# Complementarity between quantum entanglement, geometrical and dynamical
appearances in $N$ spin-$1/2$ system under all-range Ising model

Jamal Elfakir<EMAIL_ADDRESS>LPHE-Modeling and Simulation, Faculty
of Sciences, Mohammed V University in Rabat, Rabat, Morocco.  Brahim Amghar
<EMAIL_ADDRESS>LPHE-Modeling and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Centre of Physics and
Mathematics, CPM, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  Abdallah Slaoui<EMAIL_ADDRESS>LPHE-Modeling and
Simulation, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Mohammed Daoud
<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences, University
Ibn Tofail, Kenitra, Morocco.

###### Abstract

With the growth of geometric science, including the methods of exploring the
world of information by means of modern geometry, there has always been a
mysterious and fascinating ambiguous link between geometric, topological and
dynamical characteristics with quantum entanglement. Since geometry studies
the interrelations between elements such as distance and curvature, it
provides the information sciences with powerful structures that yield
practically useful and understandable descriptions of integrable quantum
systems. We explore here these structures in a physical system of $N$
interaction spin-$1/2$ under all-range Ising model. By performing the system
dynamics, we determine the Fubini-Study metric defining the relevant quantum
state space. Applying Gaussian curvature within the scope of the Gauss-Bonnet
theorem, we proved that the dynamics happens on a closed two-dimensional
manifold having both a dumbbell-shape structure and a spherical topology. The
geometric and topological phases appearing during the system evolution
processes are sufficiently discussed. Subsequently, we resolve the quantum
brachistochrone problem by achieving the time-optimal evolution. By
restricting the whole system to a two spin-$1/2$ system, we investigate the
relevant entanglement from two viewpoints; The first is of geometric nature
and explores how the entanglement level affects derived geometric structures
such as the Fubini-Study metric, the Gaussian curvature, and the geometric
phase. The second is of dynamic nature and addresses the entanglement effect
on the evolution speed and the related Fubini-Study distance. Further,
depending on the degree of entanglement, we resolve the quantum
brachistochrone problem.

Keywords: Quantum state space, Fubini-Study metric, Gaussian curvature,
Geometric phase, Quantum brachistochrone issue, Quantum entanglement.

##  I Introduction

Over the past few years, there has been growing excitement about the
application of geometric ideas to quantum physics. It is argued that the
geometrization of quantum theory provides a significant framework able to
describe, to a large extent, the physical characteristics of solvable quantum
systems [1, 2, 3, 4]. This geometrical approach has introduced the concept of
the quantum phase space, endowed naturally with the Kähler manifold structure,
on which the dynamics of quantum systems is well established [5, 6, 7, 8].
Lately, numerous studies have shown the relevance of geometric structures of
the physical state space for exploring the physical properties of quantum
systems. Indeed, It has been demonstrated that the Fubini-Study distance
traveled by a quantum system during evolution along a given curve in relevant
projective Hilbert space is related to the integral of the energy uncertainty,
which in turn is proportional to the evolution speed [9]. The quantum speed
limit time, which defines the fundamental limit on the rate of evolution of
quantum systems, is also determined by means of Bures length between mixed
quantum states [10]. Additionally, the geometric methods simplify considerably
the resolution of the quantum brachistochrone problem, which is linked to
determining the Hamiltonian that generates the time-optimal evolution between
two states [11, 12, 13]. The efficient quantum circuits in quantum computation
with $n$ qutrits are investigated by use of Riemannian geometry. Indeed, it
has been proven that the optimal quantum circuits are basically equivalent to
the shortest path between two points in a specific curved geometry of
SU($3^{n}$) [14], analogous to the qubit case wherein the geodesic in
SU($2^{n}$) is involved [15]. For other further dynamic properties explored on
the basis of geometric approaches, we recommend that readers look at the
papers [16, 17, 18, 19].

Currently, the geometric quantum mechanics, which forms the theoretical
framework of the geometric formulation of quantum theory, is the bedrock of
information geometric science, in which the quantum phenomena are handled
geometrically in the space of quantum states. One can cite, for instance, the
quantum entanglement being an intriguing physical resource in the protocols of
quantum information theory [20, 21, 22, 23, 24]. Importantly, entanglement is
shown to be closely related to the Mannoury-Fubini-Study distance separating
the entangled state and the nearest disentangled state [25]. Moreover, the
euclidean distance of an entangled state to the disentangled states is equal
to the maximal violation of a generalized Bell inequality with the tangent
functional as entanglement witness [26]. The connection between quantum
entanglement and the state space geometry has been thoroughly explored for a
spin-s system with long-range Ising interaction [27]. Further to that, the
geometrical description of entanglement is also explored within the backdrop
of the Hopf fibration, which is a topological map compactifying the related
quantum state space to an another lower-dimensional space referred to as the
Hopf bundle [28, 29, 30]. For additional findings highlighting the interplay
between quantum entanglement and geometrical characteristics, see, e.g.,
Refs.[31, 32, 33, 34].

Another significant concept that has received much attention in quantum
physics is the geometric phase, a remarkable geometric characteristic in
quantum evolution processes [35, 36, 37, 38]. It can be viewed as the holonomy
acquired by the state vector after a parallel transport along the evolution
trajectory [39, 40]. The geometric phase is now intimately related to other
geometrical features that define the quantum state spaces. In effect, it has
been proven that the geometric phase can be expressed as the line integral of
Berry-Simon connection along the Fubini-Study distance separating the two
quantum states over the corresponding projective Hilbert space [7, 41]. On the
practical side, several recent investigations have shown the most important
role of the geometric phase in the advancement of quantum information science.
Indeed, it is a valuable feature for generating quantum logic gates that are
critical to quantum computation [42, 43, 44]. Furthermore, the conditional
phase gate has been experimentally demonstrated for nuclear magnetic resonance
[45] and trapped ions [46]. The interplay between quantum entanglement and
topological and geometric phases is also extensively studied in the two-qudit
systems [47, 48, 49]. Other geometric phase applications have been realized in
Refs.[50, 51, 52, 53].

The main purpose of this work is to highlight the geometrical and dynamical
characteristics of a many-body system, represented here by $N$ spin-$1/2$
system under all-range Ising model, and their connection to quantum
entanglement. It is noteworthy that the ideas explored in this paper were
primarily inspired by the findings obtained by Krynytskyi and Kuzmak in
Ref.[27]. As a matter of fact, by performing the system dynamics, we derive
the Fubini-Study metric identifying the associated quantum state space.
Moreover, examining the Gaussian curvature (G-curvature) in the framework of
the Gauss-Bonnet theorem, we determine the topology and the structure of this
space. Afterward, we explore the acquired geometric and topological phases and
tackle the quantum brachistochrone issue. Finally, we give a detailed
explanation of the geometrical and dynamical characteristics of two
interacting spin-$1/2$ under the Ising model in connection to quantum
entanglement.

The rest of this paper is structured as follows. In Sec.II, by carrying out
the dynamics of $N$ interacting spin-$1/2$ under all-range Ising model, we
give the Fubini-Study metric and identify the associated quantum state space.
Moreover, by investigating the G-curvature within the scope of the Gauss-
Bonnet theorem, we uncover the topology and the structure of this space. The
geometric and topological phases emerging from the system evolution processes,
over the resulting state space, are thoroughly discussed in Sec.III. The
quantum brachistochrone problem is also addressed based on the evolution
velocity as well as the related Fubini-Study distance, in Sec.IV. In Sec.V, we
study the entanglement between two interacting spin-1/2 under the Ising model
from two different appearances: the first is of a geometric nature and
investigates how the entanglement degree impacts derived geometric features
such as the Fubini-Study metric, the G-curvature, and the geometric phase. The
second is of a dynamic type and addresses the entanglement effect on evolution
speed and the corresponding Fubini-Study distance. Further to this, we resolve
the quantum brachistochrone problem based on quantum entanglement. We supply
concluding remarks in Sec.VI.

