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operations can be conducted on these pairs: products, blow-ups and blow-downs,
and taking the graph of a holomorphic function on the subvariety, restricted
to the complement of its divisor. For example the cases described in Sections
8.2 and 8.3 can both be obtained by these operations. The properties of being
schön, wunderschön, and cohomologically tropical should be shellable in this
framework. We refer to [Sch21] for some results in this direction.

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# CORRECTIONS TO MODELS OF B-PHYSICS DECAY ANOMALIES

NUDŽEIM SELIMOVIĆ  Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerland

###### Abstract

Motivated by the recent anomalies observed in semileptonic decays of B-mesons,
the class of models denoted as ”4321” is introduced and analysed. The
importance of calculating next-to-leading-order (NLO) effects in performing
precise compatibility tests of data with 4321 predictions is discussed. On the
one hand, the coupling of the $U_{1}\sim(\textbf{3},\textbf{1})_{2/3}$
leptoquark appearing in the spectrum to the third-generation fermions is
large, giving rise to sizable NLO corrections in the relevant semileptonic
operators. On the other hand, the new source of flavor violation is introduced
with the vector-like fermions, resulting in the processes whose relevance can
only be quantified at the loop order.

##  1 Introduction

It has been almost a decade now since the first deviations from the Standard
Model (SM) predictions were observed in semileptonic decays of B-mesons [1].
The measurements by various collaborations that followed lead to a coherent
pattern of the SM deviations, which are now known as ”B anomalies”. The hints
collected so far are grouped into two categories: i) deviations from
$\tau/\mu$ and $\tau/e$ universality in $b\to c\ell\bar{\nu}$ charged
currents, and ii) deviations from $\mu/e$ universality in $b\to
s\ell^{+}\ell^{-}$ neutral currents. As shown in the recent study [2], the
combined significance of deviations is at the level of $4\sigma$ for both
charged and neutral current observables.
In order to provide a combined explanation, it was recognized that ingredients
of the New Physics (NP) should be: i) an approximate $U(2)^{5}$ flavour
symmetry implying dominant couplings of the new sector to the third generation
fermions [3], and ii) a TeV-scale vector leptoquark,
$U_{1}\sim(\textbf{3},\textbf{1})_{2/3}$, as a particularly successful
mediator. The different UV-complete models incorporating the two ingredients
[4, 5, 6, 7] share the TeV-scale dynamics which is always characterized by the
gauge group $\mathcal{G}_{4321}\equiv SU(4)\times SU(3)\times SU(2)\times
U(1)$ acting in a family non-universal way. As a general feature, these models
collectively denoted as ”4321”, contain new gauge fields acquiring mass in the
$\mathcal{G}_{4321}\to\mathcal{G}_{SM}$ spontaneous symmetry breaking.
Moreover, realistic realisations involve the vector-like fermions which induce
couplings between the new $SU(4)$ vectors and the light SM generations.
In this talk I want to report some of the most important results concerning
the NLO effects associated with the new fields, calculated in a series of
papers [8, 9, 10]. The motivation comes from the expectation that these
corrections are sizable since coupling of the leptoquark to SM fermions must
be large $(\mathcal{O}(3))$ in order to explain B-physics data, while being
consistent with collider searches. Additionally, vector-like fermions
introduce a new source of flavor violation which results in the flavour-
changing neutral current (FCNC) amplitudes absent at tree-level.

##  2 The 4321 models

The 4321 models are based on the $\mathcal{G}_{4321}\equiv SU(4)\times
SU(3)^{\prime}\times SU(2)_{L}\times U(1)_{X}$ gauge symmetry, with the
corresponding gauge fields denoted by $H_{\mu}^{A=1,\dots,15}$,
$C_{\mu}^{a=1,\dots 8}$, $W_{\mu}^{1,2,3}$ and $B^{\prime}_{\mu}$, and the
gauge couplings $g_{4}$, $g_{3}$, $g_{2}$ and $g_{1}$, respectively. The SM
gauge group corresponds to the $\mathcal{G}_{4321}$ subgroup $SU(3)_{c}\times
U(1)_{Y}\equiv[SU(4)\times SU(3)^{\prime}\times U(1)_{X}]_{\rm diag}$, with
$SU(2)_{L}$ being the SM one. It is useful to define the mixing angles
$\theta_{1,3}$, relating the 4321 gauge couplings to the SM ones

$g_{s}=g_{4}\,\sin\theta_{3}=g_{3}\,\cos\theta_{3}\,,\quad
g_{Y}=\sqrt{\tfrac{3}{2}}\,g_{4}\,\sin\theta_{1}=g_{1}\,\cos\theta_{1}\,,$ (1)

with $g_{s}$ and $g_{Y}$ denoting the $SU(3)_{c}$ and $U(1)_{Y}$ gauge
couplings. In terms of the 4321 gauge eigenstates, the SM gluon,
$G_{\mu}^{a}$, and hypercharge gauge boson, $B_{\mu}$, can be written as

$G_{\mu}^{a}=\cos\theta_{3}\,C_{\mu}^{a}+\sin\theta_{3}\,H_{\mu}^{a}\,,\quad
B_{\mu}=\cos\theta_{1}\,B_{\mu}^{\prime}+\sin\theta_{1}\,H_{\mu}^{15}\,,$ (2)

while the new massive vectors
$G_{\mu}^{\prime\,a}\,,Z^{\prime}_{\mu}\,,U_{\mu}^{1,2,3}$ read

$\displaystyle G_{\mu}^{\prime\,a}$
$\displaystyle=-\sin\theta_{3}\,C_{\mu}^{a}+\cos\theta_{3}\,H_{\mu}^{a}\,,\qquad
Z_{\mu}^{\prime}=-\sin\theta_{1}\,B_{\mu}^{\prime}+\cos\theta_{1}\,H_{\mu}^{15}\,,$
(3) $\displaystyle
U_{\mu}^{1,2,3}=\frac{1}{\sqrt{2}}(H_{\mu}^{9,11,13}-iH_{\mu}^{10,12,14})\,.$

