# On a Differential Intermediate Value Property

Matthias Aschenbrenner  Kurt Gödel Research Center for Mathematical Logic
Universität Wien
1090 Wien
Austria<EMAIL_ADDRESS>,  Lou van den Dries  Department
of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
U.S.A<EMAIL_ADDRESS>and  Joris van der Hoeven  CNRS, LIX, École
Polytechnique
91128 Palaiseau Cedex
France<EMAIL_ADDRESS>
(Date: May 2021)

###### Abstract.

Liouville closed $H$-fields are ordered differential fields whose ordering and
derivation interact in a natural way and where every linear differential
equation of order $1$ has a nontrivial solution. (The introduction gives a
precise definition.) For a Liouville closed $H$-field $K$ with small
derivation we show: $K$ has the Intermediate Value Property for differential
polynomials iff $K$ is elementarily equivalent to the ordered differential
field of transseries. We also indicate how this applies to Hardy fields.

The first-named author was partially supported by NSF Grant DMS-1700439. We
thank Allen Gehret for commenting on an earlier version of this paper.

## Introduction

Throughout this introduction $K$ is an ordered differential field, that is, an
ordered field equipped with a derivation $\der\colon K\to K$. (We usually
write $f^{\prime}$ instead of $\der f$, for $f\in K$.) Its constant field

$C\ :=\ \\{f\in K:\,f^{\prime}=0\\}$

yields the (convex) valuation ring

$\mathcal{O}\ :=\ \big{\\{}f\in K:\,\text{$|f|\leqslant c$ for some $c\in
C$}\big{\\}}$

of $K$, with maximal ideal

$\smallo\ :=\ {\big{\\{}f\in K:\,\text{$|f|<c$ for all $c>0$ in
$C$}\big{\\}}}.$

(It may help to think of the elements of $K$ as germs of real valued functions
and of $f\in\mathcal{O}g$ and $f\in\smallo g$ as $f=O(g)$ and $f=o(g)$,
respectively.) The above definitions exhibit $C$, $\mathcal{O}$, and $\smallo$
as definable in $K$ in the sense of model theory.

Key example: the ordered differential field ${\mathbb{T}}$ of transseries,
which contains ${\mathbb{R}}$ as an ordered subfield, and where
$C={\mathbb{R}}$. We refer to [3] for the rather elaborate construction of
${\mathbb{T}}$ and for any fact about ${\mathbb{T}}$ that gets mentioned
without proof.

Other important examples are Hardy fields. (Hardy [6] proved a striking
theorem on logarithmic-exponential functions. Bourbaki [5] put this into the
general setting of what they called Hardy fields.) Here we can give a
definition from scratch that doesn’t take much space. Notation: $\mathcal{C}$
is the ring of germs at $+\infty$ of continuous real-valued functions on
halflines $(a,+\infty)$, $a\in{\mathbb{R}}$. For $r=1,2,\dots$, let
$\mathcal{C}^{r}$ be the subring of $\mathcal{C}$ consisting of the germs at
$+\infty$ of $r$-times continuously differentiable real-valued functions on
such halflines. This yields the subring

$\mathcal{C}^{<\infty}\ :=\ \bigcap_{r\in{\mathbb{N}}^{\geqslant
1}}\mathcal{C}^{r}$

of $\mathcal{C}$, and $\mathcal{C}^{<\infty}$ is naturally a differential
ring. For a germ $f\in\mathcal{C}$ we let $f$ also denote any real valued
function representing this germ, if this causes no ambiguity. A real number is
identified with the germ of the corresponding constant function:
${\mathbb{R}}\subseteq\mathcal{C}$.

A Hardy field is by definition a differential subfield of
$\mathcal{C}^{<\infty}$. Examples:

$\quad{\mathbb{Q}},\\\
\quad{\mathbb{R}},\quad{\mathbb{R}}(x),\quad{\mathbb{R}}(x,\operatorname{e}^{x}),\quad{\mathbb{R}}(x,\operatorname{e}^{x},\log
x),\quad{\mathbb{R}}(\Gamma,\Gamma^{\prime},\Gamma^{\prime\prime},\dots),$

where $x$ denotes the germ at $+\infty$ of the identity function on
${\mathbb{R}}$. All these are actually analytic Hardy fields, that is, its
elements are germs of real analytic functions.

Let $H$ be a Hardy field. Then $H$ is an ordered differential field: for $f\in
H$, either $f(x)>0$ eventually (in which case we set $f>0$), or $f(x)=0$,
eventually, or $f(x)<0$, eventually; this is because $f\neq 0$ in $H$ implies
$f$ has a multiplicative inverse in $H$, so $f$ cannot have arbitrarily large
zeros. Also, if $f^{\prime}<0$, then $f$ is eventually strictly decreasing; if
$f^{\prime}=0$, then $f$ is eventually constant; if $f^{\prime}>0$, then $f$
is eventually strictly increasing.

In order to state the main result of this paper we need a bit more
terminology: an $H$-field is a $K$ (that is, an ordered differential field)
such that:

  * •

for all $f\in K$, if $f>C$, then $f^{\prime}>0$;

  * •

$\mathcal{O}=C+\smallo$ (so $C$ maps isomorphically onto the residue field
$\mathcal{O}/\smallo$).

We also say that $K$ has small derivation if for all $f\in\smallo$ we have
$f^{\prime}\in\smallo$. Hardy fields have small derivation, and any Hardy
field containing ${\mathbb{R}}$ is an $H$-field.

An $H$-field $K$ is said to be Liouville closed if it is real closed and for
every $f\in K$ there are $g,h\in K^{\times}$ such that
$f=g^{\prime}=h^{\prime}/h$. The ordered differential field ${\mathbb{T}}$ is
a Liouville closed $H$-field with small derivation. Any Hardy field
$H\supseteq{\mathbb{R}}$ has a smallest (with respect to inclusion) Liouville
closed Hardy field extension $\text{Li}(H)$. (The notions of “$H$-field” and
“Liouville closed $H$-field” are introduced in [1]. The capital $H$ is in
honor of Hardy, Hausdorff, and Hahn, who pioneered various aspects of our
topic about a century ago, as did Du Bois-Reymond and Borel even earlier.)

Now a very strong property: we say $K$ has DIVP (the Differential Intermediate
Value Property) if for every polynomial $P\in K[Y_{0},\dots,Y_{r}]$ and all
$f<g$ in $K$ with

$P(f,f^{\prime},\dots,f^{(r)})\ <\ 0\ <\ P(g,g^{\prime},\dots,g^{(r)})$

there exists $y\in K$ such that $f<y<g$ and $P(y,y^{\prime},\dots,y^{(r)})=0$.
(Existentially closed ordered differential fields have DIVP by [9] and [10,
Proposition 1.5]; this has limited interest for us since the ordering and
derivation in those structures do not interact.) Actually, DIVP is a bit of an
afterthought: in [3] we considered instead two robust but rather technical
properties, $\upomega$-freeness and newtonianity, and proved that
${\mathbb{T}}$ is $\upomega$-free and newtonian. (One can think of
newtonianity as a variant of differential-henselianity.) Afterwards we saw
that “$\upomega$-free + newtonian” is equivalent to DIVP, for Liouville closed
$H$-fields. Our aim is to establish this equivalence: Theorem 2.7, the main
result of this short paper.

We did not consider DIVP in [3], but it is surely an appealing property and
easier to grasp than the more fundamental notions of $\upomega$-freeness and
newtonianity. (The latter make sense in a wider setting of valued differential
fields where the valuation does not necessarily arise from an ordering, as is
the case for $H$-fields.)

Besides [3] we shall rely on [7], which focuses on a particular ordered
differential subfield of ${\mathbb{T}}$, namely
${\mathbb{T}}_{\operatorname{g}}$, consisting of the so-called grid-based
transseries; see also [3, Appendix A]. We summarize what we need from [7] as
follows:

${\mathbb{T}}_{\operatorname{g}}$ is a newtonian $\upomega$-free Liouville
closed $H$-field with small derivation, and ${\mathbb{T}}_{\operatorname{g}}$
has DIVP. We alert the reader that the terms newtonian and $\upomega$-free do
not occur in [7], and that ${\mathbb{T}}_{\operatorname{g}}$ there is denoted
by ${\mathbb{T}}$.

We call attention to the fact that $K$ is a Liouville closed $H$-field iff
$K\models\operatorname{LiH}$ for a set LiH (independent of $K$) of sentences
in the language of ordered differential fields. Also, for $H$-fields,
“$\upomega$-free” is expressible by a single sentence in the language of
ordered differential fields, and “newtonian” as well as “DIVP” by a set of
sentences in this language. The reason that “$\upomega$-free + newtonian” is
central in [3] are various theorems proved there, which are also relevant
here. To state these theorems, we consider an $H$-field $K$ below as an
$\mathcal{L}$-structure, where

$\mathcal{L}\ :=\
\\{\,0,\,1,\,{+},\,{-},\,{\times},\,\der,\,{<},\,{\preccurlyeq}\,\\}$

is the language of ordered valued differential fields. The symbols $0$, $1$,
${+}$, ${-}$, ${\times}$, , $<$ name the usual primitives of $K$, and
$\preccurlyeq$ encodes its valuation: for $a,b\in K$,

$a\preccurlyeq b\quad:\Longleftrightarrow\quad a\in\mathcal{O}b.$

We can now summarize what we need from [3, Chapters 15, 16] as follows:

The theory of newtonian $\upomega$-free Liouville closed $H$-fields is model
complete, and is the model companion of the theory of $H$-fields. The theory
of newtonian $\upomega$-free Liouville closed $H$-fields whose derivation is
small is complete and has ${\mathbb{T}}$ as a model.

For an $H$-field $K$ its valuation ring $\mathcal{O}$ and so the binary
relation $\preccurlyeq$ on $K$ can be defined in terms of the other primitives
by an existential formula independent of $K$. However, by [3, Corollary
16.2.6] this cannot be done by a universal such formula and so for the model
completeness above we cannot drop $\preccurlyeq$ from the language
$\mathcal{L}$.

######  Corollary 0.1.

Every newtonian $\upomega$-free Liouville closed $H$-field has DIVP.

###### Proof.

Let $K$ be a newtonian $\upomega$-free Liouville closed $H$-field. If the
derivation of $K$ is small, then DIVP follows from the results from [7] quoted
earlier and the above completeness result from [3]. Suppose the derivation of
$K$ is not small. Replacing the derivation of $K$ by a multiple
$\phi^{-1}\der$ with $\phi>0$ in $K$ transforms $K$ into its so-called
compositional conjugate $K^{\phi}$, which is still a newtonian $\upomega$-free
Liouville closed $H$-field, and $K$ has DIVP iff $K^{\phi}$ does. By 4.4.7 and
9.1.5 in [3] we can choose $\phi>0$ in $K$ such that the derivation
$\phi^{-1}\der$ of $K^{\phi}$ is small. ∎

This gives one direction of Theorem 2.7. In the rest of this paper we prove a
strong version, Corollary 2.6, of the other direction, without using [7] but
relying heavily on various parts of [3] with detailed references. Theorem 2.7
and the results quoted above from [3] yield the result stated in the abstract:
a Liouville closed $H$-field with small derivation is elementarily equivalent
to ${\mathbb{T}}$ iff it has DIVP.

### Connection to Hardy fields

Every Hardy field $H$ extends to a Hardy field
$H({\mathbb{R}})\supseteq{\mathbb{R}}$, and $H({\mathbb{R}})$ is in particular
an $H$-field. We refer to [4] for a discussion of the conjecture that any
Hardy field containing ${\mathbb{R}}$ extends to a newtonian $\upomega$-free
Hardy field. At the end of 2019 we finished the proof of this conjecture by
considerably refining material in [3] and [8]; this amounts to a rather
complete extension theory of Hardy fields. Note that every Hardy field extends
to a maximal Hardy field, by Zorn, and so having established this conjecture
we now know that all maximal Hardy fields are elementarily equivalent to
${\mathbb{T}}$, as ordered differential fields. Since $\mathcal{C}$ has the
cardinality $\mathfrak{c}=2^{\aleph_{0}}$ of the continuum, there are at most
$2^{\mathfrak{c}}$ many maximal Hardy fields, and we also have a proof that
there are exactly that many. (We thank Ilijas Farah for a useful hint on this
point.) These remarks on Hardy fields serve as an announcement. A rather
voluminous work containing the proof of the conjecture is currently being
prepared for publication. We also hope to include there a proof of DIVP for
newtonian $\upomega$-free $H$-fields that does not depend as in the present
paper on it being true for ${\mathbb{T}}_{\operatorname{g}}$, whose proof in
[7] uses the particular nature of ${\mathbb{T}}_{\operatorname{g}}$.