##  II Unitary evolution and the quantum state manifold of $N$ spin-$1/2$
system

###  II.1 Physical model and unitary quantum evolution

To start, the considered system is composed of $N$ qubits represented by $N$
interacting spin-$1/2$ under all-range Ising model described by the following
Hamiltonian

$\mathrm{H}=\mathtt{J}\left(\sum_{i=1}^{N}S^{z}_{i}\right)^{2},$ (1)

with $\mathtt{J}$ is the coupling constant characterizing the interaction and
$\mathtt{S}_{k}^{z}$ denotes the $z$-component of the spin operator
$\textbf{S}_{i}=(\mathtt{S}_{i}^{x},\mathtt{S}_{i}^{y},\mathtt{S}_{i}^{z})^{T}$
associated with $i$th spin-$1/2$ (i.e., $i$th qubit) which fulfills the
eigenvalues equation

$\mathtt{S}_{i}^{z}\left|{{\mathsf{m}_{i}}}\right\rangle=\hbar{\mathsf{m}_{i}}\left|{{\mathsf{m}_{i}}}\right\rangle,$
(2)

where $\mathtt{S}_{i}^{\alpha}=\frac{\hbar}{2}\sigma_{i}^{\alpha}$ and
$\sigma_{i}^{\alpha}$, $(\alpha=x,y,z)$ are the Pauli matrices,
$\mathsf{m}_{i}=\pm\hbar/2$ represent the possible values due to the
projection of the $ith$ spin over the $z$-axis, and
$\left|{{\mathsf{m}_{i}}}\right\rangle$ denote the associated eigenstates. It
is worth noting that the components of spin-$1/2$ operators
$\mathtt{S}_{i}^{x},\,\mathtt{S}_{i}^{y},$ and $\mathtt{S}_{i}^{z}$ satisfy
the algebraic structure of the Lie group SU(2):

$\left[{\mathtt{S}_{i}^{\alpha},\mathtt{S}_{j}^{\beta}}\right]=i{\delta_{ij}}\sum\limits_{\gamma=x,y,z}\epsilon^{\alpha\beta\gamma}{\mathtt{S}_{k}^{\gamma}},$
(3)

where $\delta_{ij}$ and $\epsilon^{ijk}$ designate the Kronecker and Levi-
Civita symbols, respectively. It is straightforward to see that for an even
number of spins, the above Hamiltonian has $(N/2+1)$ eigenvalues, whereas for
an odd number, it has $(N+1)/2$ eigenvalues. Explicitly, the eigenvalues and
related eigenstates are provided as follows

$\begin{array}[]{cc}\frac{\mathtt{J}\hbar^{2}}{4}N^{2}&|\uparrow\uparrow\uparrow\ldots\uparrow\uparrow\rangle,|\downarrow\downarrow\downarrow\ldots\downarrow\downarrow\rangle;\\\\[12.045pt]
\frac{\mathtt{J}\hbar^{2}}{4}(N-2)^{2}&|\downarrow\uparrow\uparrow\ldots\uparrow\uparrow\rangle,|\uparrow\downarrow\uparrow\ldots\uparrow\uparrow\rangle,\ldots,|\uparrow\uparrow\uparrow\ldots\uparrow\downarrow\rangle,\\\\[6.02249pt]
&|\uparrow\downarrow\downarrow\ldots\downarrow\downarrow\rangle,|\downarrow\uparrow\downarrow\ldots\downarrow\downarrow\rangle,\ldots,|\downarrow\downarrow\downarrow\ldots\downarrow\uparrow\rangle;\\\\[12.045pt]
\frac{\mathtt{J}\hbar^{2}}{4}(N-4)^{2}&|\downarrow\downarrow\uparrow\ldots\uparrow\uparrow\rangle,|\downarrow\uparrow\downarrow\ldots\uparrow\uparrow\rangle,\ldots,|\uparrow\uparrow\uparrow\ldots\downarrow\downarrow\rangle,\\\\[6.02249pt]
&|\uparrow\uparrow\downarrow\ldots\downarrow\downarrow\rangle,|\uparrow\downarrow\uparrow\ldots\downarrow\downarrow\rangle,\ldots,|\downarrow\downarrow\downarrow\ldots\uparrow\uparrow\rangle;\\\\[12.045pt]
\ldots&\ldots\end{array}$ (4)

Taking into account all possible combinations of spin states (i.e.,
$|\uparrow\rangle$ and $|\downarrow\rangle$), one finds that each eigenvalue
${{J\hbar^{2}{{(N-2p)}^{2}}}\mathord{\left/{\vphantom{{J{{(N-2p)}^{2}}}4}}\right.\kern-1.2pt}4}$
matches $2\mathrm{C}_{N}^{p}$ eigenstates, with $\mathrm{C}$ stands for the
binomial coefficient, while the index $p=0,...,N/2$ for $N$ even (particle
number) and $p=0,...,(N-1)/{2}$ for $N$ odd. We presume that the evolution of
the $N$ spin-$1/2$ system starts with the initial state

$|\Psi_{i}\rangle=|+\rangle^{\otimes N},$ (5)

where

$|+\rangle=\cos\frac{\theta}{2}|\uparrow\rangle+\sin\frac{\theta}{2}e^{i\varphi}|\downarrow\rangle,$

corresponds to the eigenstate of the spin-$1/2$ projection operator on the
direction denoted by the unit vector
$\textbf{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$, where
$\theta$ and $\varphi$ designate the polar and azimuthal angles, respectively.
In this respect, the initial state (5) of the system can be rewritten, using
the binomial theorem, as

$\left|\Psi_{i}\right\rangle=\sum_{p=0}^{N}\cos^{N-p}\frac{\theta}{2}\sin^{p}\frac{\theta}{2}e^{ip\varphi}\sum_{i_{1}<i_{2}<\ldots<i_{p}=1}^{N}\sigma_{i_{1}}^{x}\sigma_{i_{2}}^{x}\ldots\sigma_{i_{p}}^{x}|\uparrow\rangle^{\otimes
N},$ (6)

where we set $\hbar=1$, indicating that the energy is measured in the
frequency units. To investigate the geometrical, topological, and dynamical
features of the system under consideration in the remainder of this paper, we
need to evolve the $N$ spin-1/2 system, maintained initially in the starting
state (6), by applying the time evolution propagator
$\mathcal{P}(t)=e^{-i\mathrm{H}t}$. In this perspective, the evolving state of
the system is obtained as

$\left|\Psi(t)\right\rangle=\sum_{p=0}^{N}\cos^{N-p}\left(\frac{\theta}{2}\right)\sin^{p}\left(\frac{\theta}{2}\right)\exp\left\\{-i\left[\frac{\xi(t)}{4}(N-2p)^{2}-p\varphi\right]\right\\}\sum_{i_{1}<i_{2}<\ldots<i_{p}=1}^{\mathcal{N}}\sigma_{i_{1}}^{x}\sigma_{i_{2}}^{x}\ldots\sigma_{i_{p}}^{x}|\uparrow\uparrow\ldots\uparrow\rangle,$
(7)

with $\xi(t)=\mathtt{J}t$. In this way, we come at demonstrating the evolved
state of a collection of $N$ qubits (i.e., $N$ spin$-1/2$) that is explicitly
dependent on three parameters, namely the spherical angles $(\theta,\varphi)$
and the time $t$. It is also intriguing to note that the state (7) fulfills
the periodic requirement
$\left|\Psi(\xi)\right\rangle=\left|\Psi(\xi+2\pi)\right\rangle$ along the
temporal parameter, implying that $\xi\in[0,2\pi]$. Accordingly, one can
predict that the dynamics of the system occurs on a closed three-dimensional
manifold.

###  II.2 Geometry and topology of the resulting state manifold

After evolving the $N$ spin-$1/2$ system by means of temps-evolution operator
and determining the evolved state (7), we will now discover the geometry and
topology associated with the relevant quantum state space, which includes all
possible states that the system may reach during evolution. For this task, we
must compute the Fubini-Study metric, which defines the infinitesimal distance
$d\mathtt{S}$ between two adjoining pure quantum states
$|\Psi(\zeta^{\mu})\rangle$ and $|\Psi(\zeta^{\mu}+d\zeta^{\mu})\rangle$,
having the following form [54, 55]

$d\mathtt{S}^{2}=\mathrm{g}_{\mu\nu}d\zeta^{\mu}d\zeta^{\nu},$ (8)

where $\zeta^{\mu}$ are the physical parameters $\theta,\varphi$ and $\xi$
(i.e., representing the dynamical degrees of freedom of the considered system)
specifying the evolved state (7) and $\mathrm{g}_{\mu\nu}$ denote the
components of this metric tensor given by

$\mathrm{g}_{\mu\nu}=\mathrm{Re}\left(\left\langle\Psi_{\mu}|\Psi_{\nu}\right\rangle-\left\langle\Psi_{\mu}|\Psi\right\rangle\left\langle\Psi|\Psi_{\nu}\right\rangle\right),$
(9)

with
$\left|\Psi_{\mu,\nu}\right\rangle=\frac{\partial}{\partial\zeta^{\mu,\nu}}|\Psi\rangle$.
Using the definition (8), one finds, after a simple numerical calculation, the
explicit version of the Fubini-Study metric as follows

$\displaystyle d\mathtt{S}^{2}=$ $\displaystyle
d\mathtt{S}_{i}^{2}+\frac{1}{4}N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]d\xi^{2}$
$\displaystyle+\frac{1}{4}N(N-1)\cos\theta\sin^{2}\theta d\varphi d\xi,$ (10)

with

$d\mathtt{S}_{i}^{2}=\frac{N}{4}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (11)

corresponds to the squared line element defining the sphere of possible
initial states of the system. This can be well remarked when we take $\xi=0$
(i.e., no evolution) in the metric (II.2), we discover indeed that the state
space is reduced to a sphere of radius $2\sqrt{N}$. Furthermore, the space of
$N$ spin-$1/2$ states resulting from the temporal evolution is effectively a
closed three-dimensional manifold. Note that the components of the underlying
metric (II.2) are $\varphi$-independent, signifying that the quantum state
spaces with a predesignated azimuthal angle have the same geometry. Hence, we
draw the conclusion that the appropriate quantum state space (i.e., quantum
phase space) corresponding to the $N$ qubits (under consideration) is a two-
parametric and curved manifold, it is identified by the following metric
tensor