Table 1: Minimal matter content. Field | $SU(4)$ | $SU(3)^{\prime}$ | $SU(2)_{L}$ | $U(1)_{X}$
---|---|---|---|---
$\psi_{L}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 0
$\psi^{\pm}_{R}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\pm 1/2$
$q_{L}^{\prime\,i}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{2}$ | $1/6$
$u_{R}^{i}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | $2/3$
$d_{R}^{i}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | $-1/3$
$\ell_{L}^{\prime\,i}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $-1/2$
$e_{R}^{i}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-1$
$\Omega_{3}$ | $\mathbf{\bar{4}}$ | $\mathbf{3}$ | $\mathbf{1}$ | $1/6$
$\Omega_{1}$ | $\mathbf{\bar{4}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-1/2$

In most 4321 models, the $\mathcal{G}_{4321}\to\mathcal{G}_{SM}$ breaking is
triggered by the vacuum expectation values of two scalar fields transforming
in the antifundamental of $SU(4)$, $\Omega_{1}$ and $\Omega_{3}$, singlet and
triplet under $SU(3)^{\prime}$, respectively.111An additional scalar field,
transforming in the adjoint of $SU(4)$ and singlet under the rest, is often
introduced is some 4321 models [4, 7]. The would-be SM fermions are charged
non-universally under the 4321 gauge group, as summarised in Table 1. Here
$i=1,2$, $\psi_{L}\equiv(q_{L}^{\prime 3}\;\ell_{L}^{\prime 3})$,
$\psi_{R}^{+}\equiv(u_{R}^{3}\;\nu_{R}^{3})$ and
$\psi_{R}^{-}\equiv(d_{R}^{3}\;e_{R}^{3})$. The prime in the fields indicates
that these are not mass eigenstates.

Table 2: Additional fermion content. Here
$\chi_{L}=(Q_{L}^{\prime}\;L_{L}^{\prime})$ and $\chi_{R}=(Q_{R}\;L_{R})$. The
prime in the $\chi_{L}$ components indicates that these are not mass
eigenstates.

Model | Field | $SU(4)$ | $SU(3)^{\prime}$ | $SU(2)_{L}$ | $U(1)_{X}$
---|---|---|---|---|---
| $\chi_{L}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 0
I | $Q_{R}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{2}$ | $1/6$
| $L_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $-1/2$
II | $\chi_{L,R}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 0

In order to introduce the couplings of $SU(4)$ vectors to the light
generations, and 2-3 CKM matrix elements, we add to the minimal content one
family of left-handed fermions, transforming in the fundamental of $SU(4)$ and
$SU(2)_{L}$, and one family of right-handed partners. The massive fermions are
vector-like under the SM gauge group, therefore, the right-handed partners
should transform in the fundamental of $SU(2)_{L}$, but there is freedom in
the $SU(4)\times SU(4)^{\prime}(\supset SU(3)^{\prime})$ transformations. As
shown in Table 2, we consider two possible realisations: Model I discussed in
[6], and Model II in [7]. In both cases, having two $SU(4)$ charged fermions,
$\psi_{L}$ and $\chi_{L}$, leads to a new flavor symmetry that we denote
$U(2)_{\xi}$, with $\xi_{L}=(\psi_{L},\chi_{L})^{T}$. This symmetry is broken
by the fermion masses, causing mixing among $\psi_{L}$ and $\chi_{L}$, and
also the $SU(4)$-singlet fermions, $q_{L}^{\prime}$ and $\ell_{L}^{\prime}$.
Whatever the source of $SU(4)$ breaking is, the mass terms read

$\mathcal{L}_{\rm
mass}=\bar{\Psi}_{L}^{q\,\prime}\,M_{q}\,Q_{R}+\bar{\Psi}_{L}^{\ell\,\prime}\,M_{\ell}\,L_{R}\,,$
(4)

with the left-handed fermions arranged into the flavor vectors
$\Psi^{q\,\prime}_{L}=(q_{L}^{\prime\,2},q_{L}^{\prime\,3},Q_{L}^{\prime})^{T}$
and
$\Psi^{\ell\,\prime}_{L}=(\ell_{L}^{\prime\,2},\ell_{L}^{\prime\,3},L_{L}^{\prime})^{T}$,
and $M_{q,\ell}$ being the $3$-dimensional mass vectors. Without loss of
generality, these mass vectors can be written as

$M_{q,\ell}=\tilde{W}_{q,\ell}\,O_{q,\ell}\begin{pmatrix}0\ \ 0\ \
m_{Q,L}\end{pmatrix}^{T}\,,$ (5)

where $m_{Q,L}$ are the vector-like fermion masses. Here, the $3\times 3$
unitary matrices $\tilde{W}_{q,\ell}$ parametrize the mixing among $SU(4)$
states, while the $3\times 3$ orthogonal matrices $O_{q,\ell}$ parametrize the
mixing among different $SU(4)$ representations. Explicitly, these matrices
read

