# Transforming Radical Differential Equations to Algebraic Differential
Equations

Sebastian Falkensteiner  Research Institute for Symbolic Computation (RISC),
Johannes Kepler University Linz, Austria<EMAIL_ADDRESS>and
J.Rafael Sendra  Universidad de Alcalá, Dpto. Física y Matemáticas, Alcalá de
Henares, Madrid, Spain<EMAIL_ADDRESS>
###### Abstract.

In this paper we present an algorithmic procedure that transforms, if
possible, a given system of ordinary or partial differential equations with
radical dependencies in the unknown function and its derivatives into a system
with polynomial relations among them by means of a rational change of
variables. The solutions of the given equation and its transformation
correspond one-to-one. This work can be seen as a generalization of previous
work on reparametrization of ODEs and PDEs with radical coefficients.

###### Key words and phrases:

Algebraic ordinary differential equation, algebraic partial differential
equation, radical differential equation, rational reparametrization, radical
parametrizations.

##  1\. Introduction

Some differential equations appear to be easier solvable than others. A common
method for finding solutions is to first transform the given differential
equation into another one where solution methods are known, find its
solutions, and transform them back accordingly. A class of differential
equations which are commonly treated is that of algebraic differential
equations (shortly ADEs). They are polynomials in the unknowns and their
derivatives, and the coefficients of the polynomial are rational in the
independent variables, see e.g. [10]. For the sake of simplicity we mainly
focus on ordinary differential equations, but also give insight into the
partial case.

In [6], the authors focus on AODEs whose coefficients involve radical
expressions of the independent variable $x$, and they study the possible
transformation of the equation into another with only rational expressions of
$x$. Moreover, they show how to relate the solutions of both equations. For
instance, the method in [6] transforms

$\sqrt{x}\,y^{\prime 3}y^{\prime\prime}+y+x=0$

into

$z\,Y^{\prime 3}Y^{\prime\prime}-Y^{\prime 4}+8z^{3}\,Y+8z^{5}=0$

by using the change $r(z)=z^{2},y(z^{2})=Y(z)$. In this paper, we take an step
forward and we analyze differential equations where the unknown function and
its derivatives can appear as radical expressions as for instance

$\sqrt{y^{\prime 2}+y^{\prime}}+x\,\sqrt{y}+y=x.$

Our contribution is a method to find, if they exist, rational changes of
variables that convert systems of ODEs and PDEs of this type into systems of
algebraic differential equations. Throughout this paper, when we speak about
algebraic differential equations, we mean the equality of fractions of
differential polynomials.

Similarly as in [6], the underlying idea is to associate to the differential
system a radical parametrization such that the possible change of variable can
be derived from the tower variety; we refer to [16, 17, 18] for further
details on radical varieties.

Combining the results in this paper with that in [6], we can analyze mixed
types of equations where radical expressions occur in the differential
indeterminates and, separately, in the differential determinates. An example
of a single equation of this type is the Thomas-Fermi equation. There have
been several approaches in order to transform the equation and approximate its
solutions, see for instance [4, 21]. By our method, symbolic solutions of the
transformed equations can be computed by well-known techniques for AODEs such
as the Newton polygon method [5].

The problem that we study is the following. Define the radical tower

(1.1)
$\mathbb{F}_{0}=\mathbb{C}(z_{1},\ldots,z_{n})\subseteq\mathbb{F}_{1}\subseteq\cdots\subseteq\mathbb{F}_{m},$

where
$\mathbb{F}_{i}=\mathbb{F}_{i−1}(\delta_{i})=\mathbb{F}_{0}(\delta_{1},\ldots,\delta_{i})$
with $\delta_{i}^{e_{i}}=\alpha_{i}\in\mathbb{F}_{i−1},e_{i}\in\mathbb{N}$. In
other words, $\mathbb{F}_{m}$ is the field of functions built using several
(possibly nested) radicals. Consider the system of differential equations

(1.2)
$\mathcal{F}=\\{F_{1}(x,y_{1},\ldots,y_{\ell}^{(n_{\ell})})=0,\ldots,F_{N}(x,y_{1},\ldots,y_{\ell}^{(n_{\ell})})=0\\}$

with $n=n_{1}+\cdots+n_{\ell}+\ell$,
$F_{i}(x,z_{1},\ldots,z_{n})\in\mathbb{F}_{m}(x)$. In other words, the
independent variable $x$ appears polynomially and the dependent variables
$y_{1},\ldots,y_{\ell}$ and their derivatives possibly appear inside radicals.
We may assume that $F_{i}$ do not have any denominators. Otherwise we consider
their numerators $\operatorname{num}(F_{i})$ and we assume that the solutions
are not zeros of the denominators. Our goal is to find, if it exists, a
rational change of variables of the form

(1.3) $z_{1}=r_{1}(w_{1},\ldots,w_{n}),\ldots,z_{n}=r_{n}(w_{1},\ldots,w_{n})$

such that the new equations

(1.4)
$G_{i}(x,w_{1},\ldots,w_{n}):=F_{i}(x,r_{1}(w_{1},\ldots,w_{n}),\ldots,r_{n}(w_{1},\ldots,w_{n}))=0$

have rational expressions in $w_{1},\ldots,w_{n}$ and polynomials in $x$ as
coefficients. The denominator of $G_{i}$ is a multiple of the denominators of
$r_{1},\ldots,r_{n}$. When speaking about a solution of $G_{i}=0$, we again
consider

$\operatorname{num}(G_{i})(x,w_{1},\ldots,w_{n})=0$

and we assume that the denominator of $G_{i}$ is non-zero.

The paper is structured as follows. The next section describes the algebraic
preliminaries on radical varieties that will be used later. Section 3
describes the results on systems of ordinary differential equations. Section 4
deals with special cases of the given system. In these situations, the
procedure presented in Section 5 might get algorithmic. We underline this by
giving some examples. Section 6 treats the case of systems of partial
differential equations.

##  2\. Preliminaries on Radical Varieties

In this section we recall some results on radical varieties; for further
details see [18]. For the basic notions and results on algebraic geometry,
which are fundamental here, see for example [7].

We will express tuples of variables or functions by using bars such as
$\bar{z}=(z_{1},\ldots,z_{n})$. Let $\mathbb{F}_{m}$ and $\delta_{i}$ be
defined as in (1.1). A radical parametrization of (1.1) is a tuple

$\mathcal{P}=(P_{1}(\bar{z}),\ldots,P_{k}(\bar{z}))$

of elements of $\mathbb{F}_{m}$ whose Jacobian has rank $n$. For a suitable
election of branches we obtain a (usually non-rational) function with domain
in $\mathbb{C}^{n}$ and whose image is Zariski dense in an $n$-dimensional
variety. The Zariski closure of the image $\mathrm{Im}(\mathcal{P})$ is the
(irreducible) radical variety determined by $\mathcal{P}$ and will be denoted
by $\mathbb{V}(\mathcal{P})$ [18, Theorem 3.11 (iii)].

The function $\mathcal{P}$ can be factored as $R\circ\psi$, where
$\psi(\bar{w})=(\bar{w},\delta(\bar{w}))$, $\bar{\delta}$ are the radicals in
the construction of the tower (1.1), and $R$ is a rational function. Another
relevant algebraic variety in our construction is the tower variety of
$\mathcal{P}$, defined as the Zariski closure of $\mathrm{Im}(\psi)$, denoted
by $\mathbb{V}(\mathbb{T})$. It is also irreducible of dimension $n$.
Therefore, the map
$R:\mathbb{V}(\mathbb{T})\dashrightarrow\mathbb{V}(\mathcal{P})$ is rational.
The tower variety contains useful information about $\mathcal{P}$ and its
properties are essential in this work. One more ingredient is an incidence
variety, the Zariski closure of the map
$\bar{z}\mapsto(\bar{z},\bar{\delta},\bar{q})$, where
$\bar{q}=(q_{1},\ldots,q_{k})$ represents the coordinates of the points in
$\mathbb{V}(\mathcal{P})$, that we denote as $\mathcal{B}(\mathcal{P})$. This
provides the projections
$\pi:\mathcal{B}(\mathcal{P})\dashrightarrow\mathbb{V}(\mathcal{P})$ and
$\pi^{*}:\mathcal{B}(\mathcal{P})\dashrightarrow\mathbb{V}(\mathbb{T})$. In
[18], generators of $\mathcal{B}(\mathcal{P})$ are given and hence, using the
Closure theorem (see [7]), one may compute generators of the varieties
$\mathbb{V}(\mathcal{P})$ and $\mathbb{V}(\mathbb{T})$. We define the tracing
index of $\mathcal{P}$ as the degree of the map $\pi$, that is the cardinality
of the generic fiber of $\pi$. The calculation of the tracing index is
described in [18]. The case where the tracing index is equal to one is of
particular interest, because in this case $\mathbb{V}(\mathcal{P})$ and
$\mathbb{V}(\mathbb{T})$ are birationally equivalent [18, Theorem 4.11].