We have a second conjecture about Hardy fields in [4], whose proof is not yet
finished at this time (May 2021): for any maximal Hardy field $H$ and
countable subsets $A<B$ in $H$ there exists $y\in H$ such that $A<y<B$. This
means that the underlying ordered set of a maximal Hardy field is an
$\eta_{1}$-set in the sense of Hausdorff. Together with the (now established)
first conjecture and results from [3] it implies: all maximal Hardy fields are
back-and-forth equivalent as ordered differential fields, and thus isomorphic
assuming CH, the Continuum Hypothesis.

##  1\. Preliminaries

In order to make free use of the valuation-theoretic tools from [3] and to
make this paper self-contained modulo references to specific results from the
literature we provide more background in this section before returning to
DIVP.

### Notation and terminology

Throughout, $m$, $n$ range over ${\mathbb{N}}=\\{0,1,2,\dots\\}$. Given an
additively written abelian group $A$ we let $A^{\neq}:=A\setminus\\{0\\}$.
Rings are commutative with identity $1$, and for a ring $R$ we let
$R^{\times}$ be the multiplicative group of units (consisting of the $a\in R$
such that $ab=1$ for some $b\in R$). A differential ring will be a ring $R$
containing (an isomorphic copy of) ${\mathbb{Q}}$ as a subring and equipped
with a derivation $\der\colon R\to R$; note that then $C_{R}:=\big{\\{}a\in
R:\,\der(a)=0\big{\\}}$ is a subring of $R$, called the ring of constants of
$R$, and that ${\mathbb{Q}}\subseteq C_{R}$. If $R$ is a field, then so is
$C_{R}$. An ordered differential field is in particular a differential ring.

Let $R$ be a differential ring and $a\in R$. When its derivation is clear from
the context we denote $\der(a),{}^{2}(a),\dots,{}^{n}(a),\dots$ by
$a^{\prime},a^{\prime\prime},\dots,a^{(n)},\dots$, and if $a\in R^{\times}$,
then $a^{\dagger}$ denotes $a^{\prime}/a$, so
$(ab)^{\dagger}=a^{\dagger}+b^{\dagger}$ for $a,b\in R^{\times}$. In Section 2
we need to consider the function $\omega=\omega_{R}\colon R\to R$ given by
$\omega(z)=-2z^{\prime}-z^{2}$, and the function $\sigma=\sigma_{R}\colon
R^{\times}\to R$ given by $\sigma(y)=\omega(z)+y^{2}$ for $z:=-y^{\dagger}$.

We have the differential ring
$R\\{Y\\}=R[Y,Y^{\prime},Y^{\prime\prime},\dots]$ of differential polynomials
in an indeterminate $Y$ over $R$. We say that $P=P(Y)\in R\\{Y\\}$ has order
at most $r\in{\mathbb{N}}$ if $P\in R[Y,Y^{\prime},\dots,Y^{(r)}]$.

For $\phi\in R^{\times}$ we let $R^{\phi}$ be the compositional conjugate of
$R$ by $\phi$: the differential ring with the same underlying ring as $R$ but
with derivation $\phi^{-1}\der$ instead of . We then have an $R$-algebra
isomorphism

$P\mapsto P^{\phi}\ \colon\ R\\{Y\\}\to R^{\phi}\\{Y\\}$

with $P^{\phi}(y)=P(y)$ for all $y\in R$; see [3, Section 5.7].

For a field $K$ we have $K^{\times}=K^{\neq}$, and a (Krull) valuation on $K$
is a surjective map $v\colon K^{\times}\to\Gamma$ onto an ordered abelian
group $\Gamma$ (additively written) satisfying the usual laws, and extended to
$v\colon K\to\Gamma_{\infty}:=\Gamma\cup\\{\infty\\}$ by $v(0):=\infty$, where
the ordering on $\Gamma$ is extended to a total ordering on $\Gamma_{\infty}$
by $\gamma<\infty$ for all $\gamma\in\Gamma$.

Let $K$ be a valued field: a field (also denoted by $K$) together with a
valuation ring $\mathcal{O}$ of that field. This yields a valuation $v\colon
K^{\times}\to\Gamma$ on the underlying field such that $\mathcal{O}=\\{a\in
K:va\geqslant 0\\}$ as explained in [3, Section 3.1]. We introduce various
binary relations on the set $K$ by defining for $a,b\in K$:

$\displaystyle a\asymp b$ $\displaystyle\ :\Leftrightarrow\ va=vb,$
$\displaystyle a\preccurlyeq b$ $\displaystyle\ :\Leftrightarrow\ va\geqslant
vb,$ $\displaystyle a\prec b$ $\displaystyle\ :\Leftrightarrow\ va>vb,$
$\displaystyle a\succcurlyeq b$ $\displaystyle\ :\Leftrightarrow\
b\preccurlyeq a,$ $\displaystyle a\succ b$ $\displaystyle\ :\Leftrightarrow\
b\prec a,$ $\displaystyle a\sim b$ $\displaystyle\ :\Leftrightarrow\ a-b\prec
a.$

It is easy to check that if $a\sim b$, then $a,b\neq 0$, and that $\sim$ is an
equivalence relation on $K^{\times}$. We also let $\smallo=\\{a\in K:va>0\\}$
be the maximal ideal of $\mathcal{O}$, so $\mathcal{O}/\smallo$ is the residue
field of the valued field $K$. A convex subgroup $\Delta$ of the value group
$\Gamma$ of $v$ gives rise to the $\Delta$-coarsening of the valued field $K$;
see [ADH, 3.4].

###  $H$-fields and pre-$H$-fields

As in [3], a valued differential field is a valued field $K$ with residue
field of characteristic zero and equipped with a derivation $\der\colon K\to
K$. An ordered valued differential field is a valued differential field $K$
equipped with an ordering on $K$ making $K$ an ordered field. We consider any
$H$-field $K$ as an ordered valued differential field whose valuation ring is
the convex hull in $K$ of its constant field $C$, in accordance with
construing it as an $\mathcal{L}$-structure as specified in the introduction.

A pre-$H$-field is by definition an ordered valued differential subfield of an
$H$-field. By [3, Sections 10.1, 10.3, 10.5], an ordered valued differential
field $K$ is a pre-$H$-field iff the valuation ring $\mathcal{O}$ of $K$ is
convex in $K$, $f^{\prime}>0$ for all $f>\mathcal{O}$ in $K$, and
$f^{\prime}\prec g^{\dagger}$ for all $f,g\in K^{\times}$ with $f\preccurlyeq
1$ and $g\prec 1$. Any Hardy field $H$ is construed as a pre-$H$-field by
taking the convex hull of ${\mathbb{Q}}$ in $H$ as its valuation ring, giving
rise to the so-called “natural valuation” on $H$ as an ordered field. At the
end of Section 9.1 in [3] we give ${\mathbb{Q}}(\sqrt{2+x^{-1}})$ as an
example of a Hardy field that is not an $H$-field. Any ordered differential
field $K$ with the trivial valuation ring $\mathcal{O}=K$ is a pre-$H$-field
(so the valuation ring of a pre-$H$-field $K$ is not always the convex hull in
$K$ of its constant field, in contrast to Hardy fields and $H$-fields). If $K$
is a pre-$H$-field whose valuation ring is nontrivial, then the valuation
topology on $K$ equals its order topology, by [3, Lemma 2.4.1].

Let $K$ be a pre-$H$-field. Then the derivation of $K$ and its valuation
$v\colon K^{\times}\to\Gamma$ induce an operation
$\psi\colon\Gamma^{\neq}\to\Gamma$, given by $\psi(vf)=v(f^{\dagger})$ for
$f\not\asymp 1$ in $K^{\times}$; the pair $(\Gamma,\psi)$ is called the
$H$-asymptotic couple of $K$; see [3, Section 9.1]. Below we assume some
familiarity with $(\Gamma,\psi)$, and properties of $K$ based on it, such as
$K$ having asymptotic integration and $K$ having a gap [3, Sections 9.1, 9.2].
The flattening of $K$ is the $\Gamma^{\flat}$-coarsening of $K$ where
$\Gamma^{\flat}=\\{vf:\,f\in K^{\times},\ f^{\prime}\prec f\\}$, with
associated binary relations $\asymp^{\flat}$, $\preccurlyeq^{\flat}$ etc.; see
[ADH, 9.4].

##  2\. DIVP

In this section $K$ is a pre-$H$-field. We let $\mathcal{O}$ be its valuation
ring, with maximal ideal $\smallo$, and corresponding valuation $v\colon
K^{\times}\to\Gamma=v(K^{\times})$. Let $(\Gamma,\psi)$ be its $H$-asymptotic
couple, and $\Psi:=\big{\\{}\psi(\gamma):\gamma\in\Gamma^{\neq}\big{\\}}$.
Recall that “$K$ has DIVP” means: for all $P(Y)\in K\\{Y\\}$ and $f<g$ in $K$
with $P(f)<0<P(g)$ there is a $y\in K$ such that $f<y<g$ and $P(y)=0$.
Restricting this to $P$ of order $\leqslant r$, where $r\in{\mathbb{N}}$,
gives the notion of $r$-DIVP. Thus $K$ having $0$-DIVP is equivalent to $K$
being real closed as an ordered field. In particular, if $K$ has $0$-DIVP,
then $\Gamma=v(K^{\times})$ is divisible. From [3, Section 2.4] recall our
convention that $K^{>}=\\{a\in K:a>0\\}$, and similarly with $<$ replacing
$>$.

######  Lemma 2.1.

Suppose $\Gamma\neq\\{0\\}$ and $K$ has $1$-$\operatorname{DIVP}$. Then $\der
K=K$, $(K^{>})^{\dagger}=(K^{<})^{\dagger}$ is a convex subgroup of $K$,
$\Psi$ has no largest element, and $\Psi$ is convex in $\Gamma$.

###### Proof.

We have $y^{\prime}=0$ for $y=0$, and $y^{\prime}$ takes arbitrarily large
positive values in $K$ as $y$ ranges over $K^{>\mathcal{O}}=\\{a\in
K:a>\mathcal{O}\\}$, since by [3, Lemma 9.2.6] the set $(\Gamma^{<})^{\prime}$
is coinitial in $\Gamma$. Hence $y^{\prime}$ takes all positive values on
$K^{>}$, and therefore also all negative values on $K^{<}$. Thus $\der K=K$.
Next, let $a,b\in K^{>}$, and suppose $s\in K$ lies strictly between
$a^{\dagger}$ and $b^{\dagger}$. Then $s=y^{\dagger}$ for some $y\in K^{>}$
strictly between $a$ and $b$; this follows by noting that for $y=a$ and $y=b$
the signs of $sy-y^{\prime}$ are opposite.

Let $\beta\in\Psi$ and take $a\in K$ with $v(a^{\prime})=\beta$. Then $a\succ
1$, since $a\preccurlyeq 1$ would give $v(a^{\prime})>\Psi$. Hence for
$\alpha=va<0$ we have $\alpha+\alpha^{\dagger}=\beta$, so
$\alpha^{\dagger}>\beta$. Thus $\Psi$ has no largest element. Therefore the
set $\Psi$ is convex in $\Gamma$. ∎

Thus the ordered differential field ${\mathbb{T}}_{\log}$ of logarithmic
transseries [3, Appendix A] does not have $1$-DIVP, although it is a newtonian
$\upomega$-free $H$-field.

Does DIVP imply that $K$ is an $H$-field? No: take an $\aleph_{0}$-saturated
elementary extension of ${\mathbb{T}}$ and let $\Delta$ be as in [3, Example
10.1.7]. Then the $\Delta$-coarsening of $K$ is a pre-$H$-field with DIVP and
nontrivial value group, and has a gap, but it is not an $H$-field. On the
other hand:

######  Lemma 2.2.

Suppose $K$ has $1$-$\operatorname{DIVP}$ and has no gap. Then $K$ is an
$H$-field.

###### Proof.

In [3, Section 11.8] we defined

$\text{I}(K)\ :=\ \\{y\in K:\,\text{$y\preccurlyeq f^{\prime}$ for some
$f\in\mathcal{O}$}\\},$

a convex $\mathcal{O}$-submodule of $K$. Since $K$ has no gap, we have

$\der\smallo\ \subseteq\ \text{I}(K)\ =\ \\{y\in K:\,\text{$y\preccurlyeq
f^{\prime}$ for some $f\in\smallo$}\\}.$

Also $\Gamma\neq\\{0\\}$, and so $(\Gamma,\psi)$ has asymptotic integration by
Lemma 2.1. We show that $K$ is an $H$-field by proving
$\text{I}(K)=\der\smallo$, so let $g\in\text{I}(K)$, $g<0$. Since
$(\Gamma^{>})^{\prime}$ has no least element we can take positive
$f\in\smallo$ such that $f^{\prime}\succ g$. Since $f^{\prime}<0$, this gives
$f^{\prime}<g$. Since $(\Gamma^{>})^{\prime}$ is cofinal in $\Gamma$ we can
also take positive $h\in\smallo$ such that $h^{\prime}\prec g$, which in view
of $h^{\prime}<0$ gives $g<h^{\prime}$. Thus $f^{\prime}<g<h^{\prime}$, and so
$1$-DIVP yields $a\in\smallo$ with $g=a^{\prime}$. ∎

We refer to Sections 11.6 and 14.2 of [3] for the definitions of
$\uplambda$-freeness and $r$-newtonianity ($r\in{\mathbb{N}}$). From the
introduction we recall that $\omega(z):=-2z^{\prime}-z^{2}$. Below,
compositionally conjugating an $H$-field $K$ means replacing it by some
$K^{\phi}$ with $\phi\in K^{>}$; this preserves most relevant properties like
being an $H$-field, being $\uplambda$-free, $r$-DIVP, and $r$-newtonianity,
and it replaces $\Psi$ by $\Psi-v\phi$.

