# On Sibuya-like distributions in branching and birth-and-death processes

Lev B. Klebanov  Department of Probability and Mathematical Statistics
Faculty of Mathematics and Physics of the Charles University
Sokolovská 49/83, 186 75 Prague, Czech Republic<EMAIL_ADDRESS>and
Michal Šumbera  Nuclear Physics Institute
Czech Academy of Sciences
25068 Řež, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We report some properties of heavy-tailed Sibuya-like distributions related to
thinning, self-decomposability and branching processes. Extension of the
thinning operation of on-negative integer-valued random variables to scaling
by arbitrary positive number leads to a new class of probability distributions
with generating function $Q(w)$ expressible as a Laplace transform
$\varphi(1-w)$ and probability mass function $p_{n}$ satisfying simple one
step recurrence relation between $p_{n+1}$ and $p_{n}$. We show that the
compound Poisson-Sibuya and the shifted Sibuya distributions belong to this
class. Using the fact that the same Markov property is present in stationary
solutions of the birth and death equations we identify the Sibuya distribution
and some of its variants as particular solutions of these equations. We also
establish condition when integer-valued non-negative heavy-tailed random
variable has finite $r$-th absolute moment ($0<r<a<1$).

today

##  1\. Self-decomposability and scaling of non-negative integer-valued random
variables

Infinitely divisible distributions Feller, (1957); Norman L. Johnson, (2005);
F.W. Steutel, (2004) play nowadays an important role in several parts of the
probability theory. They are also frequently used by the models based on the
sum of several independent quantities with the same distribution because its
infinite divisibility appears to be a convenient assumption F.W. Steutel,
(2004). In the simplest case of the non-negative integer random variable (rv)
$Y\in\mathbb{Z}_{+}$ the condition of infinite divisibility means that
$\forall n\in\mathbb{N}$ it can be written as a random sum
$Y_{n}=X_{1}+X_{2}\ldots+X_{n}$ of independent identically distributed (iid)
rvs $X_{i}\in\mathbb{Z}_{+},i=1,n$. Consequently, if $\mathbb{P}(N=n)=p_{n}$
is the probability mass function (pmf) and $Q(w)=\sum_{n}p_{n}w^{n}$ the
probability generating function (pgf) of the rv $N$ and
$G(w)=\mathbb{E}[w^{X_{i}}]$ is the pgf of the rvs $X_{i}$ then the $n$th root
of the compound pgf

(1) $H(w)=G(Q(w)):=G\circ Q$

is $\forall n$ also the pgf. This is equivalent to saying that $H(w)$ can be
expressed as the pgf of compound Poisson distribution Feller, (1957):

(2) $H(w)=G\circ Q=P\circ
R\,,~{}~{}P(\lambda,w)=e^{-\lambda(1-w)}\,,~{}R(w)=1+\frac{\log
H(w)}{\lambda}$

where $P(w)$ is the pgf of the Poisson distribution and $R(w)$ is the pgf of
positive discrete rv with $R(0)=0$.

Important class of the infinitely divisible distributions on $\mathbb{Z}_{+}$
are those which are self-decomposable F.W. Steutel, (2004). Let us recall that
discrete random variable $X\in\mathbb{Z}_{+}$ with the pgf $G(w)$ is called
self-decomposable if it can be written as a sum of two independent variables
F.W. Steutel, (2004)

(3) $X=a\odot X+Y_{a};\;\;\;\forall a\in(0,1).$

The thinning operator $\odot$ Rényi, (1956); F.W. Steutel, (1979) is defined
in the following way

(4) $a\odot
X:=\sum_{i=1}^{X}Z_{i},~{}Z_{i}\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}\mathcal{B}_{a},~{}G_{a\odot
X}(w)\\!:=\\!G(\mathcal{B}_{a}(w))\\!=\\!G(1\\!-\\!a(1\\!-\\!w))$

where $\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}$ means
equality in distribution, $\mathcal{B}_{a}$ is two-valued Bernoulli-
distributed random variable with $p_{0}=1-a$ and $p_{1}=a$ and
$\mathcal{B}_{a}(w)$ is its pgf. In terms of pgfs Eq. (3) reads:

(5) $G(w)=G(\mathcal{B}_{a}(w))H_{a}(w)\,,$

where $H_{a}(w)$ is the pgf of the rv $Y_{a}$. It can be shown that the
function $H_{a}(w)$ is absolutely monotone and infinitely divisible F.W.
Steutel, (2004). Obviously if $G_{1}(w)$ and $G_{2}(w)$ are self-decomposable
pgfs then their product $G_{1}(w)G_{2}(w)$ is also self-decomposable pgf. More
general definitions of thinning operators were given in Klebanov & Slamova,
(2014); Klebanov, (2021).

The importance of the thinning operation (4) for the rvs with finite mean is
explained by the following theorem.

######  Theorem 1.

Let $Q(w)$ be a pgf of the rv $X$ such that $\mathbb{E}X=Q^{(1)}(1)<\infty$
and with scaled factorial moments $F_{j}$ of order $j$

(6)
$F_{j}:=\frac{\mathbb{E}X^{(j)}}{(\mathbb{E}X)^{j}}=\frac{\mathbb{E}X(X-1)\ldots(X-j+1)}{(\mathbb{E}X)^{j}}=\frac{Q^{(j)}(1)}{(Q^{(1)}(1))^{j}},~{}j\\!=\\!1,2,3,\ldots.$

Then the rv $Y=a\odot X$ with the pgf $G(w)=Q_{a\odot X}(w)$ has identical
scaled factorial moments as $X$.

###### Proof.

(7)
$G^{(j)}(w)=a^{j}Q^{(j)}(1-a+aw);\;\;\;\frac{G^{(j)}(1)}{(G^{(1)}(1))^{j}}=\frac{a^{j}Q^{(j)}(1)}{(aQ^{(1)}(1))^{j}}=F_{j}$

∎

######  Example 1.

Consider rv $X$ with $\langle n\rangle:=\mathbb{E}X<\infty$ which has the
negative binomial distribution (NBD) with the pgf

(8) $Q_{NBD}(w)=\Bigl{(}1+\frac{\langle
n\rangle}{k}(1-w)\Bigr{)}^{-k};\;\;k>0.$

Its factorial moments $F_{j}=(k+j-1)_{j}/k^{j}$, where
$(x)_{n}=\prod_{\ell=0}^{n-1}(x-\ell)$ is the falling factorial, do not depend
on $\langle n\rangle$ and therefore its thinned version $Y=\odot X$ with the
pgf

(9) $Q_{NBD}(\mathcal{B}_{a}(w))\\!=\\!\Bigl{(}1\\!+\\!\frac{\langle
n\rangle}{k}\Bigl{(}1\\!-\\!\mathcal{B}_{a}(w)\Bigr{)}\Bigr{)}^{-k}\\!=\\!\Bigl{(}1\\!+\\!\frac{a\langle
n\rangle}{k}(1\\!-\\!w)\Bigr{)}^{-k}$

has not only the same scaled factorial moments as the rv $X$ but is also form-
invariant w.r.t. scaling $\langle n\rangle\to a\langle n\rangle$.

Returning back to the self-decomposable distributions we note, that there is
only one distribution, discrete stable distribution (DSD), for which the
infinitely divisible component $Y_{a}$ is also self-decomposable

(10) $X=a\odot X+Y_{a}=a\odot X+b\odot
X\,,~{}~{}a^{\delta}+b^{\delta}=1,~{}~{}\forall\delta\in(0,1)\,.$

The canonical form (2) of its pgf reads

(11)
$\displaystyle\;\;Q_{DSD}(\lambda,\gamma,w)=P\circ\mathcal{S}=e^{-\lambda(1-w)^{\gamma}};\;\;\mathcal{S}(\gamma,w)=1-(1-w)^{\gamma},\;\;\gamma\in(0,1)\,,$

where $P(\lambda,w)$ and $\mathcal{S}(\gamma,w)$ are the pgfs of Poisson and
Sibuya distributions Sibuya, (1979); Devroye, (1993); Huillet & Martinez,
(2018), respectively.

Let us note that there is a wide class of self-decomposable pgfs which
similarly to Poisson (2), Bernouily (5), negative binomial (8) and discrete
stable distributions (11) depend explicitly on the argument $1-w$, i.e. allow
expansion of the type

(12) $Q(w)=\sum_{n\geq
0}p_{n}w^{n}=1+\sum_{n>0}b_{n}(1-w)^{n}\,,\;\;\;\sum_{n>0}b_{n}>-1\,.$

This category comprises also some less frequently used ones like the Mittag-
Leffler Huillet & Martinez, (2018) or Discrete Linnik Devroye, (1993); Huillet
& Martinez, (2018) distributions. Rather general construction of such pgfs is
given in Klebanov, (2021). More precisely, let $\varphi(s)$ be the Laplace
transform of a positive random variable $X$. Then

(13) $Q(w)=\varphi(1-w)$

is a probability generating function.

The change of variable $w=(1-a)+az$ in (13) gives us

(14) $Q(1-a+az)=\varphi(a(1-z)).$

In other words, application of the thinning operator $w\rightarrow 1-a+az$ is
equivalent to the scale change from $1$ to $a$ in the case of $a\in(0,1)$. We
can apply the thinning operator to arbitrary positive integer random variable
but for the case of $0<a<1$. However, $\varphi(as)$ remains to be Laplace
transform for any positive $a$. Naturally, we came to the problem: Describe
all probability generating functions $Q(w)$ such that $Q(1-a+aw)$ is again the
pgf for all $a>0$. The solution is given by the following result.

######  Theorem 2.

Let $Q(w)$ be a probability generating function. $Q(1-a+aw)$ is the pgf for
all $a>0$ if and only if there exists a Laplace transform of a positive random
variable $\varphi(s)$ such that the representation (13) holds.

###### Proof.

Without loss of generality we may consider the case $a>1$.

1\. Suppose that $\varphi(s)$ is Laplace transform of a distribution function
and $Q(w)=\varphi(1-w)$ is probability generating function. However,
$\varphi(as)$ with arbitrary $a>0$ is Laplace transform as well. According to
the result from Klebanov, (2021) mentioned above, $\varphi(a(1-w))=Q(1-a+aw)$
is probability generating function for any $a>0$.

2\. Suppose that $Q(1-a+aw)$ is probability generating function for any $a>0$.
Define $\varphi(s)=Q(1-s)$. It is necessary to proof that $\varphi(s)$ is
Laplace transform of a distribution function or, equivalently, that it is
absolutely monotone function. In other words we have to proof that

(15) $\varphi^{(k)}(s)=(-1)^{k}A_{k}(s),\quad k=0,1,\ldots,$

where the functions $A_{k}(s)$ are non-negative for $s>0$. For any $w\in(0,1)$
and any $a>0$ we have $\varphi(a(1-w))=Q(1-a+aw)$. Therefore,

(16)
$\frac{d^{k}}{dw^{k}}\varphi(a(1-w))=(-1)^{k}a^{k}\varphi^{(k)}(a(1-w))=(-1)^{k}a^{k}Q^{(k)}(1-a+aw).$

Because $Q(1-a+aw)$ is probability generating function for any $a>0$ the terms
$a^{k}Q^{(k)}(1-a+aw)$ are non-negative for all $0<w<1$ and all positive $a$.
It proves absolutely monotones of $\varphi(s)$. ∎

######  Corollary 1.

Let $\mathbb{P}(N=n)=p_{n}=Q^{(n)}(0)/n!\,,n\in\mathbb{Z}_{+}$ be a pmf of
random variable $N$ with the pgf $Q(w)$ such that

(17) $Q(w)=\varphi(1-w)=\int_{0}^{\infty}e^{-(1-w)x}f(x)dx$

where $\varphi(s)$ is the Laplace transform of a positive random variable $X$
with the probability density $f(x)=dF_{X}(x)/dx,~{}F_{X}(x)=\mathbb{P}(X<x)$.
Then $p_{n}$ satisfies the one-step recurrence relation

(18)
$p_{n+1}=\frac{p_{n}}{n+1}g(n);\;\;\;\;g(n)=\frac{Q^{(n+1)}(0)}{Q^{(n)}(0)}=\frac{(n+1)p_{n+1}}{p_{n}}=\frac{\int_{0}^{\infty}e^{-x}x^{n+1}f(x)dx}{\int_{0}^{\infty}e^{-x}x^{n}f(x)dx}.$

Let us note that if pgf $Q(w)$ has the representation (17) then for any
$b\in(0,1]$ the function $Q(bw)/Q(b)$ is pgf and has the same representation
with corresponding function $f_{b}(x)$. Namely,

(19)
$f_{b}(x)=e^{-(1-b)x/b}f(x/b)/Z(b)\,,~{}~{}Z(b)=b\int_{0}^{\infty}e^{-(1-b)u}f(u)du.$

It is also worth mentioning that the rv $Y\in\mathbb{Z}_{+}$ with the pgf
$Q(w)$ satisfying Eq. (17) can be also interpreted as Poisson–distributed
random variable whose mean $\mathbb{E}Y=X$ fluctuates according to the
probability density $f(x)$ or, equivalently, as a mixture of the Poisson pgfs
$P(x,w)=e^{-x(1-w)}$ with the mixture weight $f(x)$. Let us give few examples.

######  Example 2.

Consider pgf of the Hermite distribution Norman L. Johnson, (2005)
$Q(w)=e^{-\lambda_{1}(1-w)}e^{-\lambda_{2}(1-w)^{2}}$ describing sum of two
independent random variables $X_{1}$, $X_{2}$ having pgf
$e^{-\lambda_{1}(1-w)}$ and $e^{-\lambda_{2}(1-w)^{2}}$ correspondingly.
Values of $X_{1}$ are concentrated on $n=0,1,2,3\ldots$, of $X_{2}$ on even
values only $n=0,2,4,6\ldots$. Hermite distribution is infinitely divisible.
Its pmf satisfies recursion relation
$(n\\!+\\!1)p_{n+1}\\!=\\!\lambda_{1}p_{n}\\!+\\!2\lambda_{2}p_{n-1}$. For
$\lambda_{2}=0$ the recursion simplifies to
$(n\\!+\\!1)p_{n+1}\\!=\\!\lambda_{1}p_{n}$ and pgf $Q(w)$ can be obtained
from Eq. (13) with $f(x)=\delta(x-\lambda_{1})$. However, for $\lambda_{2}\neq
0$ the density $f(x)$ does not exist. On the opposite side is the notoriously
known example of the negative binomial distribution with the pgf (8) which can
be obtained from Eq. (13) taking for $f(x)$ the gamma distribution density
$f(x)=(\beta^{k}/\Gamma(k))x^{k-1}e^{-\beta x}$ with $\beta=(1-q)/q$.