##  3\. Differential Equations

In this section, we will use some of the notation introduced previously. In
particular, $\mathbb{F}_{m}$ is the last field of a radical tower over
$\mathbb{C}(z_{1},\ldots,z_{n})$, introduced in (1.1), and
$\bar{\delta}=(\delta_{1},\ldots,\delta_{m})$ is the tuple of algebraic
elements used in the construction of the tower. We also consider the system of
differential equations introduced in (1.2). After the change of variables

$\bar{z}=\bar{r}(\bar{w})=(\bar{r}_{1}(\bar{w}),\ldots,\bar{r}_{\ell}(\bar{w})),$

where

$\bar{r}_{i}=(r_{i,0}(\bar{w}),\ldots,r_{i,n_{i}}(\bar{w})),$

we consider $G_{i}(x,\bar{w})=F_{i}(x,\bar{r}(\bar{w}))$ as in (1.4). By a
change of variable we allow mappings $\bar{r}(\bar{w})$ such that its Jacobian
is generically non-singular, so that, via the Inverse Function Theorem, the
transformation can be inversed. Recall that $G_{i}=0$ is an algebraic equation
depending on $x,\bar{w}$ and no derivatives and involves fractional
expressions in $\bar{w}$. The differential relations among the variables are
kept by additionally imposing, for $1\leq i\leq\ell,0\leq j<n_{i}$, the
equations

$r_{i,j+1}(\bar{w})=J_{r_{i,j}}(\bar{w})\cdot(\bar{w}^{\prime})^{T}=J_{r_{i,j}}(w_{1},\ldots,w_{n})\cdot(w_{1}^{\prime},\ldots,w_{n}^{\prime})^{T}$

where $J_{r_{i,j}}$ denotes the usual Jacobian of $r_{i,j}$. Then we obtain
the differential system

(3.1)
$\mathcal{G}=\begin{cases}G_{1}(x,\bar{w})=0,\ldots,G_{N}(x,\bar{w})=0\\\
r_{i,j+1}(\bar{w})=J_{r_{i,j}}(\bar{w})\cdot(\bar{w}^{\prime})^{T},\ \ 1\leq
i\leq\ell,0\leq j<n_{i}\end{cases}$

######  Remark 3.1.

The algebraic equations $G_{i}(x,\bar{w})=0$, and the differential equations
$r_{j+1}(\bar{w})=J_{r_{j}}(\bar{w})\cdot(\bar{w}^{\prime})^{T}$ in (3.1)
might have denominators. In this case, all denominators depend only on
$\bar{w}$ and are independent of $x$. Moreover, the denominator of $G_{i}$
arises from the change of variables in the $F_{i}$ and hence, it is a multiple
of the denominators of the $r_{i,j}$. Consequently, for a given function
$\bar{w}(x)$, if $r_{j}(\bar{w}(x))$ is defined for every $0\leq j\leq n$,
then the denominator of $G_{i}(x,\bar{w}(x))$ is non-zero as well.

######  Remark 3.2.

After possibly cleaning denominators in (3.1), the system $\mathcal{G}$ might
be simplified by various differential elimination methods such as the Thomas
decomposition or differential regular chains. For a description of these two
methods see [11, 3].

The next theorem states sufficient conditions of $\bar{r}(\bar{w})$ such that
(3.1) is a system of AODEs.

######  Theorem 3.3.

Let $\mathcal{F}\subset\mathbb{F}_{m}(x)$ be as in (1.2), let
$\bar{r}(\bar{w})\in\mathbb{C}(\bar{w})^{n}$ be a rational change of
variables, and let $\bar{H}(\bar{z})\in\mathbb{F}_{m}^{\rho}$ be the tuple of
non-rational (function) coefficients of the $F(x,\bar{z})\in\mathcal{F}$
w.r.t. $x$, taken in any order. The following are equivalent.

  1. (1)

(3.1) is a system of AODEs.

  2. (2)

All the components of $\bar{H}(\bar{r}(\bar{w}))$ are rational.

###### Proof.

“(1)$\implies$(2)”: Since (3.1) is a system of AODEs, in particular, the
$G_{i}(x,\bar{w})$ are algebraic. Thus $F_{i}(x,\bar{r}(\bar{w}))=0$ is an
algebraic equation for every $F_{i}\in\mathcal{F}$. So its coefficients, and
in particular $\bar{H}(\bar{r}(\bar{w}))$, are rational functions.

“(2)$\implies$(1)”: After the substitution $\bar{z}=\bar{r}(\bar{w})$, every
$F_{i}(x,\bar{r}(\bar{w}))$ has coefficients which are rational in $\bar{w}$
and polynomial in $x$, so $G_{i}(x,\bar{w})$ is algebraic. Moreover, since
$\bar{r}$ is a tuple of rational functions, the remaining equations of (3.1)
are AODEs. ∎

In the sequel we associate to the differential system (1.2) a radical
parametrization, namely

(3.2) $\mathcal{P}=(\bar{z},\bar{H}(\bar{z}))$

with $\bar{H}$ as in Theorem 3.3, and we consider its associated radical
variety $\mathbb{V}(\mathcal{P})$ as well as its tower variety
$\mathbb{V}(\mathbb{T})$, see Section 2; observe that, as required, the rank
of the Jacobian of $\mathcal{P}$ is $n$.

######  Lemma 3.4.

Let $\mathbb{V}(\mathbb{T})$ and $\mathbb{V}(\mathcal{P})$ be as above. Then
the following statements hold.

  1. (1)

$\dim(\mathbb{V}(\mathbb{T}))=\dim(\mathbb{V}(\mathcal{P}))=n$.

  2. (2)

If $\mathbb{V}(\mathbb{T})$ is unirational, then $\mathbb{V}(\mathcal{P})$ is
unirational. Moreover, if the tracing index of $\mathcal{P}$ is one and
$\mathbb{V}(\mathbb{T})$ is rational, then $\mathbb{V}(\mathcal{P})$ is
rational.

###### Proof.

By Theorem 3.11(iii) and Theorem 4.6 in [18], the dimension of both,
$\mathbb{V}(\mathbb{T})$ and $\mathbb{V}(\mathcal{P})$, is equal to $n$. The
second item follows from Theorem 4.11 in the same reference. ∎

In the case where $\mathbb{V}(\mathbb{T})$ is a unirational curve or surface,
by Lemma 3.4, $\mathbb{V}(\mathcal{P})$ is unirational and, by Lüroth’s
Theorem for curves and surfaces, $\mathbb{V}(\mathbb{T})$ and
$\mathbb{V}(\mathcal{P})$ are rational. Indeed, there exist algorithmic
approaches to compute such birational parametrizations [20, 13]. In Lemma 3.4
we have shown that the dimension of the tower variety $\mathbb{V}(\mathbb{T})$
and $\mathbb{V}(\mathcal{P})$ is equal to $n=n_{1}+\cdots+n_{\ell}+\ell$,
where $\ell$ is the number of differential determinates and $n_{i}$ is the
order with respect to $y_{i}$, but as we will show in Section 5, in various
situations the dimension can be reduced and the case where these varieties are
curves or surfaces might occur.

######  Theorem 3.5.

Let $\mathcal{F}$ be as in (1.2), $\mathcal{G}$ be as in (3.1) and let
$\mathcal{Q}(\bar{w})=(\bar{r}(\bar{w}),\bar{\delta}(\bar{r}(\bar{w})))$ be a
rational parametrization of $\mathbb{V}(\mathbb{T})$. It holds that

  1. (1)

For every solution $\bar{w}(x)$ of $\mathcal{G}$,

$(y_{1}(x),\ldots,y_{\ell}(x)):=(r_{1,0}(\bar{w}(x)),\ldots,r_{\ell,0}(\bar{w}(x)))$

is a solution of the differential system $\mathcal{F}$.

