hskip 1.42262pt\int_{\delta}^{1-\delta}t^{-\frac{1}{p}}\hskip
1.42262pt(1-t)^{\frac{1}{p}-1}\hskip 1.42262pt\text{d}\mu(t)\\\
&\geq(1-\eta)^{3}\hskip 1.42262pta_{n}\hskip
1.42262pt\mathcal{N}_{p}^{\mu},\hskip 8.53581pt\text{for}\hskip 2.84526ptn\geq
N,\end{split}$ (2.15)

hence

$\sum_{n=0}^{\infty}b_{n}^{p}\geq\sum_{n=N}^{\infty}b_{n}^{p}\geq\sum_{n=N}^{\infty}(1-\eta)^{3\hskip
1.42262ptp}\hskip 1.42262pt\left(\mathcal{N}_{p}^{\mu}\right)^{p}\hskip
1.42262pta_{n}^{p}\geq(1-\eta)^{3\hskip 1.42262ptp+1}\hskip
1.42262pt\left(\mathcal{N}_{p}^{\mu}\right)^{p}\hskip
1.42262pt\sum_{n=0}^{\infty}a_{n}^{p}.$ (2.16)

This establishes the ”if” direction. For the ”only if” direction, assume that
$\mathcal{C}^{\mu}$ is bounded as an operator on $\ell^{p}$ for $1\leq
p<\infty$. We claim that the integral $\mathcal{N}_{p}^{\mu}$ is finite. We
shall make use of the same sequences as the ones just used. With the preceding
notation, for an arbitrarily small but fixed $\eta>0$, we have for the
sequences described above:

$b_{n}=\int_{0}^{1}e_{n}(t)\hskip 1.42262pt\text{d}\mu(t)\geq(1-\eta)\hskip
1.42262pta_{n}\hskip 1.42262pt\int_{0}^{1}(1-t)^{w-1}\hskip
1.42262ptt^{-w}\hskip 1.42262pt\text{d}\mu(t)$ (2.17)

and hence

$\left(\sum b_{n}^{p}\right)^{\frac{1}{p}}\geq\begin{bmatrix}(1-\eta)\hskip
1.42262pt\int_{0}^{1}(1-t)^{w-1}\hskip 1.42262ptt^{-w}\hskip
1.42262pt\text{d}\mu(t)\end{bmatrix}\hskip 1.42262pt\left(\sum
a_{n}^{p}\right)^{\frac{1}{p}}.$ (2.18)

From here we observe, given the boundedness of $\mathcal{C}^{\mu}$, that
$\int_{0}^{1}(1-t)^{w-1}\hskip 1.42262ptt^{-w}\hskip 1.42262pt\text{d}\mu(t)$
is finite and uniformly bounded above by the operator norm as $\eta\to 0$ .
Moreover, it is not hard to establish that, as $\eta\to 0$, the way
$\varepsilon$ is chosen means that $\varepsilon\to 0$ as well and hence the
functions

$(1-\eta)(1-t)^{w-1}\hskip
1.42262ptt^{-w}\to\frac{1}{(1-t)^{\frac{p-1}{p}}\hskip
1.42262ptt^{\frac{1}{p}}}$

pointwise as $\eta\to 0$. Fix any $z>0$ small. An application of the dominated
convergence theorem on the interval $[z,1-z]$ (on this fixed interval, the
functions $(1-\eta)(1-t)^{w-1}\hskip 1.42262ptt^{-w}$ are uniformly bounded
above) now implies that the integral

$\int_{z}^{1-z}\frac{1}{(1-t)^{\frac{p-1}{p}}\hskip
1.42262ptt^{\frac{1}{p}}}\hskip 1.42262pt\text{d}\mu(t)$

is finite and bounded above by the operator norm. As $z\to 0$, an application
of the monotone convergence theorem gives the desired result.

We finally examine the case $p=\infty$. Assume that
$\int_{0}^{1}\frac{1}{1-t}\hskip 1.42262pt\text{d}\mu(t)$ is finite and
consider a sequence $(a_{n})\in\ell^{\infty}$. We have

$\begin{split}b_{n}&=\sum_{m=0}^{\infty}\binom{n+m}{m}\int_{0}^{1}(1-t)^{n}\hskip
1.42262ptt^{m}\hskip 1.42262pta_{m}\hskip 1.42262pt\text{d}\mu(t)\\\
&\leq\begin{bmatrix}\int_{0}^{1}\sum_{m=0}^{\infty}\binom{n+m}{m}(1-t)^{n}\hskip
1.42262ptt^{m}\hskip 1.42262pt\text{d}\mu(t)\end{bmatrix}\hskip
1.42262pt\lVert(a_{n})\rVert_{\ell^{\infty}}\\\
&=\begin{bmatrix}\int_{0}^{1}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)\end{bmatrix}\lVert(a_{n})\rVert_{\ell^{\infty}}.\end{split}$
(2.19)