######  Example 3.

Let’s compare the pgf (11) of discrete stable distribution with the Laplace
integral Pollard, (1946)

(20)
$e^{-r^{\alpha}}=\int_{0}^{\infty}\\!e^{-rx}g_{\alpha}(x)dx\,,~{}~{}g_{\alpha}(x)=\frac{1}{\pi}\sum^{\infty}_{j=1}\frac{(-1)^{j+1}}{j!x^{1+\alpha
j}}\Gamma(1\\!+\\!\alpha j)\sin(\pi\alpha j)\,,$

where $g_{\alpha}(x)$ is the probability density function of the one-sided
continuous stable distribution. Exact and explicit expressions for
$g_{\alpha}(x)$, for all $\alpha=l/k<1$, with $k$ and $l$ positive integers
can be found in Penson & Górska, (2010). The substitutions $\gamma=\alpha$ and
$r=\lambda^{-\gamma}(1-w)$ in Eq. (20) enable us to express the pgf (11) as
the Laplace transform (17) of the function

(21) $f(x)=\lambda^{\gamma}g_{\gamma}(\lambda^{\gamma}x).$

Hence the pmf of the discrete stable distribution satisfies the one step
recurrence relation $(n+1)p_{n+1}=g(n)p_{n}$ with $g(n)$ given by Eq. (18) and
$f(x)$ by Eq. (21). It is worth mentioning that since a closed-form expression
for $p_{n}$ using elementary functions is unknown the proof of this relation
by the other methods is almost impossible.

######  Example 4.

Consider random variable
$Y\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}X+1$ where
$Y\in\mathbb{N}$ has Sibuya distribution. Then the rv $X\in\mathbb{Z}_{+}$ has
shifted Sibuya distribution which is self-decomposable Christoph & Schreiber,
(2000) and therefore also infinitely divisible F.W. Steutel, (2004). Its pgf
can be expressed via Eq. (17)

(22)
$\displaystyle\mathcal{S}_{0}(\gamma,w)=\frac{\mathcal{S}(\gamma,w)}{w}=\frac{1-(1-w)^{\gamma}}{w}=\int_{0}^{\infty}e^{-(1-w)x}f(x)dx;$
(23) $\displaystyle
f(x)=-\frac{e^{x}\Gamma(-\gamma,x)}{\Gamma(-\gamma)}\approx
x^{-1-\gamma}\left(\frac{\gamma}{\Gamma(1-\gamma)}+\mathcal{O}\left(\frac{1}{x}\right)\right)\,,$

where $\Gamma(a,x)$ is the incomplete gamma function with
$\Gamma(a,0)=\Gamma(a)$. From Eq. (18) we obtain

(24)
$g(n)=\frac{\int_{0}^{\infty}x^{n+1}\Gamma(-\gamma,x)dx}{\int_{0}^{\infty}x^{n}\Gamma(-\gamma,x)dx}=\frac{(n+1)(n+1-\gamma)}{n+2}.$

It is worth mentioning that the $r$-th absolute moment of the probability
density function $f(x)$ (23)

(25)
$\int_{0}^{\infty}x^{r}f(x)dx=\frac{\sin(\pi\gamma)\Gamma(r+1)}{\sin(\pi(\gamma-r))}$

diverges for $r>\gamma$. This well-known property of the continuous heavy-
tailed distributions is also shared by the family of discrete distribution
represented by the Sibuya distribution. In the latter case the mean and higher
moments $\langle n^{r}\rangle=Q^{(r)}(w=1),~{}r\geq 1$ do not exist. One may
naïvely expect that in this case also the Shannon entropy
$S=-\sum_{n}p_{n}\log p_{n}$ of these distributions diverges. However, this is
not the case. The entropy $S$ is finite either if $\langle\log
n\rangle=\sum_{n=1}^{\infty}p_{n}\log n<\infty$ or if $\exists~{}r>0$ such
that $\langle n^{r}\rangle=\sum_{n=1}^{\infty}p_{n}n^{r}<\infty$ Baccetti &
Visser, (2013). The following theorem establishes the condition when $\langle
n^{r}\rangle<\infty,~{}0<r<1$.

######  Theorem 3.

Integer-valued non-negative random variable $Z$ with probability generating
function $Q(w)$ has finite $r$-th absolute moment $(0<r<a<1)$ iff

(26) $\int_{0}^{1}\frac{1-Q(w)}{(-\log{w})^{1+r}w}dw<\infty$

###### Proof.

For continuous non-negative random variable $X$ with Laplace transform
$\varphi(u)=\mathbb{E}e^{-uX}$ its $r$-th absolute moment
$\mathbb{E}X^{r},0<r<1$ can be calculated using identity

(27)
$\displaystyle\int_{0}^{\infty}\frac{1-e^{-Xu}}{u^{1+r}}du=X^{r}\int_{0}^{\infty}\frac{1-e^{-z}}{z^{1+r}}dz=X^{r}(-\Gamma(-r));~{}z=Xu,~{}0<r<1$
(28)
$\displaystyle\mathbb{E}X^{r}=\frac{1}{-\Gamma(-r)}\int_{0}^{\infty}\frac{1-\varphi(u)}{u^{1+r}}du.$

Similarly, for the integer-valued non-negative random variable $Z$ with
Laplace transform $\varphi(u)=Q(e^{-u})$ we have

(29)
$\mathbb{E}Z^{r}=\frac{1}{-\Gamma(-r)}\int_{0}^{\infty}\frac{1-Q(e^{-u})}{u^{1+r}}du=\frac{1}{-\Gamma(-r)}\int_{0}^{1}\frac{1-Q(w)}{(-\log{w})^{1+r}w}dw.$

∎

######  Example 5.

Consider $N$ extended objects–particles each of the same unit volume
$v_{0}=1$. Particles are incompressible and densely packed occupying the total
volume $V=N$ in three dimensional space. If $N$ is the Sibuya-distributed rv
with the pgf $\mathcal{S}(\gamma,w)$ (11) then the integral (26) and hence the
corresponding moments $\mathbb{E}N^{r}$ are finite only for $0<r<\gamma<1$.
Consequently the mean $\mathbb{E}V$ of the fluctuating volume
$V\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}N$ is
nonexistent. However, for $\gamma>1/2$ the mean value of the enclosing surface
$S\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{\sim}}N^{1/2}$ as
well as of the linear extension
$R\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{\sim}}N^{1/3}$,
where $\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{\sim}}$ means
proportionality in distribution, of the 3d–space occupied by particles are
finite. Moreover for $1/2\geq\gamma>1/3$ $\mathbb{E}S$ does not exist but
$\mathbb{E}R<\infty$.

Similarly to the pgfs of Bernoulli and geometric distributions Sibuya pgfs
form a commutative semi-group under the operation of compoundig $\circ$

(30)
$\mathcal{S}(\gamma_{1})\circ\mathcal{S}(\gamma_{2})=\mathcal{S}(\gamma_{1}\gamma_{2})$

The following theorem establishes the additional similarity with Bernoulli
distribution.

######  Theorem 4.

Sapatinas, (1995) Let $Q(w)$ be the self-decomposable pgf of the random
variable $X$ on $\mathbb{Z}_{+}$ and $\mathcal{S}(\gamma,w)$ the pgf of Sibuya
distribution. Then $\forall\gamma\in(0,1)$ the function
$G(w)=Q\circ\mathcal{S}(\gamma)$ is also a self-decomposible pgf on
$\mathbb{Z}_{+}$.

###### Proof.

We need to prove that $\forall a$ the function

(31)
$\frac{G(w)}{G(\mathcal{B}_{a}(w))}=\frac{Q(\mathcal{S}(\gamma,w))}{Q(1\\!-\\!a^{\gamma}(1\\!-\\!\mathcal{S}(\gamma,w)))}\\!=\\!\frac{Q(\mathcal{S}(\gamma,w))}{Q(\mathcal{B}_{a^{\gamma}}(\mathcal{S}(\gamma,w)))}\\!=\\!Q_{a^{\gamma}}(\mathcal{S}(\gamma,w))$

is also the pgf. This is true since $Q_{\alpha^{\gamma}}(\mathcal{S}(w))$ is a
compound pgf $\forall a\in(0,1)$. ∎

######  Remark 1.

Obviously, sum of $n$ independent self-decomposable rvs $X_{1}+\ldots X_{n}$
is also self-decomposable. The non-trivial character of the above theorem
consists in the statement that this remains true also for the sum iid self-
decompolsable rvs when $n$ fluctuates according to the Sibuya distribution.

######  Remark 2.

Note that the thinned version of the Sibuya pgf appearing in Eq. 31

(32)
$1-a^{\gamma}(1-\mathcal{S}(\gamma,w))=1-\lambda(1-w)^{\gamma}:=\mathcal{S}(\lambda,\gamma,w);\;\;\;\lambda>0$

is the pgf of the scaled Sibuya distribution Christoph & Schreiber, (2000).
The latter is known to be infinitely divisible if and only if $\lambda\leq
1-\gamma$ and self-decomposable if and only if
$\lambda\leq(1-\gamma)/(1+\gamma)$ Christoph & Schreiber, (2000). In this
case, as follows from the Theorem 4, the pgf
$\mathcal{S}(\lambda,\gamma)\circ\mathcal{S}(\alpha)$ is also self-
decomposable.

######  Corollary 2.

Every Sibuya-distributed random variable $X_{1}$ with parameter $\gamma_{1}$
can be decomposed into the sum of two rvs $X_{1}=X_{2}+Y$, where $X_{2}$ has
the Sibuya distribution with parameter $\gamma_{2}<\gamma_{1}$ and $Y$ is
self-decomposable.

###### Proof.

Let $Q(w)$ be the pgf of the rv $Y$ and
$\gamma_{1}=\alpha\gamma_{2},0<\alpha<1$. Then

(33)
$\mathcal{S}(\alpha\gamma_{2},w)=\mathcal{S}(\alpha)\circ\mathcal{S}(\gamma_{2})=\mathcal{S}(\gamma_{2},w)Q(w).$

and hence

(34)
$Q(w)=\frac{\mathcal{S}(\alpha)\circ\mathcal{S}(\gamma_{2})}{\mathcal{S}(\gamma_{2},w)}=\left(\frac{\mathcal{S}(\alpha,w)}{w}\right)\circ\mathcal{S}(\gamma_{2})=\mathcal{S}_{0}(\alpha)\circ\mathcal{S}(\gamma_{2})\,,$

where $\mathcal{S}_{0}(\alpha,w)$ is the pgf of the shifted Sibuya
distribution (22). The latter is self-decomposable Christoph & Schreiber,
(2000) and so we can apply the Theorem 4 to obtain the needed result. ∎

######  Example 6.

Skipping the trivial case of the discrete stable distribution (11) which is
self-decomposible by construction let us consider the negative binomial
distribution with parameter $k=1$, so called geometric distribution, with the
pgf $Q(w)=(1+\lambda(1-w))^{-1}$, cf. Eq. (8). Then

(35) $G(w)=Q\circ\mathcal{S}(\gamma)=\frac{1}{1+\lambda(1-w)^{\gamma}}$

is the pgf of Mittag-Leffler distribution. Since $Q(w)$ is self-decomposable
it is also $G(w)$. The random variable
$Y\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}X+1$ with the
pgf $wG(w)$ where $X$ has the pgf given by Eq. (35) with
$\gamma=\frac{1}{2}{}$ and $\lambda=1$ describes the number of trials until
the first return to the origin in one-dimensional symmetric random walk on the
lattice Feller, (1957).

##  2\. Birth-and-death process

Let us consider continuous-time birth-and-death (B-D) process Feller, (1957);
Anderson, (2012) consisting of random events of the same kind – e.g. creations
and disintegrations of particles, populations of interacting species with
indistinguishable members, queueing networks (interconnected queues) etc.. The
system represents a continuous-time Markov chain with a discrete state space
$n\in\mathbb{N}$ which changes only through transitions from states to their
nearest neighbors $n\to n\pm 1$. It is described by stochastic process
$\\{X(t):t\geq 0\\}$

(36)
$\mathbb{P}(X(s+t)=j)|X(s)=i)=\mathbb{P}(X(t)=j)|X(0)=i)=P_{i,j}(t);\;\;\forall
i,j,\;t>0,s>0$

with transition probabilities $P_{i,j}(t)$ satisfying Chapman-Kolmogorov
equation

(37) $P_{i,j}(s+t)=\sum_{k}P_{i,k}(s)P_{k,j}(t).$

Evolution of the probability distribution
$p_{j}(t)\equiv\mathbb{P}(X(t)=j)=\sum_{i}\mathbb{P}(X(0=i)P_{i,j}(t)$ to find
the system in state $j$ at time $t$ if at time $t=0$ it was in state $i$ is
controlled by the time-independent transition probabilities per unit time

(38)
$\lambda_{j}=\frac{dP_{j,j+1}(t)}{dt}\Big{|}_{t=0^{+}}\,,\;\;\;\;\mu_{j}=\frac{dP_{j,j-1}(t)}{dt}\Big{|}_{t=0^{+}}.$

and satisfies differential-difference equation

(39)
$\displaystyle\frac{dp_{j}(t)}{dt}=-(\lambda_{j}+\mu_{j})p_{j}(t)+\lambda_{j-1}p_{j-1}(t)+\mu_{j+1}p_{j+1}(t),\;\;j>i\,,$
$\displaystyle\frac{dp_{i}(t)}{dt}=-\lambda_{i}p_{i}(t)+\mu_{i+1}p_{i+1}(t),\;\;i\in\mathbb{N}\,.$

The last equation in (39) reflects the initial condition
$p_{j}(0)=\delta_{ij}$.