  2. (2)

If $\mathcal{Q}$ is invertible, $(y_{1}(x),\ldots,y_{\ell}(x))$ is a solution
of $\mathcal{F}$,

$\bar{w}(x):=\mathcal{Q}^{-1}(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x),\bar{\delta}(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x)))$

and $\mathcal{Q}(\bar{w}(x))$ are defined, then $\bar{w}(x)$ is a solution of
$\mathcal{G}$.

###### Proof.

Using the differential equations in $\mathcal{G}$, one gets for $1\leq
i\leq\ell,1\leq j<n_{i}$ that $r_{i,j}(\bar{w}(x))$ is the $j$-th derivative
of $r_{i,0}(\bar{w}(x))$. So,
$\bar{r}(\bar{w}(x))=(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x))$. Moreover,
for every $1\leq k\leq N$, since $\operatorname{num}(G_{k})(x,\bar{w}(x))=0$,
and, by Remark 3.1, the denominator of $G_{k}$ specialized at $(x,\bar{w}(x))$
is non-zero, so

$F_{k}(x,\bar{r}(\bar{w}(x)))=G_{k}(x,\bar{w}(x))=0.$

Conversely, let $(y_{1}(x),\ldots,y_{\ell}(x))$ be a solution of
$\mathcal{F}$, and let $T_{1},\ldots,T_{t}$ be generators of the tower
variety; that is, $\mathbb{V}(\mathbb{T})$ is the zero-set of
$\\{T_{1},\ldots,T_{t}\\}$. Let $\bar{u}=(u_{1},\ldots,u_{n})$ and
$\bar{v}=(v_{1},\ldots,v_{m})$ be such that
$(\bar{u},\bar{v})\in\mathbb{V}(\mathbb{T})$ and such that
$\mathcal{Q}(\mathcal{Q}^{-1}(\bar{u},\bar{v}))$ is well defined. Then,

$\mathcal{Q}(\mathcal{Q}^{-1}(\bar{u},\bar{v}))=(\bar{u},\bar{v})\,\,\mod\,\,I(\mathbb{V}(\mathbb{T})).$

In particular,

$\bar{r}(\mathcal{Q}^{-1}(\bar{u},\bar{v}))=\bar{u}\,\,\mod\,\,I(\mathbb{V}(\mathbb{T})).$

Therefore, there exist polynomials $M_{i,1},\ldots,M_{i,t}$, for
$i\in\\{0,\ldots,n\\}$, such that

$\displaystyle\bar{r}(\bar{w}(x))$
$\displaystyle=(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x))$
$\displaystyle+\left(\sum_{j=1}^{t}M_{0,j}(\bar{\sigma}(x))\,T_{j}(\bar{\sigma}(x)),\ldots,\sum_{j=1}^{t}M_{n,j}(\bar{\sigma}(x))\,T_{j}(\bar{\sigma}(x))\right).$

where
$\bar{\sigma}(x):=(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x),\bar{\delta}(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x)))$.
Furthermore,
$\bar{\sigma}(x)=\psi(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x))\in\mathrm{Im}(\psi)\subset\mathbb{V}(\mathbb{T})$
for all $x$. Thus, $T_{i}(\bar{\sigma}(x))=0$ for all $i\in\\{1,\ldots,t\\}$,
and hence

$\bar{r}(\bar{w}(x))=(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x)).$

In this situation, we first observe that since $\mathcal{Q}(\bar{w}(x))$ is
defined, then the denominators of $\mathcal{Q}$ do not vanish at $\bar{w}(x)$
and, by Remark 3.1, $G_{k}(x,\bar{w}(x))$ is also defined for every $1\leq
k\leq N$. In addition,

$G_{k}(x,\bar{w}(x))=F_{k}(x,\bar{r}(\bar{w}(x)))=F_{k}(x,y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x))=0.$

Now, clearly $\operatorname{num}(G_{k})(x,\bar{w}(x))=0$. The remaining
equations in $\mathcal{G}$ hold by the chain rule applied to the components of
$\bar{r}_{i}(\bar{w}(x))$, i.e. for $1\leq i\leq\ell$, $0\leq j<n_{i}$

$r_{i,j+1}(\bar{w}(x))=\frac{d}{dx}(y_{i}^{(j)}(x))=\frac{d}{dx}(r_{i,j}(\bar{w}(x)))=J_{r_{i,j}}(\bar{w}(x))\cdot(\bar{w}^{\prime}(x))^{T}.$

∎

Note that in Theorem 3.5 we assume that $\bar{r}(\bar{w}(x))$ is defined. In
the case that the tower variety $\mathbb{V}(\mathbb{T})$ admits a polynomial
parametrization $\mathcal{Q}$, this is always the case. For details on
polynomial parametrizations see e.g. [15, 8]. In the following proposition we
analyze some particular cases when the parametrization is rational.

######  Proposition 3.6.

Let $\mathcal{F}$ be as in (1.2), $\mathcal{G}$ be as in (3.1) and let
$\mathcal{Q}(\bar{w})=(\bar{r}(\bar{w}),\bar{\delta}(\bar{r}(\bar{w})))$ be a
rational parametrization of $\mathbb{V}(\mathbb{T})$. Then the following
statements hold

  1. (1)

If $n=0$ and $\bar{w}(x)=w_{1}(x)$ is a zero of $\operatorname{num}(r_{1})$,
the denominator of $r_{1}$ does not vanish at $w_{1}(x)$.

  2. (2)

If $\mathcal{F}=\\{F\\}$ consists only of an autonomous equation, $n=1$ and
$\bar{w}(x)=(w_{1}(x),w_{2}(x))$ is a zero of the denominator of
$r_{1}(w_{1},w_{2})$ or $r_{2}(w_{1},w_{2})$ and of the numerator of $G$, then
$\bar{w}(x)$ is constant.

###### Proof.

We may assume that the denominators and numerators of $\bar{r}(\bar{w})$ are
co-prime. Then item 1 directly follows. In item 2, let
$d_{j}\in\mathbb{C}[w_{1},w_{2}]$ be the denominator of $r_{j}$ for
$j\in\\{1,2\\}$. Since $F$ is independent of $x$, also $G$ and its numerator
are autonomous. Hence, $\operatorname{num}(G)(\bar{w}(x))=d_{j}(\bar{w}(x))=0$
if and only if either $\bar{w}(x)$ is constant or
$\operatorname{num}(G),d_{j}\in\mathbb{C}[w_{1},w_{2}]$ share a common
component. In the latter case, since $\operatorname{denom}(G)$ is a multiple
of the $d_{j}$, also $\operatorname{num}(G)$ and $\operatorname{denom}(G)$
have a common factor, which can be neglected. ∎

##  4\. Special Cases

In the substitution (1.3), the number of new variables is equal to the sum of
the orders of the given differential equations in the system (1.2) with
respect to the differential indeterminates. In many cases the radicals in
$\mathcal{F}$ can be considered separately or even do not contain some of the
derivatives of the $y_{i}$. In these cases, one may take a projection of the
tower variety where the dimension is smaller than the order. In the following,
we deal with these special cases.

Special case I. We start with the case where there exist some variables
$z_{i}$ which do not occur in any of the radicals in (1.2). Let $\bar{H}$ be
as in Theorem 3.3. Let us say that $I\subsetneq\\{1,\ldots,n\\}$ is the index
set of variables $z_{i}$ appearing in $\bar{H}$ and
$I^{*}=\\{1,\ldots,n\\}\setminus I$. Say that $s=\\#(I)$. We will use the
following notation:

  * •

For an $n$–tuple $\bar{p}$ we denote by $\bar{p}^{c}$ the $s$–tuple obtained
deleting the elements in the tuple positions labeled by $I^{*}$.

  * •

For an $s$–tuple $\bar{p}$ and for an $(n-s)$–tuple $\bar{q}$ we denote by
$\bar{p}^{e}(\bar{q})$ the $n$–tuple $\bar{s}$ obtained appending to $\bar{p}$
the elements of $\bar{q}$ so that $\bar{s}^{c}=\bar{p}$.

