# SPIN SQUEEZING AND ENTANGLEMENT VIA FINITE-DIMENSIONAL DISCRETE PHASE-SPACE
DESCRIPTION

Marcelo A. Marchiolli  Avenida General Osório 414, centro, Jaboticabal, SP
14870-100, Brazil
<EMAIL_ADDRESS>Diógenes Galetti  Instituto de Física Teórica,
Universidade Estadual Paulista,
Rua Dr Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, São Paulo, SP
01140-070, Brazil
<EMAIL_ADDRESS>Tiago Debarba  Instituto de Ciências Exatas,
Universidade Federal de Minas Gerais, Departamento de Física,
Avenida Antônio Carlos 6627, Belo Horizonte, MG 31270-901, Brazil
<EMAIL_ADDRESS>
###### Abstract

We show how mapping techniques inherent to ${\rm N}^{2}$-dimensional discrete
phase spaces can be used to treat a wide family of spin systems which exhibits
squeezing and entanglement effects. This algebraic framework is then applied
to the modified Lipkin-Meshkov-Glick (LMG) model in order to obtain the time
evolution of certain special parameters related to the Robertson-Schrödinger
(RS) uncertainty principle and some particular proposals of entanglement
measure based on collective angular-momentum generators. Our results reinforce
the connection between both the squeezing and entanglement effects, as well as
allow to investigate the basic role of spin correlations through the discrete
representatives of quasiprobability distribution functions. Entropy
functionals are also discussed in this context. The main sequence correlations
$\mapsto$ entanglement $\mapsto$ squeezing of quantum effects embraces a new
set of insights and interpretations in this framework, which represents an
effective gain for future researches in different spin systems.

###### keywords:

Spin squeezing; entanglement; finite-dimensional discrete phase spaces.

##  1 Introduction

Originally introduced by E. Schrödinger in his seminal work on probability
relations between separated systems,[1] “entanglement” indeed corresponds to a
fundamental concept in physics that lies at the heart of many conceptual
problems in quantum mechanics.[2, 3] The mere comprehension of this abstract
concept and its respective changing of status, in the recent past, for
experimental measure, certainly represents a concatenation of efforts with
significant progress in theoretical and experimental physics. It is important
to stress that ‘quantum entanglement’ plays an essential role in multipartite
quantum systems,[4] since its underlying physical properties can be used as a
specific resource for determined quantum-information tasks[5] — recently,
different proposals of quantumness correlations have appeared in current
literature,[6] emphasizing the possible connections with quantum
entanglement.[7] Therefore, understanding, identifying, measuring and,
consequently, exploiting genuine quantum effects (or “quantumness”[8]) in
multipartite systems constitute a set of obligatory prerequisites for grasping
the fundamental implications of any quantum theory.[9]

Recent theoretical proposals[10] and experiments[11, 12] involving
measurements upon collective angular-momentum generators in different physical
systems fulfil, in part, the aforementioned prerequisites, as well as
corroborate the subtle match between spin-squeezing[13] and entanglement
effects. However, some necessary questions concerning the correlations among
different spin components as chief agents responsible for spin-squeezing
effects deserve be properly answered. Then, it seems reasonable to adopt a
particular algebraic framework that, within all the inherent mathematical
virtues, allows to: (i) map the kinematical and dynamical contents of a given
spin system with a finite space of states into a ${\rm N}^{2}$-dimensional
discrete phase space; (ii) comprehend the role of correlations through
discrete Wigner functions and their connections with spin-squeezing effects;
and finally, (iii) gain new insights on the effects exhibited in the Venn
diagram below which lead us to take advantage operationally of their
quantumness in many-particle experiments. Notwithstanding the appreciable
number of papers in current literature proposing similar theoretical
frameworks with different intrinsic mathematical properties,[14] let us focus
our attention upon the formalism developed in Refs. GP1–MR which complies such
demands.

CorrelationsEntanglementSpin Squeezing Figure 1: This particular Venn diagram
depicts, in a general form, the interplay among correlations, entanglement,
and spin-squeezing effects in finite-size quantum spin systems.

Based on the technique of constructing unitary operator bases initially
formulated by Schwinger,[25] this particular discrete quantum phase-space
approach embraces a well-defined algebraic structure where the ${\rm
N}$-dimensional pre-Hilbert spaces equipped with a Hilbert-Schmidt inner
product[26] endorse the finite space of states. Moreover, it leads us (i) to
exhibit and handle the pair of complementary variables related to a specific
degree of freedom we are dealing with (as well as to recognize the quantum
correlations between them), and also (ii) to obtain additional quantum
information about the physical system through the analysis of the
corresponding discrete quasiprobability distribution functions. Henceforth,
the basic idea consists in exploring this mathematical tool in order to study
the role of those correlations in connection with spin-squeezing and
entanglement processes.

Initially focused on some mathematically appealing features inherent to the
collective angular-momentum generators, the first part of this paper basically
discuss the spin coherent states and their relations with unitary
transformations through a constructive point of view. Next, we establish a
$\mbox{mod}({\rm N})$-invariant operator basis which can be immediately
employed in the mapping of quantum operators (acting into that particular
${\rm N}$-dimensional space of states) onto well-behaved functions of discrete
variables by means of a trace operation.[27] These functions correspond to the
representatives of the operators in a ${\rm N}^{2}$-dimensional phase space
labeled by a pair of discrete variables for each degree of freedom of the
physical system under investigation. Thus, all the necessary quantities for
describing its kinematical and dynamical contents can now be promptly mapped
one-to-one on such phase spaces. For completeness sake, we also present two
different prescriptions that essentially determine the time evolution for both
the discrete Wigner and Weyl functions.

These results are then applied to an extended version of the LMG model,[28]
which was initially introduced by Vidal and coworkers[29] with the aim of
investigating the statistical-mechanical properties and entanglement effects
of a particular interacting spin system. Summarizing, the modified LMG model
describes a finite set of spins $\frac{1}{2}$ with a mutual anisotropic $(XY)$
ferromagnetic interaction, and also subjected to a transverse magnetic field.
The second part of this paper shows how the quantum correlations (here
controlled by parameters related to the transverse magnetic field and
anisotropy) affect the connection between the spin-squeezing and entanglement
effects. To develop this specific task, we first employ the RS uncertainty
principle[30] in order to determine a spin-squeezing measure which
incorporates, in its definition, the covariance function — being such a
function responsible for introducing, within the Heisenberg uncertainty
principle, important contributions associated with the anticommutation
relations of the angular-momentum generators. The subsequent comparison with
the entanglement measure proposed in Ref. Hald establishes, in this way, the
aforementioned link and reinforces the fundamental role of correlations in our
initial analysis. Moreover, we also study the behaviour of the discrete Wigner
function for certain particular values of time where occur an ‘almost perfect
match’ between both the spin-squeezing and entanglement measurements (such
procedure leads us to comprehend the intricate role of correlations in a
finite-dimensional phase space). We have finalized this work with the analysis
of some entropy functionals.[31] For completeness sake, it is worth stressing
that we have worked with a number of spins which preserves basic quantum
effects, contrary to the thermodynamic limit.

This paper is structured as follows. In Section 2, we present an important set
of essential mathematical tools related to the angular-momentum generators and
spin coherent states, as well as an interesting discussion on unitary
transformations and their corresponding geometric interpretations. In Section
3, we introduce a mapping kernel for finite-dimensional discrete phase spaces
whose inherent properties allow, within other important virtues, to describe
the kinematical and dynamical contents of a given physical system with a
finite space of states. In Section 4, we apply the quantum-algebraic approach
developed in the previous section to the modified LMG model, with the aim of
analysing the influence of quantum correlations on the spin-squeezing and
entanglement effects. Section 5 summarizes the main results obtained in this
paper and discuss some possible perspectives for future research. We have
added three appendixes related to the calculational details of certain topics
and expressions used in the previous sections: Appendix A shows how the
unitary transformations associated with angular-momentum generators affect the
expressions for variances, covariances, and RS uncertainty relation; Appendix
B discuss the Kitagawa-Ueda model[13] and the inherent spin-squeezing and
entanglement effects; while Appendix C exhibits an important set of specific
numerical computations which leads us to establish a validity domain for the
entanglement criteria studied in this work.

##  2 Definitions and background for spin coherent states

In this section, we will briefly survey some mathematically appealing features
inherent to the $\mathfrak{su}(2)$ Lie algebra with emphasis on a specific
group of unitary transformations involving the generators of this algebra
which leads us to properly define the spin coherent states. For this task, let
us initially consider the standard angular-momentum generators $\left\\{{\bf
J}_{x},{\bf J}_{y},{\bf J}_{z}\right\\}$ which act on a finite-dimensional
Hilbert space $\mathcal{H}_{\mbox{\tiny${\rm N}$}}$. The familiar commutation
relation $\left[{\bf J}_{a},{\bf J}_{b}\right]=\mathrm{i}\epsilon_{abc}{\bf
J}_{c}$ for $a,b,c=x,y,z$ (letting $\hbar=1$ for convenience), where
$\epsilon_{abc}$ denotes the Levi-Civita symbol associated with the three
orthogonal directions, permits us to define the raising and lowering operators
through the particular decomposition ${\bf J}_{\pm}\mathrel{\mathop{:}}={\bf
J}_{x}\pm\mathrm{i}{\bf J}_{y}$ with $\left[{\bf J}_{+},{\bf
J}_{-}\right]=2{\bf J}_{z}$ and $\left[{\bf J}_{z},{\bf
J}_{\pm}\right]=\pm{\bf J}_{\pm}$. Moreover, $\vec{{\bf
J}}^{2}\mathrel{\mathop{:}}={\bf J}_{x}^{2}+{\bf J}_{y}^{2}+{\bf
J}_{z}^{2}={\bf J}_{z}^{2}+\frac{1}{2}\left\\{{\bf J}_{+},{\bf
J}_{-}\right\\}$ characterizes the total spin operator and constitutes an
important element of this algebra since $\vec{{\bf J}}^{2}$ (also known as a
Casimir operator) commutes with all the generators ${\bf J}_{a}$. In
particular, these results allow to construct, for example, a set of
simultaneous eigenstates of $\vec{{\bf J}}^{2}$ and ${\bf J}_{z}$, that is,
$\left\\{|j,m\rangle:|m|\leq
j\;,\;j=0,\frac{1}{2},1,\frac{3}{2},\ldots\right\\}$, with well-known
mathematical properties.[32] [(ii)]

It is interesting to note that each particular member of this representation
(here characterized by a specific eigenvalue $j$) is analogous to the Fock
state[26] of the electromagnetic field, since it can be created by the
repeated application of the raising operator (remembering that ${\bf
J}_{\pm}^{2j+1}=0$) on the ‘vaccum state’,

$|j,m\rangle\mathrel{\mathop{:}}=\mathscr{C}_{2j,j+m}^{-\frac{1}{2}}\frac{{\bf
J}_{+}^{j+m}}{(j+m)!}|j,-j\rangle\quad\mbox{for}\quad\mathscr{C}_{r,s}\equiv\frac{r!}{s!(r-s)!}.$

Thus, the set $\left\\{|j,m\rangle\right\\}$ corresponds to a discrete,
orthonormal and complete basis for the $(2j+1)$-dimensional vector space
$\mathbb{C}^{2j+1}$ of angular-momentum states,[33] whose completeness and
orthonormality relations are expressed as follow:

$\sum_{m=-j}^{j}|j,m\rangle\langle j,m|={\bf 1}\quad\mbox{and}\quad\langle
j^{\prime},m^{\prime}|j,m\rangle=\delta_{j^{\prime},j}\,\delta_{m^{\prime},m}.$

Consequently, the expansion of any quantum state $|\Psi\rangle$ belonging to
this finite-dimensional space can now be prompty obtained,

$|\Psi\rangle=\sum_{m=-j}^{j}c_{j,m}|j,m\rangle$

where $c_{j,m}\equiv\langle j,m|\Psi\rangle$ denotes the coefficients of such
a discrete expansion.

The nondiagonal matrix elements $\langle j^{\prime},m^{\prime}|{\bf
J}_{\pm}^{k}|j,m\rangle$ associated with the moments ${\bf J}_{\pm}^{k}$ show
a dependence on the binomial coefficients $\mathscr{C}_{r,s}$, that is

$\langle j^{\prime},m^{\prime}|{\bf J}_{\pm}^{k}|j,m\rangle=\frac{(j\pm
m^{\prime})!}{(j\pm
m)!}\left(\frac{\mathscr{C}_{2j,j+m^{\prime}}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\delta_{j^{\prime},j}\,\delta_{m^{\prime},m\pm
k}\qquad(0\leq k\leq 2j),$

while $\langle j^{\prime},m^{\prime}|{\bf
J}_{z}^{k}|j,m\rangle=m^{k}\delta_{j^{\prime},j}\,\delta_{m^{\prime},m}$
demonstrates an explicit connection with the discrete component $m$. These
results are extremely useful when applied to the evaluation of nondiagonal
matrix elements related to well-known families of unitary transformations
involving the linear combination of spin operators,[32, 33] and also in the
study of spin coherent states.[34, 35, 36]

###  2.1 Unitary transformations

Let ${\bf
T}(\Omega_{\pm},\Omega_{z})\mathrel{\mathop{:}}=\exp\left(\Omega_{+}{\bf
J}_{+}+\Omega_{z}{\bf J}_{z}+\Omega_{-}{\bf J}_{-}\right)$ denote a kind of
general abstract operator expressed in terms of generators of the
$\mathfrak{su}(2)$ Lie algebra and arbitrary c-number parameters
$\Omega_{\pm}$ and $\Omega_{z}$. For $\Omega_{+}=\xi$,
$\Omega_{-}=-\xi^{\ast}$ and $\Omega_{z}=\mathrm{i}\omega$, with
$\xi\in\mathbb{C}$ and $\omega\in\mathbb{R}$, such an abstract operator
represents a generator of unitary transformations whose respective generalized
‘normal’- and ‘antinormal’-order decomposition formulae display the following
expressions:[37, 38]