$d\mathtt{S}^{2}=\frac{N}{4}d\theta^{2}+\frac{1}{4}N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]d\xi^{2}.$
(12)

To further characterize this state space, we are going to determine its
topology. For this aim, we begin by assessing the corresponding G-curvature,
which can be defined in terms of the relevant metric tensor (12) in the form
[56]

$\displaystyle\mathrm{K}=\frac{1}{\left(\mathrm{g}_{\theta\theta}\mathrm{g}_{\xi\xi}\right)^{1/2}}$
$\displaystyle\left[\frac{\partial}{\partial\xi}\left(\left(\frac{\mathrm{g}_{\xi\xi}}{\mathrm{g}_{\theta\theta}}\right)^{1/2}\Gamma_{\theta\theta}^{\xi}\right)\right.$
$\displaystyle\left.-\frac{\partial}{\partial\theta}\left(\left(\frac{\mathrm{g}_{\xi\xi}}{\mathrm{g}_{\theta\theta}}\right)^{1/2}\Gamma_{\theta\xi}^{\xi}\right)\right],$
(13)

where $\Gamma_{\theta\theta}^{\xi}$ and $\Gamma_{\theta\xi}^{\xi}$ account for
the Christoffel symbols given by

$\Gamma_{\theta\theta}^{\xi}=-\frac{1}{2\mathrm{g}_{\xi\xi}}\left(\frac{\partial\mathrm{g}_{\theta\theta}}{\partial\xi}\right),\quad\text{and}\quad\Gamma_{\theta\xi}^{\xi}=\frac{1}{2\mathrm{g}_{\xi\xi}}\left(\frac{\partial\mathrm{g}_{\xi\xi}}{\partial\theta}\right).$
(14)

It is extremely intriguing to see that the temporal component
$\mathrm{g}_{\xi\xi}$ of the metric cancels out at the points $\theta=0,\pi$.
This implies that the G-curvature is not definable in these positions, hence
we conclude that it exhibits a singularity in each of these two positions.
However, it is well definable at all other positions within the space of $N$
spin-1/2 states. Reporting the metric components $\mathrm{g}_{\theta\theta}$
and $\mathrm{g}_{\xi\xi}$ in the equation (II.2), the explicit expression of
the relevant G-curvature writes

$\mathrm{K}=\frac{8}{N}\left[2-\frac{(2N-3)\cos^{2}\theta+N}{\left((2N-3)\cos^{2}\theta+1\right)^{2}}\right].$
(15)

Remark that the state space curvature (15) is mainly affected by the initial
parameters $\theta$ and $N$, while it is independent of $\xi$ containing the
temporal evolution, meaning that the state space curvature is independent of
the system dynamics. Furthermore, the G-curvature (15) verifies the following
periodic requirement $\mathrm{K}(\theta)=\mathrm{K}(\theta+\pi)$. This is
consistent with our findings because the resulting quantum phase space (12) is
a closed two-dimensional manifold.

Figure 1: The dependence of the G-curvature (15) on the initial parameter
$\theta$ for some spin-$1/2$ numbers.

From the analysis of the Fig.(1) showing the G-curvature behavior with respect
to the initial parameters $(\theta,N)$, we see that the state space geometry
is symmetric with respect to the centerline at $\theta=\pi/2$, being the
position with the minimal curvature. More specifically, in the region
$\theta\in[0,\pi/2]$, the G-curvature declines and therefore the state space
geometry takes the concave shape, whereas in the region
$\theta\in[\pi/2,\pi]$, the G-curvature increases to its maximum value, and
therefore the space geometry takes the convex shape. As a result, we infer
that the $N$ spin-$1/2$ state space has a dumbbell-shape structure.
Additionally, we notice that for $N>2$, the curvature takes negative values
for certain values of $\theta$. This is congruent with the findings given in
Ref. [27].

Considering this fact, in addition to the existence of two singularities in
the G-curvature (15), we get to the conclusion that there are two conical
flaws within the quantum phase space (12); the first one is situated near to
the location $\theta=0$, whereas the second one is situated near to the
location $\theta=\pi$. In light of these outcomes, let us now explore the
topology associated with the space of $N$ spin-$1/2$ states (12). To achieve
this, we have to compute the integer Euler characteristic $\chi(\mathrm{M})$
($\mathrm{M}$ stands for state space (12)) provided in the Gauss-Bonnet
theorem as [56]

$\frac{1}{2\pi}\left[\int_{\mathrm{M}}\mathrm{K}d\mathrm{S}+\oint_{\partial\mathrm{M}}\mathrm{k}_{\mathrm{g}}d{l}\right]=\chi(\mathrm{M}),$
(16)

where the geometric invariants $d\mathrm{S}$, $\mathrm{k}_{\mathrm{g}}$ and
$d{l}$ denote, respectively, the surface element, geodesic curvature and line
element. Additionally, the first and second terms on the left-hand side of the
equation (16) represent, respectively, the bulk and border contributions to
the Euler characteristic identifying the state space topology. Furthermore,
the Gauss-Bonnet theorem (16) can be established in terms of the state space
geometry (12) in the form

$\int_{0}^{\pi}\int_{0}^{{2\pi}}\mathrm{K}\left(\mathrm{g}_{\theta\theta}\mathrm{g}_{\xi\xi}\right)^{1/2}d\theta
d\xi+{\Lambda}=2\pi\chi(\mathrm{M}),$ (17)

here $\Lambda$ standing for the Euler border integral including the conical
flaws contribution. After a simple calculation, the Gauss-Bonnet theorem (17)
reads as

$4\pi(N-1)+\Lambda=2\pi\chi(\mathrm{M}).$ (18)

Therefore, to find the Euler characteristic $\chi$, we must first determine
the Euler border integral $\Lambda$. For this purpose, we presume that the
angular flaws are situated very near to the singular points $\theta=0,\pi$. In
this view, the underlying metric (12) can be evaluated, in the vicinity of
these two singular positions, upto the second order in $\theta$. Indeed, we
obtain

$d\mathtt{S}^{2}=\frac{N}{4}d\theta^{2}+\frac{1}{4}N(N-1)^{2}\theta^{2}d\xi^{2}.$
(19)

As a further note, the solid angle of a revolution cone having the angle at
the peak $2\theta$ is defined by $\Omega=2\pi(1-\cos\theta)$ where the second
term of this definition represents the partial solid angle sketched out by the
system around the cone apex during evolution. Taking into account the
closeness of the angular flaws at the two singular points, we can easily find

$2\pi\cos\theta\approx\frac{\mathtt{S}\left(2\pi\right)}{\mathbf{d}}=\frac{2\pi\sqrt{\mathrm{g}_{\xi\xi}}}{\sqrt{\mathrm{g}_{\theta\theta}}\theta},$
(20)

here $\mathtt{S}(2\pi)$ standing for the distance traveled by the system
within the period of time $\textbf{t}=2\pi/\mathtt{J}$ around one of the two
singular positions $(\theta=0$ or $\theta=\pi)$, while d corresponds to the
distance between evolution of the system and the relevant singular position.
As a result, the angular flaws are explicitly given by

$\Lambda=2\left[2\pi-\frac{2\pi\sqrt{\mathrm{g}_{\xi\xi}}}{\sqrt{\mathrm{g}_{\theta\theta}}\theta}\right]=4\pi\left(2-N\right),$
(21)

where we multiplied it by the factor 2 because we have two singular points.
Putting the equation (21) into (18), one finds the Euler characteristic
$\chi(\mathrm{M})=2$, showing that the quantum phase space associated with the
$N$ spin-$1/2$ system has a spherical topology. The coming section will be
devoted to a thorough analysis of the geometric phase that the $N$ spin-$1/2$
state (7) can accumulate when subjected to cyclic and arbitrary evolution
processes on the underlying quantum state space (12).

##  III Geometrical phases acquired by the $N$ spin-$1/2$ state

After investigating the geometry and topology of the $N$ spin-$1/2$ state
space identified by the metric tensor (12). Let us now focus on the geometric
phase that the evolving state (7) can acquire for both arbitrary and cyclic
evolutionary processes.