$\tilde{W}_{q,\ell}=\begin{pmatrix}1&0\\\ 0&W_{q,\ell}\end{pmatrix}\,,\quad
O_{q,\ell}=\begin{pmatrix}\cos\theta_{Q,L}&0&\sin\theta_{Q,L}\\\ 0&1&0\\\
-\sin\theta_{Q,L}&0&\cos\theta_{Q,L}\\\ \end{pmatrix}\,,$ (6)

where $\theta_{Q,L}$ are the mixing angles, and $W_{q,\ell}$ being $2\times 2$
unitary matrices. To better understand the origin of the flavor mixing
matrices, it is convenient to follow the analogy with the SM. Here, the
$SU(4)$ group puts quarks and leptons into a multiplet, which will be
misaligned after the $SU(4)$ breaking, similar to the up-down quark
misalignment inside the $SU(2)_{L}$ multiplet of the SM. This misalignment
will cause the quark-lepton flavour mixing which is parametrised by the
physical combination of $W_{q}$ and $W_{\ell}$ matrices,
$W_{q}^{\dagger}W_{\ell}=W$, appearing in the $U_{1}$ interaction, and can be
viewed as a generalization of the CKM matrix to $SU(4)$ or quark-lepton space.
In this sense, the $U_{1}$ leptoquark is analogous to the SM $W$, while the
$Z^{\prime},G^{\prime}$ are analogous to the SM $Z$. These interactions can be
written in the $SU(4)$ basis, or in the quark ($\mathcal{Q}_{L}^{i}$) and
lepton ($\mathcal{L}_{L}^{i}$) components of $\xi^{i}_{L}$, that in the mass-
eigenstate basis are given by

$\begin{pmatrix}0\\\\[2.0pt] \mathcal{Q}_{L}^{1}\\\\[2.0pt]
\mathcal{Q}_{L}^{2}\end{pmatrix}=P_{23}\,O_{q}\begin{pmatrix}q_{L}^{2}\\\\[2.0pt]
q_{L}^{3}\\\\[2.0pt] Q_{L}\end{pmatrix}\,,\quad\begin{pmatrix}0\\\\[2.0pt]
\mathcal{L}_{L}^{1}\\\\[2.0pt]
\mathcal{L}_{L}^{2}\end{pmatrix}=P_{23}\,O_{\ell}\begin{pmatrix}\ell_{L}^{2}\\\\[2.0pt]
\ell_{L}^{3}\\\\[2.0pt] L_{L}\end{pmatrix}\,,$ (7)

with $P_{23}=\rm diag(0,1,1)$ a projector onto $SU(4)$ states. The
interactions read ($i=1,2$)

$\mathcal{L}\supset\frac{g_{4}}{\sqrt{2}}\,(\mathcal{\bar{Q}}_{L}^{i}\,W_{ij}\,{U\mkern-13.5mu/}\,\mathcal{L}_{L}^{j}+{\rm
h.c.})+\frac{g_{4}}{2\sqrt{6}}\,\left(\mathcal{\bar{Q}}_{L}^{i}{Z^{\prime}\mkern-16.5mu/}\,\,\mathcal{Q}_{L}^{i}-3\,\mathcal{\bar{L}}_{L}^{i}{Z^{\prime}\mkern-16.5mu/}\,\,\mathcal{L}_{L}^{i}\right)+g_{4}\,\mathcal{\bar{Q}}_{L}^{i}{G^{\prime}\mkern-16.5mu/}\,\,^{a}\,T^{a}\mathcal{Q}_{L}^{i}\,.$
(8)

As in the SM, FCNCs proportional to the $W$ matrix are generated at the loop
level. Computing these contributions to $\Delta F=1$ semileptonic and dipole
operators, as well as $\Delta F=2$ hadronic and leptonic amplitudes,
represents one of the main results of this work.

##  3 NLO results

###  3.1 Gauge sector

In this subsection I report the main results concerning the relation between
low- and high-energy observables beyond the tree level. In particular, we
evaluate the NLO corrections in two largest couplings, $g_{4}(m_{U}^{2})$ and
$g_{s}(m_{U}^{2})$, to the Wilson coefficients of the dimension-six
semileptonic operators involving SM third-generation fermions, with $m_{U}$
being the $U_{1}$ leptoquark mass. The following normalization is adopted

$\mathcal{L}=-\frac{g_{4}^{2}}{2m_{U}^{2}}\sum_{k}\mathcal{C}_{k}(\mu)\mathcal{O}_{k}\,,$
(9)

with the most relevant operators being those generated by the $U_{1}$ tree-
level exchange

$\mathcal{O}_{LL}^{U}=(\bar{\ell}^{\prime
3}_{L}\gamma^{\mu}{q^{\prime}}_{L}^{3})(\bar{q}^{\prime
3}_{L}\gamma_{\mu}{\ell^{\prime}}_{L}^{3})\,,\quad\mathcal{O}_{LR}^{U}=-2(\bar{\ell}^{\prime
3}_{L}{e}^{\prime 3}_{R})(\bar{d}^{\prime 3}_{R}{q}^{\prime
3}_{L})+\mathrm{h.c.}\,.$ (10)

Firstly, we calculated one-loop corrections to the corresponding 4-fermion
amplitudes in the full theory, employing the dimensional regularisation with
$\overline{MS}$ renormalisation scheme as the most convenient because of the
infrared singularities and the precise knowledge of $g_{s}(\mu)$. Proceeding
this way, an additional step of expressing the (unphysical) coupling $g_{4}$
in the Lagrangian in terms of some high-energy physical observable is needed.
To this end, we chose the inclusive leptoquark width, and used it to define
the corresponding physical coupling. Secondly, the matching between the full
and the effective theory was performed. The main result is that both NP and
QCD corrections are positive, together reaching $20\%$ ($40\%$) increase in
$\mathcal{C}_{\rm LL}$ ($\mathcal{C}_{\rm LR}$), implying enhanced $U_{1}$
contribution at low-energy using the fixed high-energy inputs [8, 9]. This is
an important phenomenological consequence meaning that all collider bounds
dominated by the on-shell production of the new states will be weaker at fixed
low-energy contribution once the quantum corrections are taken into account.