$\displaystyle\\!\\!\\!\\!\\!{\bf T}(\xi,\omega)$ $\displaystyle=$
$\displaystyle\exp\left(\Lambda_{+}{\bf
J}_{+}\right)\exp\left[\ln\left(\Lambda_{z}\right){\bf
J}_{z}\right]\exp\left(\Lambda_{-}{\bf J}_{-}\right)$ (1) $\displaystyle=$
$\displaystyle\exp\left(-\Lambda_{+}^{\ast}{\bf
J}_{-}\right)\exp\left[-\ln\left(\Lambda_{z}^{\ast}\right){\bf
J}_{z}\right]\exp\left(-\Lambda_{-}^{\ast}{\bf J}_{+}\right)$

with

$\displaystyle\Lambda_{+}$ $\displaystyle=$
$\displaystyle\frac{\left(\xi/\phi\right)\sin(\phi)}{\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)},\qquad\Lambda_{-}=-\frac{\left(\xi^{\ast}/\phi\right)\sin(\phi)}{\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)},$
$\displaystyle\Lambda_{z}$ $\displaystyle=$
$\displaystyle\left[\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)\right]^{-2}\quad\mbox{and}\quad\phi=\left[|\xi|^{2}+\left(\omega/2\right)^{2}\right]^{\frac{1}{2}}.$

So, the action of ${\bf T}(\xi,\omega)$ on the generators $\\{{\bf J}_{a}\\}$
— here defined through the relation $\overline{{\bf
J}}_{a}\mathrel{\mathop{:}}={\bf T}^{\dagger}(\xi,\omega){\bf J}_{a}{\bf
T}(\xi,\omega)$ for $a=x,y,z$ — produces effectively a new set of angular-
momentum operators $\\{\overline{{\bf J}}_{a}\\}$ expressed in terms of the
old ones that, by their turn, are multiplied by determined coefficients which
depend on the parameters $\xi$ and $\omega$, namely,

$\displaystyle\overline{{\bf J}}_{x}$ $\displaystyle=$
$\displaystyle\left[\cos(2\phi)+2\mbox{${\rm
Im}$}^{2}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{x}+\left[\omega\,\frac{\sin(2\phi)}{2\phi}+2\mbox{${\rm
Re}$}(\xi)\mbox{${\rm Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{y}$ (2) $\displaystyle+\left[\omega\mbox{${\rm
Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}-\mbox{${\rm
Re}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{z},$
$\displaystyle\overline{{\bf J}}_{y}$ $\displaystyle=$
$\displaystyle\left[-\omega\,\frac{\sin(2\phi)}{2\phi}+2\mbox{${\rm
Re}$}(\xi)\mbox{${\rm Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{x}+\left[\cos(2\phi)+2\mbox{${\rm
Re}$}^{2}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf J}_{y}$ (3)
$\displaystyle+\left[\omega\mbox{${\rm
Re}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}+\mbox{${\rm
Im}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{z},$
$\displaystyle\overline{{\bf J}}_{z}$ $\displaystyle=$
$\displaystyle\left[\omega\mbox{${\rm
Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}+\mbox{${\rm
Re}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{x}+\left[\omega\mbox{${\rm
Re}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}-\mbox{${\rm
Im}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{y}$ (4)
$\displaystyle+\left[\cos(2\phi)+\frac{\omega^{2}}{2}\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{z}.$

Note that $\vec{{\bf J}}^{2}$ remains invariant under the unitary
transformation (1), which implies in the identity $\vec{\overline{{\bf
J}}}^{2}\equiv\vec{{\bf J}}^{2}$; besides, the geometric counterpart of this
particular result is directly associated with arbitrary rotations on the
surface of a sphere of radius $j(j+1)$. Indeed, let ${\bf T}(\theta,\varphi)$
denote a particular case of ${\bf T}(\xi,\omega)$ when
$\xi=\frac{\theta}{2}\exp(-\mathrm{i}\varphi)$ and $\omega=0$, which implies
in specific decomposition formulae characterized by
$\Lambda_{\pm}=\pm\tan(\theta/2)\exp(\mp\mathrm{i}\varphi)$ and
$\Lambda_{z}=\sec^{2}(\theta/2)$. The connection between $\xi$ and
$\Lambda_{+}$ reflects the stereographic projection of a two-dimensional
sphere $S^{2}$ on the complex plane $\mathbb{C}$ with one-point
compactification (in this situation, the infinite point corresponds to the
north pole of $S^{2}=\mathbb{C}\cup\\{\infty\\}$).

As a consequence, the action of ${\bf T}(\theta,\varphi)$ on the vacuum state
$|j,-j\rangle$ will define our next object of study: the spin coherent states
— also known in current literature as $\mathrm{SU}(2)$ coherent states or spin
coherent states.[36] In fact, we will establish some few important
mathematical results connected with the spin coherent states which constitute
the first basic tools for discussing the subtle link between spin squeezing
and entanglement.

###  2.2 Spin coherent states

In general, the spin coherent states can be defined by means of two different
mathematical procedures: the first one[39] basically follows the Schwinger’s
prescription of angular momentum (note that LMG model[40] represents a typical
example of this prescription) in order to produce such states, while the
second one[41] pursues an algebraic framework analogous to that adopted for
the field coherent states which allows us to describe physical systems
consisting of $N$ two-level atoms111See Refs. N1–Ag2 for a detailed discussion
on the atomic coherent-state representation and its practical applications in
multitime-correlation functions and phase-space quasidistributions. (for
instance, see the Dicke model[48]). Here, we follow closely the last way that
consists in defining the spin coherent states through the mathematical
relation $|\theta,\varphi\rangle\mathrel{\mathop{:}}={\bf
T}(\theta,\varphi)|j,-j\rangle$, with the vacuum state $|j,-j\rangle$ written
in terms of the previously mentioned basis $\\{|j,m\rangle\\}_{-j\leq m\leq
j}$. Thus, after some algebra, it is easy to show that

$\\!\\!\\!\\!\\!|\theta,\varphi\rangle=\sum_{k=0}^{2j}\left[\mathscr{C}_{2j,k}\sin^{2k}(\theta/2)\cos^{2(2j-k)}(\theta/2)\right]^{\frac{1}{2}}\exp(-\mathrm{i}k\varphi)|j,k-j\rangle$
(5)

presents a bijective mapping with the $N$-photon generalized binomial states
related to the electromagnetic field, being this correspondence properly
explored in Ref. Messina.

Next, we focus our attention on an important set of mathematical properties
inherent to the spin coherent states (5) that leads us to discuss the
squeezing effects in determined physical systems under different
circumstances. Note that a complete list of such properties can be promptly
found in Refs. Thomas and Zhang.

Non-orthogonality.

The inner product of two distinct spin coherent states can be evaluated
through the completeness relation associated with $\\{|j,m\rangle\\}$,
yielding the well-known result

$\langle\theta^{\prime},\varphi^{\prime}|\theta,\varphi\rangle=\left\\{\cos(\theta^{\prime})\cos(\theta)+\sin(\theta^{\prime})\sin(\theta)\exp[\mathrm{i}(\varphi^{\prime}-\varphi)]\right\\}^{2j}.$

For $\theta^{\prime}=\theta$ and $\varphi^{\prime}=\varphi$, we achieve
$\langle\theta,\varphi|\theta,\varphi\rangle=1$ (normalization condition).
Now, excepting for the antipodal points, the spin coherent states are in
general not orthogonal. Hence, the Majorana-Bloch sphere[51] of unit radius
represents the ideal geometric element to describe such states, since its
respective north $(\theta=\pi)$ and south $(\theta=0)$ poles correspond to the
highest/lowest states $|j,\pm j\rangle$. For completeness sake, let us briefly
mention that
$|\langle\theta^{\prime},\varphi^{\prime}|\theta,\varphi\rangle|^{2}=\cos^{4j}(\Theta/2)$
(overlap probability) with
$\cos(\Theta)\equiv\cos(\theta^{\prime})\cos(\theta)+\sin(\theta^{\prime})\sin(\theta)\cos(\varphi^{\prime}-\varphi)$
is limited to the closed interval $[0,1]$, which reinforces the overcomplete
character of the states under consideration (they do not form an orthonormal
set).

Completeness relation.

Let ${\bf P}(\theta,\varphi)=|\theta,\varphi\rangle\langle\theta,\varphi|$
denote the diagonal projector operator related to the spin coherent states
which satisfies the property[34]

$\int\mathrm{d}\Omega(\theta,\varphi){\bf P}(\theta,\varphi)={\bf
1}\quad\mbox{with}\quad\mathrm{d}\Omega(\theta,\varphi)=\frac{2j+1}{4\pi}\sin(\theta)\mathrm{d}\theta\mathrm{d}\varphi.$

In particular, such a completeness relation asserts that any quantum state
$|\Psi\rangle$ belonging to $\mathbb{C}^{2j+1}$ can be properly expanded in
this overcomplete basis, namely

$|\Psi\rangle=\int\mathrm{d}\Omega(\theta,\varphi)\Psi(\theta,\varphi)|\theta,\varphi\rangle.$

In this case, $\Psi(\theta,\varphi)$ represents a polynomial function of
degree $2j$, whose analytical expression is given by

$\Psi(\theta,\varphi)=\sum_{k=0}^{2j}\left[\mathscr{C}_{2j,k}\sin^{2k}(\theta/2)\cos^{2(2j-k)}(\theta/2)\right]^{\frac{1}{2}}\exp(\mathrm{i}k\varphi)\,\langle
j,k-j|\Psi\rangle.$

It is worth mentioning that operators acting on this Hilbert space also admit
expansions in both the nondiagonal and diagonal forms, the diagonal
representation being of particular interest to deal with statistical operators
for atoms.[41]

Minimum uncertainty states.

Note that spin coherent states constitute an important class of minimum
uncertainty states. To demonstrate this assertion, let us initially consider
the RS uncertainty principle[30] for ${\bf J}_{x}$ and ${\bf J}_{y}$, i.e.,

$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{y}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\right)^{2}\geq\frac{1}{4}\left|\langle\left[{\bf
J}_{x},{\bf J}_{y}\right]\rangle\right|^{2}=\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}$ (6)

where
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\equiv\langle\frac{1}{2}\left\\{{\bf
J}_{x},{\bf J}_{y}\right\\}\rangle-\langle{\bf J}_{x}\rangle\langle{\bf
J}_{y}\rangle$ represents the covariance function, and
$\mathscr{V}_{\mathrm{J}_{a}}\equiv\langle{\bf J}_{a}^{2}\rangle-\langle{\bf
J}_{a}\rangle^{2}$ stands for usual variance when $a=x,y$. So, if one
considers the previous mean values evaluated for the spin coherent states, it
is immediate to verify the equality sign in this equation since both the
expressions are equal to $\frac{1}{4}j^{2}\cos^{2}(\theta)$. In fact, this
particular result can be extended in order to include any non-parametrized
complex variable — see Appendix A for further results.

##  3 Mappings via finite-dimensional discrete phase spaces

In this section, we will introduce certain basic mathematical tools
constituents of the pioneering formalism initially developed in Refs. GP1 and
GP2 for physical systems with finite-dimensional space of states
$\mathcal{H}_{\mbox{\tiny${\rm N}$}}$. It is worth stressing that such tools
will represent our guidelines for the fundamentals of the formal description
of finite-dimensional phase spaces by means of discrete variables. For this
specific task, let us first establish the mod(N)-invariant operator basis[17]

${\bf G}(\mu,\nu)\mathrel{\mathop{:}}=\frac{1}{\sqrt{{\rm
N}}}\sum_{\eta,\xi=-\ell}^{\ell}\exp\left[-\frac{2\pi\mathrm{i}}{{\rm
N}}\left(\eta\mu+\xi\nu\right)\right]{\bf S}(\eta,\xi)$ (7)

expressed in terms of a discrete double Fourier transform of the symmetrized
unitary operator basis[15]

${\bf S}(\eta,\xi)\mathrel{\mathop{:}}=\frac{1}{\sqrt{{\rm
N}}}\exp\left(\frac{\pi\mathrm{i}}{{\rm N}}\eta\xi\right){\bf U}^{\eta}{\bf
V}^{\xi},$

where the labels $\eta$ and $\xi$ are associated with the dual momentum and
coordinatelike variables of an ${\rm N}^{2}$-dimensional discrete phase space
here endorsed by an underlying presymplectic structure of geometric
origin.[16] Note that these discrete labels obey the arithmetic modulo N and
assume integer values in the symmetric interval $[-\ell,\ell]$ for
$\ell=\frac{{\rm N}-1}{2}$ fixed.222Henceforth, for convenience, we assume
${\rm N}$ odd throughout this paper. However, it is important to stress that
even dimensionalities can also be dealt with simply by working on non-
symmetrized intervals. A comprehensive and useful compilation of results and
properties of the unitary operators ${\bf U}$ and ${\bf V}$ can be found in
Ref. DM1, since the primary focus of our attention is the essential features
exhibited by $\\{{\bf G}(\mu,\nu)\\}_{\mu,\nu=-\ell,\ldots,\ell}$.

The set of ${\rm N}^{2}$ operators $\\{{\bf G}(\mu,\nu)\\}$ allows, for
instance, to decompose any linear operator ${\bf O}$ acting on
$\mathcal{H}_{\mbox{\tiny${\rm N}$}}$ by means of the expansion

${\bf O}=\frac{1}{{\rm N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{O}(\mu,\nu){\bf
G}(\mu,\nu),$ (8)

where the coefficients $\mathscr{O}(\mu,\nu)\equiv\mbox{${\rm Tr}$}\left[{\bf
G}^{\dagger}(\mu,\nu){\bf O}\right]$ are evaluated through trace operation and
correspond, in this context, to a one-to-one mapping between operators and
functions embedded in a finite phase-space characterized by the discrete
variables $\mu$ and $\nu$.[27] So, if one considers the density operator
$\rho$ in such a case, we verify that

$\mbox{\boldmath$\rho$}=\frac{1}{{\rm
N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{W}_{\rho}(\mu,\nu){\bf G}(\mu,\nu)$ (9)

admits a plausible expansion with intrinsic mathematical properties since the
coefficients result in the discrete Wigner function
$\mathscr{W}_{\rho}(\mu,\nu)\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}\left[{\bf
G}^{\dagger}(\mu,\nu)\mbox{\boldmath$\rho$}\right]$. The first practical
consequence of this decomposition yields the mean value

$\langle{\bf O}\rangle\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}[{\bf
O}\mbox{\boldmath$\rho$}]=\frac{1}{{\rm
N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{O}(\mu,\nu)\mathscr{W}_{\rho}(\mu,\nu),$
(10)

whose formal expression depends explicitly on the product of both mapped forms
related to ${\bf O}$ and $\rho$. Consequently, the moments $\langle{\bf
J}_{a}^{k}\rangle$ for $a=x,y,z$ and $0\leq k\leq 2j$ can now be promptly
obtained from this formalism for any finite quantum state belonging to
$\mathcal{H}_{2j+1}$. Next, we apply such a mapping technique for the spin
coherent states in order to establish an important link between angular-
momentum operators and $(2j+1)^{2}$-dimensional discrete phase spaces.