One of the commonly used models specifying functional dependence of the
transitional probabilities $\lambda_{j}$ and $\mu_{j}$ on $j$ considers $j\to
j\pm 1$ transition as resulting from several underlying microscopic (particle-
level) processes $k\to k\pm 1$, $k=0,\ldots,j$ with the elementary transition
probabilities $\alpha_{k}$ and $\beta_{k}$, respectively. Thus one particle is
born or dies with the elementary probability per unit of time $\alpha_{0}$ or
$\beta_{0}j$, one particle can split into two particles with the rate
$\alpha_{1}j$, two particles can fuse into a single particle with the rate
$\beta_{1}j(j-1)$, etc. The relations between the transitional probabilities
$\lambda_{j}$ and $\mu_{j}$ and elementary ones are given by the following
formulas

(40)
$\lambda_{j}=\sum_{k=0}^{\ell}\alpha_{k}(j)_{k},\;\;\;\;\mu_{j}=j\sum_{k=0}^{m}\beta_{k}(j-1)_{k};\;\;\;\;\ell\leq
j;\;m\leq j;\;\;\;\;\;(j)_{k}\\!\equiv\\!\prod_{i=0}^{k-1}(j\\!-\\!i)$

For application of this approach to multiparticle production in high energy
physics see e.g. Biyajima et al., (1987); Zborovsky, (1996).

###  2.1. Stationary B-D equations

It is well known that for large $t$ solution of the B-D Eq. (39) can be well
approximated by the limiting probabilities $p_{j}=\lim_{t\to\infty}p_{j}(t)$
Feller, (1957); Anderson, (2012) which solve the Eq. (39) with
$dp_{j}(t)/dt=0$. The latter is in this case transformed into two functional
equations for one unknown function $F(j)$

(41) $\displaystyle
F(j\\!+\\!1)\equiv\mu_{j+1}p_{j+1}-\lambda_{j}p_{j}=\mu_{j}p_{j}-\lambda_{j-1}p_{j-1}\equiv
F(j),~{}~{}j>i$ $\displaystyle
F(i+1)\\!=\\!\mu_{i+1}p_{i+1}\\!-\\!\lambda_{i}p_{i}\\!=0,\;\;i\in\mathbb{N}.$

Solution of the first equation $F(j\\!+\\!1)\\!=\\!F(j)$ is fixed to zero by
the second equation $F(i+1)=0$. Consequently

(42)
$\frac{p_{j+1}}{p_{j}}=\frac{\lambda_{j}}{\mu_{j+1}}=\frac{\sum_{k=0}^{\ell}\alpha_{k}(j)_{k}}{(j+1)\sum_{k=0}^{m}\beta_{k}(j)_{k}}.$

We have thus proven the following theorem.

######  Theorem 5.

The unique solution of Eqs. (2.1) with $\lambda_{j}$ and $\mu_{j}$ given by
Eq. (40) reads:

(43)
$p_{j}\\!=\\!p_{i}\prod_{n=i}^{j-1}\frac{g(n)}{n+1};\;\;\;\;\;g(n)=\frac{(n\\!+\\!1)p_{n+1}}{p_{n}}=\frac{\sum_{k=0}^{\ell}\alpha_{k}(n)_{k}}{\sum_{k=0}^{m}\beta_{k}(n)_{k}}\,,\;\ell\leq
j\,,\;m\leq j$

where $p_{i}$ is the first non-zero value of the positive sequence defining
the pmf $p_{j}$, i.e. $\min_{i\geq 0}p_{i}>0$ and hence satisfying $g(i)>0$
and $g(i-1)\leq 0$.

######  Remark 3.

Class of discrete distributions on $\mathbb{Z}_{+}$ satisfying the recursion
formula (43) is substantially wider then usually discussed Kotz or Ord family
of distributions or their generalizations Norman L. Johnson, (2005);
Chakraborty, (2015). It is also worth nothing the similarity between Eq. (43)
and pgfs satisfying Eq. (13). However, not every pgf satisfying Eq. (13) has
its $g(n)$-function so simple that it can be written as in Eq. (43).

To illustrate this point let us give two examples which are related to Sibuya-
like distributions. The first is the thinned version of the shifted Sibuya
distribution (22) with the pgf $\mathcal{S}_{0}(\mathcal{B}_{a}(w))$. It is
self-decomposable and so it can be expressed as the Laplace integral (17).
Using Eq. (18) we can calculate its $g$–function to obtain

(44)
$g(n)=\frac{(n+1)(n+1-\gamma)}{(n+2)}\cdot\frac{\,{}_{2}F_{1}(1,\gamma+1;n+3;1-a)}{\,{}_{2}F_{1}(1,\gamma+1;n+2;1-a)}$

where ${}_{2}F_{1}(a_{1},a_{2};b_{1};z)$ is the Gauss hypergeometrical
function. Obviously only for $a=1$ the function $g(n)$ can be expressed as a
ratio of two polynomials in $n$.

Another example is the pgf of the discrete stable distribution (11). Although
its function $f(x)$ does exist, see Eq. 21, its $g(n)$

(45)
$g(n)=\frac{\sum_{i=0}^{n+1}(\lambda\gamma)^{i+1}F_{i}(\gamma)}{\sum_{i=0}^{n}(\lambda\gamma)^{i+1}F_{i}(\gamma)}\,,$

where the functions $F_{i}(\gamma)$ are some polynomials in $\gamma$
satisfying $F_{i}(1)=0,i>0$ with $F_{0}=1$, it does not allow to be written in
the form of Eq. (43).

######  Corollary 3.

The pmf (43) is the generalized hypergeometric probability distributions
Norman L. Johnson, (2005) with the pgf:

(46)
$Q(w)\\!=\\!\frac{H(zw)}{H(z)}\,,\;H(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\prod_{i=j}^{n-1}g(i)=\,_{\ell}F_{m}(a,b,z)$

where ${}_{\ell}F_{m}(a,b,z)$ represents the generalized hypergeometric
function with series expansion

(47)
$_{\ell}F_{m}(a_{1},\ldots,a_{\ell};b_{1},\ldots,b_{m};z)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\ldots(a_{\ell})_{n}}{(b_{1})_{n}\ldots(b_{m})_{n}}\frac{z^{n}}{n!}\,.$

The constants $a_{1},\ldots,a_{\ell}$ and $b_{1},\ldots,b_{m}$ in Eq. (47) are
some rational functions of $\alpha_{1},\ldots,\alpha_{\ell}$ and
$\beta_{1},\ldots,\beta_{m}$ and $z=\alpha_{\ell}/\beta_{m}$ where
$\alpha_{\ell}$ and $\beta_{m}$ are the last non-zero transition amplitudes.

###### Proof.

Using Eq. (43) together with the identity $n!=j!\prod_{i=j}^{n-1}(i+1)$ and
$z=\alpha_{\ell}/\beta_{m}$ we obtain

(48)
$Q(w)\\!=\\!\\!=\\!p_{j}\sum_{n=0}^{\infty}w^{n}\prod_{i=j}^{n-1}\frac{g(i)}{i\\!+\\!1}\\!=\\!p_{j}z^{-j}j!\sum_{n=0}^{\infty}\frac{(zw)^{n}}{n!}\prod_{i=j}^{n-1}\frac{\sum_{k=0}^{\ell}\tfrac{\alpha_{k}}{\alpha_{\ell}}(i)_{k}}{\sum_{k=0}^{m}\tfrac{\beta_{k}}{\beta_{m}}(i)_{k}}=\frac{H(zw)}{H(z)}\,.$

The last equality with $1/H(z)=p_{j}z^{-j}j!$ follows from the condition
$Q(1)=1$. Thus the pgf (48) corresponds to the power series distribution with
the function $H(z)$ which which is of the type of Eq. (47). ∎

######  Remark 4.

Note the certain arbitrariness when associating the coefficients $a_{i}$ and
$b_{i}$ with $\alpha_{i}$ and $\beta_{i}$, respectively. This is due to the
permutation symmetry of the generalized hypergeometric function (47) w.r.t.
its arguments $a_{i}$ and $b_{i}$ separately.

######  Example 7.

Consider Eq. (43) with $\ell=1,j=0$, i.e. the case with the all coefficients
$\alpha_{0},\alpha_{1},\beta_{0},\beta_{1}$ positive. Then

(49) $\displaystyle
g(n)=\frac{\alpha_{0}\\!+\\!\alpha_{1}n}{\beta_{0}\\!+\\!\beta_{1}n};\;\;\;H\left(\frac{\alpha_{1}}{\beta_{1}}\right)=\sum_{n=0}^{\infty}\frac{(\tfrac{\alpha_{1}}{\beta_{1}})^{n}}{n!}\prod_{j=0}^{n-1}\frac{\frac{\alpha_{0}}{\alpha_{1}}\\!+\\!i}{\frac{\beta_{0}}{\beta_{1}}\\!+\\!i}\\!=\\!_{1}F_{1}\left(\frac{\alpha_{0}}{\alpha_{1}};\frac{\beta_{0}}{\beta_{1}};\frac{\alpha_{1}}{\beta_{1}}\right)$

where ${}_{1}F_{1}(a;b;z)$ is Kummer’s (confluent hypergeometric) function.

Neglecting in $g(n)$ of Eq. (49) $\alpha_{1}$ and setting
$\beta_{0}=\beta_{1}$ with $z=\alpha_{0}/\beta_{1}=\theta$ yields the pmf of
the Conway-Maxwell-Poisson (CMP) distribution with parameter $r=2$ et al.,
(2005)

(50)
$g(n)=\frac{\theta}{n+1},~{}~{}p_{n}=p_{0}\prod_{i=0}^{n-1}\frac{g(i)}{i+1}=p_{0}\frac{\theta^{n}}{(n!)^{2}}\,,~{}~{}H(z)=\frac{1}{p_{0}}=\sum_{n=0}^{\infty}\frac{z^{n}}{(n!)^{2}}\,.$

Expression for the scaled factorial moments (6) of this pmf obtained in Daly &
Gaunt, (2016) reads

(51)
$F_{j}=\frac{I_{j}(2\sqrt{\theta})}{(I_{1}(2\sqrt{\theta}))^{j}};\;\;I_{j}(x)=\sum_{k=0}^{\infty}\frac{1}{k!\Gamma(j+k+1)}\left(\frac{x}{2}\right)^{j+2k}$

It is worth mentioning that the pmf (50) has appeared in Ko et al., (2001) as
a stationary solution of the kinetic master equation describing the production
of charged particles which are created or destroyed only in pairs due to the
conservation of their charge.

On the other hand neglecting in $g(n)$ of Eq. (49) the constant $\beta_{1}$ we
obtain the NBD

(52) $\displaystyle
g(n)=\frac{\alpha_{0}+\alpha_{1}n}{\beta_{0}}=q(k+n);\;\;\;p_{n}=p_{0}\frac{q^{n}(k)_{n}}{n!};\;\;\;q=\frac{\alpha_{1}}{\beta_{0}}\,,k=\frac{\alpha_{0}}{\alpha_{1}}$
(53)
$\displaystyle\sum_{n=0}^{\infty}p_{n}\\!=\\!1\\!=\\!p_{0}\left(1\\!+\\!\sum_{n=1}^{\infty}\frac{q^{n}(k)_{n}}{n!}\right)\\!=\\!(1-q)^{-k};\;\;\;p_{n}=\binom{n+k-1}{k}(1-q)^{k}q^{n}.$

######  Example 8.

Consider now more general case of Eq. (43) with $\ell=2,j=0$

(54)
$g(n)=\dfrac{\alpha_{0}+\alpha_{1}n+\alpha_{2}n(n-1)}{\beta_{0}+\beta_{1}n};$

With $z=\alpha_{2}/\beta_{1}$, $\alpha_{0}=\beta_{1}=\beta_{0}/2$ and
$\alpha_{1}=3\alpha_{2}=\beta_{0}\theta/2$ we obtain the zero-inflated
logarithmic distribution

(55)
$g(n)=\frac{(n\\!+\\!1)^{2}}{n\\!+\\!2}\theta;\;\;\;p_{n}=p_{0}\prod_{i=0}^{n-1}\frac{g(i)}{i+1}=-\ln(1\\!-\\!\theta)\dfrac{\theta^{n+1}}{n\\!+\\!1}.$

On the other hand the logarithmic distribution can be reconstructed from

(56)
$g(n)=n\theta;\;\;\;p_{n}=p_{1}\prod_{i=1}^{n-1}\frac{g(i)}{i+1};\;\;\;\frac{\alpha_{1}}{\beta_{1}}=\frac{\alpha_{2}}{\beta_{1}}=\theta,\alpha_{0}=\beta_{0}=0.$

And similarly to the above case for the geometric distribution with
$g(n)=(n+1)\theta$ the only non-zero coefficients are
$\alpha_{1}=2\beta_{1}\theta,\alpha_{2}=\beta_{1}\theta$.

######  Example 9.

While the NBD (53) is frequently used in different areas of science e.g. in
medicine Carter & Potts, (2014), biology Lloyd-Smith et al., (2005) or in
particle physics Biyajima et al., (1987); Giovannini & Van Hove, (1986);
Chliapnikov & Chikilev, (1989), the pmf obtained from Eq. (43) with non-zero
$\alpha_{0},\alpha_{2},\beta_{0},\beta_{2}$ and thus with the pgf Eq. (46)
where

(57)
$H(z)=\,_{2}F_{2}(a_{1},a_{2};b_{1},b_{2};z);\;a_{1}+a_{2}=b_{1}+b_{2}=-1,\,a_{1}a_{2}=\frac{\alpha_{0}}{\alpha_{2}},\,b_{1}b_{2}=\frac{\beta_{0}}{\beta_{2}},\,z=\frac{\alpha_{2}}{\beta_{2}}$

is seldom. Nevertheless, as shown in Zborovsky, (1996) it is important when
describing the data with a secondary maximum (bump).

##  3\. Sibuya–like solutions of the stationary birth-death equations

###  3.1. Generalized Sibuya distribution

The Sibuya distribution with the probability mass function

(58)
$\mathbb{P}(N=n)=\frac{\gamma(\gamma-1)\dots(\gamma-n+1)}{n!}(-1)^{n+1}=\binom{\gamma}{n}(-1)^{n+1},\,n\in\mathbb{N},$

first appeared in Sibuya, (1979). The generalized Sibuya distribution with
parameters $\nu\geq 0$ and $0<\gamma<\nu+1$ has been introduced in Kozubowski
& Podgorski, (2017) as the distribution of the number $N$ of trials until
series of the Bernoulli trials with the varying success probability
$\gamma/(\nu+n-1)$ and thus having the probability mass function

(59)
$p_{n}=\left(1-\frac{\gamma}{\nu+1}\right)\ldots\left(1-\frac{\gamma}{\nu+n-1}\right)\frac{\gamma}{\nu+n}.$

One connection of the generalized Sibuya distribution with the classical
Sibuya variable $N$ is through the probability of the excess
$\mathbb{P}(N-m=n|N>m)$, which has the generalized Sibuya distribution with
$\nu=m$. This extends to the generalized Sibuya variable $N$ for which
$\mathbb{P}(N-m=n|N>m)$ has also the generalized Sibuya distribution with
$\nu+m$ as its parameter.