###  3.2 Flavour changing neutral currents

The 4321 models are exposed to relevant constraints originating from FCNC
processes not present at tree-level, but receiving contributions from $U_{1}$
loops proportional to the $W$ matrix. As notable examples, I want to assess
the UV sensitivity of $B_{s}-\bar{B}_{s}$ mixing and $B\to
K^{(*)}\nu\bar{\nu}$ transition. In the case of $B_{s}-\bar{B}_{s}$ mixing,
the one-loop contribution to the Wilson coefficient parametrising this process
is

$\mathcal{C}_{bs}^{\rm
NLO}=\frac{g_{4}^{2}}{2m_{U}^{2}}\frac{\alpha_{4}}{4\pi}\left(\sin\theta_{Q}\cos^{2}\theta_{L}W_{12}W_{22}^{*}\right)^{2}\left(\frac{m_{L}^{2}}{4m_{U}^{2}}+\mathcal{O}\left(\frac{m_{L}^{4}}{m_{U}^{4}}\right)\right)\,,$
(11)

and is completely controlled by the vector-like lepton mass, $m_{L}$, for
fixed charged-current anomaly. This can be used to extract the upper bound on
$m_{L}$, and in combination with direct searches squeeze the parameter space
for these particles, putting them in the $\mathcal{O}(1\ {\rm TeV})$ ballpark.
Similarly, we performed the first complete analysis of the $U_{1}$ impact in
$\mathcal{B}(B\to K^{(*)}\nu\bar{\nu})$. Both implementations of the 4321
model (Model I and II in Table 2) predict 15% to 60% enhancement with respect
to the SM expectation, in the parameter region which provides a good fit to
the B-anomalies. The Belle II Collaboration should be able to measure this
branching fraction with a $10\%$ error assuming the SM value, probing all of
the parameter space of the 4321 models [11].

## Acknowledgments

I am grateful to my collaborators Javier Fuentes-Martín, Gino Isidori and
Matthias König. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement 833280 (FLAY), and by the Swiss National
Science Foundation (SNF) under contract 200021-175940.

## References

## References

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# Spin squeezing and concurrence

Xiaolei Yin  Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University, Hangzhou 310027, China  Xiaoqian Wang  Zhejiang Institute
of Modern Physics, Department of Physics, Zhejiang University, Hangzhou
310027, China  Department of Physics, Changchun University of Science and
Technology, Changchun 130022, China  Jian Ma  Zhejiang Institute of Modern
Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
Xiaoguang Wang<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics,
Department of Physics, Zhejiang University, Hangzhou 310027, China

###### Abstract

We study the relations between spin squeezing and concurrence, and find that
they are qualitatively equivalent for an ensemble of spin-1/2 particles with
exchange symmetry and parity, if we adopt the spin squeezing criterion given
by the recent work (G. Tóth et al. Phys. Rev. Lett. 99, 250405 (2007)). This
suggests that the spin squeezing has more intimate relations with pairwise
entanglement other than multipartite entanglement. We exemplify the result by
considering a superposition of two Dicke states.

spin squeezing,concurrence,entanglement

###### pacs:

03.65.Ud,03.67.2a

##  I Introduction

As an important resource of quantum information and computation, entanglement
Einstein ; Schrodinger  has attracted much attention in recent years
Bennett1993 ; Bennett1992 ; Ekert1991 ; Wang2001 ; WangSolomon ; Pan2001 ;
Vidal2003 ; VidalPalacios ; Leibfried ; Andre . How to measure and detect
entanglement is crucial for both theoretical investigations and potential
practical applications Bennett1996 ; Vedral1997 . The entanglement of a two-
qubit system can be well quantified by the concurrence Wootters1997 ;
Wootters1998 . However, quantification of many-body entanglement is still an
open question in quantum information.

It is well known that there are close relations between entanglement and spin
squeezing SorensenMolmer ; Sorrensen2001 ; Kitagawa1993 ; Wineland1994 ;
Kitagawa2001 ; WangSangders2003 ; Jin2007 ; Jafarpour . There are several
definitions of spin squeezing parameters Sorrensen2001 ; Kitagawa1993 ;
Wineland1994 , which are studied in different papers. The squeezing parameter
$\xi_{R}^{2}$ defined by Wineland et al. is closely related to multipartite
entanglement. It has been proven that Sorrensen2001 , for an ensemble of
spin-1/2 particles, if this squeezing parameter is less than one, the state is
entangled. The advantages of spin squeezing parameters in detecting
entanglement have been shown in both theoretical and experimental aspects.

The squeezing parameter $\xi_{S}^{2}$ defined by Kitagawa and Ueda is relevant
to pairwise entanglement Kitagawa2001 . And for states with exchange symmetry
and parity, a simple quantitative relation between $\xi_{S}^{2}$ and
concurrence was given WangSangders2003 . Furthermore, it has been shown that
the spin squeezing and pairwise entanglement are equivalent for states
generated by the one-axis twisting Hamiltonian WangSangders2003 . However,
even for states with a fixed parity, such as the states generated by one-axis
twisting Hamiltonian with a transverse field, $\xi_{S}^{2}$ is not always
equivalent to concurrence Wang2004 . Inspired by recent works Toth2007 ;
Toth2009 , where a set of generalized spin squeezing inequalities are
developed, one can define another spin squeezing parameter $\xi_{T}^{2}$ from
one of the inequalities Wang2009 . Similar to parameter $\xi_{R}^{2}$, one
advantage of the parameter $\xi_{T}^{2}$ is that we can firmly say that the
state is entangled if $\xi_{T}^{2}<1$. However, if parameter $\xi_{S}^{2}<1$,
we cannot say the state is entangled, although this parameter is closely
related to entanglement.