######  Lemma 2.3.

Suppose $K$ is an $H$-field, $\Gamma\neq\\{0\\}$, and $K$ has
$1$-$\operatorname{DIVP}$. Then $K$ is $\uplambda$-free and $1$-newtonian, and
the subset $\omega(K)$ of $K$ is downward closed.

###### Proof.

Note that $K$ has (asymptotic) integration, by Lemma 2.1. Assume towards a
contradiction that $K$ is not $\uplambda$-free. We arrange by compositional
conjugation that $K$ has small derivation, so $K$ has an element $x\succ 1$
with $x^{\prime}=1$, hence $x>C$. A construction in the beginning of [3,
Section 11.5] yields by [3, Lemma 11.5.2] a pseudocauchy sequence
$(\uplambda_{\rho})$ in $K$ with certain properties including
$\uplambda_{\rho}\sim x^{-1}$ for all $\rho$. As $K$ is not $\uplambda$-free,
$(\uplambda_{\rho})$ has a pseudolimit $\uplambda\in K$ by [3, Corollary
11.6.1]. Then $s:=-\uplambda\sim-x^{-1}$, and $s$ creates a gap over $K$ by
[3, Lemma 11.5.14]. Now note that for $P:=Y^{\prime}+sY$ we have $P(0)=0$ and
$P(x^{2})=2x+sx^{2}\sim x$, so by $1$-DIVP we have $P(y)=1$ for some $y\in K$,
contradicting [3, Lemma 11.5.12].

Let $P\in K\\{Y\\}$ of order at most $1$ have Newton degree $1$; we have to
show that $P$ has a zero in $\mathcal{O}$. We know that $K$ is
$\uplambda$-free, so by [3, Proposition 13.3.6] we can pass to an elementary
extension, compositionally conjugate, and divide by an element of $K^{\times}$
to arrange that $K$ has small derivation and $P=D+R$ where $D=cY+d$ or
$D=cY^{\prime}$ with $c,d\in C$, $c\neq 0$, and where $R\prec^{\flat}1$. Then
$R(a)\prec^{\flat}1$ for all $a\in\mathcal{O}$. If $D=cY+d$, then we can take
$a,b\in C$ with $D(a)<0$ and $D(b)>0$, which in view of $R(a)\prec D(a)$ and
$R(b)\prec D(b)$ gives $P(a)<0$ and $P(b)>0$, and so $P$ has a zero strictly
between $a$ and $b$, and thus a zero in $\mathcal{O}$. Next, suppose
$D=cY^{\prime}$. Then we take $t\in\smallo^{\neq}$ with $v(t^{\dagger})=v(t)$,
that is, $t^{\prime}\asymp t^{2}$, so

$P(t)\ =\ ct^{\prime}+R(t),\quad P(-t)\ =\ -ct^{\prime}+R(-t),\qquad R(t),\
R(-t)\ \prec\ t^{\prime}.$

Hence $P(t)$ and $P(-t)$ have opposite signs, so $P$ has a zero strictly
between $t$ and $-t$, and thus $P$ has a zero in $\mathcal{O}$.

From $\omega(z)=-z^{2}-2z^{\prime}$ we see that $\omega(z)\to-\infty$ as
$z\to+\infty$ and as $z\to-\infty$ in $K$, so $\omega(K)$ is downward closed
by $1$-IVP. ∎

For results involving $r$-DIVP for $r\geqslant 2$ we need a variant of [3,
Lemma 11.8.31]. To state this variant we introduce as in [3, Section 11.8] the
sets

$\Upgamma(K)\ :=\ \\{a^{\dagger}:\,a\in K\setminus\mathcal{O}\\}\ \subseteq\
K^{>},\qquad\Uplambda(K)\ :=\ -\Upgamma(K)^{\dagger}\ \subseteq\ K.$

The superscripts $\uparrow$, $\downarrow$ used in the statement of Lemma 2.4
below indicate upward, respectively downward, closure in the ordered set $K$,
as in [3, Section 2.1].

######  Lemma 2.4.

Let $K$ be an $H$-field with asymptotic integration. Then

$K^{>}\ =\
\operatorname{I}(K)^{>}\cup\Upgamma(K)^{\uparrow},\qquad\sigma\big{(}K^{>}\setminus\Upgamma(K)^{\uparrow}\big{)}\
\subseteq\ \omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}.$

###### Proof.

If $a\in K$, $a>\text{I}(K)$, then $a\geqslant b^{\dagger}$ for some $b\in
K^{\succ 1}$, and thus $a\in\Upgamma(K)^{\uparrow}$. Next, let $s\in
K^{>}\setminus\Upgamma(K)^{\uparrow}$; we have to show
$\sigma(s)\in\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}$. Note that
$s\in\operatorname{I}(K)^{>}$ by what we just proved. From [3, 10.2.7 and
10.5.8] we obtain an immediate $H$-field extension $L$ of $K$ and $a\in
L^{\succ 1}$ with $s=(1/a)^{\prime}$. As in the proof of [3, 11.8.31] with $L$
instead of $K$ this gives
$\sigma(s)\in\omega\big{(}\Uplambda(L)\big{)}{}^{\downarrow}$, where
$\downarrow$ indicates here the downward closure in $L$. It remains to note
that $\omega$ is increasing on $\Uplambda(L)$ by the remark preceding [3,
11.8.21], and that $\Uplambda(K)$ is cofinal in $\Uplambda(L)$ by [3,
11.8.14]. ∎

The concept of _$\upomega$ -freeness_ is introduced in [3, Section 11.7]. If
$K$ has asymptotic integration, then by [3, 11.8.30]: $K$ is $\upomega$-free
$\Leftrightarrow\
K=\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}\cup\sigma\big{(}\Upgamma(K)\big{)}{}^{\uparrow}$.

The next lemma also mentions the differential field extension $K[\imag]$ of
$K$ where $\imag^{2}=-1$, as well as linear differential operators over
differential fields like $K$ and $K[\imag]$; for this we refer to [3, Sections
5.1, 5.2].

######  Lemma 2.5.

Suppose $K$ is an $H$-field, $\Gamma\neq\\{0\\}$, $r\geqslant 2$, and $K$ has
$r$-$\operatorname{DIVP}$. Then the following hold, with (i), (ii), (iii)
using only the case $r=2$:

  1. (i)

$K=\omega(K)\cup\sigma(K^{>})=\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}\cup\sigma\big{(}\Upgamma(K)\big{)}{}^{\uparrow}$;

  2. (ii)

$K$ is $\upomega$-free and
$\omega(K)=\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}$;

  3. (iii)

for all $a\in K$ the operator ${}^{2}-a$ splits over $K[\imag]$;

  4. (iv)

$K$ is $r$-newtonian.

###### Proof.

For (i) we use the end of [3, Section 11.7] to replace $K$ with a
compositional conjugate so that $0\in\Psi$. Then $K$ has small derivation, and
we have $a\in K^{>}$ such that $a\not\asymp 1$ and $a^{\dagger}\asymp 1$.
Replacing $a$ by $a^{-1}$ if necessary this gives $a^{\dagger}=-\phi$ with
$\phi\asymp 1$, $\phi>0$, so $a\prec 1$. Then $\phi^{-1}a^{\dagger}=-1$;
replacing $K$ by $K^{\phi}$ and renaming the latter as $K$ this means
$a^{\dagger}=-1$. Let $f\in K$; to get
$f\in\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}\cup\sigma\big{(}\Upgamma(K)\big{)}{}^{\uparrow}$,
note first that $1=(1/a)^{\dagger}\in\Upgamma(K)$, so $0\in\Uplambda(K)$. Also
$\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}\subseteq\omega(K)$ by Lemma
2.3.

If $f\leqslant 0$, then $\omega(0)=0$ gives
$f\in\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}$. So assume $f>0$; we
first show that then $f\in\sigma(K^{>})$. Now for $y\in K^{>}$, $f=\sigma(y)$
is equivalent (by multiplying with $y^{2}$) to $P(y)=0$, where

$P(Y)\ :=\ 2YY^{\prime\prime}-3(Y^{\prime})^{2}+Y^{4}-fY^{2}\in K\\{Y\\}.$

See also [3, Section 13.7]. We have $P(0)=0$ and $P(y)\to+\infty$ as
$y\to+\infty$ (because of the term $y^{4}$). In view of $2$-DIVP it will
suffice to show that for some $y>0$ in $K$ we have $P(y)<0$. Now with $y\in
K^{>}$ and $z:=-y^{\dagger}$ we have

$\displaystyle P(y)\ $ $\displaystyle=\ y^{2}\big{(}\sigma(y)-f\big{)}\ =\
y^{2}\big{(}\omega(z)+y^{2}-f\big{)},\text{ hence}$ $\displaystyle P(a)\ $
$\displaystyle=\ a^{2}\big{(}\omega(1)+a^{2}-f\big{)}\ =\ a^{2}(-1+a^{2}-f)\
<\ 0,$

so $f\in\sigma(K^{>})$. By the second inclusion of Lemma 2.4 this yields
$f\in\omega\big{(}\Uplambda(K)\big{)}{}^{\downarrow}$ or
$f\in\sigma\big{(}\Upgamma(K)^{\uparrow}\big{)}$. But we have
$\sigma\big{(}\Upgamma(K)^{\uparrow}\big{)}\subseteq\sigma\big{(}\Upgamma(K)\big{)}{}^{\uparrow}$,
because $\sigma$ is increasing on $\Upgamma(K)^{\uparrow}$ by the remark
preceding [3, 11.8.30]. This concludes the proof of (i), and then (ii)
follows, using for its second part also the fact stated just before [3,
11.8.29] that we have $\omega(K)<\sigma\big{(}\Upgamma(K)\big{)}$.

Now (iii) follows from $K=\omega(K)\cup\sigma(K^{>})$ by [3, Section 5.2,
(5.2.1)]. As to (iv), let $P\in K\\{Y\\}$ of order at most $r$ have Newton
degree $1$; we have to show that $P$ has a zero in $\mathcal{O}$. For this we
repeat the argument in the proof of Lemma 2.3 so that it applies to our $P$,
using $\upomega$-freeness instead of $\uplambda$-freeness, [3, Proposition
13.3.13] instead of [3, Proposition 13.3.6], and $r$-DIVP instead of $1$-DIVP.
∎

######  Corollary 2.6.

If $K$ is an $H$-field, $\Gamma\neq\\{0\\}$, and $K$ has
$\operatorname{DIVP}$, then $K$ is $\upomega$-free and newtonian.

There are non-Liouville closed $H$-fields with nontrivial derivation that have
$\operatorname{DIVP}$; see [2, Section 14]. By Lemma 2.3 and Lemma 2.5(iii),
Liouville closed $H$-fields having $2$-$\operatorname{DIVP}$ are Schwarz
closed as defined in [3, Section 11.8].

######  Theorem 2.7.

Let $K$ be a Liouville closed $H$-field. Then

$\text{$K$ has $\operatorname{DIVP}$}\ \Longleftrightarrow\ \text{$K$ is
$\upomega$-free and newtonian.}$

###### Proof.

The forward direction is part of Corollary 2.6. The backward direction is
Corollary 0.1. ∎

## References

  * [1] M. Aschenbrenner, L. van den Dries, $H$-fields and their Liouville extensions, Math. Z. 242 (2002), 543–588.
  * [2] by same author, Liouville closed $H$-fields, J. Pure Appl. Algebra 197 (2005), 83–139.
  * [3] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, Annals of Mathematics Studies, vol. 195, Princeton University Press, Princeton, NJ, 2017.
  * [4] by same author, _On numbers, germs, and transseries_ , in: B. Sirakov et al. (eds.): Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018, vol. 2, pp. 19–42, World Scientific Publishing Co., Singapore, 2018.
  * [5] N. Bourbaki, Fonctions d’une Variable Réelle, Chapitre V, Étude Locale des Fonctions, Hermann, Paris, 1976.
  * [6] G. H. Hardy, Orders of Infinity, 2nd ed., Cambridge Univ. Press, Cambridge, 1924.
  * [7] J. van der Hoeven, Transseries and Real Differential Algebra, Lecture Notes in Math., vol. 1888, Springer-Verlag, New York, 2006.
  * [8] by same author, Transserial Hardy fields, Astérisque 323 (2009), 453–487.
  * [9] M. Singer, The model theory of ordered differential fields, J. Symbolic Logic 43 (1978), no. 1, 82–91.
  * [10] S. Spodzieja, A geometric model of an arbitrary differentially closed field of characteristic zero, Trans. Amer. Math. Soc. 374 (2021), no. 1, 663–686.