Taking the supremum over $n$, we have

$\lVert(b_{n})\rVert_{\ell^{\infty}}\leq\int_{0}^{1}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)\hskip 1.42262pt\lVert(a_{n})\rVert_{\ell^{\infty}}.$
(2.20)

By taking $(a_{n})$ to be a constant sequence different than the zero
sequence, we see that the constant is optimal. Conversely, if the operator is
bounded on $\ell^{\infty}$, picking any constant non-zero sequence $(a_{n})$,
we see that the integral $\int_{0}^{1}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)$ is finite and the result follows.

∎

###  2.1 The case where $\mu$ is allowed to have Dirac masses at $0$ and/or
$1$

To allow for the possibility of Dirac masses at $0$ or $1$, we need to be a
bit more careful with semantics and notation.

####  2.1.1 Preliminary measure-theoretic notes

Given a measure $\mu$ on $[0,1]$ and a function $f$ defined on $[0,1]$ which
is measurable with respect to $\mu$, we shall denote the integral
$\int_{0}^{1}f(t)\hskip 1.42262pt\text{d}\mu(t)$ by $\mu_{[0,1]}f$. In
general, for any measurable subset $\mathcal{A}$ of $[0,1]$, we will use
$\mu_{\mathcal{A}}(f)$ to denote the integral
$\mu_{[0,1]}(f\cdot\mathds{1}_{\mathcal{A}})$, which is well-defined as
$f\cdot\mathds{1}_{\mathcal{A}}$ is $\mu-$measurable.

For a function $g$ defined only on a subset $\mathcal{P}$ of $[0,1]$, by a
slight abuse of notation, we shall also use $\mu_{\mathcal{P}}(g)$ to denote
the integral

$\int_{\mathcal{P}}g\hskip 1.42262pt\text{d}{\mu}_{\mathcal{P}}(t),$

where $\mu_{\mathcal{P}}$ is the restriction of the measure on $\mathcal{P}$
in the obvious way.

For a given measure $\mu$ and for any function $f$ on $[0,1]$, the following
decomposition holds:

$\mu_{[0,1]}(f)=\mu_{(0,1)}(f)+c_{0}\hskip 1.42262ptf(0)+c_{1}\hskip
1.42262ptf(1),$

where $c_{1}:=\mu\left(\begin{Bmatrix}0\end{Bmatrix}\right)$ and
$c_{2}:=\mu\left(\begin{Bmatrix}1\end{Bmatrix}\right)$ are the Dirac masses of
the measure at $0$ and $1$ respectively. This, in turn, provides the following
decomposition for the matrix $\mathcal{C}^{\mu}$:

$\mathcal{C}^{\mu}_{n,k}=\tilde{\mathcal{C}}_{n,k}^{\mu}+\hat{\mathcal{C}}_{n,k}^{\mu},$
(2.21)

where $\tilde{\mathcal{C}}_{n,k}^{\mu}:=\mu_{(0,1)}\left(g_{n,k}(t)\right)$
and

$\hat{\mathcal{C}}^{\mu}:=\begin{pmatrix}c_{0}+c_{1}&c_{1}&c_{1}&c_{1}&\dots\\\
c_{0}&0&0&0&\dots\\\ c_{0}&0&0&0&\ddots\\\ c_{0}&0&0&0&\ddots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix}.$

####  2.1.2 Main Result

In this section we will show the following:

######  Theorem 2.2.

Assume that $\mu$ contains Dirac masses $c_{0}$ at $0$ and $c_{1}$ at $1$,
with $\lvert c_{0}\rvert+\lvert c_{1}\rvert>0$. For $1\leq p<\infty$, the only
case when $\mathcal{C}^{\mu}$ can be a bounded operator on $\ell^{p}$ is when
$c_{1}\neq 0=c_{0}$ and $p=1$. In that case, the operator is bounded if and
only if

$\int_{(0,1]}\frac{1}{t}\hskip
1.42262pt\text{d}\mu(t)=\int_{(0,1)}\frac{1}{t}\hskip
1.42262pt\text{d}\mu(t)+c_{1}$

is finite. Moreover, this expression, when finite, equals the norm of the
operator. Finally, when $p=\infty$, the operator is bounded if and only if
$c_{0}\neq 0=c_{1}$ and the integral

$\int_{[0,1)}\frac{1}{1-t}\hskip 1.42262pt\text{d}\mu(t)$

is finite. Moreover, the above, when finite, equals the norm of the operator
on $\ell^{\infty}$.