We consider the projection

$\begin{array}[]{lccc}\pi_{I}:&\mathbb{V}(\mathbb{T})\subset\mathbb{C}^{n+m}&\longrightarrow&\mathbb{C}^{s+m}\\\
&(\bar{u},\bar{v})&\longmapsto&(\bar{u}^{c},\bar{v})\end{array}$

where $\bar{u}=(u_{1},\ldots,u_{n}),\bar{v}=(v_{1},\ldots,v_{m})$ and
$\mathbb{V}(\mathbb{T})$ is as in (1.1). Let $\mathbb{V}(\mathbb{T})_{I}$ be
the Zariski closure of $\pi_{I}(\mathbb{V}(\mathbb{T}))$. Observe that
$\mathbb{V}(\mathbb{T})_{I}$ is the tower variety of the radical
parametrization $(\bar{z}^{c},\bar{H}(\bar{z}^{c}))$ and hence, by Lemma 3.4,
$\dim(\mathbb{V}(\mathbb{T})_{I})=s<n$. Furthermore, note that for
$\bar{p}:=(\bar{a},\bar{b})\in\pi_{I}(\mathbb{V}(\mathbb{T}))$ where
$\bar{a}\in\mathbb{C}^{s},\bar{b}\in\mathbb{C}^{m}$, it holds that

$\pi_{I}^{-1}(\bar{p})=\\{(\bar{a}^{e}(\bar{p}),\bar{b})\,|\,\bar{p}\in\mathbb{C}^{n-s}\\}.$

For an $n$–tuple $\bar{p}=(p_{1},\ldots,p_{n})$, and for
$K\subset\\{1,\ldots,n\\}$ we denote by $\bar{p}^{K}$ the tuple $(p_{i})_{i\in
K}$. Then, if
$\bar{r}(\bar{w}):=(\bar{r}^{I}(\bar{w}^{I}))^{e}(\bar{w}^{I^{*}}),$ and if

(4.1)
$\tilde{\mathcal{Q}}(\bar{w}^{I}):=(\bar{r}^{I}(\bar{w}^{I}),\bar{\delta}(\bar{r}^{I}(\bar{w}^{I})))$

is a rational (resp. birational) parametrization of
$\mathbb{V}(\mathbb{T})_{I}$, it holds that

(4.2)
$\mathcal{Q}(\bar{w}):=\pi_{I}^{-1}(\tilde{\mathcal{Q}}(\bar{w}^{I}))=(\bar{r}(\bar{w}),\bar{\delta}(\bar{r}(\bar{w})))$

is a rational (resp. birational) parametrization of $\mathbb{V}(\mathbb{T})$.
Now, since $\bar{H}$ does not depend on $\\{z_{i}\\}_{i\in I^{*}}$, Theorem
3.3 can be adapted as follows.

######  Theorem 4.1.

Let $\mathcal{F}\subset\mathbb{F}_{m}(x)$ be as in (1.2), $\mathcal{G}$ be as
in (3.1), $\bar{H}(\bar{z})\in\mathbb{F}_{m}^{\rho}$ be the tuple of non-
rational coefficients of the polynomials $F_{j}(x,\bar{z})\in\mathcal{F}$
w.r.t. $x$ depending on the variables $z_{i}$ with $i\in
I\subsetneq\\{1,\ldots,n\\}$, and let

$\bar{r}(\bar{w})=(\bar{r}^{I}(\bar{w}^{I}))^{e}(\bar{w}^{I^{*}})\in\mathbb{C}(\bar{w})^{n}$

be a rational change of variables. The following are equivalent.

  1. (1)

(3.1) is a system of AODEs.

  2. (2)

All the components of $\bar{H}(\bar{r}^{I}(\bar{w}^{I}))$ are rational.

Moreover, Theorem 3.5 (1) can be adapted directly. Let us analyze the
applicability of Theorem 3.5 (2). For this purpose, we assume that
$\tilde{\mathcal{Q}}$ is invertible (see (4.1)). This is the case if and only
if $\mathcal{Q}$ is invertible (see (4.2)). Moreover, for
$\bar{u}=(u_{1},\ldots,u_{n}),$ $\bar{v}=(v_{1},\ldots,v_{m})$ it holds that

(4.3)
$\mathcal{Q}^{-1}(\bar{u},\bar{v})=\tilde{\mathcal{Q}}^{-1}(\bar{u}^{I},\bar{v})^{e}(\bar{u}^{I^{*}}).$

In this situation, Theorem 3.5 (2) can be restated as follows.

######  Theorem 4.2.

Let $\mathcal{F}\subset\mathbb{F}_{m}(x)$ be as in (1.2) involving radicals in
the variables $z_{i}$ with $i\in I\subsetneq\\{1,\ldots,n\\}$, $\mathcal{G}$
be as in (3.1) and let $\tilde{\mathcal{Q}}(\bar{w}^{I})$ be a rational
parametrization of $\mathbb{V}(\mathbb{T})_{I}$ (see (4.1)) with the extension
$\mathcal{Q}(\bar{w})=(\bar{r}(\bar{w}),\bar{\delta}(\bar{r}(\bar{w})))$ (see
(4.2)). It holds that

  1. (1)

For every solution $\bar{w}(x)$ of $\mathcal{G}$,

$(y_{1}(x),\ldots,y_{\ell}(x)):=(r_{1,0}(\bar{w}(x))\ldots,r_{\ell,0}(\bar{w}(x)))$

is a solution of the differential system $\mathcal{F}$.

  2. (2)

If $\mathcal{Q}$ is invertible, $(y_{1}(x),\ldots,y_{\ell}(x))$ is a solution
of $\mathcal{F}$,

$\bar{w}(x):=\mathcal{Q}^{-1}(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x),\bar{\delta}(y_{1}(x),\ldots,y_{\ell}^{(n_{\ell})}(x)))$

and $\mathcal{Q}(\bar{w}(x))$ are defined, then $\bar{w}(x)$ is a solution of
$\mathcal{G}$.

Special case II. Now, let us deal with the case where the radicals in the
differential equations in $\mathcal{F}$ can be considered separately. More
precisely, let $\mathcal{F}$ be as in (1.2), and let
$\bar{H}(\bar{z})\in\mathbb{F}_{m}^{\rho}$ be the tuple of non-rational
(function) coefficients of $F(x,\bar{z})$ w.r.t. $x$. We assume that the
entries of $\bar{H}$ can be grouped separately in terms of the $z_{i}$. That
is, there exist $I_{1},\ldots,I_{s}\subset\\{1,\ldots,n\\}$, $I_{i}\cap
I_{j}=\emptyset$ for $i\neq j$, and such that the tuple
$\bar{H}=(h_{i})_{i\in\\{1,\ldots,\rho\\}}$ can be re-ordered as

$\bar{H}=(\bar{H}^{1},\ldots,\bar{H}^{s})$

where $\bar{H}^{j}$ collects the entries in $\bar{H}$ depending on
$\\{z_{d}\,|\,d\in I_{i}\\}$. Let $I_{i}^{*}=\bigcup_{j>i}I_{j}$. Furthermore
we use the notation $\bar{p}^{K}$ introduced in the special case I. In this
situation, we will proceed recursively.

Firstly, let us study the initial step. We consider the radical tower

$\mathbb{T}^{1}=\\{\mathbb{L}_{1}:=\mathbb{C}(\bar{z}^{I_{1}^{*}},\bar{H}^{2},\ldots,\bar{H}^{s}),\mathbb{F}_{0}=\mathbb{L}_{1}(\bar{z}^{I_{1}})\subseteq\mathbb{F}_{1}\subseteq\cdots\subseteq\mathbb{F}_{m_{1}}\\}$

where $\mathbb{F}_{i}=\mathbb{F}_{i−1}(\delta_{1i})$ with
$\delta_{1i}^{e_{1i}}=\alpha_{1i}\in\mathbb{F}_{i−1}\setminus\mathbb{L}_{1},e_{1i}\in\mathbb{N}$.
Let $\bar{\delta}^{1}=(\delta_{11},\ldots,\delta_{1m_{1}})$ be the tuple
collecting the $\delta_{1i}$’s involved in $\mathbb{T}^{1}$. Then, the
polynomials $F$ defining (1.2) can be seen as polynomials in
$\mathbb{F}_{m_{1}}(x)$. In addition, we consider the radical parametrization

$\mathcal{P}^{1}:=(\bar{z}^{I_{1}},H^{1}(\bar{z}^{I_{1}}))$

as well as the radical variety $\mathbb{V}(\mathcal{P}^{1})$ and its
associated tower variety

$\mathbb{V}(\mathbb{T}^{1})\subset\mathbb{C}^{\\#(I_{1})+m_{1}}.$

By Lemma 3.4, we have that
$\dim(\mathbb{V}(\mathbb{T}^{1}))=\dim(\mathbb{V}(\mathcal{P}^{1}))=\\#(I^{1})$.
Now, let us assume that $\mathbb{V}(\mathbb{T}^{1})$ is rational, and let
$\bar{r}^{I_{1}}(\bar{w}^{I_{1}})$ be the corresponding rational change of
variables. Then we consider the associated system (3.1) where $r_{k,d}$ for
$j\in I_{1}^{*},z_{j}=y_{k}^{(d)}$ are not determined yet.