# Separability and lower bounds of quantum entanglement based on realignment

Jiaxin Sun,1 Hongmei Yao,1,∗ Shao-Ming Fei,2,† and Zhaobing Fan1 1Department
of Mathematics, Harbin Engineering University, Harbin 150001, P.R. China
2School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China

###### Abstract

The detection and estimation of quantum entanglement are the essential issues
in the theory of quantum entanglement. We construct matrices based on the
realignment of density matrices and the vectorization of the reduced density
matrices, from which a family of separability criteria are presented for both
bipartite and multipartite systems. Moreover, new lower bounds of concurrence
and convex-roof extended negativity are derived. Criteria are also given to
detect the genuine tripartite entanglement. Lower bounds of the concurrence of
genuine tripartite entanglement are presented. By detailed examples we show
that our results are better than the corresponding ones in identifying and
estimating quantum entanglement as well as genuine multipartite entanglement.

###### pacs:

03.67.-a, 02.20.Hj, 03.65.-w

##  I Introduction

Quantum entanglement plays an essential role in quantum information processing
nielsen2010quantum  such as quantum computation divincenzo1995quantum ,
quantum teleportation bennett1993teleporting ; albeverio2002optimal  and
quantum cryptographic schemes ekert1991quantum ; fuchs1997optimal . However,
it is a difficult problem to distinguish entangled states from the separable
ones in general gurvits2003classical . In the last decades, a variety of
separability criteria have been presented to detect entanglement, such as the
positive partial transpose (PPT) criterion or Peres-Horodecki criterion
peres1996separability ; horodecki1996separability , computable cross-norm or
realignment criteria rudolph2003some ; rudolph2005further ; chen2002matrix ;
zhang2008entanglement , entanglement witnesses terhal2000bell ;
chruscinski2014entanglement  etc., see, e.g., horodecki2009quantum ;
guhne2009entanglement  for comprehensive surveys.

The PPT criterion peres1996separability  says that any partial transposed
bipartite separable state is still positive semidefinite. This criterion is
sufficient and necessary for the separability of $m\times n$ quantum sates
with $mn\leq 6$ horodecki1996separability . However, for higher-dimensional
cases, there exist $2\times 4$ and $3\times 3$ states which are PPT but still
entangled, termed as bound entangled states horodecki1997separability .
Another famous criterion is the realignment criterion rudolph2003some ;
rudolph2005further ; chen2002matrix ; zhang2008entanglement  which detects
many bound entangled states. In lupo2008bipartite ; li2011note , by analyzing
the symmetric functions of the singular values of the realigned matrices, the
authors presented separability criteria which are better than the realignment
criterion. In zhang2008entanglement , based on the realignment of the matrix
$\rho-\rho_{A}\otimes\rho_{B}$, improved realignment criterion has been
presented, which detects entanglement better than the realignment criterion.
In shi2023family , the authors presented a family of separable criteria for
bipartite states, which reduce to the improved realignment criterion for
particular cases. In shen2015separability , based on the realigned bipartite
density matrix, the vectorization of the reduced density matrices and a
parameterized Hermitian matrix, a family of separability criteria were
presented, which detect entanglement more efficiently than the realignment
criterion, and work also for multipartite systems. In zhang2017realignment ,
based on the sequential realignment of density matrices, a separability
criterion for multipartite quantum states has been presented. It detects the
multipartite entanglement better than the criteria given in
shen2015separability . In li2017detection , based on the realignment criterion
and PPT criterion, the authors presented a criterion to detect the genuine
multipartite entanglement of multipartite states. In qi2024detection , based
on the realigned matrix given in shi2023family , a separability criterion for
multipartite states has been presented, which detects the genuine multipartite
entangled (GME) states that are not separable with respect to any bipartition.

Another important problem in the study of quantum entanglement is the
quantification of entanglement. Various entanglement measures have been
presented in recent years horodecki2009quantum ; guhne2009entanglement ;
lee2003convex ; chen2005concurrence ; de2007lower ; huber2013entropy ;
chen2016lower ; zhu2018lower . The concurrence and the convex-roof extended
negativity (CREN) are two well known measures of entanglement. Nevertheless,
it is formidably difficult to find the analytical formulae of such
entanglement measures in general due to the optimal minimization involved. In
chen2005concurrence , based on the positive partial transposition and
realignment separability criteria, the lower bounds of concurrence were
obtained. In de2007lower  the author presented analytical lower bounds of
concurrence in terms of the local uncertainty relations and the correlation
matrix separability criteria. In ma2011measure  the GME concurrence was
presented, whose lower bound has been derived in li2017measure  based on the
norms of the correlation tensors. In li2020improved , the authors presented
improved lower bounds of concurrence and the convex-roof extended negativity
lee2003convex  based on Bloch representations.

In this paper, we first present a family of separability criteria for
bipartite systems, from which we derive tighter lower bounds of concurrence
and convex-roof extended negativity in Section II. In Section III, we
generalize our separability criteria to multipartite systems. These criteria
detect genuine multipartite entanglement as well as the multipartite full
separability. Tighter lower bound of GME concurrence is obtained. By detailed
examples we show that our separability criteria and lower bounds are better
than the correspondingly existing ones. We summarize and conclude in Section
IV.

##  II Detection and measures of entanglement for bipartite states

###  II.1 Separability criteria for bipartite systems

Let $\mathbb{C}^{m\times n}$ be the set of all $m\times n$ matrices over
complex field $\mathbb{C}$, and $\mathbb{R}$ be the real number field. For a
matrix $A=\left[a_{ij}\right]\in\mathbb{C}^{m\times n}$, the vectorization of
matrix $A$ is defined as
$\operatorname{vec}(A)=\left(a_{11},\\!\cdots\\!,a_{m1},a_{12},\\!\cdots\\!,a_{m2},\\!\cdots\\!,a_{1n},\\!\cdots\\!,a_{mn}\right)^{T}$,
where $T$ stands for the transpose.

Let $Z$ be an $m\times m$ block matrix with sub-blocks
$Z_{i,j}\in\mathbb{C}^{n\times n}$, $i,j=1,\ldots,m$. The realigned matrix
$\mathcal{R}(Z)$ of $Z$ is defined by

$\displaystyle\mathcal{R}(Z)=\left(\begin{array}[]{c}\operatorname{vec}\left(Z_{1,1}\right)^{T}\\\
\vdots\\\ \operatorname{vec}\left(Z_{m,1}\right)^{T}\\\ \vdots\\\
\operatorname{vec}\left(Z_{1,m}\right)^{T}\\\ \vdots\\\
\operatorname{vec}\left(Z_{m,m}\right)^{T}\end{array}\right).$ (8)

The realignment criterion chen2002matrix  says that any separable state $\rho$
in $\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ satisfies
$\|\mathcal{R}(\rho)\|_{tr}\leqslant 1$, where
$\|A\|_{tr}=\operatorname{tr}\left(\sqrt{A^{\dagger}A}\right)$ is the trace
norm of $A$.

For any quantum state $\rho$ in $\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$,
we define

$\displaystyle\mathcal{M}_{\alpha,\beta}^{l}(\rho)=\left(\begin{array}[]{cc}\alpha\beta
E_{l\times l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}(\rho)\right)^{T}\\\
\beta\omega_{l}\left(\operatorname{tr}_{B}(\rho)\right)&\mathcal{R}(\rho)\end{array}\right),$
(11)

where $\alpha$ and $\beta$ are arbitrary real numbers, $l$ is a natural
number, $E_{l\times l}$ is the matrix with all $l\times l$ elements being $1$,
$\operatorname{tr}_{A}$ is the partial trace over the subsystem $A$, and
$\omega_{l}(X)=\underbrace{(\operatorname{vec}(X),\cdots,\operatorname{vec}(X))}_{l}$
for any complex matrix $X$. Concerning $\mathcal{M}_{\alpha,\beta}^{l}(\rho)$
we have the following lemma, see proof in Appendix A.

######  Lemma 1.

$\left(1\right)$ For any
$\rho_{i}\in\mathbb{C}^{d_{A}}\\!\otimes\mathbb{C}^{d_{B}}$ and
$k_{i}\in\mathbb{R}$, $i=1,2,\ldots,n$, such that
$\sum\limits_{i=1}^{n}k_{i}=1$, we have

$\mathcal{M}_{\alpha,\beta}^{l}\left(\sum_{i=1}^{n}k_{i}\rho_{i}\right)=\sum_{i=1}^{n}k_{i}\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{i}\right).$

$\left(2\right)$ Let $U$ and $V$ be unitary matrices on subsystems of A and B,
respectively. For any $\rho\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$, we
have $\left\|\mathcal{M}_{\alpha,\beta}^{l}\left((U\otimes V)\rho(U\otimes
V)^{\dagger}\right)\left\|{}_{tr}=\right\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\|_{tr}\right.$.

By Lemma 1 we have the following separability criterion.

######  Theorem 1.

If a state $\rho\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ is separable,
then

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)},$

where $\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)$ is defined in (11).

###### Proof.

Since $\rho$ is separable, it can be written as a convex combination of pure
states, $\rho=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{B}^{i}$, where
$p_{i}\in[0,1]$ with $\sum\limits_{i}p_{i}=1$, $\rho_{A}^{i}$ and
$\rho_{B}^{i}$ are pure states of the subsystems $A$ and $B$, respectively.
From Lemma 1 we have

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}$
(12) $\displaystyle=$
$\displaystyle\left\|\sum_{i}p_{i}\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\right)\right\|_{tr}$
$\displaystyle\leq$
$\displaystyle\sum_{i}p_{i}\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\right)\right\|_{tr}.$

Since for any $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{p\times q}$,

$\displaystyle\mathcal{R}(A\otimes
B)=\operatorname{vec}(A)\operatorname{vec}(B)^{T},$ (13)

we obtain from the definition of
$\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)$,

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\right)\right\|_{tr}$
(16) $\displaystyle=$
$\displaystyle\left\|\left(\begin{array}[]{cc}\alpha\beta E_{l\times
l}&\alpha\omega_{l}\\!\left(\\!\operatorname{tr}_{A}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\\!\right)\\!\right)^{T}\\\
\beta\omega_{l}\left(\\!\operatorname{tr}_{B}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\\!\right)\right)&\mathcal{R}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\\!\right)\end{array}\right)\right\|_{tr}$
$\displaystyle=$ $\displaystyle\left\|\left(\begin{array}[]{cc}\alpha\beta
E_{l\times l}&\alpha\omega_{l}\left(\rho_{B}^{i}\right)^{T}\\\
\beta\omega_{l}\left(\rho_{A}^{i}\right)&\operatorname{vec}\left(\rho_{A}^{i}\right)\operatorname{vec}\left(\rho_{B}^{i}\right)^{T}\end{array}\right)\right\|_{tr}$
(19) $\displaystyle=$ $\displaystyle\|\left(\begin{array}[]{c}\alpha
E_{l\times 1}\\\
\operatorname{vec}\left(\rho_{A}^{i}\right)\end{array}\right)\left(\begin{array}[]{cc}\beta
E_{1\times
l}\quad\left.\operatorname{vec}\left(\rho_{B}^{i}\right)^{T}\right)\end{array}\|_{tr}.\right.$
(23)

As $\rho_{A}^{i}$ and $\rho_{B}^{i}$ are pure states, i.e.,
$\operatorname{tr}\left(\rho_{A}^{i}\right)^{2}=\operatorname{tr}\left(\rho_{B}^{i}\right)^{2}=1$,
we have

$\displaystyle\left\|\left(\begin{array}[]{c}\alpha E_{l\times 1}\\\
\operatorname{vec}\left(\rho_{A}^{i}\right)\end{array}\right)\left(\beta
E_{1\times
l}\quad\operatorname{vec}\left(\rho_{B}^{i}\right)^{T}\right)\right\|_{tr}$
(26) $\displaystyle=$
$\displaystyle\sqrt{l\beta^{2}+1}\operatorname{tr}\left(\left(\begin{array}[]{c}\alpha
E_{l\times 1}\\\
\operatorname{vec}\left(\rho_{A}^{i}\right)\end{array}\right)\left(\alpha
E_{1\times
l}\quad\operatorname{vec}\left(\rho_{A}^{i}\right)^{T}\right)\right)^{\frac{1}{2}}$
(29) $\displaystyle=$ $\displaystyle\sqrt{l\beta^{2}+1}\sqrt{l\alpha^{2}+1}.$
(30)

Combining (12), (16) and (26) we get

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}$
$\displaystyle\leq$
$\displaystyle\sum_{i}p_{i}\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{A}^{i}\otimes\rho_{B}^{i}\right)\right\|_{tr}$
$\displaystyle=$
$\displaystyle\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)},$

which completes the proof.