Reference Kitagawa2001  found that spin squeezing according to parameter
$\xi_{S}^{2}$ is equivalent to the minimal pairwise correlation
$\mathcal{C}_{\vec{n}_{\perp},\vec{n}_{\perp}}$ along the direction
$\vec{n}_{\perp}$ (which is perpendicular to the mean spin direction) for
symmetric states. It was further found Xiaoqian10  that for the symmetric
states, the spin squeezing defined via $\xi_{T}^{2}$ is equivalent to minimal
pairwise correlation $\mathcal{C}_{\vec{n},\vec{n}}$ along an arbitrary
direction $\vec{n}$. For states with a fixed parity, the relations between the
two parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$ are more evident. It will be
seen from Sec. 3 that $\xi_{T}^{2}$ contains the term $\xi_{S}^{2}$, and the
spin squeezing results from just the competition between pairwise correlation
along the direction $\vec{n}_{\perp}$ and that along the mean spin direction.
So, in this sense, the parameter $\xi_{T}^{2}$ is a natural generalization of
$\xi_{S}^{2}$.

We find that for states with exchange symmetry and parity, the spin squeezing
parameter $\xi_{T}^{2}$ is qualitatively equivalent to the concurrence in
characterizing pairwise entanglement. In other words, the spin squeezing
parameter and concurrence emerge and vanish at the same time. This finding is
of significance to entanglement detection in experiments. As we all know,
entanglement detectors such as entropy and concurrence are generally not easy
to measure, especially for physical systems like BEC, for which we cannot
address individual atoms. However, spin squeezing parameters are relatively
easy to measure in experiments, since they only consist of expectations and
variances of collective angular momentum operators. Nevertheless, the
traditional spin squeezing parameter $\xi_{S}^{2}$ is not always equivalent to
concurrence even for states with exchange symmetry and parity. As
$\xi_{T}^{2}$ is qualitatively equivalent to concurrence for an ensemble of
spin-1/2 particles with exchange symmetry and parity, it is better than
$\xi_{S}^{2}$ in detecting pairwise entanglement.

The paper is organized as follows: In Sec. II, we give the definitions of the
two spin squeezing parameters and concurrence. In Sec. III, we give the
concrete forms of the spin squeezing parameters and the concurrence for states
with exchange symmetry and parity. The relations between these two parameters
and concurrence were given in Sec. IV. We exemplify the result by considering
superpositions of Dicke states in Sec. V. Finally, Sec. VI is devoted to
conclusion.

##  II Spin squeezing parameters and concurrence

To study spin squeezing, we consider an ensemble of $N$ spin-1/2 particles.
For the sake of describing many-particle systems, we use the total angular
momentum operators

$J_{\alpha}=\frac{1}{2}\sum_{k=1}^{N}\sigma_{k\alpha},~{}~{}~{}\left(\alpha=x,y,z\right),$
(1)

where $\sigma_{k\alpha}$ are the Pauli matrices for the $k$-th spin. Now, we
give the definitions of the two spin squeezing parameters. The first one is
defined as Kitagawa1993 ,

$\xi_{S}^{2}=\frac{4\min(\Delta J_{\vec{n}_{\perp}})^{2}}{N},$ (2)

where $\vec{n}_{\perp}$ is perpendicular to the mean spin direction
$\vec{n}=\frac{\langle\vec{J}\rangle}{|\langle\vec{J}\rangle|}$. Since the
system has the exchange symmetry, the total angular momentum is
$j=\frac{N}{2}$. For spin coherent states Kitagawa1993 , $\Delta
J_{\vec{n}_{\perp}}=\frac{j}{2}$, and $\xi_{S}^{2}=1$. In the following, we
consider states with exchange symmetry.

The next spin squeezing parameter is based on the generalized spin squeezing
inequalities given by Tóth et al. Toth2009 , and is defined as Wang2009

$\xi_{T}^{2}=\frac{\lambda_{\min}}{\langle\vec{J}^{2}\rangle-\frac{N}{2}},$
(3)

where $\lambda_{\min}$ is the minimum eigenvalue of

$\Gamma=(N-1)\gamma+G$ (4)

with $G_{kl}=\frac{1}{2}\left\langle J_{k}J_{l}+J_{l}J_{k}\right\rangle$,
$(k,l=x,y,z)$ the correlation matrix, and $\gamma_{kl}=G_{kl}-\left\langle
J_{k}\right\rangle\left\langle J_{l}\right\rangle$ the covariance matrix. For
our states, $\langle\vec{J}^{2}\rangle-\frac{N}{2}=j\left(j+1\right)-j=j^{2}$.

The two-qubit entanglement is quantified by the concurrence, whose definition
is given by Wootters1998

$C=\max\\{\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4},0\\},$ (5)

where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$ are the
square roots of eigenvalues of $\tilde{\rho}\rho$. Here $\rho$ is the reduced
density matrix of the system, and

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{\ast}(\sigma_{y}\otimes\sigma_{y}),$
(6)

where $\rho^{\ast}$ is the conjugate of $\rho$. If $C>0$, the system displays
pairwise entanglement.

##  III States with parity

To study the relations between spin squeezing parameters and concurrence, we
consider a class of states with even (odd) parity, which means a state in the
$(2j+1)$-dimensional space with only even (odd) excitations of spins. These
kinds of states are widely studied, e.g., the states generated by the one-axis
twisting model Kitagawa1993 . The states with even parity possess important
properties, $\left\langle J_{\alpha}\right\rangle=0$, $\left\langle
J_{\alpha}J_{z}\right\rangle=\left\langle J_{z}J_{\alpha}\right\rangle=0$,
$\alpha=x,y$, which means the mean spin direction is along the $z$-axis, and
the covariances between $J_{z}$ and $J_{\alpha}$ are zero. Thus, equation (4)
becomes