###  III.1 Geometrical phase during an arbitrary evolution

In this instance, we presume that the $N$ spin-$1/2$ system evolves
arbitrarily along any evolution path on the closed two-dimensional manifold
(12). In this picture, the geometric phase gained by the evolved state (7) is
given by

$\Phi_{g}(t)=\arg\langle\Psi_{i}|\Psi(t)\rangle-{\mathrm{Im}}\int_{0}^{t}\langle\Psi(t^{\prime})|\frac{\partial}{\partial
t^{\prime}}|\Psi(t^{\prime})\rangle dt^{\prime},$ (22)

which is defined as the difference between total phase and dynamic phase [47,
57]. To calculate the geometric phase, we must first compute total the phase
acquired by the system, The overlap (i.e., transition-probability amplitude)
between the starting state (6) and the ending state (7) is obtained as

$\langle\Psi_{i}|\Psi(t)\rangle=\sum_{p=0}^{N}\mathrm{C}_{N}^{p}\cos^{2(N-p)}\left(\frac{\theta}{2}\right)\sin^{2p}\left(\frac{\theta}{2}\right)e^{-\frac{i\xi}{4}(N-2p)^{2}}.$
(23)

Inserting the expression of the overlap (23) into the first term on the right
side of the equation (22), the total phase gained by the $N$ spin-$1/2$ system
writes

$\Phi_{\operatorname{tot}}=-\arctan\left[\frac{\sum\limits_{p=0}^{N}\mathrm{C}_{N}^{p}\cos^{2(N-p)}\frac{\theta}{2}\sin^{2p}\frac{\theta}{2}\sin\left(\frac{\xi(N-2p)^{2}}{4}\right)}{\sum\limits_{p=0}^{N}\mathrm{C}_{N}^{p}\cos^{2(N-p)}\frac{\theta}{2}\sin^{2p}\frac{\theta}{2}\cos\left(\frac{\xi(N-2p)^{2}}{4}\right)}\right].$
(24)

It is interesting to observe that the total phase (24) comprises two distinct
phase components: the first is of geometrical origin (known as the geometrical
phase), and it is strongly related to the geometrical and topological features
that characterize the quantum state space of such systems [47, 56]. This
geometric component is explained by the implicit reliance of the total phase
(24) on the G-curvature (15) and the component $\mathrm{g}_{\xi\xi}$ of the
metric (12), as they all depend on the parameters $(N,\theta)$. The second one
is of dynamical origin (known as the dynamical phase), and it results from the
time evolution of Hamiltonian eigenstates (4). Furthermore, the global phase
(24) exhibits a non-linear time dependence and fulfills the following periodic
conditions :

$\Phi_{\operatorname{tot}}(\xi+4\pi)=\Phi_{\operatorname{tot}}(\xi)\qquad\text{for}\;N\;\text{integer},$
(25)

and

$\qquad\Phi_{\operatorname{tot}}(\xi+8\pi)=\Phi_{\operatorname{tot}}(\xi)\qquad\text{for}\;N\;\text{half-
integer}.$ (26)

The dynamical phase, on the other hand, can be calculated by plugging the
evolved state (7) into the second term on the right side in the equation (22).
Indeed, one finds

$\Phi_{\operatorname{dyn}}=-\frac{\xi
N}{4}\left(N\cos^{2}\theta+\sin^{2}\theta\right).$ (27)

It is proportional to the evolution time, meaning the dynamic phase primarily
instructs us about the time spent by the system during evolution. On the other
hand, the geometric phase that can be accrued by the $N$ spin-$1/2$ state (5),
during any arbitrary evolution over the quantum phase space (12), is obtained
as

$\displaystyle\Phi_{\operatorname{g}}=$
$\displaystyle-\arctan\left[\frac{\sum\limits_{p=0}^{N}\mathrm{C}_{N}^{p}\cos^{2(N-p)}\frac{\theta}{2}\sin^{2p}\frac{\theta}{2}\sin\left(\frac{\xi(N-2p)^{2}}{4}\right)}{\sum\limits_{p=0}^{N}\mathrm{C}_{N}^{p}\cos^{2(N-p)}\frac{\theta}{2}\sin^{2p}\frac{\theta}{2}\cos\left(\frac{\xi(N-2p)^{2}}{4}\right)}\right]$
$\displaystyle+\frac{\xi N}{4}\left(N\cos^{2}\theta+\sin^{2}\theta\right).$
(28)

It is clear that the resulting geometric phase (III.1) varies (i.e.,
accumulates or loses) non-linearly with the time, reflecting its dynamic
character. Otherwise, it is dependent on the freedom degrees $(\theta,\xi)$
specifying the physical states over the quantum phase space (12), which means
that the geometric phase depends on the shape of the evolution trajectory
followed by the system, while its reliance on the initial parameters
$(N,\theta)$ signifies that it is also sensitive to the state space geometry.
Accordingly, we conclude that the geometric phase (III.1) can be exploited to
parameterize the possible evolution trajectories of this system. This result
can find applications in quantum computation, because such quantum phases can
be used to design logic gates that are helpful for building good quantum
algorithms [44, 58]. Let us now turn to a special scenario in which we
investigate the geometric phase accrued by the $N$ spin-$1/2$ state (7) over a
very brief period of time. In this framework, by extending the exponential
factor given in (23) up to the second order in $\xi$, one obtains

$\langle\Psi_{i}|\Psi(t)\rangle\simeq
1+\frac{\xi^{2}N(N-1)}{64}\left[4(N-1)(N+2)\cos^{2}\theta-(N-3)(N-2)\sin^{2}2\theta+4(3N-2)\right]-i\frac{\xi
N}{4}\left(N\cos^{2}\theta+\sin^{2}\theta\right).$ (29)

In this perspective, the geometric phase (III.1) can be expressed as

$\Phi_{\operatorname{g}}\simeq-\arctan\left[\frac{{\xi
N}\left(N\cos^{2}\theta+\sin^{2}\theta\right)}{4+\frac{\xi^{2}N(N-1)}{16}\left[4(N-1)(N+2)\cos^{2}\theta-(N-3)(N-2)\sin^{2}2\theta+4(3N-2)\right]}\right]+\frac{\xi
N}{4}\left(N\cos^{2}\theta+\sin^{2}\theta\right).$ (30)

Note that for $\xi=0$, the system does not gain any quantum phase, this is
because the system state remains confined in the starting state (6) (i.e. no
evolution). As we can see that the greater the number of particles $N$, the
more the dynamic phase is dominant. Further, in the thermodynamic limit
($N\to\infty$), the total phase cancels out and therefore the geometric and
dynamic phases coincide at any moment in the evolution process. This offers us
the opportunity of measuring the geometric phase experimentally because, in
this situation, it can be obtained by the temporal integral of the average
eigenvalues of the Ising Hamiltonian (1).

###  III.2 Geometrical phase under a cyclic evolution

Here, we are focusing on the study of the geometrical phase resulting from the
cyclic evolution of the $N$ spin-$1/2$ system. In this regard, the wave
function (7) satisfies the cyclic requirement
$|\Psi({T})\rangle=e^{i\Phi_{\operatorname{tot}}}|\Psi(0)\rangle$ wherein
${T}$ represents the time span for a cyclic evolution. The AA-geometric phase
(often referred to as the Aharonov-Anandan phase) gained by the system after a
cyclic evolution (i.e., closed curve on the related parameter space) is given
by [36, 59]

$\Phi_{\operatorname{g}}^{\mathrm{AA}}=i\int_{0}^{{T}}\langle\tilde{\Psi}(t)|\frac{\partial}{\partial
t}|\tilde{\Psi}(t)\rangle dt,$ (31)

here $|\tilde{\Psi}(t)\rangle$ denotes the Anandan-Aharonov section given in
Ref. [36]. It is identified through the evolved state (7) as
$|\tilde{\Psi}(t)\rangle=e^{-i{f}(t)}|\Psi(t)\rangle$ where ${f}(t)$ is any
smooth function checking $f({T})-f(0)=\Phi_{\operatorname{tot}}$. On this
view, the AA-geometric phase (31) rewrites

$\Phi_{\operatorname{g}}^{\mathrm{AA}}=\int_{0}^{{T}}d\Phi_{\operatorname{tot}}+i\int_{0}^{{T}}\langle\Psi(t)|\frac{\partial}{\partial
t}|\Psi(t)\rangle dt.$ (32)

Reporting the equations (24) and (27) into (32), one obtains the AA-geometric
phase acquired by the $N$ spin-$1/2$ system (7) during a cyclic evolution of
the form

$\Phi_{\operatorname{g}}^{\mathrm{AA}}=-\frac{\pi}{2}N(N-1)\sin^{2}\theta.$
(33)

Thus, the obtained AA-geometric phase (33) is independent of the system
dynamics. Otherwise, it depends only on the initial parameters $\theta$ and
$N$ (i.e., the starting state), controlling the shape of the state space (12).
As a result, we conclude that the AA-geometric phase is impacted by the state
space geometry and not by the evolution path followed by the system. Hence the
cyclic evolution paths are not parameterizable by this cyclic phase. Further,
using the equation (15) into (33), one can relate the AA-geometric phase with
the G-curvature as follows

$\Phi_{\operatorname{g}}^{\mathrm{AA}}=\frac{{\pi
N(N-1)}}{2}\left[{\frac{{-56+3N\left({16-(N-1)\mathrm{K}}\right)}}{{\left({2N-3}\right)\left({N\mathrm{K}-16}\right)}}}\right].$
(34)

In Fig.(2), we depict the behavior of the cyclic geometrical phase (33) with
respect to the starting parameters $(N,\theta)$. We find that it behaves
similarly to G-curvature (see Figure 1).