###  3.1 Applications

According to a prescription adopted in Ref. GP3, let
$\\{|j,m\rangle\\}_{-j\leq m\leq j}$ denote the eigenstates of ${\bf U}$ with
eigenvalues $\\{\omega^{m}\\}$ for
$\omega=\exp\left(\frac{2\pi\mathrm{i}}{2j+1}\right)$ fixed. This assumption
leads us to establish the general properties ${\bf
U}^{\eta}|j,m\rangle=\omega^{m\eta}|j,m\rangle$ and ${\bf
V}^{\xi}|j,m\rangle=|j,m-\xi\rangle$ (together with the relations ${\bf
U}^{2j+1}={\bf 1}$, ${\bf V}^{2j+1}={\bf 1}$, and ${\bf V}^{\xi}{\bf
U}^{\eta}=\omega^{\eta\xi}{\bf U}^{\eta}{\bf V}^{\xi}$), which represent a
satisfactory mathematical connection for the present purpose. Next, we
determine some useful results related to the angular-momentum operators and
spin coherent states, remembering that ${\bf G}^{\dagger}(\mu,\nu)={\bf
G}(\mu,\nu)$ for $-\ell\leq\mu,\nu\leq\ell$.

As a first and pertinent application, let us consider the mapped expressions

$\displaystyle\left({\bf J}_{z}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j}m^{k}\underbrace{\langle j,m|{\bf
G}(\mu,\nu)|j,m\rangle}_{\mathscr{G}_{j,m|j,m}(\mu,\nu)}$
$\displaystyle\left({\bf J}_{+}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j-k}\frac{(j+m+k)!}{(j+m)!}\left(\frac{\mathscr{C}_{2j,j+m+k}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\underbrace{\langle
j,m|{\bf G}(\mu,\nu)|j,m+k\rangle}_{\mathscr{G}_{j,m|j,m+k}(\mu,\nu)}$
$\displaystyle\left({\bf J}_{-}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j+k}^{j}\frac{(j-m+k)!}{(j-m)!}\left(\frac{\mathscr{C}_{2j,j+m-k}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\underbrace{\langle
j,m|{\bf G}(\mu,\nu)|j,m-k\rangle}_{\mathscr{G}_{j,m|j,m-k}(\mu,\nu)},$

which depend on the nondiagonal matrix elements

$\mathscr{G}_{j,m|j,m^{\prime}}(\mu,\nu)=\frac{1}{2j+1}\sum_{\beta=-j}^{j}\exp\left\\{-\frac{2\pi\mathrm{i}}{2j+1}\left[\beta(\mu-m^{\prime})+(m^{\prime}-m)\left(\nu+\frac{\beta}{2}\right)\right]\right\\}$

associated with the discrete mapping kernel ${\bf G}(\mu,\nu)$ for
$m^{\prime}=m\pm k$ and $0\leq k\leq 2j$. Since the diagonal matrix element
$\mathscr{G}_{j,m|j,m}(\mu,\nu)=\delta_{m,\mu}^{[2j+1]}$ gives the Kronecker
delta function (in such a case, the superscript $[2j+1]$ denotes that this
function is different from zero when its labels are congruent modulo $2j+1$),
it seems reasonable to obtain a simple expression in the first example, that
is $\left({\bf J}_{z}^{k}\right)(\mu,\nu)=\mu^{k}$; however, this fact is not
verified for $\left({\bf J}_{\pm}^{k}\right)(\mu,\nu)$. From the kinematical
and dynamical point of view, these mapped expressions represent an
advantageous set of mathematical tools which allows us to comprehend certain
intriguing problems related to discrete symmetries of spin systems.[18]

Basically, the second application consists in evaluating the discrete Wigner
function $\mathscr{W}_{\theta,\varphi}(\mu,\nu)$ present in the expansion of
the projector ${\bf P}(\theta,\varphi)$ via Eq. (9), namely,

${\bf
P}(\theta,\varphi)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\mathscr{W}_{\theta,\varphi}(\mu,\nu){\bf
G}(\mu,\nu).$

In general, this particular quasidistribution can be written in terms of its
respective discrete Weyl function
$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}[{\bf S}(\eta,\xi){\bf P}(\theta,\varphi)]$ as follows:

$\\!\\!\\!\\!\\!\mathscr{W}_{\theta,\varphi}(\mu,\nu)=\frac{1}{\sqrt{2j+1}}\sum_{\mu,\nu=-j}^{j}\exp\left[-\frac{2\pi\mathrm{i}}{2j+1}(\eta\mu+\xi\nu)\right]\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi).$
(11)

So, after some lengthy calculations, the analytical expression for
$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)$ assumes the exact form

$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)=\frac{1}{\sqrt{2j+1}}\sum_{m=-j}^{j}\exp\left[\frac{2\pi\mathrm{i}}{2j+1}\eta\left(m-\frac{\xi}{2}\right)\right]\langle\theta,\varphi|j,m-\xi\rangle\langle
j,m|\theta,\varphi\rangle,$

where $\langle j,m|\theta,\varphi\rangle$ can be promptly obtained from Eq.
(5). It is interesting to stress that both the north $(\theta=\pi)$ and south
$(\theta=0)$ poles of the Majorana-Bloch sphere correspond to the limit cases
$\mathscr{W}_{\pi,\varphi}(\mu,\nu)=\delta_{\mu,j}^{[2j+1]}$ and
$\mathscr{W}_{0,\varphi}(\mu,\nu)=\delta_{\mu,-j}^{[2j+1]}$, which reflect
directly the highest/lowest states $|j,\pm j\rangle$ for any
$\varphi\in[0,2\pi)$. Furthermore, numerical calculations related to Eq. (11)
suggest that some relevant contributions are connected with discrete values of
the angles $\theta$ and $\varphi$ — in particular, integer multiples of
$\frac{\pi}{2j+1}$. This evidence supports the ideas of Buniy and
coworkers[52] about a ‘possible discretization’ of the Majorana-Bloch sphere,
as well as reveals some important features — and not yet properly explored —
on the discrete nature of quantum states in finite-dimensional Hilbert
spaces.333In the quantum-gravity scope,[53] it is worth mentioning that
certain theoretical approaches to the generalized uncertainty principle (GUP)
also suggest the breakdown of the spacetime continuum picture near to Planck
scale.[54] From an experimental point of view, it seems reasonable to argue
that ‘discreteness is actually less speculative than absolute continuity’.

###  3.2 Time evolution

Here, we establish a mathematical recipe that permits to investigate the
dynamics of a particular quantum system characterized by a finite space of
(discrete) states. For this task, let $\mbox{\boldmath$\rho$}(t)$ describe the
state of this physical system whose interaction with any dissipative
environment is, in principle, automatically discarded. Besides, let us
consider (for convenience) only time-independent Hamiltonians; consequently,
the time-evolution of that density operator will be governed, in such a case,
by the well-known von Neumann-Liouville equation
$\mathrm{i}\hbar\partial_{t}\mbox{\boldmath$\rho$}(t)=[{\bf
H},\mbox{\boldmath$\rho$}(t)]$. So, those initial premises represent the
constituting blocks for the aforementioned recipe.

As a first proposal in our prescription, we determine a mapped expression for
the von Neumann-Liouville equation which describes the time evolution of the
discrete Wigner function $\mathscr{W}_{\rho}(\mu,\nu;t)$. In this sense, the
mapping technique sketched in this section yields the differential equation

$\mathrm{i}\hbar\partial_{t}\mathscr{W}_{\rho}(\mu,\nu;t)=\sum_{\mu^{\prime},\nu^{\prime}=-\ell}^{\ell}\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})\mathscr{W}_{\rho}(\mu^{\prime},\nu^{\prime};t),$
(12)

where $\mathscr{L}_{\mbox{\tiny${\rm H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})$
represents the mapped form of the Liouville operator written in terms of
$\mathit{H}(\mu^{\prime\prime},\nu^{\prime\prime})=\mbox{${\rm Tr}$}\left[{\bf
G}(\mu^{\prime\prime},\nu^{\prime\prime}){\bf H}\right]$, that is,

$\displaystyle\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2\mathrm{i}}{{\rm
N}^{4}}\sum_{\Delta}\sin\left[\frac{\pi}{{\rm
N}}(\alpha\beta^{\prime}-\alpha^{\prime}\beta)\right]\exp\left\\{\frac{2\pi\mathrm{i}}{{\rm
N}}\left[\alpha(\mu-\mu^{\prime})+\beta(\nu-\nu^{\prime})\right]\right\\}$
$\displaystyle\times\exp\left\\{\frac{2\pi\mathrm{i}}{{\rm
N}}\left[\alpha^{\prime}(\mu-\mu^{\prime\prime})+\beta^{\prime}(\nu-\nu^{\prime\prime})\right]\right\\}\mathit{H}(\mu^{\prime\prime},\nu^{\prime\prime}).$

Note that $\Delta$ denotes the set
$\\{\alpha,\beta,\alpha^{\prime},\beta^{\prime},\mu^{\prime\prime},\nu^{\prime\prime}\\}\in[-\ell,\ell]$
in this expression. In the following, let us mention some few words about the
formal solution of Eq. (12): it can be expressed analytically in terms of the
series[20]

$\mathscr{W}_{\rho}(\mu,\nu;t)=\sum_{\kappa,\tau=-\ell}^{\ell}\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})\mathscr{W}_{\rho}(\kappa,\tau;t_{0}),$
(13)

whose ${\rm N}^{2}$-dimensional discrete phase-space propagator admits the
expansion

$\displaystyle\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})$ $\displaystyle=$
$\displaystyle\delta_{\kappa,\mu}^{[\mbox{\tiny${\rm
N}$}]}\delta_{\tau,\nu}^{[\mbox{\tiny${\rm
N}$}]}+\frac{\mathrm{i}}{1!\hbar}(t-t_{0})\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\kappa,\tau)$
$\displaystyle+\frac{\mathrm{i}^{2}}{2!\hbar^{2}}(t-t_{0})^{2}\sum_{\kappa^{\prime},\tau^{\prime}=-\ell}^{\ell}\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\kappa^{\prime},\tau^{\prime})\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\kappa^{\prime},\tau^{\prime},\kappa,\tau)+\cdots$

which allows to evaluate directly the time evolution of
$\mathscr{W}_{\rho}(\mu,\nu;t)$ by using the series related to the iterated
application of the mapped Liouville operator. The advantages and/or
disadvantages from this particular formal solution were adequately discussed
in Ref. GR1, and subsequently applied with great success in the discrete
Husimi-function context for the LMG model.[23]

The alternative proposal establishes a differential equation for the discrete
Weyl function analogous to Eq. (12). Then, after some calculations related to
the product of three symmetrized unitary operator bases and its respective
trace operation,[19] it is immediate to show that such an equation can be
written as

$\mathrm{i}\hbar\partial_{t}\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)=\sum_{\eta^{\prime},\xi^{\prime}=-\ell}^{\ell}\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})\widetilde{\mathscr{W}}_{\rho}(\eta^{\prime},\xi^{\prime};t),$
(14)

where
$\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}\left[{\bf S}(\eta,\xi)\mbox{\boldmath$\rho$}(t)\right]$ corresponds to
our object of study and

$\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})=\frac{2\mathrm{i}}{\sqrt{N}}\sin\left[\frac{\pi}{N}\left(\eta^{\prime}\xi-\xi^{\prime}\eta\right)\right]\underbrace{\mathcal{H}(\eta-\eta^{\prime},\xi-\xi^{\prime})}_{\mathrm{Tr}\left[{\bf
S}(\eta-\eta^{\prime},\xi-\xi^{\prime}){\bf H}\right]}$

yields the mapped forms of the Liouville and Hamiltonian operators. As
expected, Eq. (14) describes the time evolution of the discrete Weyl function
whose formal solution is given by

$\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)=\sum_{\eta^{\prime},\xi^{\prime}=-\ell}^{\ell}\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})\widetilde{\mathscr{W}}_{\rho}(\eta^{\prime},\xi^{\prime};t_{0}),$
(15)

with $\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})$ being the
discrete dual phase-space propagator which admits a time expansion similar to
$\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})$, that is

$\displaystyle\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})$
$\displaystyle=$ $\displaystyle\delta_{\eta^{\prime},\eta}^{[\mbox{\tiny${\rm
N}$}]}\delta_{\xi^{\prime},\xi}^{[\mbox{\tiny${\rm
N}$}]}+\frac{\mathrm{i}}{1!\hbar}(t-t_{0})\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})$
$\displaystyle+\frac{\mathrm{i}^{2}}{2!\hbar^{2}}(t-t_{0})^{2}\sum_{\eta^{\prime\prime},\xi^{\prime\prime}=-\ell}^{\ell}\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime\prime},\xi^{\prime\prime})\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta^{\prime\prime},\xi^{\prime\prime},\eta^{\prime},\xi^{\prime})+\cdots.$

Since the discrete Wigner and Weyl functions are connected by means of a
double Fourier transform, the low operational costs involved in this
mathematical recipe are advantageous — from a computational point of view — if
compared with the previous one. Notwithstanding the apparent advantage, it is
worth remembering that $\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)$ is a
complex function and hence it represents an intermediate step for the main
goal, namely, the time evolution of the discrete Wigner function — and
consequently, the time evolution of certain mean values via Eq. (10). The
schematic diagram shown below

(24)

represents a summary of the previous proposals for describing the dynamics of
a finite quantum system represented in a ${\rm N}^{2}$-dimensional discrete
phase space.[20]

##  4 Spin squeezing and entanglement

This section will illustrate how the mapping techniques leading to finite-
dimensional discrete phase spaces can be effectively used in the study of
important quantum effects, such as spin squeezing and entanglement. Basically,
we will invoke the LMG model already discussed in current literature (e.g.,
see Ref. Ring) which exhibits not only strong mathematical and physical
appeals,[23] but also a fundamental physical property: quantum correlations
due to different types of interactions and mediated by a transverse magnetic
field. Since quantum correlations are responsible for the aforementioned
effects, it is natural to investigate how the interactions involved in this
model affect the standard quantum noise related to the spin coherent
states.444In Appendix B, we investigate a soluble spin model with lowest-order
nonlinear interaction in ${\bf J}_{z}$,[13] whose algebraic features permit to
attain a particular set of analytical results very useful in the study of spin
squeezing and entanglement effects via quantum correlations.[11]