$\Gamma=\left(\begin{array}[]{ccc}N\left\langle
J_{x}^{2}\right\rangle&\frac{N\left\langle\left[J_{x},J_{y}\right]_{+}\right\rangle}{2}&0\\\
\frac{N\left\langle\left[J_{x},J_{y}\right]_{+}\right\rangle}{2}&N\left\langle
J_{y}^{2}\right\rangle&0\\\ 0&0&N(\Delta J_{z})^{2}+\langle
J_{z}\rangle^{2}\end{array}\right),$ (7)

where $\left[A,B\right]_{+}=AB+BA$, and equation (3) reduces to Wang2009

$\xi_{T}^{2}=\min\left\\{\xi_{S}^{2},\varsigma^{2}\right\\},$ (8)

where

$\displaystyle\varsigma^{2}$ $\displaystyle=$
$\displaystyle\frac{4}{N^{2}}\left[N(\Delta J_{z})^{2}+\langle
J_{z}\rangle^{2}\right]$ (9) $\displaystyle=$ $\displaystyle
1+\left(N-1\right)\left(\left\langle\sigma_{1z}\sigma_{2z}\right\rangle-\left\langle\sigma_{1z}\right\rangle^{2}\right)$
$\displaystyle=$ $\displaystyle 1+(N-1)C_{zz},$

with $C_{zz}$ the two-spin correlation function along $z$ direction. The
explicit form of $\xi_{S}^{2}$ could be obtained as WangSangders2003

$\displaystyle\xi_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{N}\left(\langle J_{x}^{2}+J_{y}^{2}\rangle-|\langle
J_{-}^{2}\rangle|\right)$ (10) $\displaystyle=$ $\displaystyle
1-2\left(N-1\right)$
$\displaystyle\times\left[\left|\langle\sigma_{1-}\sigma_{2-}\rangle\right|-\frac{1}{4}\left(1-\left\langle\sigma_{1z}\sigma_{2z}\right\rangle\right)\right],$

where we have used the following relations

$\displaystyle\left\langle J_{\alpha}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{2}\left\langle\sigma_{1\alpha}\right\rangle,$
$\displaystyle\langle J_{\alpha}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{4}+\frac{N(N-1)}{4}\langle\sigma_{1\alpha}\sigma_{2\alpha}\rangle,$
$\displaystyle\langle J_{-}^{2}\rangle$ $\displaystyle=$ $\displaystyle
N(N-1)\langle\sigma_{1-}\sigma_{2-}\rangle,$ (11)

which connect the local expectations with collective ones.

For such states, the significance of $\xi_{S}^{2}$ and $\xi_{T}^{2}$ and the
relations between them are clear. According to the parameter $\xi_{S}^{2}$, a
state is squeezed when the minimum variance of angular momentum in the
$\vec{n}_{\perp}$-plane is smaller than $\frac{j}{2}$, while according to
$\xi_{T}^{2}$, the variance in the mean spin direction $\vec{n}$ is also
considered. Equation (9) represents the pairwise correlation along the mean
spin direction, and this can be viewed as a complement of $\xi_{S}^{2}$, which
only considers squeezing in the $\vec{n}_{\perp}$-plane. Thus, $\xi_{T}^{2}$
can be regarded as a generalization of $\xi_{S}^{2}$, and when
$\xi_{S}^{2}<\varsigma^{2}$, the parameter $\xi_{T}^{2}$ reduces to
$\xi_{S}^{2}$.

To calculate concurrence, we first need to calculate the two-body reduced
density matrix, which can be written as WangSangders2003

$\rho=\left(\begin{array}[]{cccc}v_{+}&0&0&u^{\ast}\\\ 0&y&y&0\\\ 0&y&y&0\\\
u&0&0&v_{-}\end{array}\right),$ (12)

where

$\displaystyle v_{\pm}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(1\pm
2\langle\sigma_{1z}\rangle+\langle\sigma_{1z}\sigma_{2z}\rangle\right),$
$\displaystyle u$ $\displaystyle=$
$\displaystyle\langle\sigma_{1-}\sigma_{2-}\rangle,~{}~{}~{}y=\frac{1}{4}\left(1-\left\langle\sigma_{1z}\sigma_{2z}\right\rangle\right),$
(13)

in the basis
$\left\\{\left|00\right\rangle,\left|01\right\rangle,\left|10\right\rangle,\left|11\right\rangle\right\\}$.
Then the concurrence is given by

$C=2\max\left\\{0,~{}|u|-y,~{}y-\sqrt{v_{+}v_{-}}\right\\}.$ (14)

One key observation is that

$y^{2}-v_{+}v_{-}=-\frac{1}{4}C_{zz}.$ (15)

Thus,

$\varsigma^{2}=1-4(N-1)(y+\sqrt{v_{+}v_{-}})(y-\sqrt{v_{+}v_{-}}).$ (16)

From equations (9), (10), and (13), we obtain

$\displaystyle\xi_{S}^{2}$ $\displaystyle=$ $\displaystyle
1-2(N-1)\left(\left|u\right|-y\right),$ $\displaystyle\xi_{T}^{2}$
$\displaystyle=$ $\displaystyle\min\\{1-2(N-1)\left(\left|u\right|-y\right),$
(17) $\displaystyle 1-4(N-1)(y+\sqrt{v_{+}v_{-}})(y-\sqrt{v_{+}v_{-}})\\}.$

Now, one can see that the squeezing parameters are related to the concurrence
shown in equation (14). The relations between $\xi_{S}^{2}$ and $C$ have been
studied WangSangders2003 . In the following, we consider the squeezing
parameter $\xi_{T}^{2}$, and prove that it is qualitatively equivalent to the
concurrence in detecting pairwise entanglement.