Figure 2: The dependence of the AA-geometric phase (33) on the initial
parameter $\theta$ for some spin-$1/2$ numbers.

Specifically, the AA-geometric phase (33) decreases in the region
$\theta\in[0,\pi/2]$, in which the state space geometry has a concave shape
(i.e., the G-curvature decreases), whereas it grows in the region
$\theta\in[\pi/2,\pi]$, in which this geometry has a convex form (i.e., the
G-curvature increases). Thus, we conclude that the AA-geometric phase (33)
also has a symmetric behavior along the parameter $\theta$, this is due to the
dumbbell-shape structure of the state space. Additionally, we see that it is
interesting to discuss the topological phase that can emerge during the cyclic
evolution of the wave function (7). In reality, it constitutes the part of the
cyclic geometric phase that does not receive any dynamic contribution.
Explicitly, it is found as

$\Phi_{\operatorname{top}}^{\mathrm{AA}}=-\frac{\pi}{2}N^{2}.$ (35)

The resulting topological phase (35) is propotional to the square of the
particle number. Especially, this phase takes fractional values for $N$ odd
and multiples of $\pi$ for $N$ even. This shows that the topological phase
(resp. the particle number) is relevant to control the state space topology
given in (18). This provides the possibility to parametrize the closed
evolution paths traversed by the system through the resulting topological
phase (35). This result looks very interesting in quantum computing,
particularly in the search for efficient quantum circuits [53, 60]. This issue
can be closely connected to the determination of the optimal evolution path of
the system under consideration, by evaluating the evolution speed as well as
the related Fubini-Study distance. This is the quantum Brachistochrone
problem, which will be tackled in the succeeding section.

##  IV Speed and quantum brachistochrone issue for $N$ spin-$1/2$ system

Now, we will exploit the Riemannian geometry identifying the quantum state
space (12) to investigate some dynamical properties of the system. In
particular, we examine the evolution speed and the geodesic distance measured
by the Fubini-Study metric (12) in order to solve the related quantum
brachistochrone problem [11, 61]. This issue is often linked to achieving
time-optimal evolution, which is characterized by a maximum speed and the
trajectory between the starting state (6) and the ending state (7) is the
shortest possible. In other terms, the solution to this dynamic problem is to
find the shortest period of evolution.

###  IV.1 Speed of quantum evolution

In order to evaluate the evolution speed, we presume that the evolution of the
$N$ spin-$1/2$ system depends only on time while leaving all other parameters
unchanged. In this picture, the metric tensor (12) simplifies to

$d\mathtt{S}^{2}=\frac{1}{4}N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]d\xi^{2}.$
(36)

This shows that the dynamics of the system occurs on a circle of radius
$\sqrt{\mathrm{g}_{\theta\theta}}$. Therefore, the evolution speed of the $N$
spin-$1/2$ state (7) takes the form [9]

$\mathrm{V}=\frac{d\mathtt{S}}{dt}={2}\Delta\mathrm{E},$ (37)

where $\Delta\mathrm{E}$ designates the energy uncertainty of the above
Hamiltonian (1). From the analysis of the equation (37), we notice that the
larger the energy uncertainty, the faster the system evolves, and vice versa.
By setting the equation (36) into (37), the evolution velocity of the wave
function (7) is found as

$\mathrm{V}=\frac{\mathtt{J}}{2}\sqrt{N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]}.$
(38)

Thus, the evolution rapidity is affected by the coupling interaction
$\mathtt{J}$ and the initial parameters ($\theta$, $N$), i.e., the choice of
the starting state. From the equation (38), we remark that the larger the
particle number and the coupling constant, the faster the system evolves, with
the exception of $\theta=0$ or $\theta=\pi$, in which the evolution velocity
(38) cancels out $(\mathrm{V}=0)$ regardless of these physical parameters.
This is justified by the fact that neither the wave function (7) nor the
G-curvature (II.2) are defined in these two singular points.

Given that the speed depends parameter $\theta$, meaning that it is also
assigned both by the G-curvature (15) and the geometric phase (III.1)
including the AA-geometric-phase (33). This can be clearly seen in the figure
(3) displaying its reliance on the initial parameters $(\theta,N)$.

Figure 3: The dependence of the evolution speed (38) on the initial parameter
$\theta$ for some spin-$1/2$ numbers with $\mathtt{J}=1$.

Note that the evolution speed (38) exhibits a symmetric behavior with respect
to the state space geometry having the dumbbell-shape structure. This makes
perfect sense because the resulting phase space corresponds exactly to the
relevant quantum phase space, on which the dynamics of the $N$ spin-$1/2$
system is well established.

###  IV.2 Resolution of the quantum brachistochrone problem

To address this dynamic issue, we should determine the shortest duration
possible necessary to carry out the time-optimal evolution of the considered
system (7). To accomplish this, one begins by maximizing the evolution
velocity (38) by solving the equation $d\mathrm{V}/d\theta=0$, which yields

$N(N-1)(N-1-(2N-3)\sin^{2}\theta)\sin 2\theta=0,$ (39)

which entails that

$\sin\theta_{\max}=\sqrt{\frac{N-1}{2N-3}}.$ (40)

Therefore, the highest speed that the $N$ spin-$1/2$ system can achieve reads
as

$\mathrm{V}_{\max}=\mathtt{J}(N-1)\sqrt{\frac{N(N-1)}{8(2N-3)}},$ (41)

In this way, we manage to establish the maximum velocity, which depends only
on the particle number making up the system. Let us now investigate the
geodesic distance that the system traverses between the departure state (6)
and the arrival state (7). For this, utilizing the equation (37), one finds

$\mathtt{S}=\frac{{\xi}}{2}\sqrt{N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]}.$
(42)

As the evolution rapidity (38) is time-independent, the Fubini-Study distance
(42) traveled by the system has a linear behavior with time. Additionally, we
note that, at each instant, the evolution speed and the distance have a
similar behavior against the physical parameters $(\theta,\mathtt{J},N)$ of
the system. Specifically, at singular points $\theta=0,\pi$, the distance
vanishes, which is reasonable because at these points the state of $N$
spin-$1/2$ (7) is not defined. Thereby, one concludes that the Fubini-Study
distance (42) exhibits a local minimum at the point $\theta=\pi/2$, it given
by

$\mathtt{S}_{\min}=\frac{{\xi}}{2}\sqrt{\frac{N(N-1)}{2}}.$ (43)

Thus, the minimum feasible time required for the system to conduct any quantum
evolution writes

$\mathrm{t}_{\min}=\frac{\mathtt{S}_{\min}}{\mathrm{V}_{\max}}=\frac{t}{(N-1)}\sqrt{2N-3}.$
(44)

This is the shortest duration needed to achieve a time-optimal evolution over
the state circle (36). In specific terms, the result (44) defines the optimal
evolution condition, which is typified by the fastest evolution and the
shortest path joining the initial and final states. Therefore, the optimal
evolution states can be produced through the following unitary transformation

$\left|\Psi_{i}\right\rangle\rightarrow|\Psi(\mathrm{t}_{\min})\rangle=e^{-i\mathrm{H}\mathrm{t}_{\min}}\left|\Psi_{i}\right\rangle.$
(45)

so that the set of these states makes up an optimal state circle described by
the Fubini-Study metric of the form

$d\mathtt{S}^{2}_{\mathrm{op}}=\frac{1}{4}N(N-1)\sin^{2}\theta\left[N-1-\left(N-\frac{3}{2}\right)\sin^{2}\theta\right]d\xi^{2}_{\min}.$
(46)

with $\xi_{\min}=\mathtt{J}\mathrm{t}_{\min}$. On other hand, we remark that
the temporal condition (44) is solely influenced by the particle number, this
implies that the state circle topology affects the optimal evolution time as
well. It is also proportional to the ordinary time $t$ (i.e., matches the
evolution over the state circle (36)) with a positive proportionality factor,
meaning that the optimal and ordinary times have the same monotonicity.
Particularly, one finds that for $N=2$ (i.e., two-spin-$1/2$ system) these two
types of time coincide $(\mathrm{t}_{\min}=t)$, while for $N\geq 3$ (i.e., $N$
spin-$1/2$ system), we discover that the optimal time (44) is strictly lower
than the ordinary time, and therefore the time-optimal evolution is
achievable. Besides, in the thermodynamic limit $(N\to\infty)$, the optimal
time decreases to zero $(\mathrm{t}_{\min}\to 0)$. In this respect, the
optimal state circle (46) coincides with a straight line since its radius
$\sqrt{\mathrm{g_{\xi_{\min}\xi_{\min}}}}$ becomes infinite. As a result, we
infer that the particle number and the ordinary time are two physical
magnitudes exploitable for performing the time-optimal evolutions in such
integrable systems. At the end of this section, we note that it is intriguing
to relate the geometric and dynamic structures that we explored above with
quantum entanglement, as a physical resource of great relevance in quantum
information tasks. The next section focuses on this subject.