For $1<k\leq s$ and after $k-1$ many steps, we obtain the radical tower

$\mathbb{T}^{k}=\\{\mathbb{L}_{k}:=\mathbb{C}(\bar{z}^{I_{k}^{*}},\bar{H}^{k+1},\ldots,\bar{H}^{s}),\mathbb{F}_{0}=\mathbb{L}_{k}(\bar{z}^{I_{k}})\subseteq\mathbb{F}_{1}\subseteq\cdots\subseteq\mathbb{F}_{m_{k}}\\}$

where $\mathbb{F}_{i}=\mathbb{F}_{i−1}(\delta_{ki})$ with
$\delta_{ki}^{e_{ki}}=\alpha_{ki}\in\mathbb{F}_{i−1}\setminus\mathbb{L}_{k},e_{ki}\in\mathbb{N}$
with the radical parametrization

$\mathcal{P}^{k}:=(\bar{z}^{I_{k}},H^{k}(\bar{y}^{I_{k}})).$

Again, the radical variety $\mathbb{V}(\mathcal{P}^{k})$ and its associated
tower variety

$\mathbb{V}(\mathbb{T}^{k})\subset\mathbb{C}^{\\#(I_{k})+m_{k}}$

have dimension $\\#(I^{k})$. For a rational change of variables
$\bar{r}^{I_{k}}(\bar{w}^{I_{k}})$, the equations in system (3.1) have to be
updated.

Eventually, if all tower varieties
$\mathbb{V}(\mathbb{T}^{1}),\ldots,\mathbb{V}(\mathbb{T}^{s})$ are rational,
Theorem 3.3 and Theorem 3.5 hold by recursively applying the components in the
corresponding rational change of variables

$\bar{r}(\bar{w})=(\bar{r}^{I_{1}}(\bar{w}^{I_{1}}),\ldots,\bar{r}^{I_{s}}(\bar{w}^{I_{s}}).$

Note that by the structure of the radicals in case II, the change of variables
can be performed also in parallel.

######  Remark 4.3.

Note that special case I can be seen as a particular instance of special case
II: In the notation from above, let us define $I_{1}$ as the index set of
variables $z_{i}$ which occur in radical expressions in $\mathcal{F}$ and set
$I_{2}=I^{*}$. Then the radical tower $\mathbb{T}^{2}$ is trivial and the
change of variables is simply the identity, namely
$\bar{r}^{I_{2}}(\bar{w}^{I_{2}})=\bar{w}^{I_{2}}$.

##  5\. Algorithmic Treatment

The previous ideas are in principle algorithmic, except that for arbitrary
dimension it cannot be decided whether the tower variety
$\mathbb{V}(\mathbb{T})$ is rational or not. Therefore, we present the
following procedure in a non-algorithmic way. In some special situations,
however, the procedure is indeed an algorithm.

Let $\mathcal{F}$ be a system of ordinary differential equations as in (1.2).

  1. (1)

Collect in the tuple $\bar{H}(z_{1},\ldots,z_{n})$ all radical expressions in
(1.2).

  2. (2)

If it exists, compute a partition
$I_{1},\ldots,I_{s}\subseteq\\{1,\ldots,n\\}$ and
$\bar{H}=(\bar{H}^{1},\ldots,\bar{H}^{s})$ such that the radicals are
separated (cf. Special case II in Section 4).

  3. (3)

For every tower variety $\mathbb{V}(\mathbb{T}^{k})$, compute a rational
parametrization
$\mathcal{Q}_{k}(\bar{r}^{I_{k}},\bar{\delta}^{I_{k}}(\bar{r}^{I_{k}}))$ if
possible.

  4. (4)

Perform the change of variables $\bar{z}=\bar{r}(\bar{w})$ in order to obtain
a system of AODEs $\mathcal{G}$ as in (3.1).

######  Remark 5.1.

In the case that all the dimensions of the tower varieties
$\mathbb{V}(\mathbb{T}^{k})$, are either one or two, namely when the
$\mathbb{V}(\mathbb{T}^{k})$ defines a curve or surface, it can be decided
whether a rational parametrization $\mathcal{Q}$ exists. In the affirmative
case, compute such a proper (i.e. invertible) rational parametrization (see
[20, 13]). In this way, the above procedure is indeed an algorithm for this
special situation.

######  Remark 5.2.

Usually, one is interested in working, if possible, with real solutions. With
the method described in this paper, if the system $\mathcal{G}$ has real
solutions, and the tower variety can be parametrized over the field of the
real numbers, then the deduced solutions of the system $\mathcal{F}$ are also
real. If the tower variety is a curve, there exist algorithms to either
directly compute a real rational parametrization (see [19]) or to
reparametrize complex parametrizations into real parametrizations (see [9]).
If the tower variety is a surface the situation is not so complete as in the
case of curves, but still there exist algorithms to compute real
parametrizations for certain types of surfaces (see e.g. [2, 1, 12, 14]).

######  Example 5.3.

Let us consider the system of ordinary differential equations

$\mathcal{F}=\left\\{\begin{aligned}
F_{1}&=\sqrt[3]{\sqrt{y_{1}}+y_{1}^{\prime}}+x\,y_{2}=0,\\\
F_{2}&=\sqrt{1+y_{2}}+x^{2}\,y_{1}y_{1}^{\prime}=0\end{aligned}\right.$

The radical expressions are
$\bar{H}(z_{1},z_{2},z_{3})=(\sqrt[3]{\sqrt{z_{1}}+z_{2}},\sqrt{1+z_{3}})$. We
set $I_{1}=\\{1,2\\}$ and $I_{2}=\\{3\\}$. The tower variety
$\mathbb{V}(\mathbb{T}^{1})$ is the surface given by
$(z_{1},z_{2},\delta_{1},\delta_{2})$ where
$\delta_{1}^{2}=z_{1},\delta_{2}^{3}=\delta_{1}+z_{2}$.
$\mathbb{V}(\mathbb{T}^{1})$ can be parametrized, for example, as

$\mathcal{Q}_{1}(w_{1},w_{2})=(w_{1}^{2},w_{2}^{3}-w_{1},w_{1},w_{2}).$

The tower variety $\mathbb{V}(\mathbb{T}^{2})$ is the curve given by
$(z_{3},\delta_{3})$ where $\delta_{3}^{2}=1+z_{3}$ and has the rational
parametrization $\mathcal{Q}_{2}(w_{3})=(w_{3}^{2}-1,w_{3})$. Then the system
(1.2) is

$\mathcal{G}=\left\\{\begin{aligned} &G_{1}=w_{2}+xw_{3}^{2}-x=0,\
G_{2}=w_{3}+x^{2}w_{1}^{2}w_{2}^{3}-x^{2}w_{1}^{3}=0\\\
&w_{2}^{3}-w_{1}=2w_{1}w_{1}^{\prime}\end{aligned}\right.$

The system $\mathcal{G}$ could be triangularized by using differential Thomas
decomposition. Since the output is very lengthy, we do not present this here.