We illustrate the Theorem 1 by two examples.

######  Example 1.

Consider the state
$\rho_{x}=x\left|\xi\right\rangle\left\langle\xi\right|+\left(1-x\right)\rho_{d}$,
where $\rho_{d}$ is the $2\times 4$ bound entangled state,

$\rho_{d}=\frac{1}{1+7d}\begin{pmatrix}d&0&0&0&0&d&0&0\\\ 0&d&0&0&0&0&d&0\\\
0&0&d&0&0&0&0&d\\\ 0&0&0&d&0&0&0&0\\\
0&0&0&0&\frac{1+d}{2}&0&0&\frac{\sqrt{1-d^{2}}}{2}\\\ d&0&0&0&0&d&0&0\\\
0&d&0&0&0&0&d&0\\\ 0&0&d&0&\frac{\sqrt{1-d^{2}}}{2}&0&0&\frac{1+d}{2}\\\
\end{pmatrix},$

where $0\textless d\textless 1$, and
$\left|\xi\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$.

Set $d=0.9$ and take $\alpha=11.66$, $\beta=11.75$ and $l=5$ in Theorem 1. By
direct calculation we get from Theorem 1 that $\rho_{x}$ is entangled for
$0.233889\leq x\leq 1$. While from the realignment criterion chen2002matrix ,
the entanglement of $\rho_{x}$ is detected for $0.252758\leq x\leq 1$. The
Theorem $1$ in shi2023family  detects the entanglement of $\rho_{x}$ for
$0.233931\leq x\leq 1$, and the Corollary 2.1 in shen2015separability  detects
the entanglement of $\rho_{x}$ for $0.234778\leq x\leq 1$. Obviously our
Theorem 1 detects better the entanglement of the state $\rho_{x}$.

######  Example 2.

Consider the mixture of the bound entangled state proposed by Horodecki
horodecki1997separability ,

$\rho_{x}=\frac{1}{1+8x}\begin{pmatrix}x&0&0&0&x&0&0&0&x\\\
0&x&0&0&0&0&0&0&0\\\ 0&0&x&0&0&0&0&0&0\\\ 0&0&0&x&0&0&0&0&0\\\
x&0&0&0&x&0&0&0&x\\\ 0&0&0&0&0&x&0&0&0\\\
0&0&0&0&0&0&\frac{1+x}{2}&0&\frac{\sqrt{1-x^{2}}}{2}\\\
x&0&0&0&x&0&\frac{\sqrt{1-x^{2}}}{2}&0&\frac{1+x}{2}\\\ \end{pmatrix}$

and the $9\times 9$ identity matrix $I_{9}$,

$\rho_{p}=\frac{1-p}{9}I_{9}+p\rho_{x}.$

We take $\alpha=\beta=2$ and $l=10$. We compare among the results from our
Theorem 1, realignment criterion from chen2002matrix  and Theorem $1$ in
shi2023family  for different values of $x$, see Table 1. Table 1 shows that
our Theorem 1 detects better the entanglement of the state $\rho_{p}$ than the
criteria from chen2002matrix  and shi2023family .

Table 1: Entanglement of the state $\rho_{p}$ in Example 2 for different values of $x$ $x$ | realignment in chen2002matrix  | Theorem $1$ in shi2023family  | Our Theorem 1
---|---|---|---
0.2 | $0.9955\leq p\leq 1$ | $0.9943\leq p\leq 1$ | $0.9942\leq p\leq 1$
0.4 | $0.9960\leq p\leq 1$ | $0.9948\leq p\leq 1$ | $0.9947\leq p\leq 1$
0.6 | $0.9972\leq p\leq 1$ | $0.9964\leq p\leq 1$ | $0.9963\leq p\leq 1$
0.8 | $0.9986\leq p\leq 1$ | $0.9982\leq p\leq 1$ | $0.99815\leq p\leq 1$
0.9 | $0.9993\leq p\leq 1$ | $0.9991\leq p\leq 1$ | $0.99908\leq p\leq 1$

###  II.2 Lower bounds of concurrence and CREN for bipartite states

The concurrence of a pure state
$|\varphi\rangle\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ is defined by
rungta2001universal

$\displaystyle
C\left(|\varphi\rangle\right)=\sqrt{2\left(1-\operatorname{tr}\left(\rho_{A}^{2}\right)\right)},$
(31)

where
$\rho_{A}=\operatorname{tr}_{B}\left(|\varphi\rangle\langle\varphi|\right)$.
The concurrence of a mixed state $\rho$ is defined as

$\displaystyle
C\left(\rho\right)=\min_{\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}}\sum_{i}p_{i}C\left(\left|\varphi_{i}\right\rangle\right),$
(32)

where the minimum is taken over all possible ensemble decompositions of
$\rho=\sum\limits_{i}p_{i}\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|$,
$p_{i}\geqslant 0$ with $\quad\sum\limits_{i}p_{i}=1$.

The CREN of a pure state
$|\varphi\rangle\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ is defined by
lee2003convex

$\displaystyle\mathcal{N}\left(|\varphi\rangle\right)=\frac{\left\|\left(|\varphi\rangle\langle\varphi|\right)^{T_{B}}\right\|_{tr}-1}{d-1},$

where $d=\min\left(d_{A},d_{B}\right)$,
$\left(|\varphi\rangle\langle\varphi|\right)^{T_{B}}$ denotes the partial
transpose of $|\varphi\rangle\langle\varphi|$. For a mixed state $\rho$, its
CREN is defined via convex roof extension,

$\displaystyle\mathcal{N}\left(\rho\right)=\min_{\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}}\sum_{i}p_{i}\mathcal{N}\left(\left|\varphi_{i}\right\rangle\right),$
(33)

where the minimum is taken over all possible pure state decompositions of
$\rho$.

To derive the lower bounds of concurrence and CREN for arbitrary density
matrices, we first present the following lemma, see proof in Appendix B.

######  Lemma 2.

Let $|\varphi\rangle$ be a pure bipartite state in systems A and B, with
Schmidt decomposition
$|\varphi\rangle=\sum\limits_{i=0}^{d-1}\sqrt{\mu_{i}}\left|i_{A}i_{B}\right\rangle$,
where $d=\min\left(d_{A},d_{B}\right)$. Then

(1)
$\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\\\
\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2\sum\limits_{0\leq
i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}$,

(2)
$\left\|\mathcal{M}_{\alpha,\beta}^{l}(|\varphi\rangle\langle\varphi|)\right\|_{tr}\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+(d-1)$.

According to the above lemma, we have

######  Theorem 2.

For a bipartite state $\rho\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$,
the concurrence satisfies that

$\displaystyle
C\\!\left(\rho\right)\geq\frac{\sqrt{2}}{\sqrt{d(d-1)}}\\!\\!\left(\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}\\!\\!-\\!\sqrt{\left(l\alpha^{2}+1\\!\right)\left(l\beta^{2}+1\\!\right)}\right),$

where $d=\min\left(d_{A},d_{B}\right)$.

###### Proof.

For any $|\varphi\rangle\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ with
Schmidt form
$|\varphi\rangle=\sum\limits_{i=0}^{d-1}\sqrt{\mu_{i}}\left|i_{A}i_{B}\right\rangle$,
one has chen2005concurrence

$\displaystyle C^{2}(|\varphi\rangle)\geq\frac{8}{d(d-1)}\left(\sum_{0\leq
i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}\right)^{2}.$

From (1) of Lemma 2, we get

$\displaystyle\ \ \ \ C(|\varphi\rangle)$
$\displaystyle\geq\frac{2\sqrt{2}}{\sqrt{d(d-1)}}\sum_{0\leq i<j\leq
d-1}\sqrt{\mu_{i}\mu_{j}}$
$\displaystyle\geq\\!\\!\frac{\sqrt{2}}{\sqrt{d(d-1)}}\\!\left(\\!\left\|\mathcal{M}_{\alpha,\beta}^{l}\\!\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\\!\\!\\!\\!-\\!\sqrt{\left(l\alpha^{2}\\!+\\!1\right)\\!\left(l\beta^{2}\\!+\\!1\right)}\\!\right).$

Let $\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}$ be the optimal
decomposition of $\rho$ such that
$C\left(\rho\right)=\sum\limits_{i}p_{i}C\left(\left|\varphi_{i}\right\rangle\right)$.
From Lemma 1 we have

$\displaystyle\begin{aligned}
C\left(\rho\right)&=\sum_{i}p_{i}C\left(\left|\varphi_{i}\right\rangle\right)\\\
&\geq\frac{\sqrt{2}}{\sqrt{d(d-1)}}\sum_{i}p_{i}\left(\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|\right)\right\|_{tr}\right.\\\
&\left.\quad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}\right)\\\
&\geq\frac{\sqrt{2}}{\sqrt{d(d-1)}}\left(\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}\right.\\\
&\left.\quad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}\right).\end{aligned}$

Therefore, Theorem 2 holds.

######  Theorem 3.

For the CREN of any state
$\rho\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$, we have

$\displaystyle\mathcal{N}\left(\rho\right)\geq\frac{\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}}{d-1},$

where $d=\min\left(d_{A},d_{B}\right)$.

###### Proof.

For any pure state
$|\varphi\rangle\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ with Schmidt
form
$|\varphi\rangle=\sum\limits_{i=0}^{d-1}\sqrt{\mu_{i}}\left|i_{A}i_{B}\right\rangle$,
one has vidal2002computable

$\displaystyle\mathcal{N}\left(|\varphi\rangle\right)=\frac{2\sum\limits_{0\leq
i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}}{d-1}.$

Using (1) of Lemma 2, we get

$\displaystyle 2\sum_{0\leq i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}$
$\displaystyle\geq$
$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}.$

Let $\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}$ be the optimal
pure state decomposition of $\rho$ such that
$\mathcal{N}(\rho)=\sum\limits_{i}p_{i}\mathcal{N}\left(\left|\varphi_{i}\right\rangle\right)$.
Then using Lemma 1 we obtain

$\displaystyle\begin{aligned}
\mathcal{N}\left(\rho\right)&=\sum_{i}p_{i}\mathcal{N}\left(\left|\varphi_{i}\right\rangle\right)\\\
&\geq\sum_{i}p_{i}\frac{\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|\right)\right\|_{tr}-\sqrt{\left(l\alpha^{2}+1\right)\\!\left(l\beta^{2}+1\right)}}{d-1}\\\
&\geq\frac{\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)\right\|_{tr}-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}}{d-1},\end{aligned}$

which completes the proof.

######  Example 3.

The following $3\times 3$ $PPT$ entangled state was introduced in
bennett1999unextendible ,

$\rho=\frac{1}{4}\left(I_{9}-\sum_{i=0}^{4}\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|\right),$

where

$\displaystyle\left|\varphi_{0}\right\rangle=|0\rangle(|0\rangle-|1\rangle)/\sqrt{2}$
$\displaystyle\left|\varphi_{1}\right\rangle=(|0\rangle-|1\rangle)|2\rangle/\sqrt{2}$
$\displaystyle\left|\varphi_{2}\right\rangle=|2\rangle(|1\rangle-|2\rangle)/\sqrt{2}$
$\displaystyle\left|\varphi_{3}\right\rangle=(|1\rangle-|2\rangle)|0\rangle/\sqrt{2}$
$\displaystyle\left|\varphi_{4}\right\rangle=(|0\rangle+|1\rangle+|2\rangle)(|0\rangle+|1\rangle+|2\rangle)/3.$

Let us consider the mixture of $\rho$ with white noise,

$\rho_{t}=\frac{1-t}{9}I_{9}+t\rho.$

In Fig.1, we take $\alpha=\beta=1$. The solid green line is the bound from
Theorem 2, which shows that $\rho_{t}$ is entangled for $0.88248\leq t\leq\
1$. The dot dashed orange line is the bound from Theorem $6$ in shi2023family
, from which $\rho_{t}$ is entangled for $0.88438\leq t\leq\ 1$. And the
dashed blue line is the bound from Theorem in chen2005concurrence , from which
$\rho_{t}$ is entangled for $0.8897\leq t\leq\ 1$. Clearly, the bound in
Theorem 2 is better than the bounds from shi2023family  and
chen2005concurrence .

In Fig.2, we take $l=2$ and consider different values of $\alpha$ and $\beta$.
The solid orange line is the bound from Theorem 2 with $\alpha=\beta=10$. The
dashed blue line is the bound from Theorem 2 with $\alpha=\beta=1$. It is seen
that the lower bound of concurrence is better when $\alpha=\beta=10$.

In Fig.3, we consider the lower bound of CREN from Theorem 3 for the example
3. We take $\alpha=\beta=1$. The solid orange line is the bound from Theorem 3
with $l=10$. The dashed blue line is the bound from Theorem $7$ in
shi2023family . Clearly, the bound in Theorem 3 is better than the bound in
shi2023family .