##  V Geometric and dynamic pictures of the entanglement for two-spin system
$(N=2)$

In this section, we shall study the quantum entanglement exchanged between two
interacting spins under Ising model through two different perspectives; the
first is geometric in nature and investigates the entanglement impact on
derived geometric features such as the Fubini-Study metric, G-curvature, and
the geometric phase under arbitrary and cyclic evolutions. The second is
dynamic in nature and explores the entanglement effect on the evolution speed
as well as the Fubini-Study distance covered by the system. Importantly, we
address the quantum Brachistochrone problem depending on the entanglement
degree.

###  V.1 Entanglement degree of the two-spin system

The wave function of the global quantum system (7) is narrowed for a two-spin
system to the form

$\displaystyle|\Psi(t)\rangle=$ $\displaystyle
e^{-i\xi(t)}\cos^{2}\frac{\theta}{2}|\uparrow\uparrow\rangle+\frac{1}{2}e^{i\varphi}\sin\theta(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)$
$\displaystyle+e^{i(2\varphi-\xi(t))}\sin^{2}\frac{\theta}{2}|\downarrow\downarrow\rangle.$
(47)

Hence, the two-spin state space, on which the dynamics of the system takes
place, is defined by the following metric tensor

$d\mathtt{S}^{2}=\frac{1}{2}d\theta^{2}+\frac{1}{4}\sin^{2}\theta\left(2-\sin^{2}\theta\right)d\xi^{2}.$
(48)

Using the Wootters concurrence expression given in Ref. [62], one obtains,
after a simple calculation, the entanglement amount contained in the two-spin
state (V.1) of the form

$\boldsymbol{\mathscr{C}}=\sin^{2}\theta|\sin\xi|.$ (49)

It is the same for any two spins of the entire system (7). In other terms,
each spin pair is quantum-correlated as much as any other pair. Interestingly,
we observe that the two-spin entanglement (49) evolves periodically with time,
signifying that it is impacted by the dynamics of the system. Moreover, it
relies on the initial parameter $\theta$, showing that the entanglement degree
is also governed by the starting state choice. Notice that for $\xi=\pi/2$ and
$\theta=\pi/2$, the two-spin state (V.1) reaches its maximum entanglement
value $(\boldsymbol{\mathscr{C}}=1)$, whereas for $\theta=0$ or $\pi$, the two
spins can never be entangled $(\boldsymbol{\mathscr{C}}=0)$, because the
corresponding initial states $|\Psi_{i}\rangle=|\uparrow\uparrow\rangle$ or
$|\downarrow\downarrow\rangle$ are the Hamiltonian eigenstates. This can be
also justified topologically and geometrically by the existence of a conical
defect close to these two singular points.

###  V.2 Geometrical picture of the entanglement

In order to evoke the geometric aspect of the quantum correlations between the
two spins under study, we suggest a thorough description illustrating the
nexus between the entanglement and the geometric structures derived above.
Setting the equation (49) into (48), we give the Fubini-Study metric
identifying the two-spin state space in terms of the concurrence as

$d{\mathtt{S}^{2}}=\frac{{d{\boldsymbol{\mathscr{C}}^{2}}}}{{8\boldsymbol{\mathscr{C}}(\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})}}-\frac{{d\boldsymbol{\mathscr{C}}d\xi}}{{4\tan\xi(\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})}}+\frac{\boldsymbol{\mathscr{C}}}{4}\left({\frac{1}{{2{{\tan}^{2}}\xi(\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})}}+\frac{{2\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}}}}{{{{\sin}^{2}}\xi}}}\right)d{\xi^{2}}.$
(50)

which can be transformed in its diagonal form

$d\mathtt{S}^{2}=\frac{1}{8\boldsymbol{\mathscr{C}}_{r}\left(1-\boldsymbol{\mathscr{C}}_{r}\right)}d\boldsymbol{\mathscr{C}}_{r}^{2}+\frac{1}{4}\boldsymbol{\mathscr{C}}_{r}\left(2-\boldsymbol{\mathscr{C}}_{r}\right)d\xi^{2},$
(51)

with $\boldsymbol{\mathscr{C}}_{r}={\boldsymbol{\mathscr{C}}}/{|\sin\xi|}$
denotes the reduced concurrence varying in the interval $[0,1]$. Thereby, we
have managed to reparameterize the relevant phase space (48) according to the
amount of entanglement shared between the two spins and the evolution time,
which are two measurable physical magnitudes. This demonstrates the
feasibility of examining, experimentally, all the geometrical, topological and
dynamical characteristics of this state space, namely the phase space
geometry, the quantum phases, the evolution speed, and the geodesic distance
covered by the two-spin system (V.1) during its evolution. Importantly, the
quantum entanglement serves in shrinking the state space dimension. For
instance, the states of two spins with the same entanglement level
$(\text{i.e.,}\,\boldsymbol{\mathscr{C}}=\text{constant})$ are located on
closed one-dimensional manifolds defined by

$d{\mathtt{S}^{2}}=\frac{\boldsymbol{\mathscr{C}}}{4}\left({\frac{1}{{2{{\tan}^{2}}\xi(\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})}}+\frac{{2\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}}}}{{{{\sin}^{2}}\xi}}}\right)d{\xi^{2}}.$
(52)

They are, in fact, closed curves, along the metric component
$\mathrm{g}_{\xi\xi}$, on the whole state space (50). On the other hand, the
states with the same degree of reduced entanglement
$(\text{i.e.,}\,\boldsymbol{\mathscr{C}}_{r}=\text{constant})$ are located on
circles identified by

$d\mathtt{S}^{2}=\frac{1}{4}\boldsymbol{\mathscr{C}}_{r}\left(2-\boldsymbol{\mathscr{C}}_{r}\right)d\xi^{2},$
(53)

whose radii
$\mathtt{R}=\sqrt{\boldsymbol{\mathscr{C}}_{r}\left(2-\boldsymbol{\mathscr{C}}_{r}\right)}/2$
depend on the level of reduced entanglement taken into account. In this way,
we demonstrate the relevance of utilizing quantum correlations to minimize the
dimensionality of the state space (50). This finding is applicable to all
integrable quantum systems.

In the same framework, we can also investigate the influence of entanglement
on the G-curvature of the two-spin phase space (50). As a matter of fact,
putting the equation (49) into (15), we derive the curvature according to the
concurrence under the form

$\mathrm{K}=4\left[{2+\frac{{\left|{\sin\xi}\right|\left({\boldsymbol{\mathscr{C}}-3\left|{\sin\xi}\right|}\right)}}{{{{\left({\boldsymbol{\mathscr{C}}-2\left|{\sin\xi}\right|}\right)}^{2}}}}}\right].$
(54)

This outcome demonstrates again the explicit reliance of the state space
geometry on the entanglement amount exchanged between the two interacting
spins. From the equation (54), we observe that for $\xi=0$ (i.e., no
evolution), the G-curvature is $\boldsymbol{\mathscr{C}}$-independent and
takes a constant value $(\mathrm{K}=8)$, which corresponds to the curvature of
the initial state sphere (55), while for $\xi>0$ (i.e., evolution case) it is
$\boldsymbol{\mathscr{C}}$-dependent and its behavior versus the entanglement
is shown in the figure (4). We notice that the G-curvature diminishes as the
entanglement amount exchanged between the two spins increases. This can be
explained by the fact that the existence of quantum correlations causes a
decrease in the state space curvature. Further, the G-curvature reaches
negative values for the entanglement degrees verifying the condition

${\left|{\sin\xi}\right|\left({\boldsymbol{\mathscr{C}}-3\left|{\sin\xi}\right|}\right)}<-2{{{\left({\boldsymbol{\mathscr{C}}-2\left|{\sin\xi}\right|}\right)}^{2}}}.$
(55)

This elucidates the quantum correlation impact between the two particles in
the compactification of the related state space (50) when the requirement (55)
is met. It is interesting to note that the separable states
$(\boldsymbol{\mathscr{C}}=0)$ are housed in the regions of highest curvature
$\mathrm{K}_{\max}=5$,

Figure 4: The G-curvature (54) versus the concurrence (49) for some values of
$\xi$.

whereas the states of maximum entanglement $(\boldsymbol{\mathscr{C}}=1)$ are
housed in the regions of lowest curvature

${{\mathrm{K}}_{\min}}=4\left[{2-\frac{{\left|{\sin\xi}\right|\left({3\left|{\sin\xi}\right|-1}\right)}}{{{{\left({2\left|{\sin\xi}\right|-1}\right)}^{2}}}}}\right].$
(56)

Thus, we conclude that the information about the entanglement degree of the
two-spin system (V.1) allows to determine its localization on the
corresponding phase space (50). This clearly proves the deterministic
character of geometric quantum mechanics, this latter is based on the
geometrization of Hilbert space by introducing the concept of quantum phase
space analogous to that of classical mechanics [4, 6].