Writing the pmf (59) as

(60)
$p_{n}=p_{1}\prod_{i=1}^{n-1}\frac{g(i)}{i+1};\;\;\;p_{1}=\frac{\gamma}{\nu+1};\;\;\;g(i)=\frac{(i+1)(i+\nu-\gamma)}{i+\nu+1}.$

we can see that function $g(i)$ of the generalized Sibuya distribution is a
special case of Eq. (54) with $\beta_{2}=0$ and thus solves the stationary B-D
equations (2.1). The corresponding non-zero coefficients are

(61)
$\frac{\alpha_{2}}{\beta_{1}}=1,\;\;\frac{\alpha_{1}}{\beta_{1}}=2+\nu-\gamma,\;\;\frac{\alpha_{0}}{\beta_{1}}=\nu-\gamma,\;\;\frac{\beta_{0}}{\beta_{1}}=\nu+1\,.$

Moreover for $\nu=0$ we have $p_{1}=\gamma$ and Eq. (60) reduces to the pmf of
Sibuya distribution (11).

The pgf of the generalized Sibuya distribution calculated from Eq. (60) reads

(62) $Q(w)=\frac{w\gamma}{\nu+1}\cdot\,_{2}F_{1}(1,\nu-\gamma+1;\nu+2;w)\,,$

where ${}_{2}F_{1}(a_{1},a_{2};b_{1};z)$ is the Gauss hypergeometric function.

###  3.2. Extended Sibuya distribution

Consider now Eq. (43) with $\ell=2,j=1$ and hence with the function $g(n)$
given by Eq. (54) with $\alpha_{0}=\beta_{0}=0$

(63)
$\displaystyle\;\;\;g(n)=\frac{\alpha_{2}n(n\\!-\\!1)+\alpha_{1}n}{\beta_{1}n}=b(n-\gamma);\;\;\;\;\frac{\alpha_{2}}{\beta_{1}}=b,\;\frac{\alpha_{1}}{\beta_{1}}=b(1-\gamma),\;\gamma=1-\frac{\alpha_{1}}{\alpha_{2}}.$

Using Corollary 3 it is easy to verify that the corresponding pgf

(64)
$\mathcal{R}(b,\gamma,w)=\frac{H(bw)}{H(b)}=\frac{1-(1-bw)^{\gamma}}{1-(1-b)^{\gamma}}=\frac{\mathcal{S}(\gamma,bw)}{\mathcal{S}(\gamma,b)}$

represents yet another generalization of the Sibuya pgf
$\mathcal{S}(\gamma,w)$. For $\gamma=-k<0$ and fixed $b=q$ the pgf (64)
coincides with the NBD pgf (8) truncated at zero

(65)
$\mathcal{R}(w)=\frac{1-(1-qw)^{-k}}{1-(1-q)^{-k}}=\frac{Q_{NBD}(w)-Q_{NBD}(0)}{1-Q_{NBD}(0)};~{}~{}~{}Q_{NBD}(w)=\left(\frac{1-q}{1-qw}\right)^{k}\,.$

For $\gamma\to 0$, i.e. when $\alpha_{2}\to\alpha_{1}$, the pgf (64)
approaches the logarithmic distribution

(66) $\lim_{\gamma\to
0}\frac{1-(1-bw)^{\gamma}}{1-(1-b)^{\gamma}}=\frac{\log(1-bw)}{\log(1-b)}\,$

and for $\gamma\to 1$ it converges to distribution with the pmf
$p_{n}=\delta_{1n}$.

Let us note that the pgf (64) was already mentioned in Letac, (2019) (see
Eq.(6) therein) as the natural exponential family extension of the Sibuya
distribution. Since no name was given to it nor its relation with the other
known distributions was discussed we take a liberty and call it the pgf of
extended Sibuya distribution stressing its smooth transition in $\gamma$ to
the pgfs of several fundamental discrete distributions.

Let us now study the factorial moments $\mathbb{E}X^{(j)}=\langle
n^{(j)}\rangle:=\langle n(n-1)\ldots(n-j+1)\rangle,~{}j\geq 1$, see Eq.(6), of
the pgf $\mathcal{R}(w)$

(67) $\langle
n^{(j)}\rangle=\mathcal{R}^{(j)}(1)=\frac{(-b)^{j}(1-b)^{\gamma-j}\gamma^{(j)}}{(1-b)^{\gamma}-1}\,;~{}~{}\gamma^{(j)}=\frac{\Gamma(\gamma\\!+\\!1)}{\Gamma(\gamma\\!+\\!1\\!-\\!j)}\,,~{}~{}j\geq
1.$

When approaching the Sibuya distribution, i.e. for $b\to 1^{-}$ we obtain
expected results

(68) $\langle
n\rangle=\mathcal{R}^{(1)}(1)=\frac{p_{1}}{(1\\!-\\!b)^{1-\gamma}}\xrightarrow[b\to
1^{-}]{}\infty;\;\;p_{1}=\mathcal{R}^{(1)}(0)=\frac{b\gamma}{1-(1-b)^{\gamma}}~{}\xrightarrow[b\to
1^{-}]{}~{}\gamma.$

For the second scaled factorial moment $F_{2}$, cf. Eq. (6) we have

(69) $F_{2}=\frac{\langle n(n-1)\rangle}{\langle
n\rangle^{2}}=\frac{(1\\!-\\!\gamma)\left(1\\!-\\!(1-b)^{\gamma}\right)}{\gamma(1-b)^{\gamma}}=\frac{1\\!-\\!\gamma}{\gamma}\cdot\left(e^{\delta\gamma}\\!-\\!1\right);\;\;\delta:=-\log(1\\!-b).$

Ignoring the region with $\gamma<0$ which corresponds to the zero-truncated
NBD let us analyze the behavior of

(70)
$\frac{dF_{2}}{d\gamma}=\frac{e^{\gamma\delta}(\gamma\delta(1\\!-\\!\gamma)\\!-\\!1)\\!+\\!1}{\gamma^{2}}\,.$

For $\gamma\to 0$, i.e. for the logarithmic distribution, $F_{2}$ has a
minimum at $\delta=2$. Further up, in the region $(\gamma>0,\delta>2)$, the
extreme of $F_{2}(\gamma)$ moves along the trajectory
$dF_{2}/d\gamma|_{\gamma_{0}}=0$ or equivalently $\psi(\gamma_{0},\delta)=0$,
where $\psi(\gamma,\delta)=(e^{\gamma\delta}(\gamma\delta(1-\gamma)-1)+1$.
While $F_{2}(\gamma)$ increases for $0<\gamma<\gamma_{0}$ it decreases for
$\gamma_{0}<\gamma<1$. When approaching the singularity at $b=1$ the second
scaled factorial moment $F_{2}$ and hence also its maximum goes to infinity.
Let us add that the same maxima appear also in the higher scaled factorial
moments $F_{j},j>2$.

###  3.3. The shifted variants of extended and generalized Sibuya
distributions

Consider random variable
$Y\stackrel{{\scriptstyle\scriptsize\mathclap{\mbox{d}}}}{{=}}X+1$ where
$Y\in\mathbb{N}$ has either extended or generalized Sibuya distribution. Then
the rv $X\in\mathbb{Z}_{+}$ has shifted extended or generalized Sibuya
distribution, respectively. The case when $Y$ has ordinary Sibuya distribution
was already discussed in Example 4. Note that contrary to the original
extended or generalized Sibuya distributions their shifted variants are now
defined on $\mathbb{Z}_{+}$ and so they may be self-decomposable.

####  3.3.1. The shifted extended Sibuya distribution

If $p_{n}$ and $\mathcal{R}(b,\gamma,w)$ are the pmf the pgf of the extended
Sibuya distribution (63), respectively, then $r_{n}=p_{n+1}$ and
$\mathcal{R}_{0}(b,\gamma,w)=\mathcal{R}(b,\gamma,w)/w$, cf. Eq. (22), are the
pmf and the pgf of the shifted extended Sibuya distribution. Corresponding
function $g(n)$ thus reads

(71)
$g_{r}(n)=\frac{(n+1)r_{n+1}}{r_{n}}=\frac{n+1}{n+2}\cdot\frac{(n+2)p_{n+2}}{p_{n+1}}=\frac{n+1}{n+2}b(n+1-\gamma);\;\;b\leq
1\,$

cf. Eq. (24). The elementary transition amplitudes (43) derived from Eq. (71)
are

(72)
$\frac{\alpha_{2}}{\beta_{1}}=b,\;\;\frac{\alpha_{1}}{\beta_{1}}=b(3-\gamma),\;\;\frac{\alpha_{0}}{\beta_{1}}=b(1-\gamma),\;\;\frac{\beta_{0}}{\beta_{1}}=2\,.$

######  Theorem 6.

The shifted extended Sibuya distribution is self-decomposable.

###### Proof.

Following Christoph & Schreiber, (2000) we use

######  Lemma 1.

Bondesson, (1992) A strictly decreasing pmf $r_{n},n=0,1,2,\dots$ such that

(73) $\max_{0\leq n\leq
j}\frac{r_{n+1}}{r_{n}}\leq\frac{j+2}{j+1}\cdot\frac{r_{j+1}-r_{j+2}}{r_{j}-r_{j+1}}\,,\;\;\forall
j=0,1,2,\ldots$

is discrete self-decomposable.

In our case the sequence

$\frac{r_{n+1}}{r_{n}}=\frac{p_{n+2}}{p_{n+1}}=b\left(1-\frac{1+\gamma}{n+2}\right)$

is strictly decreasing and therefore the left hand side of Eq. (73) gives:

$\max_{0\leq n\leq
j}\frac{r_{n+1}}{r_{n}}=b\left(1-\frac{1+\gamma}{j+2}\right).$

For the right hand side of the inequality (73) we have

$\frac{j+2}{j+1}\cdot\frac{r_{j+1}-r_{j+2}}{r_{j}-r_{j+1}}=\frac{(j+2)}{(j+1)}\cdot\frac{b(j+1-\gamma)(j+3-b(j+2-\gamma))}{(j+3)(j+2-b(j+1-\gamma))}.$

Dividing now the r.h.s. of the Eq. (73) by its l.h.s. we obtain

$R(b,j)=\frac{j+2}{j+1}\cdot\frac{r_{j+1}-r_{j+2}}{r_{j}-r_{j+1}}\Big{/}\max_{0\leq
n\leq
j}\frac{r_{n+1}}{r_{n}}=\frac{(j+2)^{2}(j+3-b(j+2-\gamma))}{(j+1)(j+3)(j+2-b(j+1-\gamma))}.$

Now we need to prove that $R(b,j)\geq 1$. It is easy to verify that $R(b,j)$
is decreasing function with respect to $b$ for each fixed $j$. Therefore, its
minimum is attained at point $b=1$ where

$\frac{j^{2}+4j+4}{j^{2}+4j+2}>1.$

Hence Eq. (73) is satisfied and the shifted extended Sibuya distribution is
indeed self-decomposable and thus also infinitely divisible F.W. Steutel,
(2004). ∎

For $b<1$ the Theorem 1 tells us that the scaled factorial moments of the
shifted extended Sibuya distribution with the pgf
$\mathcal{R}_{0}(b,\gamma,w)$ and of its thinned version
$\mathcal{R}_{0}(b,\gamma,1-a+aw)$ are the same. Moreover, since
$\mathcal{R}_{0}(b,\gamma,w)=\mathcal{S}_{0}(\gamma,bw)/\mathcal{S}_{0}(\gamma,b)$,
see Eq. (64), it is easy to check using Eq. 19 with $f(x)$ given by Eq. (23)
that the pgf of the shifted extended Sibuya distribution satisfies the Theorem
2 and hence the scaling variable $a$ can be any positive number.

####  3.3.2. The shifted generalized Sibuya distribution

If $p_{n}$ is the pmf of the generalized Sibuya distribution (59) then
$r_{n}=p_{n+1}$ is the pmf of the shifted generalized Sibuya distribution with
the $g$-function

(74)
$g_{r}(n)=\frac{(n+1)r_{n+1}}{r_{n}}=\frac{n+1}{n+2}\cdot\frac{(n+2)p_{n+2}}{p_{n+1}}=\frac{(n+1)(n+1+\nu-\gamma)}{n+2+\nu}$

and hence

(75)
$\frac{\alpha_{2}}{\beta_{1}}=1,\;\;\frac{\alpha_{1}}{\beta_{1}}=(3+\nu-\gamma),\;\;\frac{\alpha_{0}}{\beta_{1}}=(1+\nu-\gamma),\;\;\frac{\beta_{0}}{\beta_{1}}=\nu+2\,.$

Interestingly, the only difference between the coefficients of the original
(61) and the shifted distribution (75) consists in replacement $\nu\to\nu+1$.

To check the self-decomposability we once again use the Lemma 1 to find that

$\frac{r_{n+1}}{r_{n}}=\frac{p_{n+2}}{p_{n+1}}=\frac{n+1+\nu-\gamma}{n+2+\nu}=1-\frac{1+\gamma}{n+2+\nu}$

is strictly decreasing. Therefore the l.h.s. of the Eq. (73) gives:

$M=\max_{0\leq n\leq
j}\frac{r_{n+1}}{r_{n}}=\left(1-\frac{1+\gamma}{j+2+\nu}\right).$

Using Eq. (74) we rewrite the r.h.s. of the Eq. (73) as

$N=\frac{j+2}{j+1}\cdot\frac{r_{j+1}-r_{j+2}}{r_{j}-r_{j+1}}=\frac{g_{r}(j)}{j+1}\cdot\frac{j+2-g_{r}(j+1)}{j+1-g_{r}(j)}=-\frac{(j+2)(\gamma+2\nu-1)^{2}}{(j+1)(j-\nu+2)(j-\nu+3)}$

Dividing now the r.h.s. of the Eq. (73) by its l.h.s. and neglecting terms
$O\left(j^{-3}\right)$ we observe that there always $\exists j_{0}>0$ such
that $\forall j\geq j_{0}$ the ratio

$\frac{N}{M}\approx\frac{(\gamma+2\nu-1)^{2}}{j^{2}}<\frac{(3\nu)^{2}}{j^{2}}<1$

where in that last equation we have used the inequality $0<\gamma<\nu+1$. Thus
contrary to the previous case the shifted generalized Sibuya distribution is
not self-decomposable.