######  Example 5.4.

Let us consider the Thomas-Fermi equation

$\sqrt{x}\,y^{\prime\prime}-y^{3/2}=0.$

By the methods from [6], the given differential equation gets transformed by
$x=z^{2}$ into the differential equation

$F=z\,Y^{\prime\prime}-Y^{\prime}-4z^{2}\,Y^{3/2}=0.$

The only radical expression in $F$ is $Y^{1/2}$ (special case I), leading to a
tower variety $\mathbb{V}(\mathbb{T})$ parametrized by $(W^{2},W)$. By using
the transformation $Y=W^{2}$, and directly set
$Y^{\prime}=2WW^{\prime},Y^{\prime\prime}=2(WW^{\prime\prime}+W^{\prime 2})$,
the above equation is equivalent to

$G=z\,W\,W^{\prime\prime}+z\,W^{\prime 2}-W\,W^{\prime}-2z^{2}\,W^{3}=0.$

By using the Newton polygon method [5], we obtain for the given initial value
$y(0)=W(0)=0$ the solution family

$W(z)=cz+\frac{c^{2}z^{5}}{12}+\frac{c^{3}z^{9}}{360}+\mathcal{O}(z^{13}),$

where $c>0$ is an arbitrary constant, and hence,

$y(x)=\sqrt{W(\sqrt{x})}=\sqrt{c}\sqrt{x}+\frac{c^{3/2}x^{9/2}}{24}+\frac{c^{5/2}x^{17/2}}{1920}+\mathcal{O}(x^{25/2}).$

For given $c>0$, the terms of order $25/2$ are uniquely given and no field
extension of the coefficients is necessary anymore. Hence, existence and
uniqueness of the solution is guaranteed and $y(x)$ is a convergent Puiseux
series in $\mathbb{Q}(\sqrt{c})((x^{1/2}))$.

Let us note that for the frequently used initial values $y(0)=1,y(\infty)=0$
there is no formal Puiseux series solution and other solution methods as the
Newton polygon method have to be used for finding solutions of $G=0$ and
hence, of the Thomas-Fermi equation itself.

##  6\. Partial Differential Equations

In this section we treat systems of partial differential equations
$\mathcal{F}$ in the differential indeterminates $x_{1},\ldots,x_{q}$. More
precisely, let $\mathcal{F}$ consist of differential equations of order at
most $s$ and possibly involving radicals in
$\Theta(\alpha)y_{k}:=\frac{\partial^{|\alpha|}y_{k}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{q}^{\alpha_{q}}}$ with $1\leq k\leq\ell$,
$|\alpha|\leq s$. Let $z_{1},\ldots,z_{n}$ enumerate the $\Theta(\alpha)y_{k}$
and consider a radical tower (1.1). We assume that
$\mathcal{F}\subset\mathbb{F}_{m}(x_{1},\ldots,x_{q})$.

We are looking for a change of variables (1.3) such that
$F(\bar{x},\bar{r}(\bar{w}))$ is an algebraic equation for every $F$ in
$\mathcal{F}$. Let $z_{i}=\Theta(\alpha)y_{k},z_{j}=\Theta(\alpha+e_{p})y_{k}$
for some $|\alpha|<s$ and a unit vector $e_{p}$ which is one exactly at the
$p$-th position. The differential relations are kept by additionally imposing
all such

(6.1) $\sum_{t=1}^{n}\frac{\partial\,r_{i}}{\partial
w_{t}}(\bar{w})\cdot\frac{\partial\,w_{t}}{\partial x_{p}}=r_{j}(\bar{w}).$

In this way, Theorem 3.3 and Theorem 3.5 hold in the expected way as for the
case of ODEs (where $q=1$). Let us illustrate this in the following example.

######  Example 6.1.

Let us consider the system of partial differential equations

$\mathcal{F}=\left\\{\begin{aligned}
F_{1}&=\sqrt[3]{\sqrt{z_{1}-1}+z_{3}}+x_{1}\,(z_{1}+\sqrt{z_{5}})=0,\\\
F_{2}&=\sqrt{z_{2}+1}^{3}+x_{2}^{2}\,z_{1}z_{4}=0,\\\
F_{3}&=x_{1}\,\sqrt{z_{1}z_{4}-z_{2}z_{3}}+x_{2}\,z_{5}^{2}-1=0\end{aligned}\right.$

where $(z_{1},z_{2},z_{3},z_{4},z_{5})=(y_{1},\frac{\partial\,y_{1}}{\partial
x_{1}},\frac{\partial\,y_{1}}{\partial
x_{2}},\frac{\partial^{2}\,y_{1}}{\partial x_{1}\partial x_{2}},y_{2})$. The
radical coefficients are collected in

$\bar{H}=\left(\sqrt[3]{\sqrt{z_{1}-1}+z_{3}},\sqrt{z_{5}},\sqrt{z_{2}+1}^{3},\sqrt{z_{1}z_{4}-z_{2}z_{3}}\right).$

We set $I_{1}=\\{1,2,3,4\\}$ and $I_{2}=\\{5\\}$. The tower variety
$\mathbb{V}(\mathbb{T}^{1})$ is given by
$(\bar{z}^{I_{1}},\delta_{1},\ldots,\delta_{4})$ where
$\delta_{1}^{2}=z_{1}-1,\delta_{2}^{3}=\delta_{1}+z_{3},\delta_{3}^{2}=z_{1}z_{4}-z_{2}z_{3},\delta_{4}^{2}=z_{2}+1$.
A parametrization of $\mathbb{V}(\mathbb{T}^{1})$ is

$\displaystyle\mathcal{Q}_{1}(\bar{w}^{I_{1}})=\big{(}$ $\displaystyle
w_{1}^{2}+1,w_{4}^{2}-1,w_{2}^{3}-w_{1},\frac{w_{2}^{3}w_{4}^{2}-w_{1}w_{4}^{3}-w_{2}^{3}+w_{3}^{2}+w_{1}}{w_{1}^{2}+1},$
$\displaystyle w_{1},w_{2},w_{3},w_{4}\big{)}.$

Additionally considering the transformation
$\mathcal{Q}_{2}(w_{5})=(w_{5}^{2},w_{5})$, and the differential relations, we
obtain the system

$\mathcal{G}=\left\\{\begin{aligned}
G_{1}&=w_{2}+x_{1}\,(w_{1}^{2}+1+w_{5})=0,\\\
G_{2}&=w_{4}^{3}+x_{2}^{2}(w_{2}^{3}w_{4}^{2}-w_{1}w_{4}^{3}-w_{2}^{3}+w_{3}^{2}+w_{1})=0,\\\
G_{3}&=x_{1}w_{3}+x_{2}w_{5}^{4}-1=0,\\\
&2w_{1}\cdot\frac{\partial\,w_{1}}{\partial x_{1}}=w_{4}^{2}-1,\quad
2w_{1}\cdot\frac{\partial\,w_{1}}{\partial x_{2}}=w_{2}^{3}-w_{1},\\\
&\left(3w_{2}^{2}\cdot\frac{\partial\,w_{2}}{\partial
x_{1}}-\frac{\partial\,w_{1}}{\partial
x_{1}}\right)\cdot(w_{1}^{2}+1)=w_{2}^{3}w_{4}^{3}-w_{1}w_{4}^{3}-w_{2}^{3}+w_{3}^{2}+w_{1},\\\
&2w_{4}(w_{1}^{2}+1)\cdot\frac{\partial\,w_{4}}{\partial
x_{2}}=w_{2}^{3}w_{4}^{3}-w_{1}w_{4}^{3}-w_{2}^{3}+w_{3}^{2}+w_{1}\end{aligned}\right.$

## Acknowledgements

Authors partially supported by the grant PID2020-113192GB-I00 (Mathematical
Visualization: Foundations, Algorithms and Applications) from the Spanish
MICINN. Second author also supported by the Austrian Science Fund (FWF): P
31327-N32.