The connection between the geometric phase and the quantum entanglement can be
also explored here. Indeed, reporting the equation (49) into (III.1), we
obtain the geometric phase gained by the two-spin state (V.1) in terms of the
concurrence as

$\Phi_{\operatorname{g}}=-\arctan\left[\dfrac{(2\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})\sin\xi}{(2\left|{\sin\xi}\right|-\boldsymbol{\mathscr{C}})\cos\xi+\boldsymbol{\mathscr{C}}}\right]+\xi\begin{pmatrix}1-\dfrac{\boldsymbol{\mathscr{C}}}{2\left|{\sin\xi}\right|}\end{pmatrix}.$
(57)

Notice that the geometric phase is determined by the two new physical degrees
of freedom : entanglement and time. This means that it depends on each point
(i.e., system physical state) of the underlying phase space (50).
Consequently, we can say that the geometric phase depends both on the path
followed by the system and the geometry of the state space. Since the
geometrical phase is defined in terms of these two measurable physical
magnitudes (i.e., $\boldsymbol{\mathscr{C}}$ and $\xi$), this offers us the
ability to measure it experimentally for any arbitrary evolution process of
the system. This result is extremely important because it can be exploited to
build efficient quantum circuits based on the amount of entanglement exchanged
between the two interacting spins. To further highlight the interplay between
the geometric phase and entanglement, we have graphed the reliance of the Eq.
(57) with respect to the concurrence for some values of $\xi$ in Figure (5),
we observe that the geometric phase (57) gained by the two-spin system (V.1)
during its evolution from the separable state $(\boldsymbol{\mathscr{C}}=0)$
to the maximum entanglement state $(\boldsymbol{\mathscr{C}}=1)$ exhibits
approximately a parabolic behavior. From here, we can divide its evolvement
into two main stages : the first stage involves the geometric phase decrease
along the concurrence interval
$\boldsymbol{\mathscr{C}}\in[0,\boldsymbol{\mathscr{C}}_{\text{c}}]$ with
$\boldsymbol{\mathscr{C}}_{\text{c}}$ denotes the critical entanglement degree
in which this phase reaches its minimum value (see the figure (5)), it is
given explicitly as

$\boldsymbol{\mathscr{C}}_{\text{c}}=\sin\xi-\cot\frac{\xi}{2}\sqrt{\frac{{\sin\xi}}{\xi}\left({2-\xi\sin\xi-2\cos\xi}\right)}.$
(58)

Figure 5: The geometric phase (57) versus the concurrence (49) for some values
of $\xi$.

In this stage, the evolving state (V.1) acquires a negative geometric phase,
which can be interpreted as the geometric phase part lost by the system.
Geometrically, we can say that during parallel transport, the state vector
(V.1) rotates clockwise (i.e., makes an angle of negative sign) with respect
to the separable state (i.e., starting state). Thereby, we discover that, in
the region $[0,\boldsymbol{\mathscr{C}}_{\text{c}}]$, the quantum correlations
favors the loss of the geometric phase. The second stage concerns the
geometric phase increase along the interval
$\boldsymbol{\mathscr{C}}\in[\boldsymbol{\mathscr{C}}_{\text{c}},1]$ (i.e.,
reverse behavior), the evolving state (V.1) accumulates a positive geometric
phase, which can be viewed as the geometric phase part gained by the system.
We can say, geometrically, that during parallel transport, the state vector
(V.1) rotates counterclockwise (i.e., makes an angle of positive sign) with
respect to the separable state. In this way, we find that, in the region
$[\boldsymbol{\mathscr{C}}_{\text{c}},1]$, the quantum correlations favors the
gain of the geometric phase. Accordingly, the geometric phase behavior versus
the entanglement is approximately symmetric with respect to the critical value
$\boldsymbol{\mathscr{C}}_{\text{c}}$, this is mainly due to the dumbbell-
shape structure of the underlying phase space (50). On the practical side, the
quantum entanglement is then an interesting physical resource that can be
exploited experimentally to control the geometric phase resulting from the
evolution processes of the two-spin system.

In the cyclic evolution scenario, the geometric phase can be also investigated
in connection to the entanglement. Indeed, inserting the equation (49) into
(33), we give the AA-geometric phase accumulated by the evolved state (V.1) in
relation to the concurrence as

$\Phi_{\operatorname{g}}^{\mathrm{AA}}=-{\pi}\frac{\boldsymbol{\mathscr{C}}}{\left|{\sin\xi}\right|}.$
(59)

It is proportional to the entanglement level between the two spins with a
negative proportionality factor, implying that the AA-geometric phase
decreases linearly as entanglement increases. For the sake of clarity, we
display this behavior in Figure (6), we observe indeed that the more the
system is entangled, the more it accumulates an AA-geometric phase of negative
sign.

Figure 6: The AA-geometric phase (59) versus the concurrence (49) for some
values of $\xi$.

This is roughly the same behavior as we observed for the geometric phase (57)
in the first stage (i.e., in the region
$[0,\boldsymbol{\mathscr{C}}_{\text{c}}]$), and hence the same interpretations
can be provided for the AA-geometric phase. Regarding the topological phase
resulting from the cyclic evolutions of the two-spin system, it is given by
$\Phi_{\operatorname{top}}^{\mathrm{AA}}=-2\pi$. Thus, it is unaffected by the
entanglement, because it constitutes the AA-geometrical phase part receiving
no contribution from the dynamic part.

###  V.3 Dynamical picture of the entanglement

To close this section, it would be interesting to examine the dynamical aspect
of entanglement by highlighting the link between the entanglement amount
exchanged between the two spins and the relevant dynamical properties, such as
the evolution speed and the traveled geodesic distance during a given
evolution process over the resulting phase space (50). As a result, we address
the quantum brachistochrone problem by relying on the entanglement level of
the two-spin system. To accomplish this, using the equation (49) into (38),
the related evolution speed can be expressed in relation to the concurrence as
follows

$\mathrm{V}=\frac{\mathtt{J}}{2\left|\sin\xi\right|}\sqrt{{\boldsymbol{\mathscr{C}}}\begin{pmatrix}2\left|\sin\xi\right|-{\boldsymbol{\mathscr{C}}}\end{pmatrix}}.$
(60)

In this way, we manage to relate the rapidity of the two-spin system with its
entanglement degree. In other words, the result (60) reflects the explicit
relatedness between the system dynamics and the evolvement of the quantum
correlations. Consequently, we infer that the dynamical characteristics of
such a system are determinable through its entanglement state. The evolution
speed reliance according to the concurrence is shown in Figure (7).

Interestingly, the variation of the evolution velocity (60) is split into two
different parts : the first part shows the the evolution velocity increase of
the two-spin state until it attains its highest value
$\mathrm{V}_{\max}={\mathtt{J}}/2$, matching the critical entanglement level
$\boldsymbol{\mathscr{C}}=\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime}=\left|\sin\xi\right|$.
This

Figure 7: The evolution speed (60) verus the concurrence (49) for some values
of $\xi$ with $\mathtt{J}=1$.

proves that, in this part, the existence of quantum correlations speed up the
system evolution over the related phase space (50). The second part concerns
the concurrence interval
$\boldsymbol{\mathscr{C}}\in[\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime},1]$,
in which the evolution speed has reversed its monotonicity, it diminishes
continuously until it achieves its local minimum
$\mathrm{V}(\boldsymbol{\mathscr{C}}=1)$. This signifies that, in this second
part, the quantum correlations slow down this evolution. As a result, we
conclude that the system dynamics is controllable by its entanglement level.
This outcome can be usefully exploited in quantum information protocols.
Utilizing the equation (37), we can also establish the Fubini-Study distance
traveled by the two-spin state (V.1) in relation to the concurrence, it is
found as

$\mathtt{S}=\frac{\xi}{2\left|\sin\xi\right|}\sqrt{{\boldsymbol{\mathscr{C}}}\begin{pmatrix}2\left|\sin\xi\right|-{\boldsymbol{\mathscr{C}}}\end{pmatrix}}.$
(61)

Thereby, we arrive at expressing the Fubini-Study distance (i.e., dynamical
observable), relating any two quantum states over the two-spin phase space
(50), in terms of the entanglement level and the evolution time. This proves
again the feasibility to investigate experimentally the dynamical properties,
which will motivate their exploitation in the novel applications of quantum
technology [63].

Figure 8: The Fubini-Study distance (61) versus the concurrence (49) for some
values of $\xi$.