###  3.4. Beyond the Sibuya distribution

Let us consider the following four-parameter generalization of the Sibuya pgf

(76)
$Q(w)=\frac{1-(1-(1-(1-bw)^{\gamma})^{\ell})^{k}}{1-(1-(1-(1-b)^{\gamma})^{\ell})^{k}}=\frac{1-(1-(H(b,\gamma,w))^{\ell})^{k}}{1-(1-(H(b,\gamma,1))^{\ell})^{k}}.$

In Klebanov, (2021) it was shown that Eq. (76) represents a probability
generating function in the case of $\gamma=1/m$, $k=m$, $b=1$ where
$m,\;\ell\in\mathbb{N}$. It is not difficult to prove this statement remains
true for the case of positive integer $k\leq m$ and $b\in(0,1]$. For the case
of $k=\ell=1$ the function $Q(w)$ coincides with probability generating
function $\mathcal{R}(w)$ of the extended Sibuya distribution (64). For the
case of $k=1$ the pgf Eq. (76) represents the sum of $\ell$ extended Sibuya–
distributed random variables. In particular, for $\ell=2$, the function $g(n)$
is easily calculable

(77)
$g(n)=\frac{b(n+1)\Bigl{(}\binom{2\gamma}{n+1}-2\binom{\gamma}{n+1}\Bigr{)}}{2\binom{\gamma}{n}-\binom{2\gamma}{n}}\,.$

Let us note that for $\gamma\geq 1/2$ the function (77) is positive and for
$n\geq 2$ it is close to linear function of $n$ $g(n)\approx
a_{1}(\gamma)(n+a_{2}(\gamma))$. In particular for $\gamma=1/2$ it coincides
with the extended Sibuya distribution (63).

On the other hand for $k=\ell=2$, $m\geq 2$ the ratio $g(n)$ (43) takes the
form

(78)
$g(n)=-b\frac{(1+n)\left(4\binom{2/m}{1+n}-4\binom{3/m}{1+n}+\binom{4/m}{1+n}\right)}{2\left(4\binom{2/m}{n}-4\binom{3/m}{n}+\binom{4/m}{n}\right)}.$

Most simple case is $m=2$ with $g(n)=\frac{b}{4}(2n-3)$ which is positive for
$n\geq 2$. The pmf then reads

(79)
$p_{n}=p_{2}\prod_{i=2}^{n-1}\frac{b}{4}\frac{2i-3}{i+1}=\frac{\alpha_{3}n(n\\!-\\!1)(n\\!-\\!2)+\alpha_{2}n(n\\!-\\!1)}{\beta_{2}n(n\\!-\\!1)};\;\;\frac{\alpha_{3}}{\beta_{2}}=\frac{b}{2},\;\frac{\alpha_{2}}{\beta_{2}}=\frac{7b}{4}.$

###  3.5. The extended Sibuya as progeny

Consider branching process $(Z_{n})_{n\geq 0}^{\infty}$ governed by the pmf
$p_{k},k\in\mathbb{Z}_{+}$ Feller, (1957); Harris, (2012). Markov property of
the process determines how the direct descendants of the $n$th generation form
the $(n\\!+\\!1)$st generation. If $Z_{n}$ is the size of the $n$th generation
and $H_{n}(w)$ is its pgf then rv $Y_{n}=1+Z_{1}+\ldots+Z_{n}$ with the pgf
$Q_{n}(w)$ represents the total number of descendants up to the $n$th
generation including the ancestor (zero generation). By assumption $Z_{0}=1$
and $H_{1}(w)=H(w)=\sum_{k=0}^{\infty}p_{k}w^{k}$ and therefore

(80) $Q_{1}(w)=wH(w);~{}~{}~{}Q_{n}(w)=wH(Q_{n-1}(w)).$

If $H^{(1)}(1)=\sum_{0}^{\infty}kp_{k}\leq 1$ then the rv
$Y=\sum_{n=0}^{\infty}Z_{n}$ is finite and is called progeny Feller, (1957).
The pgf of the progeny then corresponds to a fixed point of Eq. (80)

(81) $\lim_{n\to\infty}Q_{n}(w)=Q(w)=wH(Q(w)).$

By inverting the last equation we can express $H(w)$ in terms of $Q(w)$

(82) $Q(w)=wH(Q(w));~{}~{}w=Q^{-1}(u)=g(u);\;\;H(u)=\frac{u}{g(u)}.$

Note that although such inversion is in principle always possible the rv $Y$
is the progeny only when all the expansion coefficients of the function $H(u)$
in powers of $u$ are positive. In particular, for the case of Sibuya pgf (11)
Eq. (82) with $Q(w)=\mathcal{S}(\gamma,w)$ the progeny was shown to exists
only for $\frac{1}{2}\leq\gamma<1$ Letac, (2019).

For the extended Sibuya distribution Eq. (82) with $Q(w)=\mathcal{R}(w)$
yields

(83)
$\displaystyle~{}~{}H(u,b,\gamma)=\frac{ub}{1-(1-uc)^{1/\gamma}};~{}c=1-(1-b)^{\gamma};~{}~{}\gamma\neq
0.$

Expanding $H(u,b,\gamma)$ in powers of $u$

(84)
$H(u,b,\gamma)=\frac{b\gamma}{c}-\frac{1}{2}b(\gamma-1)u-\frac{\left(bc\left(\gamma^{2}-1\right)\right)}{12\gamma}u^{2}-\frac{\left(bc^{2}\left(\gamma^{2}-1\right)\right)}{24\gamma}u^{3}+O\left(u^{4}\right).$

we observe that for $\gamma<-1$ the coefficient $\sim-(\gamma^{2}-1)u^{2}$
becomes negative (taking into account that $bc/\gamma>0$) and for
$-1<\gamma<0$ the same is true for the coefficient
$\sim-(bc^{2}\left(\gamma^{2}-1\right))u^{3}$. On the other hand for
$\gamma=-1$ with $H(u)=-\frac{b}{c}+bu$ the corresponding Geometric
distribution pgf $\mathcal{R}(w,b,-1)$ is progeny.

Further on at $\gamma=0$

(85)
$H(u,b,0)=\frac{bu}{1-(1-b)^{u}}=-\frac{b}{\log(1-b)}+\frac{bu}{2}-\frac{1}{12}u^{2}(b\log(1-b))+\frac{1}{720}bu^{4}\log^{3}(1-b)+O\left(u^{5}\right)$

it is now coefficient $\sim\log^{3}(1-b)u^{4}$ which is negative. Finally for
$\gamma>0$ substitution $t=bw\to w$ in Eq. (64) yields the pgf of Sibuya
distribution (11) $\mathcal{S}(t)$ multiplied by positive factor
$1/(1-(1-b)^{\gamma})$ and hence with the identical signs of all expansion
coefficients. Therefore the results obtained in Letac, (2019) for $\gamma>0$
are unchanged and extended Sibuya pgf is progeny for
$\gamma\in\langle\tfrac{1}{2},1)$ while for $0<\gamma<\tfrac{1}{2}$ it is not.
In terms of first and second (factorial) moments of the pgf (83)

(86) $\displaystyle\langle
k\rangle=H^{(1)}(1,b,\gamma)\\!=\\!1\\!+\\!\frac{(1\\!-\\!b)c}{b\gamma}\leq 1$
(87) $\displaystyle\langle
k(k-1)\rangle=H^{(2)}(1,b,\gamma)\\!=\\!\frac{c(1-b)^{1-2\gamma}\left[2(1\\!-\\!b)(c-b(\gamma+1)(c-\gamma)\right]}{b^{2}\gamma^{2}}.$

While in the regions $-1\gamma<0$ and $0<\gamma<\frac{1}{2}$ the function
$H^{(2)}(1,b,\gamma)$ has non-monotonic dependence on the parameter $b$ with a
maximum at $\frac{3}{4}<b_{max}<1$ rapidly moving towards 1, for
$\gamma\geq\frac{1}{2}$ it becomes a monotonously increasing function of $b$.
The change in the vicinity of $\gamma=\frac{1}{2}$ is quite dramatic:

(88) $\lim_{b\to 1^{-}}H^{(2)}(1,b,\gamma<\tfrac{1}{2})=0;~{}~{}\lim_{b\to
1^{-}}H^{(2)}(1,b,\gamma\geq\tfrac{1}{2})=\infty.$

##  4\. Conclusion

The goal of this article was to discuss the properties of Sibuya-like random
variables related to their scaling and birth-death processes. We have shown
that their pmf satisfy a one-step recurrence relation
$g(n)=(n+1)p_{n+1}/p_{n}$ with some of them representing the stationary
solutions of the birth-death equations and most of them being self-
decomposable. In addition to already known Sibuya-like distribution (shifted
Sibuya, generalized Sibuya, disctere stable) we have found a new one -
extended Sibuya distribution. The latter is very similar to ordinary Sibuya
distribution having the same bounds of its validity as a progeny in branching
process.

## Acknowledgements

The work of Lev B. Klebanov was partially supported by Grant 19-28231X GA ČR.
Michal Šumbera is partially supported by the grants LTT17018 and LTT18002 of
the Ministry of Education of the Czech Republic.

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# A Genuine Multipartite Entanglement Measure Generated by the Parametrized
Entanglement Measure

Xian Shi<EMAIL_ADDRESS>College of Information Science and Technology,
Beijing University of Chemical Technology, Beijing 100029, China

###### Abstract

In this paper, we investigate a genuine multipartite entanglement measure
based on the geometric method. This measure arrives at the maximal value for
the absolutely maximally entangled states and has desirable properties for
quantifying the genuine multipartite entanglement. We present a lower bound of
the genuine multipartite entanglement measure. At last, we present some
examples to show that the genuine entanglement measure is with distinct
entanglement ordering from other measures, and we also present the advantages
of the measure proposed here with other measures.

###### pacs:

03.65.Ud, 03.67.Mn

##  I introduction

Quantum entanglement is an essential feature of quantum mechanics. It plays an
important role in quantum information and quantum computation theory
horodecki2009quantum , such as superdense coding bennett1992communication ,
teleportation bennett1993teleporting  and the speedup of quantum algorithms
shimoni2005entangled .
One of the most important problems is to quantify the entanglement in a
composite quantum system. Vedral $et$ $al.$ in vedral1997quantifying
presented the condition that the amount of entanglement cannot increase under
local operation and classical communication (LOCC) is necessary for an
entanglement measure. Then Vidal considered the entanglement measures with
stronger properties and proposed a general mathematical framework to build
entanglement monotone with functions satisfying some properties for pure
statesvidal2000entanglement . The other interesting approach with operational
significance to quantify the entanglement is proposed in gour2020optimal ;
shi2021extension . Compared with the bipartite entanglement systems, the
complexity of a multipartite entanglement system grows remarkly with the
increasing number of parties and the increasing dimension of the systems. The
notion of multipartite entanglement measure can be refined into the so-called
genuine multipartite entanglement (GME) guhne2005multipartite ;
plenio2014introduction . Substantial results have been achiveved on the GME
measures in the last few decades meyer2002global ; blasone2008hierarchies ;
hiesmayr2009two ; ma2011measure ; jungnitsch2011taming ;
rafsanjani2012genuinely ; xie2021triangle ; beckey2021computable ;
guo2022genuine ; li2022geometric . In blasone2008hierarchies , the authors
proposed a genuinely entangled measure defined as the shortest distance from a
given state to the k-separable states, which was denoted as generalized
geometric measure (GGM). The other GME measure, the genuinely multipatite
concurrence (GMC) was defined as the minimal bipartite concurrence among all
bipartitions ma2011measure . However, the above two measures cannot show all
the conditions of entanglement among the parties, both two are defined on the
minimizations of the partitions. Concurrence fill was proposed as a three-
qubit GME measure xie2021triangle , it is the square of the area of the three-
qubit concurrence triangle. The authors in guo2022genuine  generalized the
above method to build a multipartite entanglement measure, however, it is hard
when the parties are bigger. Hence there is much work to do on how to
understand and quantify the multipartite entanglement. Recently, another
method to investigate the GME measure is proposed, it was based on the
geometric mean of all bipartite entanglement measure concurrence
li2022geometric .
In this paper, we investigate a GME measure which was based on the geometric
mean in terms of a bipartite entanglement measure, $F_{q}$
yang2021parametrized . This measure satisfies the following properties,
subadditivity, continuity for pure states. We also present the bound of this
GME measure based on the method proposed in dai2020experimentally . At last,
we present some examples to show that the GME measure proposed here is with
different orders from GMC and GGM, we also prsent some advantages of the
measure when comparing with GMC and GGM.
This paper is organised as follows. In Sec. II, we present the preliminary
knowledge needed here. In Sec. III, we present the main results. We show that
the measure is a GME measure, then we consider the measure for the W states
and GHZ states in $n$-qubit systems and present that the GHZ states is more
entangled than the W states. We also present the measure satifies the
subadditivity and continuity for pure states. And then we present a lower
bound of the measure for multipartite mixed states. We also make some
comparisons between the measure proposed here and the GMC, GGM by considering
some examples. In Sec. IV, we end with a conclusion.

##  II Preliminary Knowledge

Concurrence is one of the most important entanglement measures for bipartite
quantum systems hill1997entanglement , it has attracted much attention of the
relevant researchers since the end of the last century
wootters1998entanglement ; coffman2000distributed ; mintert2004concurrence ;
chen2005concurrence ; zhang2016evaluation ; dai2020experimentally ;
li2020improved . For a bipartite pure state $|\psi\rangle_{AB},$ its
concurrence is defined as

$\displaystyle
C(|\psi\rangle_{AB})=\sqrt{2(1-Tr_{A}|\psi\rangle\langle\psi|)},$ (1)

when $\rho_{AB}$ is a mixed state, its concurrence is defined as

$\displaystyle C(\rho_{AB})=\mathop{\rm min}\sum_{i}p_{i}C(|\psi\rangle_{i}),$
(2)

where the minimum takes over all the decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$ For two qubit mixed
states, there exists a direct link between the concurrence and the
entanglement of formation wootters1998entanglement .