In Fig.4, we take $l=2$ and consider different values of $\alpha$ and $\beta$.
The solid orange line is the bound from Theorem 3 with $\alpha=\beta=10$. And
the dashed blue line is the bound from Theorem 3 with $\alpha=\beta=1$. It can
be seen that the lower bound of CREN is better when $\alpha=\beta=10$.

Figure 1: Lower bounds of concurrence for $\rho_{t}$ with $\alpha=\beta=1$,
solid green line for the bound from Theorem 2, dot dashed orange line for the
bound from Theorem $6$ in shi2023family , dashed blue line for the bound from
Theorem in chen2005concurrence . Figure 2: Lower bounds of concurrence for
$\rho_{t}$ with $l=2$, solid orange (dashed blue) line for the bound from
Theorem 2 with $\alpha=\beta=10$ ($\alpha=\beta=1$). Figure 3: Lower bounds of
CREN for $\rho_{t}$ with $\alpha=\beta=1$, solid orange line for the bound
from Theorem 3 with $l=10$, dashed blue line for the bound from Theorem $7$ in
shi2023family . Figure 4: Lower bounds of CREN for $\rho_{t}$ with $l=2$,
solid orange (dashed blue) line for the bound from Theorem 3 with
$\alpha=\beta=10$ ($\alpha=\beta=1$).

##  III Detection and measures of multipartite entanglement

###  III.1 Separability criteria for multipartite states

We first consider tripartite case. Denote the three bipartitions of a
tripartite quantum state
$\rho\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\mathbb{C}^{d_{3}}$
as $1|23$, $2|13$ and $3|12$. If a tripartite state $\rho$ is biseparable
jing2022criteria , then

$\displaystyle\begin{aligned}
&\rho=\sum_{i}p_{i}|\varphi_{i}\rangle^{1|23}\langle\varphi_{i}|+\sum_{j}p_{j}|\varphi_{j}\rangle^{2|13}\langle\varphi_{j}|\\\
&\quad\quad+\sum_{k}p_{k}|\varphi_{k}\rangle^{3|12}\langle\varphi_{k}|,\end{aligned}$

where $p_{i},p_{j},p_{k}\geq 0$, with
$\sum\limits_{i}p_{i}+\sum\limits_{j}p_{j}+\sum\limits_{k}p_{k}=1$. Otherwise,
$\rho$ is called genuinely tripartite entangled. We define

$\displaystyle\mathscr{M}(\rho)\\!\\!=\\!\frac{1}{3}\left(\left\|\mathscr{M}_{1|23}\\!\left(\rho\right)\\!\right\|_{tr}\\!\\!+\\!\left\|\mathscr{M}_{2|13}\\!\left(\rho\right)\\!\right\|_{tr}\\!\\!+\\!\left\|\mathscr{M}_{3|12}\\!\left(\rho\right)\\!\right\|_{tr}\\!\right),$
(34)

where $\mathscr{M}_{i|jk}$ stands for the matrix (11) under bipartition $i$
and $jk$, $\\{i,j,k\\}=\\{1,2,3\\}$.

######  Theorem 4.

If a tripartite state
$\rho\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\mathbb{C}^{d_{3}}$
is biseparable, then

$\displaystyle\mathscr{M}(\rho)\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+\frac{2(d-1)}{3},$

where $d_{i}=d$ $\left(i=1,2,3\right)$.

###### Proof.

Let
$|\varphi\rangle\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\mathbb{C}^{d_{3}}$
be biseparable under bipartition $1|23$, i.e.,
$|\varphi\rangle=\left|\varphi_{1}\right\rangle\otimes\left|\varphi_{23}\right\rangle$.
From Theorem 1 and (2) of Lemma 2 we have

$\displaystyle\begin{aligned}
&\quad\mathscr{M}(|\varphi\rangle\langle\varphi|)\\\
&=\frac{1}{3}\left(\left\|\mathscr{M}_{1|23}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}+\left\|\mathscr{M}_{2|13}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\right.\\\
&\left.\quad+\left\|\mathscr{M}_{3|12}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\right)\\\
&\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+\frac{2(d-1)}{3},\end{aligned}$
(35)

which holds also for biseparable states under bipartitions $2|13$ and $3|12$.

For biseparable mixed state,
$\rho=\sum_{i}p_{i}\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|$,
$p_{i}\geq 0$, $\sum_{i}p_{i}=1$, we have

$\displaystyle\mathscr{M}(\rho)\leq\sum_{i}p_{i}\mathscr{M}(|\varphi_{i}\rangle\langle\varphi_{i}|),$

which gives rise to that

$\displaystyle\begin{aligned}
\mathscr{M}(\rho)&\leq\sum_{i}p_{i}\left(\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+\frac{2(d-1)}{3}\right)\\\
&=\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+\frac{2(d-1)}{3}.\end{aligned}$

by using from (35).

We provide an example to illustrate the Theorem 4.

######  Example 4.

Consider the state $\rho_{W}^{q}$ in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$,

$\rho_{W}^{q}=\frac{1-q}{27}I_{27}+q\left|\varphi_{W}\right\rangle\left\langle\varphi_{W}\right|,$

where
$\left|\varphi_{W}\right\rangle=\frac{1}{\sqrt{6}}(|001\rangle+|010\rangle+|100\rangle+|112\rangle+|121\rangle+|211\rangle)$,
$I_{27}$ is the $27\times 27$ identity matrix.

In this case $d=3$. When $\alpha=\beta=1$ and $l=2$, Theorem 4 detects the
genuine tripartite entanglement of $\rho_{W}^{q}$ for $0.805211\leq q\leq 1$.
When $\alpha=\beta=1$ and $l=1$, Theorem 4 reduces to the Theorem $2$ in
qi2024detection , which detects the genuine tripartite entanglement of
$\rho_{W}^{q}$ for $0.805321\leq q\leq 1$. The Theorem $1$ in
de2011multipartite  detects the genuine tripartite entanglement of
$\rho_{W}^{q}$ for $0.917663\textless q\leq 1$. Obviously, our Theorem 4 is
more effective in detecting the genuine tripartite entanglement.

Next we consider the fully separability of general multipartite states. Any
multipartite state
$\rho\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_{n}}$
can be written as

$\displaystyle\rho=\sum_{i}Y_{1}^{i}\otimes Y_{2}^{i}\otimes\cdots\otimes
Y_{n}^{i},$

where $Y_{j}^{i}\in\mathbb{C}^{d_{j}\times d_{j}}$, $j=1,2,\cdots,n$. We
define

$\displaystyle\mathcal{M}\mathcal{R}_{\alpha_{q},\ldots,\alpha_{n}}^{l,q}(\rho)$
(36) $\displaystyle=$
$\displaystyle\sum_{i}Y_{1}^{i}\otimes\\!\cdots\\!\otimes
Y_{q-1}^{i}\otimes\\!\mathcal{M}_{\alpha_{q},\ldots,\alpha_{n}}^{l}\left[Y_{q}^{i},\ldots,Y_{n}^{i}\right],$

where $q=1,\ldots,n-1$, $\alpha_{q},\ldots,\alpha_{n}$ are nonnegative real
numbers, $l$ is a natural number,

$\displaystyle\mathcal{M}_{\alpha_{q},\ldots,\alpha_{n}}^{l}\left[Y_{q}^{i},\ldots,Y_{n}^{i}\right]$
(39) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\alpha_{q}E_{l\times 1}\\\
\operatorname{vec}\left(Y_{q}^{i}\right)\end{array}\right)\bigotimes_{j=0}^{n-q-1}\left(\alpha_{n-j}E_{1\times
l}\quad\operatorname{vec}\left(Y_{n-j}^{i}\right)^{T}\right),$

with $E_{l\times 1}\in\mathbb{C}^{l}$ a vector with all elements being 1 and
$E_{1\times l}=E_{l\times 1}^{T}$. We have the following separability
criterion for multipartite states.

######  Theorem 5.

If a multipartite state
$\rho\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_{n}}$
is fully separable, then for any $1\leq{q}\leq{n}-1$,

$\displaystyle\left\|\mathcal{M}\mathcal{R}_{\alpha_{q},\ldots,\alpha_{n}}^{l,q}(\rho)\right\|_{tr}\leq\prod_{k=q}^{n}\sqrt{l\alpha_{k}^{2}+1},$

where $\mathcal{M}\mathcal{R}_{\alpha_{q},\ldots,\alpha_{n}}^{l,q}(\rho)$ is
defined in (36).

###### Proof.

Since any fully separable state $\rho$ can be written as
$\rho=\sum_{i}p_{i}\rho_{1}^{i}\otimes\rho_{2}^{i}\otimes\cdots\otimes\rho_{n}^{i}$,
where $p_{i}\in[0,1]$ with $\sum\limits_{i}p_{i}=1$, we have

$\displaystyle\begin{aligned}
&\quad\left\|\mathcal{M}\mathcal{R}_{\alpha_{q},\ldots,\alpha_{n}}^{l,q}(\rho)\right\|_{tr}\\\
&=\left\|\sum\limits_{i}p_{i}\rho_{1}^{i}\otimes\cdots\otimes\rho_{q-1}^{i}\otimes\mathcal{M}_{\alpha_{q},\cdots,\alpha_{n}}^{l}\left[\rho_{q}^{i},\ldots,\rho_{n}^{i}\right]\right\|_{tr}\\\
&\leq\sum\limits_{i}p_{i}\left\|\rho_{1}^{i}\otimes\cdots\otimes\rho_{q-1}^{i}\otimes\mathcal{M}_{\alpha_{q},\ldots,\alpha_{n}}^{l}\left[\rho_{q}^{i},\ldots,\rho_{n}^{i}\right]\right\|_{tr}\\\
&=\sum\limits_{i}p_{i}\operatorname{tr}\left(\rho_{1}^{i}\right)\cdots\operatorname{tr}\left(\rho_{q-1}^{i}\right)\prod_{k=q}^{n}\left\|\left(\begin{array}[]{c}\alpha_{k}E_{l\times
1}\\\
\operatorname{vec}\left(\rho_{k}^{i}\right)\end{array}\right)\right\|_{tr}\\\
&=\sum\limits_{i}p_{i}\prod_{k=q}^{n}\sqrt{l\alpha_{k}^{2}+1}\\\
&=\prod_{k=q}^{n}\sqrt{l\alpha_{k}^{2}+1},\end{aligned}$

which proves Theorem 5.

The following example illustrates the power of the Theorem 5.

######  Example 5.

Consider the quantum state $\rho_{GHZ}^{p}$ gittsovich2010multiparticle ,

$\rho_{GHZ}^{p}=\frac{1-p}{8}I_{8}+p\left|\psi_{GHZ}\right\rangle\left\langle\psi_{GHZ}\right|,$

where $0\leq p\leq 1$ and
$\left|\psi_{GHZ}\right\rangle=\frac{1}{x}(|000\rangle+\varepsilon|110\rangle+|111\rangle)$
with the real parameter $\varepsilon$ and the normalization factor $x$.

Set $\alpha_{i}=10^{-1}$ $(i=1,2,3)$, $l=2$ and $q=1$. We use Theorem 5 to
detect the entanglement of state $\rho_{GHZ}^{p}$ and compare the results with
Theorem $2.1$ in zhang2017realignment  and Theorem $3.1$ in
shen2022optimization . From Table 2, it is seen that our Theorem 5 is more
efficient than the Theorem $2.1$ in zhang2017realignment  and the Theorem
$3.1$ in shen2022optimization  for different values of $\varepsilon$.

Table 2: Entanglement of state $\rho_{GHZ}^{p}$ in Example 5 with respect to different values of $\varepsilon$ $\varepsilon$ | Theorem $2.1$ in zhang2017realignment  | Theorem $3.1$ in shen2022optimization  | Theorem 5
---|---|---|---
$10^{-1}$ | $0.4118\leq p\leq 1$ | $0.4039\leq p\leq 1$ | $0.4026\leq p\leq 1$
1 | $0.4256\leq p\leq 1$ | $0.4200\leq p\leq 1$ | $0.4194\leq p\leq 1$
10 | $0.7727\leq p\leq 1$ | $0.7665\leq p\leq 1$ | $0.7652\leq p\leq 1$

###  III.2 Lower bounds of GME concurrence for tripartite systems

The GME concurrence of a pure state
$|\varphi\rangle\in\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$
is defined by ma2011measure ,

$\displaystyle
C_{GME}(|\varphi\rangle)=\sqrt{\\!\min\left\\{1-\operatorname{tr}\left(\rho_{1}^{2}\right),1-\operatorname{tr}\left(\rho_{2}^{2}\right),1-\operatorname{tr}\left(\rho_{3}^{2}\right)\right\\}},$

where $\rho_{i}$ is the reduced density matrix of subsystem $i$. The GME
concurrence of a mixed state $\rho$ is defined as

$\displaystyle
C_{GME}(\rho)=\min_{\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}}\sum_{i}p_{i}C_{GME}\left(\left|\varphi_{i}\right\rangle\right),$
(40)

where the minimum is taken over all possible ensemble decompositions of
$\rho=\sum\limits_{i}p_{i}\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|$,
$p_{i}\geqslant 0$ with $\quad\sum\limits_{i}p_{i}=1$. We have the following
theorem about the lower bound of GME concurrence, see proof in Appendix C.