###### Proof.

We begin by establishing the following two lemmata:

######  Lemma 2.3.

If the measure $\mu$ contains a Dirac mass at $0$, then the matrix is
unbounded on $\ell^{p}$ for $1\leq p<\infty$.

This holds for the following reason. The matrix $\hat{\mathcal{C}}^{\mu}$ for
$c_{0}\neq 0$ does not even map $\ell^{p}$ to itself, since

$\begin{pmatrix}c_{0}+c_{1}&c_{1}&c_{1}&c_{1}&\dots\\\ c_{0}&0&0&0&\dots\\\
c_{0}&0&0&0&\ddots\\\ c_{0}&0&0&0&\ddots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix}\begin{pmatrix}a_{0}\\\
a_{1}\\\ a_{2}\\\ a_{3}\\\
\vdots\end{pmatrix}=\begin{pmatrix}(c_{0}+c_{1})\hskip
1.42262pta_{0}+c_{1}\hskip 1.42262pt(a_{1}+a_{2}+\dots)\\\ c_{0}\hskip
1.42262pta_{0}\\\ c_{0}\hskip 1.42262pta_{0}\\\ c_{0}\hskip 1.42262pta_{0}\\\
\vdots\end{pmatrix}$ (2.22)

and clearly, as the sequence on the right-hand side eventually consists of the
same non-zero (as long as $a_{0}\neq 0$) constant, it cannot be in $\ell^{p}$.
Finally, since the matrix $\tilde{\mathcal{C}}^{\mu}$ comprises only positive
entries, the sum $\tilde{\mathcal{C}}^{\mu}+\hat{\mathcal{C}}^{\mu}$ also
fails to map $\ell^{p}$ to $\ell^{p}$.

######  Lemma 2.4.

If the measure $\mu$ contains a Dirac mass at $1$ and no Dirac mass at $0$,
then the matrix $\mathcal{C}^{\mu}$ is unbounded as an operator on $\ell^{p}$
for $1<p<\infty$.

To establish this, notice that the first term of the sequence on the right-
hand side of (2.22) is $c_{1}(a_{0}+a_{1}+\dots)$. In general, the $\ell^{p}$
spaces are nested, in the sense that $\ell^{p}\subset\ell^{q}$ for $p<q$ and
hence for each $p>1$ there exists a sequence in $\ell^{p}$ which is not in
$\ell^{1}$. Since $\tilde{\mathcal{C}}^{\mu}$ consists of only positive
entries, it again follows that $\mathcal{C}^{\mu}$ does not map $\ell^{p}$ to
itself when $c_{1}\neq 0=c_{0}$ and $p>1$. The only case left to examine, when
the measure contains a Dirac mass, is when $c_{1}\neq 0=c_{0}$ and
$\tilde{\mathcal{C}}^{\mu}$ is viewed as an operator on $\ell^{1}$. In this
case we claim that the operator is bounded if and only if

$\mu_{(0,1]}\left(\frac{1}{t}\right)=\int_{(0,1)}\frac{1}{t}\hskip
1.42262pt\text{d}\mu(t)+c_{1}$

is finite and that, when finite, the expression above equals the operator
norm.

Indeed, assume first that the expression above is finite. Proceeding as in
(2.6), we get, for $0<t<1$:

$\sum_{n=0}^{\infty}e_{n}(t)\leq\frac{1}{t}\sum_{n=0}^{\infty}a_{n}(t)$ (2.23)

for all $n$. We now notice that

$b_{0}=\int_{(0,1)}e_{0}(t)\hskip
1.42262pt\text{d}\mu(t)+c_{1}\lVert(a_{n})\rVert_{\ell^{1}}$ (2.24)

and

$b_{n}=\int_{(0,1)}e_{n}(t)\hskip 1.42262pt\text{d}\mu(t)$ (2.25)

for $n\geq 1$. Adding (2.24) and (2.25) together, we arrive at

$\lVert(b_{n})\rVert_{\ell^{1}}\leq\left(\int_{(0,1)}\frac{1}{t}\hskip
1.42262pt\text{d}\mu(t)+c_{1}\right)\hskip
1.42262pt\lVert(a_{n})\rVert_{\ell^{1}}$ (2.26)