By scrutinizing the two figures (7) and (8), we find that the Fubini-Study
distance (61) exhibit, with respect to the entanglement degree, the same
behavior as the evolution velocity (60), and thus the same conclusions can be
achieved. Let us now solve the quantum brachistochrone issue for the two
interacting spins based on their entanglement. To realize this, maximizing the
evolution speed (60) with respect to the concurrence, the shortest time
required to achieve a time-optimal evolution, over the relevant phase space
(50), reads as

$\boldsymbol{\tau}=\frac{\mathtt{S}}{\mathrm{V}_{\max}}=\frac{\xi}{\mathtt{J}\left|\sin\xi\right|}\sqrt{{\boldsymbol{\mathscr{C}}}\begin{pmatrix}2\left|\sin\xi\right|-{\boldsymbol{\mathscr{C}}}\end{pmatrix}}.$
(62)

So, the optimal time (62) depends on both the ordinary time, the coupling
constant, and the entanglement level of the system. Specifically, we observe
that for $\boldsymbol{\mathscr{C}}=0$ the optimal time cancels out
$(\boldsymbol{\tau}=0)$, this because the evolving state (V.1) coincides with
the disentangled starting state $|\Psi_{i}\rangle=|++\rangle$ (i.e., no
evolvement). For the critical entanglement level
$\boldsymbol{\mathscr{C}}=\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime}$ the
optimal time attains its highest value $(\boldsymbol{\tau}=t)$, signifying
that the optimal and ordinary evolutions of the two-spin system coincide,
whereas for
$\boldsymbol{\mathscr{C}}\in\,]0,\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime}[\,\cup\,]\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime},1]$
the optimal time is strictly less than the ordinary time
$(\boldsymbol{\tau}<t)$. In this respect, the optimal evolution states can be
generated via the unitary operation given by

$\left|\Psi_{i}\right\rangle\rightarrow|\Psi(\boldsymbol{\tau})\rangle=e^{-i\mathrm{H}\boldsymbol{\tau}}\left|\Psi_{i}\right\rangle.$
(63)

In fact, they make up one-dimensional space of optimal states, over the whole
space (50), defined by the metric tensor

$d{{\mathtt{S}}}^{2}_{\text{opt}}=\frac{{\boldsymbol{\mathscr{C}}}}{4\sin^{2}\xi}\left(2\left|\sin\xi\right|-\boldsymbol{\mathscr{C}}\right)d\boldsymbol{\xi}^{2},$
(64)

with $\boldsymbol{\xi}=\mathtt{J}\boldsymbol{\tau}$. The behavior of the
optimal time (62) according to the entanglement is illustrated in Figure (9),
we discover that the lower the entanglement level (resp. the ordinary time) of

Figure 9: The optimal time (62) versus the concurrence (49) for some values of
$\xi$ with $\mathtt{J}=1$.

the two spins ${\boldsymbol{\mathscr{C}}}\to 0$ (resp. $t\to 0$), the shorter
the optimal time $\boldsymbol{\tau}\to 0$. Accordingly, we conclude that the
entanglement and the ordinary time are two physical magnitudes exploitable for
realizing the optimal evolutions in such integrable systems. These kinds of
evolutions are of paramount importance in quantum computing for designing good
quantum algorithms [64, 65].

##  VI Conclusion and outlook

To summarize, we investigated a physical system consisting of $N$ interacting
spin-$1/2$ under all-range Ising model. We assumed that the starting state is
the tensor product of $N$ eigenstates of the spin-$1/2$ projection operator on
the positive direction denoted by the unit vector. After applying the time
evolution propagator, the obtained evolved state is identified by three
dynamical degrees of freedom, which are the spherical angles
$(\theta,\varphi)$ and the evolution time $t$. We established the Fubini-Study
metric and explored that the related quantum phase space is a closed two-
dimensional manifold. Moreover, by examining the G-curvature in the framework
of the Gauss-Bonnet theorem, we demonstrated that this state space has both a
dumbbell-shape structure and a spherical topology. The geometric phase
acquired by the $N$ spin-$1/2$ system is also studied within the arbitrary and
cyclic evolution processes. This is achieved by evaluating the difference
between the total and dynamic phases. We found that the geometric phase, in
the arbitrary evolution case, varies nonlinearly with the time, reflecting its
dynamic character. Geometrically, we showed that it is affected both by the
evolution trajectory taken by the system and the associated state space
geometry. On this view, we concluded that the geometric phase can be exploited
to parameterize the possible evolution trajectories of the system. This result
may be interesting for building robust quantum gates. Further, in the
thermodynamic limit ($N\to\infty$), the total phase cancels out and therefore
the geometric and dynamic phases coincide at any moment in the evolution
process. This offers the opportunity of measuring the geometric phase
experimentally. In the cyclic evolution case, we have calculated the AA-
geometric phase and discovered that it is independent of the system dynamics.
Otherwise, it depends only on the initial state choice (i.e., the initial
parameters), signifying that it is influenced by the state space geometry and
not by the evolution path followed by the system. Hence the cyclic evolution
paths are not parameterizable by the AA-geometric phase. We have also derived
the topological phase appearing naturally during cyclic evolutions. We found
that it is proportional to the square of the particle number $N^{2}$,
especially it takes fractional values for $N$ odd and multiples of $\pi$ for
$N$ even. This signifies that the number of spins influences the topology of
the state space. The evolution speed and the Fubini-Study distance separating
the quantum states are also well examined. As a result, we resolved the
quantum Brachistochrone problem for the $N$ spin-$1/2$ system by determining
the shortest time (i.e., optimal time) required to perform an time-optimal
evolution, and defined the underlying optimal state space. In this
perspective, we discovered that for $N=2$ (i.e., two spin$-1/2$ system) the
optimal and ordinary times coincide $(\mathrm{t}_{\min}=t)$, while for $N\geq
3$ (i.e., $N$ spin-$1/2$ system), the optimal time (44) is strictly lower than
the ordinary time, and thus the time-optimal evolution is achievable. Besides,
in the thermodynamic limit $(N\to\infty)$, the optimal time decreases to zero
$(\mathrm{t}_{\min}\to 0)$. In this scheme, the optimal state circle coincides
with a straight line since its radius becomes infinite. Thereby, we inferred
that the particle number and the ordinary time are two physical magnitudes
exploitable for performing the time-optimal evolutions in such integrable
systems.

On other hand, by restricting the whole system to a two-spin system, i.e., two
interacting spin under the Ising model, we have studied the quantum
entanglement via two approaches. The first approach is of geometric nature, in
which we give the Fubini-Study metric in connection to the Wootters
concurrence as a quantum correlation quantifier. This outcome may be
interesting for the experimental handling of the geodesic distance between
entangled states and also in the state space geometry adjustment. We proved
that an increase in the entanglement degree between the two spins causes a
decrease in the state space curvature until it reaches negative values,
showing the quantum correlation effect in the compactification of the related
state space. Additionally, the entanglement can be used to identify the
quantum states over the space of states, for example, the states of maximum
entanglement are housed in the regions of lowest curvature, whereas the
separable states are housed in the regions of highest curvature. The geometric
phase acquired by the two-spin system is sufficiently discussed in relation to
the entanglement. We explored that the geometric phase exhibits two different
behaviors with respect to the critical entanglement level
$\boldsymbol{\mathscr{C}}_{\text{c}}$
$(\Phi_{\operatorname{g}}(\boldsymbol{\mathscr{C}}_{\text{c}})={\Phi_{\operatorname{g}}}_{\min})$:
the first one is in the interval $[0,\boldsymbol{\mathscr{C}}_{\text{c}}]$,
wherein the quantum correlations favors the loss of the geometric phase, while
the second one is in the interval $[\boldsymbol{\mathscr{C}}_{\text{c}},1]$,
wherein the quantum correlations favors the gain of the geometric phase. In
the cyclic evolution scenario, the AA-geometric phase behaves similarly to the
geometric phase in the first interval
$[0,\boldsymbol{\mathscr{C}}_{\text{c}}]$. This highlights the significance of
quantum entanglement for controlling the geometric phase evolvement in such
spin systems. The second approach is of dynamic nature, we linked the
evolution speed with the concurrence, we observed that the speed displays two
different behaviors with respect to the critical entanglement level
$\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime}$
$(\mathrm{V}(\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime})=\mathrm{V}_{\max})$:
the first one is in the interval
$[0,\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime}]$, wherein the quantum
correlations speed up the system evolution over the relevant phase space,
while the second one is in the interval
$[\boldsymbol{\mathscr{C}}_{\text{c}}^{\prime},1]$, wherein the quantum
correlations slow down this evolution. Accordingly, we concluded that the
system dynamics is controllable by its entanglement level. The same behavior
is noticed for the Fubini-Study distance separating the entangled states.
Finally, we solved the quantum barchistochrone problem based on the
entanglement amount exchanged between the two spins. We inferred that the
quantum entanglement and the ordinary time are two physical magnitudes
exploitable to realize the time-optimal evolution in such a spin system. Thus,
we were able to illustrate, to a significant extent, the connection between
quantum entanglement and the geometric and dynamical characteristics
characterizing the considered two-spin phase space.

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