###  4.1 The modified Lipkin-Meshkov-Glick model

Here, we adopt the prescription established by Vidal and coworkers[29] for the
modified LMG model through the Hamiltonian operator (written in the spin
language)

${\bf H}=-2h{\bf J}_{z}-2\wp_{+}\left[\vec{{\bf J}}^{2}-{\bf
J}_{z}^{2}-(N/2){\bf I}\right]-\wp_{-}\left({\bf J}_{+}^{2}+{\bf
J}_{-}^{2}\right),$ (25)

which describes a set of $N$ spins half mutually interacting in the $xy$ plane
subjected to a transverse magnetic field $h$. The coefficients
$\wp_{\pm}=\frac{\lambda}{2N}(1\pm\gamma)$ have an important role in this
description: they allow to (i) investigate the different anti-ferromagnetic
$(\lambda<0)$ and ferromagnetic $(\lambda>0)$ cases inherent to the model for
any anisotropy parameter $|\gamma|\leq 1$ ($\gamma=1$ refers to the isotropic
case), and also (ii) ensure that the free energy per spin is finite in the
thermodynamical limit. In particular, such a model presents a second-order
quantum phase transition at $\lambda=|h|$ for $\lambda>0$ fixed, whose
symmetric $(\lambda<|h|)$ and broken $(\lambda>|h|)$ phases are well-defined
within the mean-field approach; furthermore, its ground-state entanglement
properties exhibit a rich structure which reflects the internal symmetries of
the Hamiltonian operator ${\bf H}$.[29, 55] Indeed, such an operator preserves
the magnitude of the total spin operator since $[{\bf H},\vec{{\bf J}}^{2}]=0$
for all $\gamma$, and does not couple states having a different number of
spins pointing in the field direction, namely,

$\biggl{[}{\bf
H},\prod_{j=1}^{N}\mbox{\boldmath$\sigma$}_{j,z}\biggr{]}=0\qquad(\mbox{spin-
flip symmetry}).$

Consequently, it is immediate to verify that Eq. (25) can be diagonalized
within each $(2j+1)$-dimensional multiplet labelled by the eigenvalues of
$\vec{{\bf J}}^{2}$ and ${\bf J}_{z}$, this fact being responsible for the
soluble character of the associated spin model.555It is important to stress
that the prescritpion here adopted does not coincide with the original Lipkin-
Meshkov-Glick model[28], which was initially introduced over 40 years ago in
nuclear physics[40] for treating certain fermionic systems. In fact, this new
modified version brings to scene the modern statistical mechanics point of
view, where the collective properties of spin systems can be worked out with
great success.[29, 55] The extended original version of the LMG model, which
explores the parity symmetry via finite-dimensional discrete phase spaces, can
be found in Ref. GP3.

Next, let us mention some few words about two discrete conserved quantities
inherent to the spin model under investigation which reflect certain
additional symmetry properties for $\gamma=1$ fixed. In this case, the
simplest operator commuting with ${\bf H}$, therefore giving a constant of
motion, is the parity operator $\Pi$ — here defined as
$\mbox{\boldmath$\Pi$}\mathrel{\mathop{:}}={\bf
R}_{z}(\pi)\equiv\exp(\mathrm{i}\pi{\bf J}_{z})$. This result tells us that
the Hamiltonian matrix, in the ${\bf J}_{z}$ representation, breaks into two
disjoint blocks involving only even and odd eigenvalues of ${\bf J}_{z}$,
respectively. The second interesting quantity comes from the anticommutation
relation $\\{{\bf H},{\bf R}\\}=0$ for ${\bf R}(\pi,0,\pi/2)\equiv{\bf
R}_{x}(\pi){\bf R}_{z}(\pi/2)$: it corresponds to a particular rotation of the
angular-momentum quantization frame by the Euler angles $(\pi,0,\pi/2)$,
transforming, in this way, ${\bf H}\rightarrow-{\bf H}$ for the isotropic
case. Thus, if $|E_{j}\rangle$ is an energy eigenstate with eigenvalue
$E_{j}$, then ${\bf R}|E_{j}\rangle$ is also an eigenstate of ${\bf H}$ with
eigenvalue $-E_{j}$. This specific symmetry property gives rise to an energy
spectrum that is symmetric about zero.[18, 23, 29]

After this condensed review, we establish below a sequence of steps that
allows us to calculate the time evolution of several quantities necessary for
investigating both the squeezing and entanglement effects associated with the
aforementioned spin model in terms of the parameters $h$ and $\wp_{\pm}$. The
first one consists in adopting the theoretical framework described in the
previous section for the time-dependent discrete Wigner function
$\mathscr{W}_{\rho}(\mu,\nu;t)$ defined upon a $(2j+1)^{2}$-dimensional
discrete phase space labeled by the angular-momentum and angle pair
$(\mu,\nu)\in[-j,j]$ — in particular, we adopt the theoretical prescription
established in Ref. GR1 for the angle variable, that is
$\theta_{\nu}=\frac{2\pi}{2j+1}\nu$. Since this approach depends on the Wigner
function evaluated at time $t_{0}=0$, the second step consists in fixing the
spin coherent states as initial state — see Eq. (11). The last step refers to
the numerical calculation of the moments

$\langle{\bf
J}_{a}^{k}\rangle_{\theta,\varphi}(t)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\left({\bf
J}_{a}^{k}\right)(\mu,\nu)\mathscr{W}_{\theta,\varphi}(\mu,\nu;t)$ (26)

and covariance functions in the Schrödinger picture fixing the initial state
in $\theta=\frac{\pi}{2}$ and $\varphi=0$ (in this way, the uncertainties are
redistributed between the orthogonal components in the $yz$-plane), and also
considering (for convenience, not necessity) only the ferromagnetic case
$\lambda=1$. Henceforth, Eq. (25) will assume the simplified form ${\bf
H}^{\prime}=-h{\bf J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf
J}_{y}^{2}\right)$, where the constant term $\frac{1}{4}(1+\gamma){\bf I}$ was
suppressed at this initial stage since it represents only a phase for the time
evolution operator with null contribution (as expected) within our
computational approach.

###  4.2 The match between squeezing and entanglement effects

In this subsection, we adopt the theoretical framework established for the
Kitagawa-Ueda model (see Appendix B) concerning the match between squeezing
and entanglement effects. In particular, let us initially consider the results
obtained in Table 4.2 for the RS uncertainty principle related to the angular-
momentum generators and its connections with the $\mathcal{S}$-inequalities.
Since the covariance function $\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}(t)$
has a central role in this algebraic approach and requires only numerical
computations of $\langle{\bf J}_{a}^{k}\rangle_{\theta,\varphi}(t)$ and
$\langle\\{{\bf J}_{a},{\bf J}_{b}\\}\rangle_{\theta,\varphi}(t)$, it is
convenient to clarify certain subtle steps that are inherent to the exact
analytical calculation of the last quantity. According to the mapping
technique discussed in Section 3, the expression for

$\\!\\!\\!\\!\\!\langle\\{{\bf J}_{a},{\bf
J}_{b}\\}\rangle_{\theta,\varphi}(t)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\left(\\{{\bf
J}_{a},{\bf J}_{b}\\}\right)(\mu,\nu)\mathscr{W}_{\theta,\varphi}(\mu,\nu;t)$
(27)

basically depends on the mapped form of the anticommutation relation, that is

$\left(\\{{\bf J}_{a},{\bf
J}_{b}\\}\right)(\mu,\nu)=\frac{1}{(2j+1)^{2}}\sum_{\Omega}\Gamma_{{\rm
A}}(\mu,\nu|\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime})\left({\bf
J}_{a}\right)(\mu^{\prime},\nu^{\prime})\left({\bf
J}_{b}\right)(\mu^{\prime\prime},\nu^{\prime\prime})$

where $\Omega$ stands for
$\\{\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime}\\}\in[-j,j]$.
It is worth stressing that[19]

$\displaystyle\Gamma_{{\rm
A}}(\mu,\nu|\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime})=\frac{2}{(2j+1)^{2}}\sum_{\Omega^{\prime}}\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta^{\prime}(\mu-\mu^{\prime})+\xi^{\prime}(\nu-\nu^{\prime})\right]\right\\}$
$\displaystyle\qquad\times\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta^{\prime\prime}(\mu-\mu^{\prime\prime})+\xi^{\prime\prime}(\nu-\nu^{\prime\prime})\right]\right\\}\cos\left[\frac{\pi}{2j+1}(\eta^{\prime}\xi^{\prime\prime}-\xi^{\prime}\eta^{\prime\prime})\right]$

shows explicitly the embryonic structure of the continuous cosine function
presents in the well-known Weyl-Wigner-Moyal phase space approach[56] — here,
$\Omega^{\prime}$ denotes the set
$\\{\eta^{\prime},\xi^{\prime},\eta^{\prime\prime},\xi^{\prime\prime}\\}\in[-j,j]$.
Thus, the $\mathcal{S}$-inequalities can be properly estimated for the
modified LMG model, which lead us to investigate the squeezing effects. For
completeness sake, let us briefly mention that $\langle[{\bf J}_{a},{\bf
J}_{b}]\rangle_{\theta,\varphi}(t)$ yields an expression analogous to Eq.
(27), but with two minor modifications: the function $\cos(z)$ in
$\Gamma_{{\rm A}}$ should be replaced by $\mathrm{i}\sin(z)$ (keeping constant
the argument of the trigonometric functions), which implies in the change
$\Gamma_{{\rm A}}\rightarrow\Gamma_{{\rm C}}$.

The $\mathcal{S}$-inequalities exhibit unique mathematical virtues since they
yield a direct connection with the RS uncertainty principle related to the
non-commuting pair $\\{{\bf J}_{a},{\bf J}_{b}\\}_{a,b=x,y,z}$ of angular-
momentum operators, where the covariance functions have an important role.
Indeed, such inequalities lead us to investigate the squeezing effects through
a well-established criterion in literature: for $\mathcal{S}_{a}^{(c)}<1$
(squeezing condition) and $\mathcal{S}_{b}^{(c)}>1$, the inequality
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}>1$ is always preserved; moreover,
the saturation $\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}=1$ describes
minimum uncertainty states. In this table, we show all the possible links
among RS uncertainty principles and $\mathcal{S}$-inequalities, with
$\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}$ restricted to the closed interval
$\left[-\sqrt{\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}},\sqrt{\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}}\,\right]$;
moreover, the superscript $(c)$ of the product
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}$ denotes the angular-momentum
component resulting from the commutation relation between ${\bf J}_{a}$ and
${\bf J}_{b}$.  RS uncertainty principles $\mathcal{R}$-denominators
$\mathcal{S}$-inequalities $\\{{\bf J}_{x},{\bf J}_{y},{\bf
J}_{z}\\}\in\mathcal{H}_{\mbox{\tiny${\rm N}$}}$ $\mathcal{R}_{abc}\neq
0\;\forall\mbox{\boldmath$\rho$}\in\mathcal{H}_{\mbox{\tiny${\rm N}$}}$
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}/\mathcal{R}_{abc}^{2}$
$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{y}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}$
$\mathcal{R}_{xyz}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}})^{2}+\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{x}^{(z)}\mathcal{S}_{y}^{(z)}\geq 1$
$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{z}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{y}\rangle|^{2}$
$\mathcal{R}_{xzy}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}})^{2}+\frac{1}{4}|\langle{\bf
J}_{y}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{x}^{(y)}\mathcal{S}_{z}^{(y)}\geq 1$
$\mathscr{V}_{\mathrm{J}_{y}}\mathscr{V}_{\mathrm{J}_{z}}-\left(\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{x}\rangle|^{2}$
$\mathcal{R}_{yzx}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}})^{2}+\frac{1}{4}|\langle{\bf
J}_{x}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{y}^{(x)}\mathcal{S}_{z}^{(x)}\geq 1$

In order to illustrate the theoretical framework developed in this work, let
us now consider the $\mathcal{S}$-inequality
$\mathcal{S}_{y}^{(x)}\mathcal{S}_{z}^{(x)}\geq 1$ and the
$\mathcal{E}$-inequalities $\\{\mathcal{E}_{y},\mathcal{E}_{z}\\}$ described
in Appendix B by means of Eq. (42). In particular, this procedure allows not
only to establish a direct link between squeezing and entanglement effects for
the modified LMG model, but also to investigate how these effects are affected
by the transverse magnetic field $h$ and anisotropy parameter $\gamma$ with
$N=20$ fixed. For instance, Fig. 2(a,c) represents the plots of
$\mathcal{S}_{y}^{(x)}(t)$ (dot-dashed line) and $\mathcal{S}_{z}^{(x)}(t)$
(solid line) versus $t\in[0,50]$ for different values of $(h,\gamma)$: (a)
$(-0.1,0.2)$ and (c) $(-0.13,0.1)$. It is interesting to observe at a first
glance how the squeezing effect is sensitive to small variations in the
parameters $|h|$ and $\gamma$. Further numerical computations corroborate this
sensitivity and allow us to produce the following proper description about
such an evidence as time goes on: for $\gamma>|h|$, both the parameters
$\mathcal{S}_{y}^{(x)}(t)$ and $\mathcal{S}_{z}^{(x)}(t)$ exhibit squeezing
effect and oscillatory behaviour due to the prevalence of quantum correlation
effects (here introduced by the anisotropy parameter $\gamma$) on the
transverse magnetic field $h$; however, if one considers $\gamma<|h|$, the
squeezing effect is strongly reduced (in fact, it always survives for small
values of time) in both the parameters, which reveals the relative strong
influence of $h$ on the quantum correlation effects. With this in mind, let us
now consider Fig. 2(b,d) where the plots of $\mathcal{E}_{y}(t)$ (dot-dashed
line) and $\mathcal{E}_{z}(t)$ (solid line) are exhibited for the same values
of $(h,\gamma)$ used, respectively, in 2(a,c). The ‘almost perfect match’
between squeezing and entanglement effects demonstrate the previous assertions
and indeed reinforce the fundamental sequence correlation $\mapsto$
entanglement $\mapsto$ squeezing of quantum effects.