As a generalized von Neumann entropy, Tsallis-$q$ entropy tsallis1988possible
; landsberg1998distributions  can present more properties of the entangled
states tsallis2001peres ; rossignoli2002generalized . For a pure state
$|\psi\rangle_{AB}=\sum_{i}\sqrt{\lambda_{i}}|ii\rangle,$ then its Tsallis-$q$
entanglement measure san2010tsallis  is defined as

$\displaystyle T_{q}(|\psi\rangle_{AB})=\frac{1-\rho_{A}^{q}}{q-1},$

here $\rho_{A}=Tr_{B}\rho_{AB},$ $q\in(0,1)\cup(1,\infty).$

Motived by the Tsallis-$q$ entanglement entropy, Yang $et$ $al.$ proposed a
parametrized entanglement measure, $q$-concurrence, for a bipartite
entanglement systems yang2021parametrized . When $|\psi\rangle_{AB}$ is a pure
state, this measure is defined as

$\displaystyle\mathcal{C}_{q}(|\psi\rangle_{AB})=F_{q}(\rho_{A}),$ (3)

here $F(\rho)=1-\mathop{\rm Tr}\rho^{q}$, $q\geq 2.$ When $\rho_{AB}$ is a
mixed state, its $q$-concurrence is defined as

$\displaystyle\mathcal{C}_{q}(\rho_{AB})=\mathop{\rm
min}\sum_{i}p_{i}\mathcal{C}_{q}(|\psi_{i}\rangle),$ (4)

where the minimum takes over all the decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$

Next we recall some properties of the function $F_{q}(\rho)$ proposed in
yang2021parametrized .

###### Lemma 1

For any density matrix $\rho$ and $q\geq 2$, $F_{q}(\rho)$ satisfies the
following properties:

  * (i).

_Non-negativity:_ $F_{q}(\rho)\geq 0.$

  * (ii).

_Subadditivity:_ For a general bipartite state $\rho_{AB}$, $F_{q}(\rho_{AB})$
satisfies the inequalies:

$\displaystyle|F_{q}(\rho_{A})-F_{q}(\rho_{B})|\leq F_{q}(\rho_{AB})\leq
F_{q}(\rho_{A})+F_{q}(\rho_{B}).$ (5)

  * (iii).

_Concavity_

$\displaystyle\sum p_{i}F_{q}(\rho_{i})\leq F_{q}(\sum_{i}p_{i}\rho_{i}),$ (6)

where $p_{i}\in(0,1],$ and $\sum_{i}p_{i}=1$, $\rho_{i}$ are density matrices.
Furthermore, The equality holds if and only if $\rho_{i}$ are identical for
all $p_{i}>0.$

  * (iv)

_quasiconvex:_

$\displaystyle F_{q}(\sum_{i}p_{i}\rho_{i})\leq\sum
p_{i}^{q}F_{q}(\rho_{i})+1-\sum_{i}p_{i}^{q},$ (7)

where the inequality holds if and only if
$\rho_{i}=|\psi_{i}\rangle\langle\psi_{i}|,$ and $\\{|\psi_{i}\rangle\\}$ are
orthogonal.

Due to the properties of $F_{q}(\rho)$ above, we have that the maximum of
$F_{q}(|\psi\rangle_{AB})$ is attained when its reduced density matrix of the
smaller subsystem is the maximally mixed state, that is,

$\mathop{\rm
max}_{|\psi\rangle_{AB}}F_{q}(|\psi\rangle_{AB})=\frac{d^{q-1}-1}{d^{q-1}},$

here $d$ is the dimension of the smaller system.

Next we review the knowledge needed on multipartite entanglement.

An $n$-partite pure state $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is full
product if it can be written as

$\displaystyle|\psi\rangle_{A_{1}A_{2}\cdots
A_{n}}=|\phi_{1}\rangle_{A_{1}}|\phi_{2}\rangle_{A_{2}}\cdots|\phi_{n}\rangle_{A_{n}},$
(8)

otherwise, it is entangled. A multipartite pure state is called genuinely
entangled if

$\displaystyle|\psi\rangle_{A_{1}A_{2}\cdots
A_{n}}\neq|\phi\rangle_{S}|\zeta\rangle_{\overline{S}},$ (9)

for any partite $S|\overline{S}$, here $S$ is a subset of
$\boldsymbol{A}=\\{A_{1},A_{2},\cdots,A_{n}\\}$, and
$\overline{S}=\boldsymbol{A}-S$. An $n$ partite mixed state $\rho$ is
biseparable if it can be written as a convex combination of biseparable pure
states $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|,$ where the pure
states in $\\{|\psi_{i}\rangle\\}$ can be biseparable with respect to some
bipartitions. If an $n$-partite state is not biseparable, then it is genuinely
entangled.

Then we recall the necessary conditions of a genuine multipartite entanglement
(GME) measure $E$ should satisfy ma2011measure :

  * 1

. it is entanglement monotone.

  * 2

. $E(\rho)=0,$ if $\rho$ is biseparable.

  * 3

. $E(\rho)>0$, if $\rho$ is genuinely entangled state.

Based on the $q$-concurrence, we present a GME measure in terms of the
geometric methods.

###### Definition 2

Assume $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is an $n$-partite pure state,
the geometric mean of $q$-concurrence $(GqC)$ is defined as

$\displaystyle\mathcal{G}_{q}(|\psi\rangle)=[\mathcal{P}_{q}(|\psi\rangle)]^{\frac{1}{c(\alpha)}},$
(10)

where $\alpha=\\{\alpha_{i}\\}$ is the set that denotes all possible
bipartitions $\\{A_{\alpha_{i}}|B_{\alpha_{i}}\\}$ of the $n$ parties,
$c(\alpha)$ is the cardinality of $\alpha$, and
$\mathcal{P}_{q}(|\psi\rangle)$ is

$\mathcal{P}_{q}(|\psi\rangle)=\Pi_{\alpha_{i}\in\alpha}{\mathcal{C}_{q}}_{A_{\alpha_{i}}B_{\alpha_{i}}}(|\psi\rangle),$
$\displaystyle\begin{split}c(\alpha)=\left\\{\begin{array}[]{lr}\sum_{m=1}^{\frac{n-1}{2}}C_{n}^{m},\hskip
8.53581pt\textit{if $n$ is odd,}\\\
\sum_{m=1}^{\frac{n-2}{2}}C_{n}^{m}+\frac{1}{2}C_{n}^{\frac{n}{2}},\hskip
8.53581pt\textit{if $n$ is even.}\end{array}\right.\end{split}$

When $\rho$ is an $n$-partite mixed state,

$\displaystyle\mathcal{G}_{q}(\rho)=\mathop{\rm
min}\sum_{i}p_{i}\mathcal{G}_{q}(|\psi_{i}\rangle),$ (11)

where the minimum takes over all the decompositions of
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$

##  III MAIN RESULTS

In this section, we present the main results of this article. In Sec. III.1,
we present the properties of the GqC. In Sec. III.2, a lower bound of the GqC
for an $n$-partite mixed state $\rho$ was presented. In Sec. III.3, we make a
comparison of the GqC with other GME measures.

###  III.1 The properties of GqC

Here we first show that the GqC is a GME measure.

###### Theorem 3

For an arbitrary $n$-partite quantum state $|\psi\rangle_{A_{1}A_{2}\cdots
A_{n}}$, the GqC is a GME measure.

###### Proof.

Here we show that the GqC satisfies the properties $2$ and $3$. The proof of
the property 1 is similar to the proof in li2022geometric , and we omit it
here.

Assume $|\psi\rangle$ is a biseparable pure state, then according to the
definition of biseparable states in Sec. II, there exists a partition
$S|\overline{S}$ of $\boldsymbol{A}$ such tat
$|\psi\rangle=|\phi_{1}\rangle_{S}|\phi_{2}\rangle_{\overline{S}}$,
$\mathcal{C}_{q}(|\psi\rangle_{S|\overline{S}})=0$, thus we have
$\mathcal{G}_{q}(|\psi\rangle)=0.$ And due to the definition of the GqC for
mixed states, when a mixed state $\rho$ is biseparable,
$\mathcal{G}_{q}(\rho)=0,$ Hence we present the proof of condition 1.

As a GME pure state $|\psi\rangle$ can be written as (9), that is,
$|\psi\rangle$ cannot be written as product states with respect to any
bipartition, then we have all the bipartite $\mathcal{C}_{q}$ is bigger than
0, so $\mathcal{G}_{q}(|\psi\rangle)>0.$ And due to the definition of a mixed
GME state $\rho$ and the definition of GqC, we have when $\rho$ is GME,
$\mathcal{G}_{q}(\rho>0),$ then we prove the condition $2.$ $\sqcap$$\sqcup$

A pure multipartite entangled state is called absolutely maximally entangled
state (AMES) if all reduced density operators obtained by tracing out at least
half of the particles of the pure state are maximally mixed helwig2012absolute
. The AMES can be used to develop the quantum secret sharing shemes
helwig2013absolutely  and quantum error correction codes grassl2015quantum ;
alsina2021absolutely . And the GHZ state is the only AMES up to the local
unitary operations in three qubit systems goyeneche2015absolutely ;
shi2021multilinear . Due to the definitions and properties of
$\mathcal{C}_{q}(|\psi\rangle)$, when $|\psi\rangle$ is an AMES, then
$\mathcal{G}_{q}(\cdot)$ gets the maximum, that is, it can also be seen as a
proper GME measure li2022geometric .
Next we consider two pure states in multipartite systems that are inequivalent
in terms of stochastic LOCC (SLOCC), the W states and GHZ states. The authors
in joo2003quantum  showed that in three qubit systems, a perfect teleportation
can be performed via the GHZ state, while the W state cannot. Moreover, for a
$k$-partite W and GHZ state ($k\geq 3$), the infimum asymptotic ratio from GHZ
to W is 1, however, the infimum asymptotic ratio from W to GHZ is bigger than
1 vrana2015asymptotic . Thus the GHZ states can be thought more entangled than
the W states.

###### Example 4

$\displaystyle|W_{n}\rangle=$ $\displaystyle\frac{1}{\sqrt{n}}(|10\cdots
0\rangle+|01\cdots 0\rangle+\cdots+|00\cdots 1\rangle),$
$\displaystyle|GHZ_{n}\rangle=$ $\displaystyle\frac{1}{\sqrt{2}}(|00\cdots
0\rangle+|11\cdots 1\rangle),$

Here we place the results on $\mathcal{G}_{q}(|W_{n}\rangle)$ and
$\mathcal{G}_{q}(|GHZ_{n}\rangle)$ in Sec. V.1. In Fig. 1, we can see the
values of
$\frac{\mathcal{G}_{3}(|W_{n}\rangle)}{\mathcal{G}_{3}(|GHZ_{n}\rangle)}$
tends to 1 with the increase of $n$.

Figure 1: In this figure, we plot the $\mathcal{G}_{3}(|W_{n}\rangle)$ with
the red circle and
$\frac{\mathcal{G}_{3}(|W_{n}\rangle)}{\mathcal{G}_{3}(|GHZ_{n}\rangle)}$ with
the star when $n$ from $5$ to 21.

Next we show the GqC satisfies the subadditivity and continuity for pure
states. First we prove the subadditivity.

Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are pure states in a
bipartite system $\mathcal{H}_{d_{1}}\otimes\mathcal{H}_{d_{2}},$ then we have

$\displaystyle\mathcal{C}_{q}(|\psi_{1}\rangle\otimes|\psi_{2}\rangle)-\mathcal{C}_{q}(|\psi_{1}\rangle)-\mathcal{C}_{q}(|\psi_{2}\rangle)$
$\displaystyle=1-\mathop{\rm Tr}\rho_{A}^{q}\mathop{\rm
Tr}\sigma_{A}^{q}-1+\mathop{\rm Tr}\rho_{A}^{q}-1+\mathop{\rm
Tr}\sigma_{A}^{q}$ $\displaystyle=-(1-\mathop{\rm
Tr}\rho_{A}^{q})(1-\mathop{\rm Tr}\sigma_{A}^{q})\leq 0,$

here we denote $\rho_{A}=tr_{B}|\psi_{1}\rangle_{AB}\langle\psi_{1}|,$
$\sigma_{A}=tr_{B}|\psi_{2}\rangle_{AB}\langle\psi_{2}|$. Due to the above
inequality, we have $\mathcal{C}_{q}(\cdot)$ is subadditivity for pure states
in arbitrary dimensional bipartite systems.

$\displaystyle\mathcal{C}_{q}(|\psi_{1}\rangle\otimes|\psi_{2}\rangle)\leq\mathcal{C}_{q}(|\psi_{1}\rangle)+\mathcal{C}_{q}(|\psi_{2}\rangle).$
(12)

Then we show that $\mathcal{G}_{q}(\cdot)$ satisfies the subadditivity
property for pure states.

Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are two pure states in
$n$-partite systems, then we have

$\displaystyle(\mathcal{G}_{q}(|\psi_{1}\rangle\otimes|\psi_{2}\rangle)^{c(\alpha)}-(\mathcal{G}_{q}(|\psi_{1}\rangle)+\mathcal{G}_{q}(|\psi_{2}\rangle))^{c(\alpha)}$
$\displaystyle=$
$\displaystyle\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{1}\rangle\otimes|\psi_{2}\rangle)-[(\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{1}\rangle))^{\frac{1}{c(\alpha)}}+(\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{2}\rangle))^{\frac{1}{c(\alpha)}}]^{c(\alpha)}$
$\displaystyle\leq$
$\displaystyle\Pi_{\alpha_{i}\in\alpha}(\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{1}\rangle)+\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{2}\rangle))-[(\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{1}\rangle))^{\frac{1}{c(\alpha)}}+(\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{2}\rangle))^{\frac{1}{c(\alpha)}}]^{c(\alpha)}$
$\displaystyle\leq$ $\displaystyle 0,$ (13)

here the first inequality is due to the subadditivity of the bipartite
entanglement measure $(\ref{sub}),$ the second inequality is due to the
Mahler’s inequality. Due to the inequality (13), we have
$\mathcal{G}_{q}(\cdot)$ satisfies the subadditivity for pure states.

At last, we present that the GqC satisfies continuity for pure states. First
we present a lemma on the $\mathcal{C}_{q}(\cdot)$ of pure states.

###### Lemma 5

Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are pure states in
$\mathcal{H}_{d}\otimes\mathcal{H}_{d},$ when
$\lVert|\psi_{1}\rangle-|\psi_{2}\rangle\rVert_{1}\leq\epsilon$, then we have

$\displaystyle|\mathcal{C}_{q}(|\psi_{1}\rangle)-\mathcal{C}_{q}(|\psi_{2}\rangle)|\leq
d[(1+\frac{\epsilon}{d})^{q}-1].$ (14)

Then we can generalize the results to the GqC for the pure states.