######  Theorem 6.

For any tripartite state
$\rho\in\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, we have

$\displaystyle\begin{aligned}
C_{GME}(\rho)&\geq\frac{1}{\sqrt{d(d-1)}}{\bigg{(}\mathscr{M}(\rho)\bigg{.}}\\\
&{\bigg{.}-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}-\frac{2(d-1)}{3}\bigg{)}},\end{aligned}$

where $\mathscr{M}(\rho)$ is defined in (34).

######  Example 6.

Consider the following state
$\rho_{x}\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$,

$\rho_{x}=\frac{x}{8}I_{8}+(1-x)\left|\phi_{GHZ}\right\rangle\left\langle\phi_{GHZ}\right|,0\leq
x\leq 1,$

where
$\left|\phi_{GHZ}\right\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$,
$I_{8}$ is the $8\times 8$ identity matrix.

Set $\alpha=\beta=1$ and $l=2$. From Theorem 6 we get
$C_{GME}(\rho_{x})\geq\frac{3\sqrt{2}\left(1-x\right)}{4}+\frac{\sqrt{5\left(x^{2}-2x+10\right)}}{4}-\frac{11\sqrt{2}}{6}$.
Fig.5 shows that $\rho_{x}$ is genuine tripartite entangled for $0\leq x\leq
0.192912$. When $\alpha=\beta=1$ and $l=1$, Theorem 6 reduces to the Theorem
$3$ in qi2024detection , which detects the genuine tripartite entanglement of
$\rho_{x}$ for $0\leq x\textless 0.192758$. The Corollary $1$ in
zhao2023detecting  shows that $\rho_{x}$ is genuine tripartite entangled for
$0\leq x\textless 0.1919$. It can be seen that our Theorem 6 detects better
the genuine tripartite entanglement.

Figure 5: Lower bound of GME concurrence for $\rho_{x}$

##  IV Conclusions and discussions

The detection and estimation of quantum entanglement are of great significance
in the quantum information processing. By constructing matrices based on the
realignment of density matrices and the vectorization of the reduced density
matrices, we have presented a family of separability criteria for both
bipartite and multipartite systems. From these criteria we have derived new
lower bounds of concurrence and convex-roof extended negativity. As for
tripartite systems, we have also obtained the criteria to detect the genuine
tripartite entanglement, and the lower bounds of the GME concurrence. We have
shown by detailed examples that our separability criteria are more efficient
than the known realignment criteria chen2002matrix  and the separability
criterion given in shi2023family . For multipartite cases, examples show that
our criteria detect genuine tripartite entanglement and multipartite fully
separable states better than the ones in qi2024detection ; de2011multipartite
; zhang2017realignment ; shen2022optimization ; zhao2023detecting .

## ACKNOWLEDGMENTS

This work is supported by JCKYS2024604SSJS001, JCKYS2023604SSJS017,
G2022180019L, the National Natural Science Foundation of China (NSFC) under
Grants 12075159 and 12171044, and the specific research fund of the Innovation
Platform for Academicians of Hainan Province under Grant No. YSPTZX202215.

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## APPENDIX

###  IV.1 Proof of Lemma 1

###### Proof.

(1) The vectorization of matrices has the following properties,

$\displaystyle\operatorname{vec}\left(k_{1}A+k_{2}B\right)=k_{1}\operatorname{vec}(A)+k_{2}\operatorname{vec}(B)$
(A1)

for any $A,B\in\mathbb{C}^{m\times n}$, $k_{i}\in\mathbb{R}$ $(i=1,2)$ and

$\displaystyle\operatorname{vec}(ABC)=\left(C^{T}\otimes
A\right)\operatorname{vec}(B)$ (A2)

for any $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times p}$ and
$C\in\mathbb{C}^{p\times q}$.

From (A1), it yields that for any $A,B\in\mathbb{C}^{mn\times mn}$,
$k_{i}\in\mathbb{R}$ $(i=1,2)$,

$\displaystyle\mathcal{R}\left(k_{1}A+k_{2}B\right)=k_{1}\mathcal{R}(A)+k_{2}\mathcal{R}(B).$
(A3)

Clearly, from (A1) for any $A,B\in\mathbb{C}^{m\times n}$,
$k_{i}\in\mathbb{R}$ $(i=1,2)$, we have

$\displaystyle\omega_{l}\left(k_{1}A+k_{2}B\right)=k_{1}\omega_{l}(A)+k_{2}\omega_{l}(B).$
(A4)

From the definition of $\mathcal{M}_{\alpha,\beta}^{l}\left(\rho\right)$ in
(11), (A3) and (A4), we get

$\displaystyle\mathcal{M}_{\alpha,\beta}^{l}\left(\sum_{i=1}^{n}k_{i}\rho_{i}\right)$
(A7) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}\alpha\beta
E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}\left(\sum\limits_{i=1}^{n}k_{i}\rho_{i}\right)\right)^{T}\\\
\beta\omega_{l}\left(\operatorname{tr}_{B}\left(\sum\limits_{i=1}^{n}k_{i}\rho_{i}\right)\right)&\mathcal{R}\left(\sum\limits_{i=1}^{n}k_{i}\rho_{i}\right)\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sum\limits_{i=1}^{n}k_{i}\alpha\beta
E_{l\times
l}&\alpha\omega_{l}\left(\sum\limits_{i=1}^{n}k_{i}\operatorname{tr}_{A}\left(\rho_{i}\right)\right)^{T}\\\
\beta\omega_{l}\left(\sum\limits_{i=1}^{n}k_{i}\operatorname{tr}_{B}\left(\rho_{i}\right)\right)&\sum\limits_{i=1}^{n}k_{i}\mathcal{R}\left(\rho_{i}\right)\end{array}\right)$
(A10) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sum\limits_{i=1}^{n}k_{i}\alpha\beta
E_{l\times
l}&\alpha\sum\limits_{i=1}^{n}k_{i}\omega_{l}\left(\operatorname{tr}_{A}\left(\rho_{i}\right)\right)^{T}\\\
\beta\sum\limits_{i=1}^{n}k_{i}\omega_{l}\left(\operatorname{tr}_{B}\left(\rho_{i}\right)\right)&\sum\limits_{i=1}^{n}k_{i}\mathcal{R}\left(\rho_{i}\right)\end{array}\right)$
(A13) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}k_{i}\left(\begin{array}[]{cc}\alpha\beta
E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}\left(\rho_{i}\right)\right)^{T}\\\
\beta\omega_{l}\left(\operatorname{tr}_{B}\left(\rho_{i}\right)\right)&\mathcal{R}\left(\rho_{i}\right)\end{array}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}k_{i}\mathcal{M}_{\alpha,\beta}^{l}\left(\rho_{i}\right),$

which proves (1) of Lemma 1.

(2) A general state $\rho$ can be written as rudolph2005further
$\rho=\sum\limits_{i}\xi_{i}\otimes\eta_{i}$, where
$\xi_{i}\in\mathbb{C}^{d_{A}\times d_{A}}$ and
$\eta_{i}\in\mathbb{C}^{d_{B}\times d_{B}}$. Denote $\sigma=(U\otimes
V)\rho\left(U\otimes V\right)^{\dagger}$. We have

$\displaystyle\mathcal{M}_{\alpha,\beta}^{l}(\sigma)=\left(\begin{array}[]{cc}\alpha\beta
E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}(\sigma)\right)^{T}\\\
\beta\omega_{l}\left(\operatorname{tr}_{B}(\sigma)\right)&\mathcal{R}(\sigma)\end{array}\right),$
(A19)

where

$\displaystyle\alpha\omega_{l}\left(\operatorname{tr}_{A}(\sigma)\right)^{T}$
$\displaystyle=$
$\displaystyle\alpha\omega_{l}\left(\operatorname{tr}_{A}\left((U\otimes
V)\sum_{i}\left(\xi_{i}\otimes\eta_{i}\right)\left(U\otimes
V\right)^{\dagger}\right)\right)^{T}$ $\displaystyle=$
$\displaystyle\alpha\omega_{l}\left(\sum_{i}\operatorname{tr}_{A}\left(U\xi_{i}U^{\dagger}\otimes
V\eta_{i}V^{\dagger}\right)\right)^{T}$ $\displaystyle=$
$\displaystyle\alpha\omega_{l}\left(\sum_{i}\operatorname{tr}\left(\xi_{i}\right)V\eta_{i}V^{\dagger}\right)^{T}.$

From (A2) and (A4), it yields that

$\displaystyle\alpha\omega_{l}\left(\operatorname{tr}_{A}(\sigma)\right)^{T}$
(A20) $\displaystyle=$
$\displaystyle\alpha\sum_{i}\operatorname{tr}\left(\xi_{i}\right)\omega_{l}\left(V\eta_{i}V^{\dagger}\right)^{T}$
$\displaystyle=$
$\displaystyle\alpha\sum_{i}\operatorname{tr}\left(\xi_{i}\right)\left(\operatorname{vec}\left(V\eta_{i}V^{\dagger}\right),\ldots,\operatorname{vec}\left(V\eta_{i}V^{\dagger}\right)\right)^{T}$
$\displaystyle=$
$\displaystyle\alpha\sum_{i}\operatorname{tr}\left(\xi_{i}\right)\left(\left(\left(V^{\dagger}\right)^{T}\otimes
V\right)\omega_{l}\left(\eta_{i}\right)\right)^{T}$ $\displaystyle=$
$\displaystyle\alpha\sum_{i}\operatorname{tr}\left(\xi_{i}\right)\left((\overline{V}\otimes
V)\omega_{l}\left(\eta_{i}\right)\right)^{T}$ $\displaystyle=$
$\displaystyle\alpha\sum_{i}\operatorname{tr}\left(\xi_{i}\right)\omega_{l}\left(\eta_{i}\right)^{T}(\overline{V}\otimes
V)^{T}$ $\displaystyle=$
$\displaystyle\alpha\omega_{l}\left(\sum_{i}\operatorname{tr}_{A}\left(\xi_{i}\otimes\eta_{i}\right)\right)^{T}(\overline{V}\otimes
V)^{T}$ $\displaystyle=$
$\displaystyle\alpha\omega_{l}\left(\operatorname{tr}_{A}(\rho)\right)^{T}(\overline{V}\otimes
V)^{T}.$

Similarly, we have

$\displaystyle\beta\omega_{l}\left(\operatorname{tr}_{B}(\sigma)\right)=\beta(\overline{U}\otimes
U)\omega_{l}\left(\operatorname{tr}_{B}(\rho)\right).$ (A21)

Using (A2), (A3) and (13) we obtain

$\displaystyle\mathcal{R}(\sigma)$ (A22) $\displaystyle=$
$\displaystyle\sum_{i}\mathcal{R}\left((U\otimes
V)\left(\xi_{i}\otimes\eta_{i}\right)\left(U\otimes V\right)^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\sum_{i}\mathcal{R}\left(U\xi_{i}U^{\dagger}\otimes
V\eta_{i}V^{\dagger}\right)$ $\displaystyle=$
$\displaystyle\sum_{i}\operatorname{vec}\left(U\xi_{i}U^{\dagger}\right)\operatorname{vec}\left(V\eta_{i}V^{\dagger}\right)^{T}$
$\displaystyle=$
$\displaystyle\sum_{i}\left(\left(U^{\dagger}\right)^{T}\otimes
U\right)\operatorname{vec}\\!\left(\xi_{i}\right)\\!\left(\\!\left(\left(V^{\dagger}\right)^{T}\otimes
V\right)\operatorname{vec}\\!\left(\\!\eta_{i}\\!\right)\\!\right)^{T}$
$\displaystyle=$ $\displaystyle\sum_{i}(\overline{U}\otimes
U)\operatorname{vec}\left(\xi_{i}\right)\operatorname{vec}\left(\eta_{i}\right)^{T}(\overline{V}\otimes
V)^{T}$ $\displaystyle=$ $\displaystyle\sum_{i}(\overline{U}\otimes
U)\mathcal{R}\left(\xi_{i}\otimes\eta_{i}\right)(\overline{V}\otimes V)^{T}$
$\displaystyle=$ $\displaystyle(\overline{U}\otimes
U)\mathcal{R}(\rho)(\overline{V}\otimes V)^{T}.$

Therefore, combing (A19), (A20), (A21) and (A22) we have

$\displaystyle\mathcal{M}_{\alpha,\beta}^{l}(\sigma)$ (A25) $\displaystyle=$
$\displaystyle\left(\\!\\!\begin{array}[]{cc}\alpha\beta E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}(\rho)\right)^{T}(\overline{V}\otimes
V)^{T}\\\ \beta(\overline{U}\otimes
U)\omega_{l}\left(\operatorname{tr}_{B}(\rho)\right)&(\overline{U}\otimes
U)\mathcal{R}(\rho)(\overline{V}\otimes V)^{T}\end{array}\\!\\!\right)$
$\displaystyle=$ $\displaystyle\mathaccent
869{U}\left(\begin{array}[]{cc}\alpha\beta E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}(\rho)\right)^{T}\\\
\beta\omega_{l}\left(\operatorname{tr}_{B}(\rho)\right)&\mathcal{R}(\rho)\end{array}\right)\mathaccent
869{V}$ $\displaystyle=$ $\displaystyle\mathaccent
869{U}\mathcal{M}_{\alpha,\beta}^{l}(\rho)\mathaccent 869{V},$

where $\overline{U}=\left(U^{\dagger}\right)^{T}$,

$\mathaccent 869{U}=\left(\begin{array}[]{cc}I_{l\times l}&0\\\
0&\overline{U}\otimes U\end{array}\right),~{}~{}\mathaccent
869{V}=\left(\begin{array}[]{cc}I_{l\times l}&0\\\
0&V\otimes\overline{V}\end{array}\right)^{\dagger}.$

Hence,

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}\left((U\otimes
V)\rho\left(U\otimes V\right)^{\dagger}\right)\right\|_{tr}$ $\displaystyle=$
$\displaystyle\left\|\mathaccent
869{U}\mathcal{M}_{\alpha,\beta}^{l}(\rho)\mathaccent 869{V}\right\|_{tr}$
$\displaystyle=$
$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}(\rho)\right\|_{tr},$

which proves (2) of Lemma 1.