Figure 2: The ‘almost perfect match’ between squeezing and entanglement
effects is clearly visible in this selection of pictures, where the parameters
(a,c) $\mathcal{S}_{y}^{(x)}(t)$ (dot-dashed line) and
$\mathcal{S}_{z}^{(x)}(t)$ (solid line) as well as (b,d) $\mathcal{E}_{y}(t)$
(dot-dashed line) and $\mathcal{E}_{z}(t)$ (solid line) are plotted as
functions of time and placed side by side for immediate comparision.
Furthermore, this set of plots corresponds to two different values of
$(h,\gamma)$, namely, (a,b) $(-0.1,0.2)$ and (c,d) $(-0.13,0.1)$, which leads
us to investigate how the quantum correlations are affected by the transverse
magnetic field $h$ and anisotropy parameter $\gamma$ via
$(2j+1)^{2}$-dimensional discrete phase space inherent to the modified LMG
model.

At this moment, we pay special attention to the results obtained by Tóth and
co-workers[10] where a full set of generalized spin-squeezing inequalities for
entanglement detection was derived and studied in detail. For such task, let
us formally establish an entanglement criterion for separable states through a
set of inequalities involving the mean values $\left\\{\langle{\bf
J}_{a}\rangle,\langle{\bf J}_{a}^{2}\rangle\right\\}_{a=x,y,z}$ that
summarizes those aforementioned results.

Entanglement criterion.

Let $\rho$ describe an $N$-particle density operator, as well as
$\left\\{\langle{\bf J}_{x}\rangle,\langle{\bf J}_{y}\rangle,\langle{\bf
J}_{z}\rangle,\langle{\bf J}_{x}^{2}\rangle,\langle{\bf
J}_{y}^{2}\rangle,\langle{\bf J}_{z}^{2}\rangle\right\\}$ characterize the
mean values of collective angular-momentum operators associated with the
physical system of interest which, by hypothesis, are previously known. For
separable states represented by the convex sum
$\mbox{\boldmath$\rho$}=\sum_{k}p_{k}\mbox{\boldmath$\rho$}_{k}^{(1)}\otimes\mbox{\boldmath$\rho$}_{k}^{(2)}\otimes\ldots\otimes\mbox{\boldmath$\rho$}_{k}^{(N-1)}\otimes\mbox{\boldmath$\rho$}_{k}^{(N)}$,
where $p_{k}$ features the probability distribution for a given $k$, one
verifies that

$\displaystyle\langle{\bf J}_{x}^{2}\rangle+\langle{\bf
J}_{y}^{2}\rangle+\langle{\bf J}_{z}^{2}\rangle$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}N(N+2)$ (28)
$\displaystyle\frac{1}{2}(N+2)\left(\mathscr{V}_{\mathrm{J}_{x}}+\mathscr{V}_{\mathrm{J}_{y}}+\mathscr{V}_{\mathrm{J}_{z}}\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{4}N(N+2)$ (29)
$\displaystyle\frac{1}{2}(N+2)\left[\langle{\bf J}_{a}^{2}\rangle+\langle{\bf
J}_{b}^{2}\rangle-(N-1)\mathscr{V}_{\mathrm{J}_{c}}\right]$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}N(N+2)$ (30)
$\displaystyle(N-1)\left(\mathscr{V}_{\mathrm{J}_{a}}+\mathscr{V}_{\mathrm{J}_{b}}\right)-\langle{\bf
J}_{c}^{2}\rangle+N$ $\displaystyle\geq$ $\displaystyle\frac{1}{4}N(N+2)$ (31)

are always fulfilled, $\frac{1}{4}N(N+2)$ being considered a lower/upper bound
in all cases, with $\\{a,b,c\\}$ labelling all the possible permutations of
$\\{x,y,z\\}$. Hence, the violation of any inequality (29)-(31) leads, in this
framework, to entangled states since Eq. (28) is valid for all quantum states
— indeed, such inequalities define a polytope in the three-dimensional
$(\langle{\bf J}_{x}^{2}\rangle,\langle{\bf J}_{y}^{2}\rangle,\langle{\bf
J}_{z}^{2}\rangle)$-space, where the separable states lie inside this
geometric representation. It is important to stress that physical systems with
particle-number fluctuations (i.e., the particle number $N$ is not constant,
such as in BEC experiments) are discarded in this criterion.[57]

######  Remark 4.1.

Let us now apply these inequalities to the modified LMG model with the main
aim of corroborating the previous results and presenting alternative forms for
detection of spin-squeezing and entanglement effects. Numerical computations
show that Eqs. (30) and (31) are not satisfied for determined values of time,
which leads us to choose one of them whose particular time evolution combines
with those exhibited by $\mathcal{S}$ and $\mathcal{E}$-parameters — see Fig.
2(a)-(d). So, let us define the parameter

$E_{a}\mathrel{\mathop{:}}=\frac{(N-1)\mathscr{V}_{\mathrm{J}_{a}}}{\langle{\bf
J}_{b}^{2}\rangle+\langle{\bf J}_{c}^{2}\rangle-\frac{N}{2}}\geq
1\qquad(a,b,c=x,y,z)$ (32)

based on Eq. (30), where the violation $E_{a}<1$ directly implies in the
quantum effects under investigation. Note that, compared with Eq. (42),
$E_{a}$ has subtle differences: it requires the sum of mean values related to
the angular-momentum-squared operators (see denominator of both expressions),
while the previous one involves only squared mean values of the angular-
momentum operators. Figure 3 shows the plots of $E_{y}(t)$ (dot-dashed line)
and $E_{z}(t)$ (solid line) versus $t\in[0,50]$ for the same set of values
associated with the transverse magnetic field and anisotropy parameters used
in the previous figure, i.e., (a) $(-0.1,0.2)$ and (b) $(-0.13,0.1)$. It is
important to stress that the functional characteristics of the involved
parameters in the study of spin-squeezing and entanglement effects, as well as
the associated entanglement dynamics to the modified LMG model, represent two
essential factors that could explain (since the time evolution also depends on
the initial state chosen for the physical system), in part, the similarities
of behaviour related to the $\mathcal{S}$, $\mathcal{E}$, and $E$-parameters
viewed in Figs. 2 and 3. In summary, the entanglement criterion conceived by
Tóth et al.[10] reinforces the ‘almost perfect match’ between spin-squeezing
and entanglement effects, which leads us to investigate, in this particular
case, the quantum correlation rules via discrete Wigner and Husimi
functions.666For completeness sake, we present a case study for $h=0$ (absence
of transverse magnetic field) in appendix C, whose numerical computations lead
us to discuss the real necessity of demanding the respective validity domains
of the entanglement criteria used in this work.

Figure 3: Plots of $E_{y}(t)$ (dot-dashed line) and $E_{z}(t)$ (solid line) as
a function of time for the same values of $(h,\gamma)$ used in the previous
figure, that is, (a) $(-0.1,0.2)$ and (b) $(-0.13,0.1)$. Here, two factors
immediately emerge from our considerations on these pictures: (i) their
similarities with the previous cases (emphasizing, mainly, the strong
influence of $h$ on the entanglement effects), and consequently, (ii) the
important role of quantum correlations in such measurements. It is worth
mentioning that the particle number $N$ used in the numerical calculations
remains the same (i.e., $N=20$), which preserves important basic quantum
effects (contrary to the thermodynamic limit).

Henceforth, let us mention some few words on the entanglement dynamics of the
Hamiltonian operator ${\bf H}^{\prime}$ (Schrödinger picture): it basically
involves the action of collective spin operators responsible for introducing
the quantum effects attributed to two-body correlations — here mediated by the
anisotropy parameter $\gamma$ — upon a nonseparable initial state, that is,
the collective spin-coherent state (5) with $\theta=\frac{\pi}{2}$ and
$\varphi=0$ fixed. In this sense, since the discrete Wigner function reflects
the action of the time-evolution operator ${\bf
U}(t)\mathrel{\mathop{:}}=\exp\left(-\frac{\mathrm{i}}{\hbar}{\bf
H}^{\prime}t\right)$ upon the initial state $|\frac{\pi}{2},0\rangle$ in a
$(2j+1)^{2}$-dimensional discrete phase space, it seems natural to investigate
its behaviour for those specific values of time where effectively occur the
spin-squeezing and entanglement effects.

###  4.3 Finite-dimensional discrete phase spaces: a case study of quantum
correlations via Wigner and Husimi functions

Initially, let us consider the time evolution of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ upon a $21^{2}$-dimensional phase
space labeled by the pair $(\mu,\nu)$ of dimensionless discrete angular-
momentum and angle variables,[20] where the particular operator ${\bf
H}^{\prime}=-h{\bf J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf
J}_{y}^{2}\right)$ will take place at this description. As usual, the
parameters employed in the numerical computations are the same as those
mentioned in figure 2(b), i.e., $N=20$ and $(h,\gamma)=(-0.1,0.2)$.
Furthermore, we choose six representative time values of $\mathcal{E}_{z}(t)$
which reflect the first local minimum and maximum points exhibited in figure
2(b). In this way, figure 4 shows the contour plots of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ versus $(\mu,\nu)\in[-10,10]$
where, in particular, (a) $t=0$ (initial Wigner function), (b) $t=2.15$ (first
local minimum point), (c) $t=4.75$ (first local maximum point), (d) $t=7.10$
(second local minimum point), (e) $t=9.05$ (second local maximum point), and
(f) $t=9.95$ (corresponds to $\mathcal{E}_{z}(9.95)\approx 1$). In all these
pictures, it is interesting to observe how quantum correlations associated
with two-body interaction term modify the initial correlations present in the
collective spin-coherent state $|\frac{\pi}{2},0\rangle$; besides, note that
both $\mu$ and $\nu$ were conveniently shifted in the snapshots. Next, let us
describe certain specific points inherent to the time evolution of the
discrete Wigner function.

Figure 4: (Color online) Time evolution of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for the modified LMG model with
$N=20$ and $(h,\gamma)=(-0.1,0.2)$ fixed, where the labels $\mu$ and $\nu$
characterize, respectively, the dimensionless angular-momentum and angle pair.
In particular, these pictures show how the two-body interaction term present
in the Hamiltonian operator ${\bf H}^{\prime}$ affects the initial Wigner
distribution $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$ (via contour plots)
for determined values of time. We have adopted in our numerical computations
the values (a) $t=0$, (b) $t=2.15$, (c) $t=4.75$, (d) $t=7.10$, (e) $t=9.05$,
and (f) $t=9.95$, which illustrate a representative but not complete evolution
of the discrete Wigner function. Moreover, these values correspond to the
first local minimum and maximum points of $\mathcal{E}_{z}(t)$ — see solid
curve in figure 2(b) — while (f) represents the situation
$\mathcal{E}_{z}(9.95)\approx 1$. Although this phase space appears to the
eyes as continuous (due to the large number of points associated with $N=20$),
it is important to stress that such a particular phase space is (by
construction) genuinely discrete.

  * •

4(a). The function $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$ exhibits a
rotational symmetry with negative values located at the proximities of
$(-10,0)$ and $(10,0)$ (see small orange zones); as well as different widths
in the respective $\mu$ and $\nu$-directions. This important fact suggests, if
one considers both the directions, that marginal distributions related to the
angular momentum and angle variables can be somehow relevant and necessary in
the analysis of quantum correlations. The approximate format of an ellipse —
white zone in the middle part of the contour plot — represents a region where
there exists raised probability peaks which feature, by their turn, the
initial correlations associated with $|\frac{\pi}{2},0\rangle$.

  * •

4(b). This first case of time evolution corresponds to
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=2.15)$ and exemplifies the first
local minimum point of $\mathcal{E}_{z}(t)$ (see solid curve in figure 2(b)
when $t=2.15$). The entanglement effects (associated with the two-body
interaction, and synonymous of quantum correlation) introduce additional
symmetries that modify the statistical weights of the initial quantum state
$|\frac{\pi}{2},0\rangle$, originating, in this way, new interference patterns
which should explain the changed widths of the discrete Wigner distribution
function in $t=2.15$, as well as the appearance of new zones where negative
probabilities occur. Note that $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$
exhibits a motion towards the frontier of the discrete phase space labeled by
$\mu=-10$, the transverse magnetic field $h$ being the agent responsible for
such behaviour. Numerical computations indeed corroborate this assertion and
allow us to show that, for $h=0$, the Wigner function remains frozen in the
discrete phase space, having its widths modified as time goes by.

  * •

4(c). This particular case depicts
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=4.75)$ and represents the first local
maximum point of $\mathcal{E}_{z}(t)$. Here,
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ begins its motion towards the
centre of such a discrete phase space, where the interference patterns
associated with the components of ${\bf U}(t=4.75)|\frac{\pi}{2},0\rangle$
have specifically yielded the portrait verified in this situation. The small
white ‘island’ located in the right hand side of this picture and quite near
to the raised probability peaks (main white zone) depicts a small region with
negative probabilities.

  * •

4(d). The second minimum point of $\mathcal{E}_{z}(t)$ has as discrete phase-
space representative the contour plot of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=7.10)$. These unique interference
patterns (once ${\bf H}^{\prime}$ describes for $h\neq 0$ a nonperiodic
system) with small ‘islands’ of positive and negative probabilities represent
the accumulated effects attributed to the transverse magnetic field and
anisotropy parameter. Moreover, let us briefly mention that ${\bf
U}(t=7.10)|\frac{\pi}{2},0\rangle$ has a coincidence probability (also known
as fidelity and/or time correlation) with the original state close to $89\%$,
while 4(b) results in $55\%$.

  * •

4(e). Note that $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=9.05)$ corresponds to
the second local maximum point of $\mathcal{E}_{z}(t)$, i.e., we are
describing a situation where the entanglement does not occur. For the sake of
comparison, the coincidence probabilities associated with 4(c) and 4(e) assume
the respective percentages of $48\%$ and $78\%$. This difference can be
justified, in principle, by means of a simple inspection between both the
pictures: while 4(e) is located near the centre of discrete phase space, 4(c)
stays in the proximity of $\mu=-10$. However, if one considers the previous
case 4(d) for subsequent comparison, different contributions should be taken
into account in the analysis.