## References

  * [1] Andradas, C., Recio, T., Sendra, J. R., and Tabera, L. F. On the simplification of the coefficients of a parametrization.  Journal of symbolic computation 44, 2 (2009), 192–210.
  * [2] Andradas, C., Recio, T., Tabera, L. F., Sendra, J. R., and Villarino, C. Proper real reparametrization of rational ruled surfaces.  Computer aided geometric design 28, 2 (2011), 102–113.
  * [3] Boulier, F., Lemaire, F., Poteaux, A., and Maza, M. M. An equivalence theorem for regular differential chains.  Journal of Symbolic Computation 93 (2019), 34–55.
  * [4] Burrows, B., and Core, P. A variational-iterative approximate solution of the Thomas-Fermi equation.  Quarterly of Applied Mathematics 42, 1 (1984), 73–76.
  * [5] Cano, J. The Newton Polygon Method for Differential Equations.  In Proceedings of the 6th International Conference on Computer Algebra and Geometric Algebra with Applications (Berlin, Heidelberg, 2005), IWMM’04/GIAE’04, pp. 18–30.
  * [6] Caravantes, J., Sendra, J. R., Sevilla, D., and Villarino, C. Transforming odes and pdes from radical coefficients to rational coefficients.  Mediterranean Journal of Mathematics 18, 3 (2021), 1–14.
  * [7] Cox, D., Little, J., and O’Shea, D. Ideals, Varieties, and Algorithms: an Introduction to Computational Algebraic Geometry and Commutative Algebra.  Springer Science & Business Media, 2013.
  * [8] Pérez-Díaz, S., and Sendra, J. R. Computing birational polynomial surface parametrizations without base points.  Mathematics 8, 12 (2020), 2224.
  * [9] Recio, T., and Sendra, J. R. Real reparametrizations of real curves.  Journal of Symbolic Computation 23, 2-3 (1997), 241–254.
  * [10] Ritt, J. Differential Algebra, vol. 33.  American Mathematical Society, 1950.
  * [11] Robertz, D. Formal algorithmic elimination for PDEs, vol. 2121.  Springer, 2014.
  * [12] Schicho, J. Rational parametrization of real algebraic surfaces.  In ISSAC ’98 (1998).
  * [13] Schicho, J. Rational parametrization of surfaces.  Journal of Symbolic Computation 26, 1 (1998), 1–29.
  * [14] Schicho, J. Proper parametrization of real tubular surfaces.  J. Symb. Comput. 30 (2000), 583–593.
  * [15] Sendra, J. Normal parametrizations of algebraic plane curves.  Journal of Symbolic Computation 33, 6 (2002), 863–885.
  * [16] Sendra, J. R., and Sevilla, D. Radical parametrizations of algebraic curves by adjoint curves.  Journal of Symbolic Computation 46, 9 (2011), 1030–1038.
  * [17] Sendra, J. R., and Sevilla, D. First steps towards radical parametrization of algebraic surfaces.  Computer Aided Geometric Design 30, 4 (2013), 374–388.
  * [18] Sendra, J. R., Sevilla, D., and Villarino, C. Algebraic and algorithmic aspects of radical parametrizations.  Computer Aided Geometric Design 55 (2017), 1–14.
  * [19] Sendra, J. R., and Winkler, F. Algorithms for rational real algebraic curves.  Fundamenta Informaticae 39, 1, 2 (1999), 211–228.
  * [20] Sendra, J. R., Winkler, F., and Pérez-Díaz, S. Rational Algebraic Curves.  Algorithms and Computation in Mathematics. Springer-Verlag Berlin Heidelberg, 2008.
  * [21] Turkyilmazoglu, M. Solution of the Thomas–Fermi equation with a convergent approach.  Communications in Nonlinear Science and Numerical Simulation 17, 11 (2012), 4097–4103.


# Single-photon-added coherent state based measurement transition and its
advantages in precision measurement

Yusuf Turek<EMAIL_ADDRESS>Akbar Islam  Ahmad Abliz<EMAIL_ADDRESS>School of Physics and Electronic Engineering, Xinjiang Normal University,
Urumqi, Xinjiang 830054, China

(August 28, 2024)

###### Abstract

In this work, the measurement transition as well as precision measurement
advantages of single-photon-added coherent state after postselected von
Neumann measurement are investigated. We noticed that the weak-to-strong
measurement transition characterized by the shifts of pointer’s position and
momentum variables occurred in continuously by controlling a dimensionless
parameter associated with system-pointer coupling. We calculate the ratio
between the signal-to-noise ratios of nonpostselected and postselected
measurements, and the latter is used to find the quantum Fisher information.
We found that the single-photon-added coherent pointer state can improve the
precision of the measurement processes such as signal-to-noise ratio and
parameter estimation after postselected von Neumann measurement characterized
by postselection and weak value. Furthermore, contrary to the results of
several previous studies, we found that the anomalous large weak values can’t
improve the precision measurement processes related to single-photon-added
coherent state.

###### pacs:

03.65.-w, 03.65.Ta, 42.50.−p, 03.67.−a

##  I introduction

As we know, although the theory of quantum mechanics and its applications are
extremely successful, but its fundamental problems associated with the
measurement are still unsolved puzzles that the scientists must to face.
Handling the measurement problem requires to understand the unique role of
measurement in quantum mechanics. As a foundation of quantum theory the
quantum measurement problem were elaborated in the 1930s by Bohr, Shrodinger,
Heisenberg, von Neumann and Mandelstam. Especially, in 1932 the von Neumann
formulated (von Neumann J, 1955) the first standard version of quantum
measurement characterized by strong projective operation to the measured
system so that it can cause the wave packet collapse.

According to the von Neumann’s model, we can get the desired information of
the system observable by single trial since the coupling between the measured
system and pointer or meter is strong enough to distinguish its different
eigenvalues. This strong quantum measurement model promotes the exploring of
the quantum theory from its fundamentals (Li  _et al._ , 2019; Brodutch and
Cohen, 2016; Zurek, 2003; Wheeler and Zurek, 1983; Braginsky and Khalili,
1992; Jacobs, 2014) to quantum art technologies (Debnath  _et al._ , 2016;
Barends  _et al._ , 2014; Ofek  _et al._ , 2016; Kielpinski  _et al._ , 2002;
Biamonte  _et al._ , 2017; LeCun  _et al._ , 2015; Monz  _et al._ , 2016;
Gyongyosi and Imre, 2019). However, because of the collapse of wave function
owing to the decoherence, the conventional strong quantum measurements cannot
be directly used in some hot topics of quantum information science such as
quantum state based high precision measurements (Xu  _et al._ , 2020; Chen
_et al._ , 2018a; Brunner and Simon, 2010; Xu  _et al._ , 2013),
reconstruction of unknown quantum state (Yu  _et al._ , 2019; Huang  _et al._
, 2011; Gaikwad  _et al._ , 2021; Řeháček  _et al._ , 2001), etc. On the other
hand, this strong measurement model becomes useless if the coupling between
measured system and pointer is too weak to distinguish the different
eigenvalues of the observable. In the latter case, the sub-wave-packets
corresponding to different eigenvalues of system observable are overlapped so
that we can’t get the enough information about the system beacause our
subsequent trial may cause the wave-packet’s collapse (Zurek, 1981) . To solve
this issue, in 1988 Aharonov and co-workers proposed a generalized quantum
measurement technique called weak measurement which is characterized by
postselection and weak value (Aharonov  _et al._ , 1988). In this measurement
model, even though the coupling strength is too small compared to traditional
strong measurement but still can get the required system information
statistically as accurate as strong projective measurement by taking many
measurement trials since it doesn’t cause wave function’s collapse after
measurement (Tollaksen  _et al._ , 2010). This new measurement model solved
plenty of fundamental problems in quantum theory and is widely accepted as a
new measurement technique (see (Dressel  _et al._ , 2014) and references
therein). However, one may ask some questions like what is the connection
between weak and strong measurement? How can the weak-to-strong measurement
transition be characterized? In recent years, the above questions raised wide
interests among the physicists.