To show that this is the optimal constant, we again use the sequences
described in the previous section. With the same notation as before, with
$a_{n}=(n+1)^{-w}$, we have

$\begin{split}\sum_{n=0}^{\infty}b_{n}&=b_{0}+\sum_{n=1}^{\infty}b_{n}=\int_{(0,1)}e_{0}(t)\hskip
1.42262pt\text{d}\mu(t)+c_{1}\lVert(a_{n})\rVert_{\ell^{1}}+\sum_{n=1}^{\infty}\int_{(0,1)}e_{n}(t)\hskip
1.42262pt\text{d}\mu(t)\\\
&\geq\left((1-\eta)^{4}\int_{(0,1)}\frac{1}{t}\hskip
1.42262pt\text{d}\mu(t)+c_{1}\right)\hskip
1.42262pt\lVert(a_{n})\rVert_{\ell^{1}}.\end{split}$ (2.27)

Letting $\eta\to 0$, we obtain the desired result.

Finally, let us suppose that the operator $\mathcal{C}^{\mu}$ is bounded on
$\ell^{1}$. The same argument as in the previous case shows the finiteness of
$\int_{(0,1]}\frac{1}{t}\hskip 1.42262pt\text{d}\mu(t)$.

We finally examine the case $p=\infty$. The proof is similar in spirit to the
previous results, although a bit simpler. Indeed, if $c_{1}\neq 0$ then it is
clear from (2.21) that $\mathcal{C}^{\mu}$ does not map $\ell^{\infty}$ to
itself.

Assume that $\int_{[0,1)}\frac{1}{1-t}\hskip 1.42262pt\text{d}\mu(t)$ is
finite and consider a sequence $(a_{n})\in\ell^{\infty}$. We have

$\begin{split}b_{n}&=\sum_{m=0}^{\infty}\binom{n+m}{m}\int_{(0,1)}(1-t)^{n}\hskip
1.42262ptt^{m}\hskip 1.42262pta_{m}\hskip 1.42262pt\text{d}\mu(t)+c_{0}\hskip
1.42262pta_{n}\\\
&\leq\begin{bmatrix}\int_{(0,1)}\sum_{m=0}^{\infty}\binom{n+m}{m}(1-t)^{n}\hskip
1.42262ptt^{m}\hskip 1.42262pt\text{d}\mu(t)\end{bmatrix}\hskip
1.42262pt\lVert(a_{n})\rVert_{\ell^{\infty}}+c_{0}a_{n}\\\
&=\begin{bmatrix}\int_{(0,1)}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)\end{bmatrix}\lVert(a_{n})\rVert_{\ell^{\infty}}+c_{0}a_{n}.\end{split}$
(2.28)

Taking the supremum of (2.28) over $n$, we have

$\lVert(b_{n})\rVert_{\ell^{\infty}}\leq\int_{[0,1)}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)\hskip 1.42262pt\lVert(a_{n})\rVert_{\ell^{\infty}}.$
(2.29)

By taking $(a_{n})$ to be a constant sequence different than the zero
sequence, we see that the constant is optimal.

Conversely, assuming that the operator is bounded from $\ell^{\infty}$ to
itself, again picking a non-zero constant sequence, we see that

$\begin{split}b_{n}&=\sum_{m=0}^{\infty}\binom{n+m}{m}\int_{(0,1)}(1-t)^{n}\hskip
1.42262ptt^{m}\hskip 1.42262pta_{m}\hskip 1.42262pt\text{d}\mu(t)+c_{0}\hskip
1.42262pta_{n}\\\ &=\begin{bmatrix}\int_{[0,1)}\frac{1}{1-t}\hskip
1.42262pt\text{d}\mu(t)\end{bmatrix}\hskip 1.42262pta_{n}.\end{split}$ (2.30)

From this, the finiteness of the operator norm implies the finiteness of

$\int_{[0,1)}\frac{1}{1-t}\hskip 1.42262pt\text{d}\mu(t)$

and the result follows.

∎

Acknowledgements

I would like to thank Aristomenis Siskakis for introducing me to this problem
and for various beneficial suggestions regarding the format of the paper. I
would also like to thank him, as well as Vasilis Daskalogiannis, Petros
Galanopoulos and Giorgos Stylogiannis for several fruitful discussions on the
topic. Finally, I thank Vasilis for useful comments and suggestions on the
introduction.

## References

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