###### Theorem 6

Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are two pure states in
$n$-partite systems
$\mathcal{H}_{d}\otimes\mathcal{H}_{d}\otimes\cdots\otimes\mathcal{H}_{d}$,
here $\lVert|\psi_{1}\rangle-|\psi_{2}\rangle\rVert\leq\epsilon,$ then we have

$\displaystyle|\mathcal{G}_{q}(|\psi_{1}\rangle)-\mathcal{G}_{q}(|\psi_{2}\rangle)|\leq[\sum_{i=1}^{\frac{n-1}{2}}C_{n}^{i}d^{i}[(1+\frac{\epsilon}{d^{i}})^{q}-1]]^{\frac{1}{c(\alpha)}}$
(15)

Here we place the proof of the lemma 5 and Theorem 6 in Sec. V.2.

###  III.2 A lower bound of GqC

In this subsection, we first present a lower bound of the entanglement measure
$\mathcal{C}_{q}(\rho_{AB})$ for a bipartite mixed state $\rho_{AB}$, then we
extend the results to the GqC for multipartite mixed states.

###### Theorem 7

For a bipartite mixed state $\rho$ on the system
$\mathcal{H}_{m}\otimes\mathcal{H}_{n}(m\leq n)$, $|\phi\rangle$ is an
arbitrary pure state in $\mathcal{H}_{m}\otimes\mathcal{H}_{n}$ its revised
parametrized entanglement measure $\mathcal{C}_{q}(\rho)$ satisfies

$\displaystyle\mathcal{C}_{q}(\rho)\geq{co[R(\Lambda)]},$ (16)

where $\Lambda=\mathop{\rm
max}\\{\frac{\langle\phi|\rho|\phi\rangle}{s_{1}m},\frac{1}{m}\\}$,
$R(\Lambda)=1-\gamma(\Lambda)^{q}-\frac{[1-\gamma(\Lambda)]^{q}}{(m-1)^{q-1}},$
with $\gamma(\Lambda)=\frac{\sqrt{\Lambda}+\sqrt{(m-1)(1-\Lambda)}}{m}$, and

$co[R(\Lambda)]=\frac{m^{q-1}-1}{m^{q-2}(m-1)}(\Lambda-\frac{1}{m}).$

We place the proof of this theorem in the Appendix V.3.

Before generalizing the bound for $\mathcal{C}_{q}(\rho_{AB})$ of a bipartite
state $\rho_{AB}$ to $\mathcal{G}_{q}(\rho_{A_{1}A_{2}\cdots A_{n}})$ of a
multipartite mixed state $\rho_{A_{1}A_{2}\cdots A_{n}}$, we will denote some
definitions. Let $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ be an arbitrary
$n$-partite pure state in
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\otimes\mathcal{H}_{A_{n}},$
$\alpha_{i}$ be a possible bipartition of $\\{A_{1},A_{2},\cdots,A_{n}\\},$
then by the Schmidt decomposition, $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ can
be written as $|\psi\rangle=U_{\alpha}\otimes
U_{\overline{\alpha}}\sum_{i=1}^{q_{\alpha}}\sqrt{s_{i}^{(\alpha)}}|ii\rangle$
under the bipartition $\alpha|\overline{\alpha},$ here
$\\{\sqrt{s_{i}^{(\alpha)}}\\}$ are the Schmidt coefficients in decreasing
order, $m_{\alpha}$ denotes the number of nonzero Schmidt coefficients. Next
let $s_{1}=\mathop{\rm max}_{\alpha}\\{s_{1}^{\alpha}\\},$ $m=\mathop{\rm
max}_{\alpha}\\{m_{\alpha}\\}.$

###### Theorem 8

Assume $\rho_{A_{1}A_{2}\cdots A_{n}}$ is a mixed state on an $n$-paritite
system. Then we have

$\displaystyle\mathcal{G}_{q}(\rho)\geq\frac{m^{q-1}-1}{m^{q-2}(m-1)}(\Lambda^{{}^{\prime}}-\frac{1}{m}),$
(17)

here we denote $\Lambda^{{}^{\prime}}=\mathop{\rm
max}\\{\frac{\langle\phi|\rho|\phi\rangle}{s_{1}m},\frac{1}{m}\\}$

We place the proof of Theorem 8 in the Sec. V.4. Next we present an example to
show the results.

###### Example 9

Consider a $3$-qubit $W$ state with the white noise,

$\displaystyle\rho_{W}=p|W\rangle\langle W|+\frac{1-p}{8}\mathbb{I},\hskip
8.53581ptp\in(0,1).$ (18)

here we present the bound of $\mathcal{G}_{2}(\cdot)$ for $\rho_{W}$ in Fig.
2. There we plot the lower bound of G$2$C for $\rho_{W}$.

Figure 2: In this figure, the red line denotes the lower bound of G2C for
$\rho_{W}$, here we use $|\phi\rangle=|W\rangle$ and $s_{1}=\frac{2}{3}.$

###### Example 10

Let $\rho_{GHZ}$ be a $3$-qubit mixed state,

$\displaystyle\rho_{GHZ}=$ $\displaystyle c|GHZ\rangle\langle
GHZ|+\frac{1-c}{8}\mathbb{I},$ $\displaystyle|GHZ\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle),\hskip
8.53581ptc\in(0,1).$ (19)

here we present the bound of $\mathcal{G}_{q}(\cdot)$ for $\rho_{W}$ in Fig.
2. There we plot the lower bound of G$q$C for $\rho_{GHZ}$ when $q\in[2,12]$.
From the Fig. 3, we see that the lower bound we present is monotone in terms
of $q$.

Figure 3: In this figure, we plot the lower bound of $G_{q}(\rho_{GHZ})$ when
$q\in[2,12]$, here we use $|\phi\rangle=|GHZ\rangle.$

###  III.3 Comparisons with other GME measures

In this section, we present some examples on the $\mathcal{G}_{q}(\cdot)$ and
compare them with other GME measures, GBC and GGM. Through comparison, we can
get the difference between GqC and other GME measures, also we can get the
advantages of GqC.
Entanglement ordering is meaningful when considering the entanglement measures
virmani2001optimal ; zyczkowski2002relativity . It means that if $E_{1}$ and
$E_{2}$ are two entanglement measures, for any pair of $\sigma_{1}$ and
$\sigma_{2}$, $E_{1}(\sigma_{1})\geq E_{1}(\sigma_{2})$ derives
$E_{2}(\sigma_{1})\geq E_{2}(\sigma_{2}).$ The GqC can lead to different
entanglement ordering when comparing with other GME measures. Next we consider
the following two classes of states in three-qubit systems to show that the
entanglement ordering of G4C is different from GMC and GGM.

###### Example 11

Class I:
$\displaystyle|\psi\rangle=\frac{1}{\sqrt{2}}(\cos\theta|000\rangle+\sin\theta|001\rangle)+\frac{1}{\sqrt{2}}|111\rangle,$
Class II:
$\displaystyle|\phi\rangle=\cos\theta|000\rangle+\sin\theta|111\rangle.$

(a) The G4C and GMC for the Class I states and Class II states.

(b) The G4C and GGM for the Class I states and Class II states.

Figure 4: The G4C, GMC and GGM for the Class I states and Class II states.

From the Fig. 4(a), we see that the G4C owns different entanglement order from
the GMC. For a given state belonging to the Class I state, there are many
states in Class II with larger GMC but with smaller G4C. This can be shown by
drawing vertical or horizontal lines, then compare the states at the
intersection point. Similarly, from the Fig. 4(b), the entanglement order of
G4C is different from GGM. For a given state of the Class I, there are many
states in Class II that with larger GGM but with smaller G4C. The opposite
results can be arrived at when given a Class II pure state.
At last, we present another class of 4-qubit pure states, which shows the
advantages of GqC when comparing with GGM and GMC.

$\displaystyle|\psi\rangle=$
$\displaystyle\cos\theta|\phi_{1}\rangle+\sin\theta|0111\rangle,\hskip
8.53581pt\theta\in(0,\frac{\pi}{2}),$ (20) $\displaystyle|\phi_{1}\rangle=$
$\displaystyle\cos\frac{2\pi}{3}|0100\rangle+\sin\frac{2\pi}{3}|1000\rangle.$

Figure 5: The G3C,GMC and GGM for the four qubit pure states (20).

As presented in Fig.5, when $\theta$ increases from $[\theta_{1},\theta_{3}],$
the values of G3C increases, while the GGM decreases from
$[\theta_{2},\theta_{3}]$ and GMC decreases from $[\theta_{1},\theta_{3}].$
Next when $\theta$ ranges in $[0,\theta_{3}],$ each G3C value corresponds to a
unique state in the class $(\ref{e12}),$ while there exists pairs of states in
the class $(\ref{e12})$ with the same GGM or GBC, that is, the GME measure G3C
can detects the robustness between some states while the GGM and GBC cannot.
Moreover, Fig. 5 show the smoothness of G3C, a sharp peak appears with the
varying $\theta$ when considering the GMC and GGM.

##  IV Conclusion

In this paper, we have proposed and investigated a GME measure based on the
geometric mean method. First we have presented the GqC is a GME measure and
satisfies the subadditivity and continuity for pure states. We have also
presented the analytical expressions of GqC for the W states and GHZ states in
$n$-qubit systems. From the analytical expressions, we can see that the
entanglement of the GHZ state is stronger than the W state. Next we have
presented a lower bound of the GqC based on the method proposed in
dai2020experimentally . At last, we have presented the advantages of the GqC
by comparing with the GMC and GGM through some examples. Due to the importance
of the study of GME measures, our results can provide a reference for future
work on the study of multiparty quantum entanglement.

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##  V Appendix

###  V.1 Some results on $\mathcal{G}_{q}(|W_{n}\rangle)$ and
$\mathcal{G}_{q}(|GHZ_{n}\rangle)$

An $n$-qubit W state can be represented as

$\displaystyle|W_{n}\rangle=\frac{1}{\sqrt{n}}(|00\cdots 1\rangle+|01\cdots
0\rangle+\cdots+|00\cdots 1\rangle).$

Through computation, we have

$\displaystyle\mathcal{C}_{q1,n-1}(|W_{n}\rangle)=$ $\displaystyle
1-\frac{1}{n^{q}}-\frac{(n-1)^{q}}{n^{q}},$
$\displaystyle\mathcal{C}_{q,2,n-2}(|W_{n}\rangle)=$ $\displaystyle
1-\frac{2^{q}}{n^{q}}-\frac{(n-2)^{q}}{n^{q}},$ $\displaystyle\cdots,$
$\displaystyle\hskip 5.69054pt\cdots,\hskip 5.69054pt\cdots$
$\displaystyle\mathcal{C}_{q,k,n-k}(|W_{n}\rangle)=$ $\displaystyle
1-\frac{k^{q}}{n^{q}}-\frac{(n-k)^{q}}{n^{q}},$

An $n$-qubit GHZ state can be represented as

$\displaystyle|GHZ_{n}\rangle=\frac{1}{\sqrt{2}}(|00\cdots 0\rangle+|11\cdots
1\rangle).$

Through computation, we have

$\displaystyle\mathcal{C}_{q,k,n-k}(|GHZ_{n}\rangle)=1-\frac{1}{2^{q-1}},$

then we have

$\displaystyle\mathcal{G}_{q}(|W_{n}\rangle)=\begin{split}\left\\{\begin{array}[]{lr}(\Pi_{k=1}^{\frac{n-1}{2}}[1-\frac{k^{q}}{n^{q}}-\frac{(n-k)^{q}}{n^{q}}]^{C_{n}^{k}})^{\frac{1}{c(\alpha)}},\hskip
8.53581pt\textit{if $n$ is odd,}\\\
(\Pi_{k=1}^{\frac{n}{2}-1}[1-\frac{k^{q}}{n^{q}}-\frac{(n-k)^{q}}{n^{q}}]^{C_{n}^{k}}[1-\frac{1}{2^{q-1}}]^{\frac{C_{n}^{\frac{n}{2}}}{2}})^{\frac{1}{c(\alpha)}},\hskip
8.53581pt\textit{if $n$ is even.}\end{array}\right.\end{split}$
$\displaystyle\frac{\mathcal{G}_{q}(|W_{n}\rangle)}{\mathcal{G}_{q}(|GHZ_{n}\rangle)}=\begin{split}\left\\{\begin{array}[]{lr}\frac{\exp[\frac{\sum_{k=1}^{\frac{n-1}{2}}C_{n}^{k}}{c(\alpha)}(\ln(1-\frac{k^{q}}{n^{q}}-\frac{(n-k)^{q}}{n^{q}}))]}{1-\frac{1}{2^{q-1}}},\hskip
8.53581pt\textit{if $n$ is odd,}\\\
\frac{\exp[\frac{\sum_{k=1}^{\frac{n}{2}-1}C_{n}^{k}}{c(\alpha)}\ln(1-\frac{k^{q}}{n^{q}}-\frac{(n-k)^{q}}{n^{q}})+\frac{\frac{C_{n}^{\frac{n}{2}}}{2}\ln(1-\frac{1}{2^{q-1}})}{c(\alpha)}]}{1-\frac{1}{2^{q-1}}},\hskip
8.53581pt\textit{if $n$ is even.}\end{array}\right.\end{split}$

Then by using the Stolz-Cresaro theorem, we have

$\lim_{k\rightarrow\infty}\frac{\mathcal{G}_{q}(|W_{2k}\rangle)}{\mathcal{G}_{q}(|GHZ_{2k}\rangle)}=1,$
$\lim_{k\rightarrow\infty}\frac{\mathcal{G}_{q}(|W_{2k+1}\rangle)}{\mathcal{G}_{q}(|GHZ_{2k+1}\rangle)}=1.$

###  V.2 The proof on the continuity of $\mathcal{G}_{q}(\cdot)$ for pure
states

Here we present the proof of Lemma 5.