##  IV Relations between spin squeezing parameters and concurrence

Firstly, we prove that for a state with exchange symmetry and parity, if
concurrence $C>0$, it must be spin squeezed according to the criterion
$\xi_{T}^{2}<1$. From equation (14) we note that when $C>0$, there are two
cases, $C=\left|u\right|-y>0$ or $C=y-\sqrt{v_{+}v_{-}}>0$. However, since the
density matrix $\rho$ is positive, we find
$\sqrt{v_{+}v_{-}}\geq\left|u\right|$, then immediately

$\left(|u|-y\right)\left(y-\sqrt{v_{+}v_{-}}\right)\leq 0,$ (18)

which means $\left|u\right|-y$ and $y-\sqrt{v_{+}v_{-}}$ cannot be positive
simultaneously. Therefore, if $C>0$, we have Vidal2006

$C=\left\\{\begin{array}[]{ll}2\left(\left|u\right|-y\right),{}{}{}&\left|u\right|>y,\\\
2\left(y-\sqrt{v_{+}v_{-}}\right),{}{}{}&y>\sqrt{v_{+}v_{-}}.\end{array}\right.$
(19)

According to equations (8) and (17), we get the following relations

$\xi_{T}^{2}=\left\\{\begin{array}[]{ll}1-\left(N-1\right)C,{}{}{}&\left|u\right|>y,\\\
1-2\left(N-1\right)\left(y+\sqrt{v_{+}v_{-}}\right)C,{}{}{}&y>\sqrt{v_{+}v_{-}},\end{array}\right.$
(20)

since $C>0$, there always be $\xi_{T}^{2}<1$.

Now, we prove that if the state is spin squeezed $\left(\xi_{T}^{2}<1\right)$,
concurrence $C>0$. If $\xi_{T}^{2}<1$, there are two cases,
$\xi_{T}^{2}=\xi_{S}^{2}<1$ or $\xi_{T}^{2}=\varsigma^{2}<1$. As discussed
above, according to equations (17) and (18), $\xi_{S}^{2}<1$ and
$\varsigma^{2}<1$ could not occur simultaneously. Therefore, if
$\xi_{T}^{2}=\xi_{S}^{2}<1$, we have Vidal2004

$C=\frac{1-\xi_{T}^{2}}{N-1},$ (21)

while if $\xi_{T}^{2}=\varsigma^{2}<1$, we have

$C=\frac{1-\xi_{T}^{2}}{2\left(N-1\right)\left(y+\sqrt{v_{+}v_{-}}\right)}.$
(22)

Therefore, if the state is squeezed, concurrence $C>0$.

Table 1: Spin squeezing parameters and concurrence for states with exchange symmetry and parity. | Pairwise entangled ($C>0$) |  Unentangled
---|---|---
Concurrence |  $C=2(\left|u\right|-y)>0$ | $C=2(y-\sqrt{v_{+}v_{-}})>0$ | $C=0$
$\xi_{S}^{2}$ |  $\xi_{S}^{2}=1-(N-1)C<1$ | $\xi_{S}^{2}>1$ | $\xi_{S}^{2}\geq 1$
$\xi_{T}^{2}$ |  $\xi_{T}^{2}=1-\left(N-1\right)C<1$ |  $\xi_{T}^{2}=1-2(N-1)(y+\sqrt{v_{+}v_{-}})\times{C<1}$ | $\xi_{T}^{2}\geq 1$

The relations between spin squeezing and concurrence is displayed in Table 1,
and we can see that, for a symmetric state, $\xi_{T}^{2}<1$ is qualitatively
equivalent to $C>0$, that means spin squeezing according to $\xi_{T}^{2}$ is
equivalent to pairwise entanglement. Although $\xi_{S}^{2}<1$ indicates $C>0$,
when $C=2(y-\sqrt{v_{+}v_{-}})>0$, we find $\xi_{S}^{2}>1$. Therefore, a spin-
squeezed state ($\xi_{S}^{2}<1$) is pairwise entangled, while a pairwise
entangled state may not be spin-squeezed according to the squeezing parameter
$\xi_{S}^{2}$. Then, we come to the conclusion that for states with exchange
symmetry and parity, the spin squeezing parameter $\xi_{T}^{2}$ is
qualitatively equivalent to the concurrence in characterizing pairwise
entanglement. In the following, we will give some examples and applications of
our result.

##  V Examples and Applications

We first consider a superposition of Dicke states with parity, and then
consider states without a fixed parity. The states under consideration are all
based on Dicke states Dicke1954 , and are defined as

$|n\rangle_{N}\equiv|\frac{N}{2},-\frac{N}{2}+n\rangle,~{}~{}~{}n=0,\ldots,N,$
(23)

where $|0\rangle_{N}\equiv|\frac{N}{2},-\frac{N}{2}\rangle$ denotes a state
for which all spins are in the ground states, and $n$ is the excitation number
of spins. Such states are elementary in atomic physics, and may be
conditionally prepared in experiments with quantum non-demolition technique
Molmer1998 ; Mandel ; Lemer2009 .

As we consider the state with even parity, we choose a simple superposition of
Dicke states as

$|\psi_{D}\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+2\rangle_{N},~{}~{}~{}n=0,\ldots,N-2$
(24)

with the angle $\theta\in[0,\pi)$ and the relative phase $\varphi\in[0,2\pi)$.
We can easily check that, for the superposition state in equation (24), the
mean spin direction is along the $z$-axis. The expressions for the relevant
spin expectation values can be obtained as

$\displaystyle\left\langle J_{z}\right\rangle$ $\displaystyle=$ $\displaystyle
m+2\sin^{2}\theta,$ $\displaystyle\langle J_{z}^{2}\rangle$ $\displaystyle=$
$\displaystyle m^{2}+\left(4m+1\right)\sin^{2}\theta,$ $\displaystyle\langle
J_{+}^{2}\rangle$ $\displaystyle=$ $\displaystyle\langle
J_{-}^{2}\rangle=\frac{1}{2}e^{i\varphi}\sin 2\theta\sqrt{\mu_{n}},$ (25)

where $m=n-\frac{N}{2},$ and
$\mu_{n}=\left(n+1\right)\left(n+2\right)\left(N-n\right)\left(N-n-1\right)$.