  * •

4(f). This last case depicts $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=9.95)$
for $\mathcal{E}_{z}(9.95)\approx 1$. It is interesting to observe how the
nonperiodic dynamics related to the Hamiltonian operator ${\bf H}^{\prime}$
affects any reconstruction process of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$: some numerical estimates of
$\mathfrak{F}(t)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}[\mbox{\boldmath$\rho$}(0)\mbox{\boldmath$\rho$}(t)]$ for $t=9.95$
results in $51\%$ of time correlation.

Now, let us show an essential formal result that explores the connection
between discrete Husimi and Wigner distribution functions,[58]

$\mathscr{H}_{\rho}(\mu,\nu;t)=\frac{1}{2j+1}\sum_{\mu^{\prime},\nu^{\prime}=-j}^{j}E(\mu,\nu|\mu^{\prime},\nu^{\prime})\,\mathscr{W}_{\rho}(\mu^{\prime},\nu^{\prime};t).$
(33)

Here, $E(\mu,\nu|\mu^{\prime},\nu^{\prime})$ defines a smoothing process
characterized by the discrete phase-space function[21]

$E(\mu,\nu|\mu^{\prime},\nu^{\prime})=\frac{1}{2j+1}\sum_{\eta,\xi=-j}^{j}\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta(\mu^{\prime}-\mu)+\xi(\nu^{\prime}-\nu)\right]\right\\}\mathscr{K}(\eta,\xi)$

that closely resembles the role of a usual Gaussian function in the continuous
phase space, since
$\mathscr{K}(\eta,\xi)\mathrel{\mathop{:}}=\mathscr{M}(\eta,\xi)/\mathscr{M}(0,0)$
is here responsible for the sum of products of Jacobi $\vartheta$-functions
evaluated at integer arguments, namely,

$\displaystyle\mathscr{M}(\eta,\xi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mathfrak{a}}}{2}\bigl{\\{}\vartheta_{3}(\mathfrak{a}\eta|\mathrm{i}\mathfrak{a})\left[\vartheta_{3}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})+\exp(\mathrm{i}\pi\eta)\vartheta_{4}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})\right]$
$\displaystyle+\exp(\mathrm{i}\pi\xi)\vartheta_{4}(\mathfrak{a}\eta|\mathrm{i}\mathfrak{a})\left[\vartheta_{3}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})+\exp[\mathrm{i}\pi(\eta+2j+1)]\vartheta_{4}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})\right]\bigr{\\}}$

where $\mathfrak{a}=(4j+2)^{-1}$ — in particular, see Appendix A of Ref. MR
for definitions and technical details on Jacobi $\vartheta$-functions. So, in
order to illustrate how the smoothing process acts on the summand of Eq. (33),
let us consider $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ and also the same
set of parameters used in the previous figure. In addition, it should be
noticed that $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ is strictly positive
as well as limited to the closed interval $[0,1]$ for any $t\geq 0$;
consequently, all the negative values and non-regular patterns described in
figure 4 will be washed out by $E(\mu,\nu|\mu^{\prime},\nu^{\prime})$. Figure
5 exhibits the contour plots of $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$,
where such effects are effectively checked along the pictures 5(a-f). In these
cases, the width changes observed for the $(\mu,\nu)$-directions are related
to the two-body interaction term (here mediated by means of $N$ and $\gamma$),
while the motion towards frontier located at $\mu=-10$ is associated with the
transverse magnetic field $h$. Since the widths associated with the angular-
momentum and angle distributions are also affected by those effects, it seems
convenient at this moment to employ the Wehrl-type entropy functionals[31] for
understanding how the correlations between the discrete variables $\mu$ and
$\nu$ of a finite-dimensional phase space change for $t\geq 0$.

Figure 5: (Color online) Time evolution of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for the same set of parameters
employed in the previous figure. In what concerns the discrete Husimi
function, it must be stressed that such a function is particularly obtained
from $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ through a smoothing process
characterized by a discrete phase-space function that closely resembles the
role of the Gaussian function in the continuous phase space. This process
essentially explains the disappearance of all irregular regions and negative
values exhibited by the discrete Wigner distribution, which implies in a well-
behaved function along its time evolution and limited to the closed interval
$[0,1]$. Similarly to the previous case, it is worth mentioning that such a
particular phase space is genuinely discrete (by ab initio construction), even
though it appears to the eyes as contionuous.

To this end, let us now consider the mutual correlation functional
$\mathrm{I}[\mathrm{H};t]$ defined as follows:[22]
$\mathrm{I}[\mathrm{H};t]\mathrel{\mathop{:}}=\mathrm{E}[\mathrm{Q};t]+\mathrm{E}[\mathrm{R};t]-\mathrm{E}[\mathrm{H};t]\geq
0$. In this particular definition,

$\mathrm{E}[\mathrm{H};t]\equiv-\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t)\ln\left[\mathscr{H}_{\rho}(\mu,\nu;t)\right]$
(34)

corresponds to the time-dependent joint entropy functional here expressed in
terms of the discrete Husimi function (33), while

$\mathrm{E}[\mathrm{Q};t]\equiv-\frac{1}{\sqrt{2j+1}}\sum_{\mu=-j}^{j}\mathscr{Q}_{\rho}(\mu;t)\ln\left[\mathscr{Q}_{\rho}(\mu;t)\right]$
(35)

and

$\mathrm{E}[\mathrm{R};t]\equiv-\frac{1}{\sqrt{2j+1}}\sum_{\nu=-j}^{j}\mathscr{R}_{\rho}(\nu;t)\ln\left[\mathscr{R}_{\rho}(\nu;t)\right]$
(36)

represent the marginal entropies — which are directly related, by their turn,
to the respective marginal distribution functions (for technical details, see
Ref. Ruzzi)

$\mathscr{Q}_{\rho}(\mu;t)=\frac{1}{\sqrt{2j+1}}\sum_{\nu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t)\quad\mbox{and}\quad\mathscr{R}_{\rho}(\nu;t)=\frac{1}{\sqrt{2j+1}}\sum_{\mu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t).$

Note that $\mathrm{E}[\mathrm{H};t]$, $\mathrm{E}[\mathrm{Q};t]$ and
$\mathrm{E}[\mathrm{R};t]$ constitute some basic mathematical elements for
describing functional correlations between $\mu$ and $\nu$, and consequently,
the width changes associated with the discrete Husimi function
$\mathscr{H}_{\rho}(\mu,\nu;t)$. Within several important properties inherent
to the entropy functionals, the Araki-Lieb inequality[59]
$\left|\mathrm{E}[\mathrm{Q};t]-\mathrm{E}[\mathrm{R};t]\right|\leq\mathrm{E}[\mathrm{H};t]\leq\mathrm{E}[\mathrm{Q};t]+\mathrm{E}[\mathrm{R};t]$
has a central role in such a description: for instance, it leads us to
investigate how the dynamic correlations (introduced by means of the
Hamiltonian operator ${\bf H}^{\prime}$) affect the underlying correlations of
the initial state $\mbox{\boldmath$\rho$}(t_{0})$ — here mapped onto
$\mathscr{H}_{\rho}(\mu,\nu;t_{0})$. So, if one computes the entropy
functionals from $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$, some interesting
results can be promptly achieved.

Figure 6: Plots of (a) $\mathrm{E}[\mathrm{H};t]$ and (b)
$\mathrm{I}[\mathrm{H};t]$ versus $t\in[0,50]$ for two different values of
transverse magnetic field $h$ and anisotropy parameter $\gamma$ with $N=20$
fixed, namely, the solid curves in both pictures correspond to $(-0.1,0.2)$,
while the dot-dashed curves are related to $(-0.13,0.1)$. In each case, the
orange region describes particular situations of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ where the entropy functionals are
restricted to (a) $\mathrm{E}[\mathrm{H};t]\leq\mathrm{E}[\mathrm{H};0]\approx
0.1994$ and (b) $\mathrm{I}[\mathrm{H};t]\leq\mathrm{I}[\mathrm{H};0]\approx
0.7144$. It is important to stress that $\mathrm{E}[\mathrm{H};t]$ and
$\mathrm{I}[\mathrm{H};t]$ are sensitive to small variations of $h$.

Figure 6 illustrates the time evolution of (a) $\mathrm{E}[\mathrm{H};t]$ and
(b) $\mathrm{I}[\mathrm{H};t]$ as a function of $t\in[0,50]$ for the same
values of $(h,\gamma)$ used in figure 2. Then, let us initially focus on
$(-0.1,0.2)$ (see solid curves in both the pictures) since this case coincides
with that used in the contour plots of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for specific times. Those
concentric ellipses observed in figure 5(a) can be considered as a phase-space
signature of the initial state $|\frac{\pi}{2},0\rangle$ at $t=0$, here
endorsed by $\mathrm{E}[\mathrm{H};0]\approx 0.1994$ and
$\mathrm{I}[\mathrm{H};0]\approx 0.7144$, which means that such a state
presents an initial correlation between $\mu$ and $\nu$ featured by a spread
angular-momentum distribution and ‘localized’ angle distribution. Indeed, the
sum $\mathrm{E}[\mathrm{Q};0]+\mathrm{E}[\mathrm{R};0]\approx 0.5150$
justifies such an assertion and (as expected) also corroborates the right-hand
side of the Araki-Lieb inequality. Already at $t=2.15$, one observes the
displacement of the discrete Husimi function towards the frontier $\mu=-10$
accompanied, in this situation, by a decrease of asymmetry between its
intrinsic widths in the $(\mu,\nu)$-directions. Since
$\mathrm{E}[\mathrm{H};t=2.15]\approx 0.1958$ implies in effective information
gain upon the discrete angular-momentum and angle collective variables, the
extra correlations introduced through the two-body interaction term explain,
in principle, the effects depicted in pictures 2(a,b). However, the deformed
widths viewed in 5(c-f) after the first rebound from that frontier and
subsequent return towards the centre of the finite-dimensional discrete phase
space are responsible for the inscreased values of both entropy functionals
when $4.75\leq t\leq 9.95$. Let us now briefly discuss the case $(-0.13,0.1)$
here characterized by a significant reduction of the effects related to
$\gamma$. Although the information gain happens more frequently in such a
case, the spin squeezing and entanglement effects exhibited in pictures 2(c,d)
are associated with frequent motions of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ towards the frontier $\mu=-10$,
followed by small variations of its widths.

A last pertinent question then emerges from our considerations about discrete
Husimi function and Wehrl-type entropy functionals: “How the smoothing process
characterized by Eq. (33) affects the description of correlation strength
between the discrete labels $\mu$ and $\nu$ here used to describe the finite-
dimensional phase space”? To answer this question, let us initially comprehend
the role of $\mathscr{K}(\eta,\xi)$ present in the function
$E(\mu,\nu|\mu^{\prime},\nu^{\prime})$: it was basically written as a sum of
products of Jacobi $\vartheta$-functions evaluated at integer arguments, which
plays, in such an aforementioned discrete phase space, the role reserved to
the Gaussian functions in the continuous case ($N\rightarrow\infty$); besides,
$\sqrt{4j+2}$ represents its respective width and it assumes a constant value,
in this situation, for a given $j$. Hence, any sub-Planck structures[60] or
even relevant correlations within this range are smeared in the smoothing
process, which directly affects the description of correlation strength via
Wehrl-type entropy functionals. Another important restriction associated with
$\mathrm{E}[\mathrm{H};t]$ emerges from the mathematical property
$\mathrm{S}[\mbox{\boldmath$\rho$}(t)]=-\mbox{${\rm
Tr}$}[\mbox{\boldmath$\rho$}(t)\ln\mbox{\boldmath$\rho$}(t)]\leq\mathrm{E}[\mathrm{H};t]$,
i.e., Eq. (34) consists of an upper bound for the von Neumann entropy;[31]
consequently, “if there are small distance fluctuations or if
$\mathscr{H}_{\rho}(\mu,\nu;t)$ is concentrated on small regions of discrete
phase space”, then the percent error estimated $\delta=\mathrm{S}/\mathrm{E}$
will be very bad. However, these apparent limitations can be circumvented by:
(i) adopting the prescription of Manfredi and Feix[61] for quantum entropy
based on Wigner functions in continuous phase-space, (ii) reformulating the
function $\mathscr{K}(\eta,\xi)$ in order to modify its respective width,[22]
and finally (iii) including a parallel study on von Neumann entropy[62] and
quantum discord[7] which permits us to increase our knowledge base on the
intricate mechanisms of correlation strength related to the spin-squeezing and
entanglement effects for any spin systems. In many ways the analysis presented
in this work of the modified LMG model is complementary to that provided in
Ref. Dusuel; moreover, in what concerns the discrete phase-space approach and
its implications for different physical systems, our results sound promising
at a first glance.

##  5 Conclusions

In this non-trivial quantum mechanical scenario of correlations, entanglement
and spin-squeezing effects, as well as their connections, any associated
theoretical and/or experimental proposals for measures of the aforementioned
effects shall necessarily be accomplished by exhaustive tests of confidence
within a wide class of analogous physical systems. The mere comprehension of
these effects by means of a ‘specific theoretical/experimental measure applied
to a particular physical system’ does not change its respective status of
proposal per se for standard measure: such transition demands ‘time, patience,
and efforts’ to understand the subtle role of correlations and their intrinsic
mechanisms in quantum mechanics. In this paper, we have used a theoretical
framework of finite-dimensional discrete phase space as an alternative
approach for the study of the spin-squeezing and entanglement effects. The
emphasis on covariances and its fundamental role in the investigation of spin-
squeezing effects is not accidental: additional correlations related to the
anticommutation relations of the angular-momentum generators are now included
in the analysis of spin systems. This first effective gain does not imply in
the match between the previous effects, since both the measures here adopted
for describing entanglement exhibit ‘slightly different functional relations’
and also present ‘speculative features’, as expected.

In what concerns the connection between the spin-squeezing and entanglement
effects, both time-evolution operator and initial state of a given spin system
play a fundamental role in such process: they describe the short-range and/or
long-range correlations of the multipartite system under investigation, which
could explain the similarity degree of the spin-squeezing and entanglement
measures studied in this paper, as well as its ‘almost perfect match’. In this
sense, the modified LMG model fulfils a reasonable set of mathematical and
physical prerequisites that lead us to corroborate, by means of numerical
computations, the aforementioned link, for then establishing, subsequently,
its inherent limitations. Besides, these results open new possibilities for
future investigations in different physical systems which encompass underlying
$\mathfrak{su}(2)$ structures with distinct physical properties — in
particular, those multipartite systems where contributions related to the
long-range correlations are indeed necessary in the effective description of
entanglement.