Lemma 5: Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are pure states in
$\mathcal{H}_{d}\otimes\mathcal{H}_{d},$ when
$\lVert|\psi_{1}\rangle-|\psi_{2}\rangle\rVert_{1}\leq\epsilon$, then we have

$\displaystyle|\mathcal{C}_{q}(|\psi_{1}\rangle)-\mathcal{C}_{q}(|\psi_{2}\rangle)|\leq
d[(1+\frac{\epsilon}{d})^{q}-1].$ (21)

###### Proof.

As partial trace is trace-preserving, then
$\lVert\rho_{A}-\sigma_{A}\rVert_{1}\leq\epsilon,$ here
$\rho_{A}=Tr_{B}|\psi\rangle\langle\psi|$,
$\sigma_{A}=Tr_{B}|\sigma\rangle\langle\sigma|.$

Next as $\lVert\cdot\rVert_{1}$ is unitarily invariant norm, then

$\displaystyle\lVert
Eig^{\downarrow}(\rho_{A})-Eig^{\downarrow}(\sigma_{A})\rVert\leq\lVert\rho_{A}-\sigma_{A}\rVert_{1},$
$\displaystyle\lVert\rho_{A}-\sigma_{A}\rVert_{1}\leq\lVert
Eig^{\downarrow}(\rho_{A})-Eig^{\uparrow}(\sigma_{A})\rVert,$

that is, we only need to consider the classical case. Readers who are
interesting in the above two inequalities please refer to bhatia2013matrix .

Assume $\rho_{A}$ and $\sigma_{A}$ are two diagonal density matrices with
their diagonal elements are $\\{p_{i}\\}_{i=1}^{d}$ and
$\\{r_{i}\\}_{i=1}^{d},$ respectively. And $\\{p_{i}\\}_{i=1}^{d}$ and
$\\{r_{i}\\}_{i=1}^{d}$ satisfy $\sum_{i}p_{i}=\sum_{i}r_{i}=1,$ and
$\sum_{i}|p_{i}-r_{i}|\leq\epsilon.$ Then

$\displaystyle|\mathcal{C}_{q}(\rho_{A})-\mathcal{C}_{q}(\sigma_{A})|=$
$\displaystyle|\mathop{\rm Tr}\rho_{A}^{q}-\mathop{\rm Tr}\sigma_{A}^{q}|$
$\displaystyle\leq$ $\displaystyle\sum_{i}|p_{i}^{q}-r^{q}_{i}|$ (22)

Next let $|p_{i}-r_{i}|=\epsilon_{i}$, $\sum_{i}\epsilon_{i}\leq\epsilon,$
when $p_{i}=r_{i}+\epsilon_{i},$ then

$\displaystyle|p_{i}^{q}-r_{i}^{q}|=$
$\displaystyle|(r_{i}+\epsilon_{i})^{q}-r_{i}^{q}|$ $\displaystyle\leq$
$\displaystyle(1+\epsilon_{i})^{q}-1,$

the last inequality is due to that $(r_{i}+\epsilon_{i})^{q}-r_{i}^{q}$ is
increasing in terms of $r_{i}$, and $r_{i}\in(0,1).$ And due to that
$(r_{i}+\epsilon_{i})^{q}-r_{i}^{q}$ is increasing in terms of $r_{i}$, when
$p_{i}=r_{i}-\epsilon_{i}$, the above inequality is also valid. Then the
inequality $(\ref{in1})$ becomes

$\displaystyle(\ref{in1})=\sum_{i}|p_{i}^{q}-r^{q}_{i}|\leq$
$\displaystyle\epsilon^{q}\frac{\sum_{i}[(1+\epsilon_{i})^{q}-1]}{\epsilon^{q}}$
$\displaystyle\leq$ $\displaystyle d[(1+\frac{\epsilon}{d})^{q}-1],$ (23)

when all $\epsilon_{i}=\frac{\epsilon}{d}$, the equality in the last
inequality is valid.  $\sqcap$$\sqcup$

Theorem 6: Assume $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are two pure
states in $n$-partite systems
$\mathcal{H}_{d}\otimes\mathcal{H}_{d}\otimes\cdots\otimes\mathcal{H}_{d}$,
here $\lVert|\psi_{1}\rangle-|\psi_{2}\rangle\rVert\leq\epsilon,$ then we have

$\displaystyle|\mathcal{G}_{q}(|\psi_{1}\rangle)-\mathcal{G}_{q}(|\psi_{2}\rangle)|\leq[\sum_{i=1}^{\frac{n-1}{2}}C_{n}^{i}d^{i}[(1+\frac{\epsilon}{d^{i}})^{q}-1]]^{\frac{1}{c(\alpha)}}$
(24)

###### Proof.

When $n$ is odd, then we have

$\displaystyle|\mathcal{G}_{q}(|\psi_{1}\rangle)-\mathcal{G}_{q}(|\psi_{2}\rangle)|$
$\displaystyle=$
$\displaystyle|\mathcal{P}_{q}(|\psi_{1}\rangle)^{\frac{1}{c(\alpha)}}-\mathcal{P}_{q}(|\psi_{2}\rangle)^{\frac{1}{c(\alpha)}}|$
$\displaystyle\leq$
$\displaystyle|\mathcal{P}_{q}(|\psi_{1}\rangle)-\mathcal{P}_{q}(|\psi_{2}\rangle)|^{\frac{1}{c(\alpha)}}$
$\displaystyle=$
$\displaystyle|\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{1}\rangle)-\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\psi_{2}\rangle)|^{\frac{1}{c(\alpha)}}$
$\displaystyle\leq$
$\displaystyle[\sum_{i=1}^{\frac{n-1}{2}}C_{n}^{i}d^{i}[(1+\frac{\epsilon}{d^{i}})^{q}-1]]^{\frac{1}{c(\alpha)}}.$
(25)

the first inequality is due to that when $p,q,x\in(0,1)$,
$|p^{x}-q^{x}|\leq|p-q|^{x}$, the second inequality is due to the following
inequality, when $x_{i},y_{i}\in(0,1),$ $i=1,2,\cdots,n,$ then we have

$\displaystyle|x_{1}x_{2}\cdots x_{n}-y_{1}y_{2}\cdots y_{n}|$
$\displaystyle=$ $\displaystyle|(x_{1}-y_{1})x_{2}x_{3}\cdots
x_{n}+y_{1}(x_{2}-y_{2})x_{3}\cdots x_{n}$ $\displaystyle\hskip 2.84526pt+$
$\displaystyle y_{1}y_{2}(x_{3}-y_{3})x_{4}x_{5}\cdots
x_{n}\cdots+y_{1}y_{2}\cdots y_{n-1}(x_{n}-y_{n})|$ $\displaystyle\leq$
$\displaystyle\sum_{i}|x_{i}-y_{i}|,$ (26)

here the last inequality is due to the triangle inequality and
$x_{i},y_{i}\in(0,1).$ $\sqcap$$\sqcup$

###  V.3 The proof of Theorem 7

Here we prove Theorem 7 based on the method in zhang2016evaluation .

Theorem 7: _For a bipartite mixed state $\rho$ on the system
$\mathcal{H}_{m}\otimes\mathcal{H}_{n}(m\leq n)$, its revised parametrized
entanglement measure $\mathcal{C}_{q}(\rho)$ satisfies_

$\displaystyle\mathcal{C}_{q}(\rho)\geq co[R(\Lambda)],$ (27)

_where $\Lambda=\mathop{\rm
max}\\{\frac{\langle\phi|\rho|\phi\rangle}{s_{1}m},\frac{1}{m}\\}$,
$R(\Lambda)=1-\gamma(\Lambda)^{q}-\frac{[1-\gamma(\Lambda)]^{q}}{(m-1)^{q-1}},$
with $\gamma(\Lambda)=\frac{\sqrt{\Lambda}+\sqrt{(m-1)(1-\Lambda)}}{m}$, and_

$co[R(\Lambda)]=\frac{m^{q-1}-1}{m^{q-2}(m-1)}(\Lambda-\frac{1}{m}).$

###### Proof.

Here we consider the following function

$\displaystyle R(\lambda)=\mathop{\rm
min}_{\vec{\mu}}\\{L(\vec{\mu})|\lambda=\frac{1}{m}(\sum_{i}\sqrt{\mu_{i}})^{2}\\}.$
(28)

Here $L(\vec{\mu})=1-\sum_{i}\mu_{i}^{2}$. Due to the results in
terhal2000entanglement ; berry2003bounds , the minimum $L(\vec{\mu})$ versus
$\lambda$ to $\vec{\mu}$ in the form

$\displaystyle\vec{\mu}=\\{t,\frac{1-t}{m-1},\frac{1-t}{m-1},\cdots,\frac{1-t}{m-1}\\}\hskip
8.53581pt\textit{for $t\in[\frac{1}{m},1],$}$ (29)

Therefore, we have the minimum $L(\vec{\mu})$ and the function $t(\lambda)$
are

$\displaystyle L(t)=$ $\displaystyle 1-t^{q}-\frac{(1-t)^{q}}{(m-1)^{q-1}},$
(30) $\displaystyle t(\lambda)=$
$\displaystyle\frac{1}{m}(\sqrt{\lambda}+\sqrt{(m-1)(1-\lambda)})^{2},\hskip
8.53581pt\textit{with $\lambda\in[\frac{1}{m},1].$}$ (31)

Substituting $(\ref{t1})$ into (30), we have

$\displaystyle L^{{}^{\prime}}_{\lambda}=L^{{}^{\prime}}_{t}\times
t^{{}^{\prime}}_{\lambda},$ (32)

through computation, we have

$\displaystyle
L^{{}^{\prime}}_{t}=-qt^{q-1}+\frac{q(1-t)^{q-1}}{(m-1)^{q-1}},$ (33)
$\displaystyle
L_{t}^{{}^{\prime\prime}}=-q(q-1)t^{q-2}-\frac{q(q-1)(1-t)^{q-2}}{(m-1)^{q-1}}\leq
0,$ (34) $\displaystyle
t_{\lambda}^{{}^{\prime}}=\frac{1}{m}(\sqrt{\lambda}+\sqrt{(m-1)(1-\lambda)})$
$\displaystyle\hskip
56.9055pt\times(\frac{1}{\sqrt{\lambda}}+\frac{1-m}{\sqrt{(m-1)(1-\lambda)}})\leq
0,$ (35)

as $L_{t}^{{}^{\prime\prime}}\leq 0,$ $\mathop{\rm
max}_{t}L^{{}^{\prime}}_{t}=L^{{}^{\prime}}(\frac{1}{m})=0,$ that is
$L_{\lambda}^{{}^{\prime}}\geq 0,$ $L(\lambda)$ is an increasing function.

Next we prove $L(\lambda)$ is concave.

$\displaystyle
L_{\lambda}^{{}^{\prime\prime}}=L_{t}^{{}^{\prime\prime}}{t_{\lambda}^{{}^{\prime}}}^{2}+L^{{}^{\prime}}_{t}{t_{\lambda}^{{}^{\prime\prime}}}$
(36)

and

$\displaystyle
t_{\lambda}^{{}^{\prime\prime}}=-\frac{\sqrt{(m-1)(1-\lambda)}}{2m(1-\lambda)^{2}\lambda^{\frac{3}{2}}}.$
(37)

Through rough computation,

$\displaystyle L_{\lambda}^{{}^{\prime\prime}}\leq 0,$

then we have

$\displaystyle
co[R(\lambda)]=\frac{m^{q}-1}{m^{q-1}(m-1)}(\lambda-\frac{1}{m}).$ (38)

Next we have

$\displaystyle\mathcal{C}_{q}(\rho)=$
$\displaystyle\sum_{i}p_{i}\mathcal{C}_{q}(|\psi_{i}\rangle)=\sum_{i}p_{i}L(\vec{\mu^{j}})$
$\displaystyle\geq$ $\displaystyle\sum_{i}p_{i}co[R(\lambda^{j})]\geq
co[R(\sum_{j}p_{j}\lambda^{j})]\geq co[R(\Lambda)],$ (39)

where $|\psi_{j}\rangle=U_{A}\otimes
U_{B}\sum_{i}\sqrt{\mu_{i}^{j}}|ii\rangle$ with $\sqrt{\mu_{i}^{j}}$ being its
Schmidt coefficients in decreaing order. The last inequality is due to that
$co{R(\lambda)}$ is an increasing function.  $\sqcap$$\sqcup$

###  V.4 The proof of Theorem 8

Here we present the proof of Theorem 8.

Theorem 8: Assume $\rho_{A_{1}A_{2}\cdots A_{n}}$ is a mixed state on an
$n$-paritite system. Then we have

$\displaystyle\mathcal{G}_{q}(\rho)\geq\frac{m^{q-1}-1}{m^{q-2}(m-1)}(\Lambda^{{}^{\prime}}-\frac{1}{m}),$
(40)

here we denote $\Lambda^{{}^{\prime}}=\mathop{\rm
max}\\{\frac{\langle\phi|\rho|\phi\rangle}{s_{1}m},\frac{1}{m}\\}$

###### Proof.

Assume $\\{p_{j},|\phi_{j}\rangle\\}$ is the optimal decompostion of the state
$\rho$ in terms of $\mathcal{G}_{q}(\rho),$ then we have

$\displaystyle\mathcal{G}_{q}(\rho)=$
$\displaystyle\sum_{j}p_{j}[\Pi_{\alpha_{i}\in\alpha}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\phi_{j}\rangle)]^{\frac{1}{c(\alpha)}}$
$\displaystyle=$
$\displaystyle\sum_{j}[\Pi_{\alpha_{i}\in\alpha}p_{j}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\phi_{j}\rangle)]^{\frac{1}{c(\alpha)}}$
$\displaystyle\geq$ $\displaystyle\sum_{j}\mathop{\rm
min}_{\alpha_{i}\in\alpha}p_{j}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\phi_{j}\rangle)$
$\displaystyle\geq$ $\displaystyle\mathop{\rm
min}_{\alpha_{i}\in\alpha}\sum_{j}p_{j}\mathcal{C}_{qA_{\alpha_{i}}B_{\alpha_{i}}}(|\phi_{j}\rangle)$
$\displaystyle\geq$
$\displaystyle\frac{m^{q-1}-1}{m^{q-2}(m-1)}(\Lambda^{{}^{\prime}}-\frac{1}{m}),$
(41)

here we denote $\Lambda^{{}^{\prime}}=\mathop{\rm
max}\\{\frac{\langle\phi|\rho|\phi\rangle}{s_{1}m},\frac{1}{m}\\}.$ The first
inequality is due to that $\mathop{\rm min}(x,y)\leq\sqrt{xy},$ $\forall
x,y\in[0,1].$ The last inequality is due to the Lemma 5.  $\sqcap$$\sqcup$