By substituting equations (25) to equation (9) and (10), it is easy to get

$\displaystyle\xi_{S}^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{2}{N}\\{\left|\sin\theta\cos\theta\right|\sqrt{\mu_{n}}$ (26)
$\displaystyle-4[m^{2}+4(m+1)\sin^{2}\theta]-N^{2}\\}$

and

$\displaystyle\varsigma^{2}$ $\displaystyle=$
$\displaystyle\frac{4}{N}\left[m^{2}+4\left(m+1\right)\sin^{2}\theta\right]$
(27) $\displaystyle-\frac{4(N-1)}{N^{2}}\left[m+2\sin^{2}\theta\right]^{2}.$

From the results in Wang2002  we can easily get Vidal2006

$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{e^{i\varphi}\sin
2\theta}{2N(N-1)}\sqrt{\mu_{n}},$ $\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{N}{4(N-1)}-\frac{[m^{2}+4(m+1)\sin^{2}\theta]}{N(N-1)},$
$\displaystyle\sqrt{v_{+}v_{-}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(N^{2}-2N+4\left\langle
J_{z}^{2}\right\rangle)^{2}-16(N-1)^{2}\left\langle
J_{z}\right\rangle^{2}}}{4N(N-1)}.$

Insert equation (LABEL:rho_results) to equation (14), one can get the
expression of concurrence.

Figure 1: Spin squeezing parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$, and
concurrence as functions of $\theta$ for $N=3$ and $n=0.$ Figure 2: Spin
squeezing parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$, and concurrence as
functions of $\theta$ for $N=3$ and $n=0.$ The numerical result give
$\xi_{S}^{2}=7/3$, $\xi_{T}^{2}=1/9$, and $C=2/3$, when $\theta=\pi/2.$

In figure 1, we plot these two spin squeezing parameters and concurrence
versus $\theta$ in one period. We observe that for
$\theta\in(0,\pi/3)\cup(2\pi/3,\pi)$, $\xi_{T}^{2}=\xi_{S}^{2}<1$, therefore
the state is spin squeezed in the $x$-$y$ plane, moreover, as $C>0$, the state
is pairwise entangled. For $\theta\in(\pi/3,2\pi/3)$, it is obviously that the
state is also pairwise entangled, since $C>0,$ while spin squeezing occurs in
the $z$-axis since $\xi_{T}^{2}<1$ while $\xi_{S}^{2}>1$. The results show
clearly that $\xi_{T}^{2}<1$ is equivalent to $C>0$. But if we adopt
$\xi_{S}^{2}<1$ as squeezing parameter, the spin squeezing is not
qualitatively equivalent to concurrence.

The equivalence of $\xi_{T}^{2}<1$ and $C>0$ for states with parity has been
demonstrated above. Here, we discuss states without parity to see the
relations between spin squeezing and entanglement. For simplicity, we choose

$|\psi_{D}\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+1\rangle_{N},~{}~{}~{}n=0,\ldots,N-1.$
(29)

Specifically, if $\theta=\frac{\pi}{2}$, $n=0$ or $n=N-2$, the above state
degenerates to the W state. Moreover, when $N=3$, equation (29) reduces to

$|\psi_{D}\rangle=\frac{1}{\sqrt{3}}(|110\rangle+|101\rangle+|011\rangle).$
(30)

The two-qubit reduced density matrix becomes

$\rho=\frac{1}{3}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&1&1&0\\\ 0&1&1&0\\\
0&0&0&1\end{array}\right),$ (31)

and using equation (14) we find $C=\frac{2}{3}$. We can also get the
expectations of spin components, $\left\langle
J_{z}\right\rangle=-\frac{1}{2}$, $\left\langle
J_{x}^{2}\right\rangle=\left\langle J_{y}^{2}\right\rangle=\frac{7}{4}$,
$\left\langle J_{y}^{2}\right\rangle=\frac{1}{4}$, and then we can easily get
the spin squeezing parameters, $\xi_{S}^{2}=\frac{7}{3}$ and
$\xi_{T}^{2}=\frac{1}{9}$. The numerical results for $\xi_{T}^{2}$ is
displayed in figure 2, which coincide with the special result. It is
interesting to see that, although $|\psi_{D}\rangle$ has no parity, the state
is entangled $\left(C>0\right)$ and is spin squeezed according to
$\xi_{T}^{2}$ in the entire interval. However, according to parameter
$\xi_{S}^{2}$ the state is not squeezed in the middle region. Therefore, we
find that $\xi_{T}^{2}$ is more effective than $\xi_{S}^{2}$ in detecting
pairwise entanglement.

##  VI Conclusion

In conclusion, we have studied the relations between spin squeezing and
pairwise entanglement. We have considered two types of spin squeezing
parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$ , and the pairwise entanglement is
characterized by concurrence $C$. We find that, for states with exchange
symmetry and parity, spin squeezing according to $\xi_{T}^{2}$ is
qualitatively equivalent to pairwise entanglement. In detecting pairwise
entanglement, parameter $\xi_{T}^{2}$ is more effective than parameter
$\xi_{S}^{2}$.

It is important to emphasize that, the above conclusion can be extended to the
states without (even or odd) parity. For states with properties $\left\langle
J_{\alpha}\right\rangle=0$, $\left\langle
J_{\alpha}J_{z}\right\rangle=\left\langle J_{z}J_{\alpha}\right\rangle=0$,
$\alpha=x,y$, we can have the same conclusion that spin squeezing and pairwise
entanglement are qualitatively equivalent. The following superposition of
Dicke states are examples:
$\left|\psi_{D^{\prime}}\right\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+n^{\prime}\rangle_{N},$
$n=0,\ldots,N-n^{\prime}$, for all $n^{\prime}\geq 3$. As we have seen,
parameter $\xi_{S}^{2}$ is a key factor in $\xi_{T}^{2}$ for our states. The
present results imply that the spin squeezing has more intimate relations with
pairwise entanglement.

## Acknowledgements

This work is supported by NSFC with grant No. 10874151 and 10935010; and the
Fundamental Research Funds for the Central Universities.

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