Our particular phase-space approach, nevertheless, also reveals its drawbacks.
To begin with, Eq. (12) and its constituting blocks coming from the numerical
recipe described in subsection 3.2 do not satisfy the basic criterion ‘easy-
to-compute’. The large number of sums that appeared in the time evolution of
discrete Wigner function and mean values obligatorily implies in high
computational and operational costs, whose complexities grow as $N$ increases.
Moreover, the modulo $N$ extraction phase here adopted for the discrete
variables still follows that mathematical prescription discussed in Ref. DM1,
which represents a necessary operational cost inherent to the
mod($N$)-invariant operator basis ${\bf G}(\mu,\nu)$. However, such apparent
limitations can be circumvented in this context by adopting the theoretical
framework exposed in Ref. MR for the mod($N$)-invariant unitary operator basis
$\mbox{\boldmath$\Delta$}(\mu,\nu)$, and reformulating the content associated
with the time evolution of the discrete Wigner function. This procedure will
allow a real computational gain in the numerical calculations.

Now, let us briefly mention some possibilities for future research that stem
from the present paper. As a first example, we recall from Ref. Modi those
considerations on the concept of quantum discord and its important link with
the subtle boundary between entanglement and classical correlations. Given
that the Wehrl-type entropy functionals failed in the description of
correlations related to the entanglement and spin-squeezing effects, it seems
reasonable to introduce such a measure in this context since quantum discord
can lead us to a more efficient analysis on quantum and classical correlations
in multipartite physical systems. Another possible example of research
consists in employing that finite-dimensional discrete phase-space framework
in order to corroborate the Lieb’s conjecture for the spin coherent states[63]
via quantum dynamics of spin systems.

## Acknowledgements

The authors thank Maurizio Ruzzi and anonymous referee for providing valuable
suggestions on an earlier version of this manuscript. Tiago Debarba is
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq), Brazil.

##  Appendix A Variances, covariances, and uncertainty relation

There are interesting results relating covariance functions and variances
associated with certain summations of two non-commuting observables ${\bf X}$
and ${\bf Y}$, namely,

$\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\pm\mbox{\scriptsize${\rm
Y}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}}\pm
2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}\quad\mbox{and}\quad\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}\pm\mathrm{i}\mbox{\scriptsize${\rm
Y}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}}\pm
2\mathrm{i}\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}.$

Note that such connections can be generalized for a large number of non-
commuting observables. For instance, if one considers the set $\\{{\bf X},{\bf
Y},{\bf Z}\\}$, it is easy to show that

$\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}$ $\displaystyle=$
$\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Z}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Z}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}\mbox{\scriptsize${\rm Z}$}}$
$\displaystyle=$ $\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}+\mbox{\scriptsize${\rm Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}+\mbox{\scriptsize${\rm Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm Z}$}}$

yields the statistical balance equation

$\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Z}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}+\mbox{\scriptsize${\rm Z}$}}.$
(37)

This particular ‘conservation law’ describes, if applied to the angular-
momentum generators, how the statistical fluctuations are distributed among
the observables $\\{{\bf J}_{x},{\bf J}_{y},{\bf J}_{z}\\}$ in the measurement
process for a given initial quantum state. Table A shows some analytical
expressions obtained for the variances and covariances related to the angular-
momentum generators and spin coherent states, which exemplify Eq. (37).
Following, let us discuss how the unitary transformations modify the signal-
to-noise ratio for an ideal experimental situation where the deleterious
effects of a low-efficiency detection process are absent.[64]

The generator of unitary transformations ${\bf T}(\xi,\omega)$ maintains, by
definition, the linearity of the angular-momentum operators. Indeed, the new
set

$\left\\{\overline{{\bf J}}_{a}\mathrel{\mathop{:}}={\bf
T}^{\dagger}(\xi,\omega){\bf J}_{a}{\bf
T}(\xi,\omega)\;:\;\xi\in\mathbb{C}\;\,\mbox{and}\;\,\omega\in\mathbb{R}\right\\}_{a=x,y,z}$

shows explicitly such a mathematical property through the relations

$\displaystyle\overline{{\bf J}}_{x}$ $\displaystyle=$ $\displaystyle
A_{11}{\bf J}_{x}+A_{12}{\bf J}_{y}+A_{13}{\bf J}_{z}$
$\displaystyle\overline{{\bf J}}_{y}$ $\displaystyle=$ $\displaystyle
A_{21}{\bf J}_{x}+A_{22}{\bf J}_{y}+A_{23}{\bf J}_{z}$
$\displaystyle\overline{{\bf J}}_{z}$ $\displaystyle=$ $\displaystyle
A_{31}{\bf J}_{x}+A_{32}{\bf J}_{y}+A_{33}{\bf J}_{z},$ (38)

whose coefficients $\\{A_{ij}\\}$ can be immediately determined from the
results obtained in subsection 2.1. Since the commutation relation
$[\overline{{\bf J}}_{a},\overline{{\bf
J}}_{b}]=\mathrm{i}\epsilon_{abc}\overline{{\bf J}}_{c}$ keeps its invariable
form, some restrictions on the aforementioned coefficients should be
established in this context, that is,

$\displaystyle A_{22}A_{33}-A_{23}A_{32}$ $\displaystyle=$ $\displaystyle
A_{11},\quad A_{13}A_{32}-A_{12}A_{33}=A_{21},\quad
A_{12}A_{23}-A_{13}A_{22}=A_{31},$ $\displaystyle A_{23}A_{31}-A_{21}A_{33}$
$\displaystyle=$ $\displaystyle A_{12},\quad
A_{11}A_{33}-A_{13}A_{31}=A_{22},\quad A_{13}A_{21}-A_{11}A_{23}=A_{32},$
$\displaystyle A_{21}A_{32}-A_{22}A_{31}$ $\displaystyle=$ $\displaystyle
A_{13},\quad A_{12}A_{31}-A_{11}A_{32}=A_{23},\quad
A_{11}A_{22}-A_{12}A_{21}=A_{33}.$

It is worth stressing that these general equations are valid for any unitary
transformations which preserve Eq. (A); moreover, after certain proper
manipulations of such equations, this set can be reduced to

$\displaystyle A_{11}A_{12}+A_{21}A_{22}+A_{31}A_{32}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle A_{11}A_{13}+A_{21}A_{23}+A_{31}A_{33}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
A_{12}A_{13}+A_{22}A_{23}+A_{32}A_{33}$ $\displaystyle=$ $\displaystyle 0.$

As mentioned in the main part of the text, the identity $\vec{\overline{{\bf
J}}}^{2}\equiv\vec{{\bf J}}^{2}$ also preserves the original form of the total
spin operator, which implies in the additional relations

$A_{11}^{2}+A_{21}^{2}+A_{31}^{2}=1,\quad
A_{12}^{2}+A_{22}^{2}+A_{32}^{2}=1,\quad A_{13}^{2}+A_{23}^{2}+A_{33}^{2}=1.$

Note that
$\mathscr{V}_{\overline{\mathrm{J}}_{x}}+\mathscr{V}_{\overline{\mathrm{J}}_{y}}+\mathscr{V}_{\overline{\mathrm{J}}_{z}}\equiv\mathscr{V}_{\mathrm{J}_{x}}+\mathscr{V}_{\mathrm{J}_{y}}+\mathscr{V}_{\mathrm{J}_{z}}=j$
represents an important by-product in this process since the sum of such
variances remains invariant under the unitary transformation described by Eq.
(A).

The explicit results for the variances and covariances — as shown on the table
below — allow us, in principle, to illustrate both the Eqs. (6) and (37).
Besides, they also lead us to verify the uncertainty relation
$\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}-\left(\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}\right)^{2}\geq\frac{1}{4}\left|\langle\left[{\bf
J}_{a},{\bf J}_{b}\right]\rangle\right|^{2}$ for any $a,b=x,y,z$. Note that
the saturation is reached in this case for $\theta=0$ and $\pi$, which
correspond to the fiducial states $|j,-j\rangle$ and $|j,j\rangle$.  Variances
and covariances associated with $\\{{\bf J}_{x},{\bf J}_{y},{\bf J}_{z}\\}$
for the spin coherent states
$\mathscr{V}_{\mathrm{J}_{x}}=(j/2)\left[1-\cos^{2}(\varphi)\sin^{2}(\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{y}}=(j/2)\left[1-\sin^{2}(\varphi)\sin^{2}(\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{z}}=(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}=-(j/4)\sin(2\varphi)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}=(j/4)\cos(\varphi)\sin(2\theta)$
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}=(j/4)\sin(\varphi)\sin(2\theta)$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{y}}=(j/2)\left\\{2-\sin^{2}(\theta)\left[1+\sin(2\varphi)\right]\right\\}$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{z}}=(j/2)\left[1+\sin^{2}(\varphi)\sin^{2}(\theta)+\cos(\varphi)\sin(2\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{y}+\mathrm{J}_{z}}=(j/2)\left[1+\cos^{2}(\varphi)\sin^{2}(\theta)+\sin(\varphi)\sin(2\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{y}+\mathrm{J}_{z}}=(j/2)\left\\{2-\sin(2\varphi)\sin^{2}(\theta)+[\cos(\varphi)+\sin(\varphi)]\sin(2\theta)\right\\}$
See Ref. Inomata for certain mean values involving some quadratic forms of
${\bf J}_{\pm}$ and ${\bf J}_{z}$.

The evaluation of the signal-to-noise ratio (SNR) associated with collective
and intrinsinc degrees of freedom is crucial in any experimental data
analysis: indeed, it provides quantitative information on the measured signal
and the noise inherent to the experiment under investigation. Thus, let us
define SNR through the expression
$\mathfrak{R}_{a}\mathrel{\mathop{:}}=|\langle{\bf
J}_{a}\rangle|/\sqrt{\mathscr{V}_{\mathrm{J}_{a}}}$, where each component of
$\vec{{\bf J}}$ presents an important role in the measurement process.
However, if one considers experimental situations — rotations and/or time
evolutions — that transform ${\bf J}_{a}\mapsto\overline{{\bf J}}_{a}$, it
seems natural to analyse the case
$\mathfrak{R}_{a}\mapsto\overline{\mathfrak{R}}_{a}\mathrel{\mathop{:}}=|\langle\overline{{\bf
J}}_{a}\rangle|/\sqrt{\mathscr{V}_{\overline{\mathrm{J}}_{a}}}$. In this
sense, the variances

$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}}$ $\displaystyle=$
$\displaystyle
A_{11}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{11}A_{12}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{11}A_{13}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{12}A_{13}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{y}}$ $\displaystyle=$
$\displaystyle
A_{21}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{22}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{23}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{21}A_{22}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{21}A_{23}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{22}A_{23}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{z}}$ $\displaystyle=$
$\displaystyle
A_{31}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{32}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{33}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{31}A_{32}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{31}A_{33}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{32}A_{33}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$

exhibit an explicit dependence on all the previous variance and covariance
functions, which modifies substantially the estimate of
$\\{\mathfrak{R}_{a}\\}$. Besides, the uncertainty relation

$\mathscr{V}_{\overline{\mathrm{J}}_{a}}\mathscr{V}_{\overline{\mathrm{J}}_{b}}-\left(\mathscr{V}_{\overline{\mathrm{J}}_{a}\overline{\mathrm{J}}_{b}}\right)^{2}\geq\frac{1}{4}\left|\langle[\overline{{\bf
J}}_{a},\overline{{\bf J}}_{b}]\rangle\right|^{2}$ (39)

is also modified in this context, since the covariances

$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}\overline{\mathrm{J}}_{y}}$
$\displaystyle=$ $\displaystyle
A_{11}A_{21}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}A_{22}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}A_{23}\mathscr{V}_{\mathrm{J}_{z}}+(A_{11}A_{22}+A_{12}A_{21})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{11}A_{23}+A_{13}A_{21})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{12}A_{23}+A_{13}A_{22})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}\overline{\mathrm{J}}_{z}}$
$\displaystyle=$ $\displaystyle
A_{11}A_{31}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}A_{32}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}A_{33}\mathscr{V}_{\mathrm{J}_{z}}+(A_{11}A_{32}+A_{12}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{11}A_{33}+A_{13}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{12}A_{33}+A_{13}A_{32})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{y}\overline{\mathrm{J}}_{z}}$
$\displaystyle=$ $\displaystyle
A_{21}A_{31}\mathscr{V}_{\mathrm{J}_{x}}+A_{22}A_{32}\mathscr{V}_{\mathrm{J}_{y}}+A_{23}A_{33}\mathscr{V}_{\mathrm{J}_{z}}+(A_{21}A_{32}+A_{22}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{21}A_{33}+A_{23}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{22}A_{33}+A_{23}A_{32})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$

possess a similar dependence if one compares them with the previous case.
Finally, it is worth stressing that such results are extremely relevant in the
investigative process of squeezing and entanglement effects for spin
systems.[10, 11, 12, 13]

For instance, Muñoz and Klimov[65] have recently introduced a particular set
of discrete displacement generators upon a $2^{n}\otimes 2^{n}$ discrete phase
space, which allows us to establish a specific family of discrete spin
coherent states for $n$-qubit systems with interesting mathematical and
physical properties. Generated by application of the discrete displacement
generators to a symmetric fiducial state, such states have isotropic
fluctuations in a tangent plane whose geometric features are well-defined: in
general, its direction does not coincide with that chosen for the mean value
$\langle\vec{{\bf J}}\rangle$. Besides, these reference states constitute the
essential basic elements necessary for investigating the squeezing effects
associated with the non-symmetric $n$-qubit states (resulting from the
application of XOR gates) through the covariances
$\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}$. It is important to emphasize
that the number of XOR gates applied to the discrete spin coherent states in
order to minimize fluctuations in the original homogeneous plane strongly
depends on the number of qubits. Since
$\\{\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}\\}_{a,b=x,y,z}$ is also
modified by the action of XOR gates, we believe that our results can be
somehow useful for the detailed examination of such squeezing effects.