###  IV.2 Proof of Lemma 2

###### Proof.

$\left(1\right)$ Since
$|\varphi\rangle=\sum\limits_{i=0}^{d-1}\sqrt{\mu_{i}}\left|i_{A}i_{B}\right\rangle$,
one has

$|\varphi\rangle\langle\varphi|=\sum\limits_{i=0}^{d-1}\sum\limits_{j=0}^{d-1}\sqrt{\mu_{i}\mu_{j}}\left|i_{A}i_{B}\right\rangle\left\langle
j_{A}j_{B}\right|.$

Hence,

$\displaystyle\mathcal{M}_{\\!\alpha,\beta}^{l}(|\varphi\rangle\langle\varphi|)\\!=\left(\begin{array}[]{cc}\\!\\!\alpha\beta
E_{l\times
l}&\alpha\omega_{l}\left(\operatorname{tr}_{A}(|\varphi\rangle\langle\varphi|)\right)^{T}\\\
\\!\\!\beta\omega_{l}\left(\operatorname{tr}_{B}(|\varphi\rangle\langle\varphi|)\right)&\\!\\!\mathcal{R}(|\varphi\rangle\langle\varphi|)\end{array}\right).$
(A31)

Let

$\displaystyle\mathrm{M}_{1}=\left(\begin{array}[]{cc}\alpha\beta E_{l\times
l}&\alpha A\\\ \beta
A^{T}&\Delta_{1}\end{array}\right),~{}~{}\mathrm{M}_{2}=\left(\begin{array}[]{cc}0&0\\\
0&\Delta_{2}\end{array}\right),$ (A36)

where

$\displaystyle
A=\left(\begin{array}[]{c}\mu_{0},\underbrace{0,\ldots,0}_{d-1},0,\mu_{1},\underbrace{0,\ldots,0}_{d-2},0,0,\mu_{2},\underbrace{0,\ldots,0}_{d-3},\ldots,0,\ldots,0,\mu_{d-1}\\\
\vdots\\\
\mu_{0},\underbrace{0,\ldots,0}_{d-1},0,\mu_{1},\underbrace{0,\ldots,0}_{d-2},0,0,\mu_{2},\underbrace{0,\ldots,0}_{d-3},\ldots,0,\ldots,0,\mu_{d-1}\end{array}\right)_{l\times
d^{2}},$ (A40)
$\displaystyle\Delta_{1}=\operatorname{diag}(\mu_{0},\underbrace{0,\ldots,0}_{d-1},0,\mu_{1},\underbrace{0,\ldots,0}_{d-2},0,0,\mu_{2},\underbrace{0,\ldots,0}_{d-3},\ldots,0,\ldots,0,\mu_{d-1})\in\mathbb{C}^{d^{2}\times
d^{2}},$ $\displaystyle\begin{aligned}
&\Delta_{2}=\operatorname{diag}\left(0,\sqrt{\mu_{0}\mu_{1}},\ldots,\sqrt{\mu_{0}\mu_{d-1}},\sqrt{\mu_{1}\mu_{0}},0,\sqrt{\mu_{1}\mu_{2}},\ldots,\sqrt{\mu_{1}\mu_{d-1}}\right.\\\
&\left.\sqrt{\mu_{2}\mu_{0}},\sqrt{\mu_{2}\mu_{1}},0,\sqrt{\mu_{2}\mu_{3}},\ldots,\sqrt{\mu_{2}\mu_{d-1}},\ldots,\sqrt{\mu_{d-1}\mu_{0}},\ldots,\sqrt{\mu_{d-1}\mu_{d-2}},0\right)\in\mathbb{C}^{d^{2}\times
d^{2}}.\end{aligned}$

Using the definition of trace norm we get

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}(|\varphi\rangle\langle\varphi|)\right\|_{tr}=\left\|\mathrm{M}_{1}\right\|_{tr}+\left\|\mathrm{M}_{2}\right\|_{tr},$

where $\left\|\mathrm{M}_{2}\right\|_{tr}=2\sum\limits_{0\leq i<j\leq
d-1}\sqrt{\mu_{i}\mu_{j}}$. Since
$\Delta_{1}=\mathcal{R}\left(\Delta_{1}\right)$ and $\Delta_{1}$ is a
separable state, according to Theorem 1 we get

$\displaystyle\left\|\mathrm{M}_{1}\right\|_{tr}=\left\|\mathcal{M}_{\alpha,\beta}^{l}\left(\Delta_{1}\right)\right\|_{tr}\leq\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}.$

Therefore,

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}(|\varphi\rangle\langle\varphi|)\right\|_{tr}$
$\displaystyle\leq$
$\displaystyle\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2\sum_{0\leq
i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}.$

$\left(2\right)$ It has been proved in qi2024detection  that for any
$\mu_{i}>0$, $i=0,1,\ldots,d-1$, such that
$\mu_{0}+\mu_{1}+\cdots+\mu_{d-1}=1$, one has

$\displaystyle 2\sum\limits_{0\leq i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}\leq d-1.$

Based on above relation and (1) of Lemma 2, we get

$\displaystyle\left\|\mathcal{M}_{\alpha,\beta}^{l}(|\varphi\rangle\langle\varphi|)\right\|_{tr}$
$\displaystyle\leq$
$\displaystyle\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2\sum_{0\leq
i<j\leq d-1}\sqrt{\mu_{i}\mu_{j}}$ $\displaystyle\leq$
$\displaystyle\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+(d-1).$

Therefore, we complete the proof of (2) in Lemma 2.

###  IV.3 Proof of Theorem 6

###### Proof.

For a pure state $|\varphi\rangle$ in
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\mathbb{C}^{d_{3}}$, using
Theorem 2 and (31) we get

$\displaystyle\begin{aligned}
&\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\geq\frac{1}{\sqrt{d(d-1)}}{\Big{(}\left\|\mathscr{M}_{1|23}\left(|\varphi\rangle\langle\varphi|\right)\right\|\bigg{.}}\\\
&{\Big{.}\qquad\quad\qquad\qquad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}\Big{)}},\end{aligned}$
(A41) $\displaystyle\begin{aligned}
&\sqrt{1-\operatorname{tr}\left(\rho_{2}^{2}\right)}\geq\frac{1}{\sqrt{d(d-1)}}{\Big{(}\left\|\mathscr{M}_{2|13}\left(|\varphi\rangle\langle\varphi|\right)\right\|\Big{.}}\\\
&{\Big{.}\qquad\quad\qquad\qquad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}\Big{)}}\end{aligned}$
(A42)

and

$\displaystyle\begin{aligned}
&\sqrt{1-\operatorname{tr}\left(\rho_{3}^{2}\right)}\geq\frac{1}{\sqrt{d(d-1)}}{\Big{(}\left\|\mathscr{M}_{3|12}\left(|\varphi\rangle\langle\varphi|\right)\right\|\Big{.}}\\\
&{\Big{.}\qquad\quad\qquad\qquad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}\Big{)}}.\end{aligned}$
(A43)

Combining (A41), (A42) and (A43) we get

$\displaystyle\begin{aligned} &\quad
3\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}-3\mathscr{M}(|\varphi\rangle\langle\varphi|)\\\
&\quad+\left(3\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2(d-1)\right)\\\
&=3\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}-\left(\left\|\mathscr{M}_{1|23}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\right.\\\
&\left.\quad+\left\|\mathscr{M}_{2|13}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}+\left\|\mathscr{M}_{3|12}\left(|\varphi\rangle\langle\varphi|\right)\right\|_{tr}\right)\\\
&\quad+\left(3\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2(d-1)\right)\\\
&\geq
3\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\\!\\!-\\!\sqrt{d(d-1)}\left(\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\right.\\\
&\left.\quad+\sqrt{1-\operatorname{tr}\left(\rho_{2}^{2}\right)}+\sqrt{1-\operatorname{tr}\left(\rho_{3}^{2}\right)}\right)+2(d-1)\\\
&=2\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}-\sqrt{d(d-1)}\left(\sqrt{1-\operatorname{tr}\left(\rho_{2}^{2}\right)}\right.\\\
&\left.\quad+\sqrt{1-\operatorname{tr}\left(\rho_{3}^{2}\right)}\right)+2(d-1).\end{aligned}$

Since $\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\geq 0$ and
$\sqrt{1-\operatorname{tr}\left(\rho_{i}^{2}\right)}\leq\sqrt{1-\frac{1}{d}}(i=2,3)$,
we obtain

$\displaystyle\begin{aligned} &\quad
3\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}-3\mathscr{M}(|\varphi\rangle\langle\varphi|)\\\
&\quad+\left(3\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}+2(d-1)\right)\\\
&\geq
2\sqrt{d(d-1)}\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\\!-\\!\\!\sqrt{d(d-1)}\\!\left(\sqrt{1-\operatorname{tr}\left(\rho_{2}^{2}\right)}\right.\\\
&\left.\quad+\sqrt{1-\operatorname{tr}\left(\rho_{3}^{2}\right)}\right)+2(d-1)\\\
&\geq 2(d-1)-2\sqrt{d(d-1)}\sqrt{1-\frac{1}{d}}\\\ &=0.\end{aligned}$

Therefore

$\displaystyle\begin{aligned}
&\sqrt{1-\operatorname{tr}\left(\rho_{1}^{2}\right)}\geq\frac{1}{\sqrt{d(d-1)}}{\bigg{(}\mathscr{M}(|\varphi\rangle\langle\varphi|)\bigg{.}}\\\
&{\bigg{.}\qquad\quad\qquad\qquad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}-\frac{2(d-1)}{3}\bigg{)}}.\end{aligned}$

Similarly, for $i=2,3$, we have

$\displaystyle\begin{aligned}
&\sqrt{1-\operatorname{tr}\left(\rho_{i}^{2}\right)}\geq\frac{1}{\sqrt{d(d-1)}}{\bigg{(}\mathscr{M}(|\varphi\rangle\langle\varphi|)\bigg{.}}\\\
&{\bigg{.}\qquad\quad\qquad\qquad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}-\frac{2(d-1)}{3}\bigg{)}}.\end{aligned}$

Now let $\left\\{p_{i},\left|\varphi_{i}\right\rangle\right\\}$ be the optimal
decomposition for $\rho$ such that
$C_{GME}(\rho)=\sum\limits_{i}p_{i}C_{GME}\left(\left|\varphi_{i}\right\rangle\right)$.
Then

$\displaystyle\begin{aligned}
C_{GME}(\rho)&=\sum\limits_{i}p_{i}C_{GME}\left(\left|\varphi_{i}\right\rangle\right)\\\
&\geq\frac{1}{\sqrt{d(d-1)}}\sum\limits_{i}p_{i}{\bigg{(}\mathscr{M}(|\varphi_{i}\rangle\langle\varphi_{i}|)\bigg{.}}\\\
&{\bigg{.}\quad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}-\frac{2(d-1)}{3}\bigg{)}}\\\
&\geq\frac{1}{\sqrt{d(d-1)}}{\bigg{(}\mathscr{M}(\rho)\bigg{.}}\\\
&{\bigg{.}\quad-\sqrt{\left(l\alpha^{2}+1\right)\left(l\beta^{2}+1\right)}-\frac{2(d-1)}{3}\bigg{)}},\end{aligned}$

which completes proof of Theorem 6.