##  Appendix B The Kitagawa-Ueda model

Let us initiate our investigation on spin squeezing and entanglement through
the Kitagawa-Ueda model[13] — here described by the Hamiltonian ${\bf
H}=\chi{\bf J}_{z}^{2}$ — which describes a nonlinear interaction proportional
to ${\bf J}_{z}^{2}$. The unitary transformations generated by the time-
evolution operator ${\bf U}(\tau)=\exp\left(-\mathrm{i}\tau{\bf
J}_{z}^{2}\right)$ for $\tau=\chi t$ represents a basic set of mathematical
tools that allows us, in principle, to calculate certain important quantities
necessary to the investigative process. Indeed, if one considers the unitary
transformations

$\displaystyle{\bf J}_{x}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{x}{\bf U}(\tau)=\exp(\mathrm{i}\tau)\left[{\bf
J}_{x}\cos(2\tau{\bf J}_{z})-{\bf J}_{y}\sin(2\tau{\bf J}_{z})\right]$
$\displaystyle{\bf J}_{y}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{y}{\bf U}(\tau)=\exp(\mathrm{i}\tau)\left[{\bf
J}_{x}\sin(2\tau{\bf J}_{z})+{\bf J}_{y}\cos(2\tau{\bf J}_{z})\right]$
$\displaystyle{\bf J}_{z}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{z}{\bf U}(\tau)={\bf
J}_{z}\;\rightleftharpoons\;\left[{\bf J}_{z},{\bf U}(\tau)\right]=0,$ (40)

it is not so hard to yield the mean values (Heisenberg picture)

$\displaystyle\langle{\bf J}_{x}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{x}(\tau)|\theta,\varphi\rangle=\mathscr{A}_{\tau}^{2j-1}j\cos\left[\varphi-(2j-1)\delta_{\tau}\right]\sin(\theta)$
$\displaystyle\langle{\bf J}_{y}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{y}(\tau)|\theta,\varphi\rangle=\mathscr{A}_{\tau}^{2j-1}j\sin\left[\varphi-(2j-1)\delta_{\tau}\right]\sin(\theta)$
$\displaystyle\langle{\bf J}_{z}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{z}(\tau)|\theta,\varphi\rangle=-j\cos(\theta)$ (41)

for the spin coherent states, where
$\mathscr{A}_{\tau}=\left[\cos^{2}(\tau)+\sin^{2}(\tau)\cos^{2}(\theta)\right]^{\frac{1}{2}}$
denotes the amplitude function, and
$\delta_{\tau}=\arctan\left[\tan(\tau)\cos(\theta)\right]$ the phase function.
Note that such results lead us to conclude that unitary transformations
involving nonlinear forms of the angular-momentum generators do not preserve,
in general, the linear feature of such operators — for instance, see Eqs. (B)
and (A). This particularity is directly associated with the generators of the
$\mathfrak{su}(2)$ Lie algebra and does not apply to the squeeze operator
${\bf
S}(\zeta)\mathrel{\mathop{:}}=\exp\left[\frac{1}{2}\left(\zeta^{\ast}{\bf
a}^{2}-\zeta{\bf a}^{\dagger 2}\right)\right]$, since ${\bf S}(\zeta)$ is
responsible for unitary transformations in the Heisenberg-Weyl algebra which
preserve the linearity of the boson annihilation and creation operators.[37,
38]

Another interesting point inherent to the time-evolution operator ${\bf
U}(\tau)$ establishes the relation $\langle\vec{{\bf
J}}^{2}(\tau)\rangle_{\theta,\varphi}=\langle\vec{{\bf
J}}^{2}(0)\rangle_{\theta,\varphi}=j(j+1)$, which preserves the mean value of
the total spin operator in $\tau=0$. In fact, this result can be interpreted
as a direct consequence of the unitary transformations originated from the
action of ${\bf U}(\tau)$ on the angular-momentum operators. Following, let us
discuss some important points related to uncertainty relation, squeezing
effects and their links with entanglement, considering the spin coherent
states as initial states of the physical system.

The time-dependent variance and covariance functions, as shown on the table
below for the spin coherent states, constitute an important group of formal
mathematical results related to the Kitagawa-Ueda model that permits us to
investigate not only the Robertson-Schrödinger uncertainty principle, but also
both the squeezing and entanglement effects associated with the collective
angular-momentum operators.  Time-dependent mean values, variances and
covariance function for the Kitagawa-Ueda model $\langle{\bf
J}_{x}^{2}(\tau)\rangle_{\theta,\varphi}=j/2+(j/4)(2j-1)\left[1+\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\langle{\bf
J}_{y}^{2}(\tau)\rangle_{\theta,\varphi}=j/2+(j/4)(2j-1)\left[1-\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\langle{\bf J}_{z}^{2}(\tau)\rangle_{\theta,\varphi}\equiv\langle{\bf
J}_{z}^{2}(0)\rangle_{\theta,\varphi}=j^{2}\cos^{2}(\theta)+(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}}(\tau)=j/2+(j/4)(2j-1)\left[1+\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\qquad\qquad\quad-j^{2}\mathscr{A}_{\tau}^{4j-2}\cos^{2}\left[\varphi-(2j-1)\delta_{\tau}\right]\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{y}}(\tau)=j/2+(j/4)(2j-1)\left[1-\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\qquad\qquad\quad-j^{2}\mathscr{A}_{\tau}^{4j-2}\sin^{2}\left[\varphi-(2j-1)\delta_{\tau}\right]\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{z}}(\tau)\equiv\mathscr{V}_{\mathrm{J}_{z}}(0)=(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}(\tau)=(j/4)(2j-1)\mathscr{A}_{2\tau}^{2j-2}\sin[2\varphi-(2j-2)\delta_{2\tau}]\sin^{2}(\theta)$
$\qquad\qquad\quad-(j^{2}/2)\mathscr{A}_{\tau}^{4j-2}\sin[2\varphi-(4j-2)\delta_{\tau}]\sin^{2}(\theta)$
See Refs. SS1 and Hald for some measurement criteria involving such physical
quantities.

  * •

Initially, let us introduce the parameters
$\mathcal{S}_{x}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{x}}/\mathcal{R}_{xyz}$
and
$\mathcal{S}_{y}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{y}}/\mathcal{R}_{xyz}$
with
$\mathcal{R}_{xyz}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}})^{2}+\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}\right]^{\frac{1}{2}}$ (in this definition,
$\mathcal{R}_{xyz}\neq 0$ represents a condition sine qua non), which lead us
to rewrite Eq. (6) as follows: $\mathcal{S}_{x}\mathcal{S}_{y}\geq 1$. In
principle, this simplified form allows us to investigate the squeezing effects
related to the aforementioned model through exact expressions obtained in
Table B for the time-dependent variance and covariance functions. Figure 7(a)
shows the plots of $\mathcal{S}_{x}$ (dot-dashed line) and $\mathcal{S}_{y}$
(dashed line) versus $\tau\in[0,2\pi]$ for $j=2$ and
$\theta=\varphi=\frac{\pi}{4}$ fixed. The hachured area exhibited in the
picture describes, in this case, that region where the squeezing effects occur
for $\mathcal{S}_{x(y)}(\tau)<1$ (but not both simultaneously). Such a
particular evidence of squeezing effect generalizes, through an effective way,
those results obtained by Kitagawa and Ueda[13] in the absence of one-axis
twisting mechanism.777In fact, this mathematical procedure introduces
artificially additional quantum correlations (by means of time-dependent
unitary transformations that consist of rotations around the $x$-axis) for the
angular-momentum operators, which allow to reduce the standard quantum noise
related to the spin coherent states down to
$\frac{1}{2}\sqrt[3]{\frac{j}{3}}$. It is important to emphasize that such an
estimate does not consider the contributions originated from the covariance
function.

  * •

How the squeezing and entanglement effects can be connected? Sørensen et
al[12] have proposed an interesting experiment for the Bose-Einstein
condensates (and attainable with present technology) which allows us, in
principle, to answer this intriguing question through a fundamental effect in
quantum mechanics: the many-particle entanglement. Since correlations among
spins yield the squeezing effect, it is natural to build a bridge connecting
both the entanglement and spin squeezing effects by means of a solid
mathematical framework which allows us to establish certain reliable
measurement criteria.[10] In this sense, let us comment some few words on the
entanglement criterion adopted in such an experiment: it is basically focussed
on the separability criterion for the $N$-particle density operator involving
the variances and squared mean values associated with each orthogonal
component of the collective angular-momentum operators, that is,

$\mathcal{E}_{a}\equiv\frac{N\mathscr{V}_{\mathrm{J}_{a}}}{\langle{\bf
J}_{b}\rangle^{2}+\langle{\bf J}_{c}\rangle^{2}}<1\qquad(a,b,c=x,y,z).$ (42)

Thus, the parameters $\\{\mathcal{E}_{a}\\}$ characterize the atomic
entanglement in this context, and quantum states with $\mathcal{E}_{a}<1$ are
referred to as spin squeezed states.[12] For instance, Figure 7(b) shows the
plots of $\mathcal{E}_{x}$ (dot-dashed line), $\mathcal{E}_{y}$ (dashed line),
as well as $\mathcal{E}_{z}$ (solid line) versus $\tau\in[0,2\pi]$ for the
same values of $\\{j,\theta,\varphi\\}$ adopted in the squeezing criterion —
see Fig. 7(a). Note that $\mathcal{E}_{x}$ and $\mathcal{E}_{y}$ exhibit
values less than one, except $\mathcal{E}_{z}$, which implies in entanglement
effect (see hachured region) for the orthogonal components ${\bf J}_{x}(\tau)$
and ${\bf J}_{y}(\tau)$, as expected, since $[{\bf H},{\bf J}_{z}]=0$. The
similarities between both the figures and the evidence of the squeezing and
entanglement effects for the same values of $\tau$ justify, in such a case,
the above connection per se.

Figure 7: (a) represents the plots of $\mathcal{S}_{x}$ (dot-dashed line) and
$\mathcal{S}_{y}$ (dashed line) versus $\tau\in[0,2\pi]$ with $j=2$ ($N=4$
spins) and $\theta=\varphi=\frac{\pi}{4}$ fixed. Note that both the parameters
$\mathcal{S}_{x}(\tau)$ and $\mathcal{S}_{y}(\tau)$ assume values less than
one in different intervals of $\tau$ (see hachured region), which implies in
the squeezing effect for each case obeying the relation
$\mathcal{S}_{x}(\tau)\mathcal{S}_{y}(\tau)\geq 1$ (the saturation is reached
for $\tau=m\pi$ with $m\in\mathbb{N}$). (b) corresponds to the plots of
$\mathcal{E}_{x}$ (dot-dashed line), $\mathcal{E}_{y}$ (dashed line), and
$\mathcal{E}_{z}$ (solid line) versus $\tau\in[0,2\pi]$, for the same values
of $\\{j,\theta,\varphi\\}$ adopted in the previous picture. The hachured area
denotes, in this case, that region where the entanglement effect occurs.

##  Appendix C A case study for $h=0$

To what extent the anisotropy parameter $\gamma$ can be effectively employed
in order to validate the entanglement criteria here adopted in the modified
LMG model? Since the two-body term of the Hamiltonian ${\bf H}^{\prime}=-h{\bf
J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf J}_{y}^{2}\right)$ plays an
essential role in such a case, we cannot refrain from studying it per se in a
deeper perspective — in this context, we will assume $h=0$ hereafter. Note
that the plethora of results exposed and discussed along the text for
assigning the existence of entanglement in a multi-spin system can then be
directly compared in the particular case of the two-body correlations
associated with the model at hand. This important procedure will produce an
effective range for $\gamma$ with $N=20$ fixed, via extensive numerical
computations of Eqs. (42) and (32), whose physical implications permit, within
other features, to answer (in part, at least) the aforementioned question.

Figure 8: Plots of (a,c) $\mathcal{E}_{z}(t)$ (solid line), $E_{z}(t)$ (dot-
dashed line), $\mathcal{S}_{z}^{(x)}(t)$ (dashed line) and (b,d)
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ versus $t\in[0,50]$ for (a,b)
$\gamma=0.5$ and (c,d) $\gamma=0.948$ (critical value), with $N=20$ fixed and
$h=0$ (absence of transverse magnetic field). The transition between the
different values of $\gamma$, here depicted by pictures (a) and (c),
illustrates how the squeezing and entanglement criteria (as well as the
covariance function) are affected by the anysotropy parameter $\gamma$. Such a
numerical evidence suggests the existence of a validity domain for the
aforementioned criteria, where the ‘almost perfect match’ between squeezing
and entanglement effects should be preserved for any $t\geq 0$.

Now, we discuss some essential points inherent to the numerical results
obtained through the discrete Wigner function (13) evaluated at equally spaced
time intervals (i.e., $0.05$ time units) and $|\gamma|\leq 1$. The first one
confirms the link between squeezing and entanglement effects verified in
section 4 for values up to $\gamma=0.5$; however, if one considers the
interval $0.5<\gamma<0.9$, small differences concerning the parameters
$\\{\mathcal{S}_{a}^{(x)}(t),\mathcal{E}_{a}(t),E_{a}(t)\\}$ for $a=y,z$ begin
to appear, this fact being considered by us as a first numerical evidence
about the ‘limitation’ of the particular entanglement measures
$\mathcal{E}_{a}(t)$ and $E_{a}(t)$ here studied. The second essential point
establishes that, for $\gamma\approx 0.948$ (critical value),
$\mathcal{E}_{a}(t)$ does not predict entanglement effects anymore, while
$E_{a}(t)$ still does — in this case, it is worth stressing that
$\mathcal{S}_{a}^{(x)}(t)$ also indicates the presence of the squeezing effect
for determined values of time.

With respect to the proposals of squeezing and entanglement measures discussed
in this work, let us draw attention to the presence of the covariance function
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ in the expression for
$\mathcal{S}_{a}^{(x)}(t)$ (see table 4.2 and appendix A) which depends
explicitly on the mean value (27). Our numerical calculations allow to show,
in particular, that $\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ is
responsible for the occurrence of squeezing effect in the entire domain of
$\gamma$, since it carries essential additional information, when we take into
account the RS inequality (instead of the Heisenberg one), associated with the
anticommutation relation of the angular-momentum operators $\mathbf{J}_{y}$
and $\mathbf{J}_{z}$.

Therefore, our results reinforce the real necessity of a deeper discussion on
the validity domain related to the entanglement criteria here used to
characterize the important link between squeezing and entanglement
effects.[13] Thus, it seems that we cannot assure beforehand the viability of
such criteria without also especifying their respective validity domains in
order to be employed with a consistent degree of confidence.

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