In general, the measurement results can be readout from the final state of the
pointer. In other words, the information of system’s observable in both strong
and weak measurement schemes can be obtained in the shifts of pointer’s
position and momentum after measurement. Thus, the problems related to weak-
to-strong measurement transition could be explained by investigating the
average values of the observable showed in pointer’s final shifts. To study
the measurement transitions in weak-to-strong regimes, we have to take into
account the full order evolution of unitary operator of interaction
Hamiltonian (Turek  _et al._ , 2015a; Nakamura  _et al._ , 2012). Zhu et al.
(Zhu  _et al._ , 2011) studied the quantum measurement transition for a qubit
system without any restriction on the coupling strength between the system and
the pointer, and they found the transitions from the weakest to strongest
regimes. In a recent work (Pan  _et al._ , 2020), the weak-to-strong
measurement transition is demonstrated experimentally by using a single
trapped ${}^{40}Ca^{+}$ ion. They found that the weak-to-strong measurement
transition predicts the continuous relation between the weak value and
expectation value of system’s observable showed in pointer’s position shift,
and the transition can be controlled by modulating the coupling parameter
$\Gamma=g/\sigma$. In this transition factor, $g$ is the coupling constant
between the measured system and pointer and $\sigma$ is the width of an
initial Gaussian pointer, which characterizes the measurement strengths, i.e.,
$\Gamma<1$ ($\Gamma>1$) represents weak measurement (strong measurement ). In
the above two theoretical and experimental works both considered the
fundamental Gaussian wave packet as a pointer. One of the authors of the
current work (Turek  _et al._ , 2015b) studied the average shifts of pointer’s
position and momentum for a system observable satisfying the property
$\hat{A}^{2}=1$ by considering the full order evolution of unitary operator of
nonclassical pointer states including squeezed vacuum state and Schrodinger
cat state. They predicted that the two extreme limits fit well with the weak
and strong measurement correspondences. Most recently, Orszag et al. (Araya-
Sossa and Orszag, 2021) examined the weak-to-strong measurement transition for
coherent squeezed state, and theoretically proposed its implementation in
${}^{40}Ca^{+}$ ion stored inside the Paul trap interacting with two laser
fields as specific frequencies. They showed that the shift in the pointer’s
position and momentum variables establishes a relationship with a new value
defined as the transition value, which can generalize the weak value (weak
measurement regime) as well as the conditional expectation value (strong
measurement regime). They extend the Josa’s theorem (Jozsa, 2007), and found
that weak-to-strong measurement transition can occur in continuously by
modulating the coupling parameter $\Gamma$. In (Pan  _et al._ , 2020), they
used the fundamental Gaussian mode as a pointer that correspond to the vacuum
state case of the measuring device in (Araya-Sossa and Orszag, 2021). As we
know, the fundamental Gaussian mode is the special case of coherent state and
its preparation is experimentally well established. The coherent state is a
semi-classical one with a Gaussian Wigner function, and if we add one photon
to the coherent state, one can obtain the non-Gaussian quantum state called
single-photon-added coherent (SPAC) state which possesses nonclassical
properties (Agarwal and Tara, 1991; Zavatta  _et al._ , 2004, 2005). As an
intermediate state, the SPAC state covers both the single photon state and the
coherent state, which have potential applications in quantum metrology
(Giovannetti  _et al._ , 2011; Vitelli  _et al._ , 2010; Joo  _et al._ , 2011,
2012; Hu  _et al._ , 2014; Magnitskii  _et al._ , 2017; Huang  _et al._ ,
2015; Liu  _et al._ , 2016; Huang  _et al._ , 2018; Chen  _et al._ , 2018b;
Maleki  _et al._ , 2021; Maleki, 2021; Solntsev  _et al._ , 2021).
Furthermore, since the SPAC state has no vacuum component and contains large
single-photon probability, it bears more profound applications in many fields
including quantum state engineering (Dai and Jing, 2008), quantum
communication (Pinheiro and Ramos, 2013), quantum key distribution (Wang  _et
al._ , 2017; Miranda and Mundarain, 2017; Srikara  _et al._ , 2020; Zhu  _et
al._ , 2018; Wang  _et al._ , 2014), and quantum metrology (Filippov  _et al._
, 2013; Gard  _et al._ , 2016; Braun  _et al._ , 2014; Schnabel, 2017).
Especially, in a recent work (Chen  _et al._ , 2020) the authors showed that
the performance of quantum digital signature can be improved significantly by
using the SPAC state compared with the one using weak coherent state. Thus,
the SPAC state may be a promising candidate for the implementation of digital
signature in the near future.

In this paper, motivated by aforementioned works, we study weak-to-strong
measurement transition for the SPAC state. We also investigate the advantages
of SPAC state in some precision measurement processes such as signal-to-noise
ratio (SNR) and unknown parameter estimation after postselected von Nuemann
measurement. To achieve this, we take the spatial and internal degrees of
freedom of SPAC state as pointer and measured system, respectively. We
introduce the transition value of system observable, and obtain the final
normalized state of SPAC state after postselection. We find that the weak-to-
strong measurement transition characterized by the shifts of pointer’s
position and momentum variables occurred in continuously by controlling a
dimensionless parameter $\Gamma$ associated with system-pointer coupling. We
also assertain that the SPAC state based postselection measurement can improve
the SNR and Fisher information significantly for moderate weak values and
coupling parameter $\Gamma$. We anticipate that the results of this work may
help the further development of theoretical and experimental schemes of weak-
to-strong measurement transition based on SPAC pointer state and their using
in the above mentioned quantum information processing and quantum metrology.

The rest of this paper is organized as follows. In Sec. II, we give the final
pointer state after postseleceted measurement and introduce some concepts
related to the weak-to-strong measurement transition. In Sec. III, we give the
details of weak-to-strong measurement transition controlled by coupling
parameter $\Gamma$ between measured system and pointer. Subsequently, in Sec.
IV and Sec. V, we investigate the advantages of SPAC state in improving the
efficiency of SNR and parameter estimations processes. Finally, we summarize
our findings of this study in Sec. VI.

##  II  Basic Concepts

In quantum measurement theory, the related discussions usually begin with the
interaction Hamiltonian of the system and pointer since it contains the main
information between the ingredients of pointer (measuring device) and measured
system. According to the measurement theory, the explicit expressions of
Hamiltonian of the pointer and measured system doesn’t affect the results of
the measurement. In our case, the interaction Hamiltonian of a measurement is
taken as the standard von Neumann Hamiltonian

$H=g(t)\hat{\text{$\sigma$}}_{x}\otimes\hat{P},$ (1)

where $g\left(t\right)$ is the interaction coupling function between the
pointer and measured system, $\hat{P}$ denotes the conjugate momentum operator
to the position operator $\hat{X}$ of the pointer with
$[\hat{X},\hat{P}]=i\hat{I}$, and
$\hat{\sigma}_{x}=|\text{$\uparrow_{x}$}\rangle\langle\downarrow_{x}|-|\downarrow_{x}\rangle\langle\uparrow_{x}|$
is Pauli $x$ operator of the system to be measured. Here,
$|\uparrow_{x}\rangle$ and $|\downarrow_{x}\rangle$ are the eigenstates of
$\hat{\sigma}_{x}$ with corresponding eigenvalues $1$ and $-1$, respectively.
The coupling $g\left(t\right)$ is a nonzero function in a finite interaction
time interval $t-t_{0}$, i.e.,
$\int_{t_{0}}^{t}g(\tau)d\tau=g\delta\left(t-t_{0}\right)$. For this kind of
impulsive interaction, the time evolution operator $e^{-\frac{i}{\hbar}\int
Hd\tau}$ of our total system becomes as
$e^{-\frac{1}{\hbar}ig\hat{\sigma}_{x}\otimes\hat{P}}$. Hereafter, we put
$\hbar=1$ and assume all factors in $g\hat{\text{$\sigma$}}_{x}\otimes\hat{P}$
are dimensionless.

In this work, we take the polarization and spatial degrees of freedom of SPAC
state as the measured system and pointer, respectively. We assume that
initially the system and pointer state are prepared in

$|\Phi_{in}\rangle=|\psi_{i}\rangle\otimes|\phi\rangle,$ (2)

where

$|\psi_{i}\rangle=\cos\frac{\varphi}{2}|\uparrow_{z}\rangle+e^{i\delta}\sin\frac{\varphi}{2}|\downarrow_{z}\rangle$
(3)

and

$|\phi\rangle=\gamma a^{\dagger}|\alpha\rangle,\ \ \ \
\gamma=\frac{1}{\sqrt{1+|\alpha|^{2}}}$ (4)

are the initial states of the measured system and pointer, respectively. Here,
$\varphi\in[0,\pi]$ , $\delta\in[0,2\pi)$, $\alpha=re^{i\theta}$ is the
parameter of the coherent state $|\alpha\rangle$, and $|\phi\rangle$ is the
mathematical definition of SPAC state.

Since the operator $\hat{\sigma}_{x}$ satisfies
$\hat{\sigma}_{x}^{2}=\hat{I}$, we can write the unitary evolution operator
$e^{-ig\hat{\sigma}_{x}\otimes\hat{P}}$ as

$\displaystyle e^{-ig\hat{\sigma}_{x}\otimes\hat{P}}$
$\displaystyle=\frac{1}{2}(\hat{I}+\hat{\sigma}_{x})\otimes
D\left(\frac{\Gamma}{2}\right)+\frac{1}{2}(\hat{I}-\hat{\sigma}_{x})\otimes
D\left(-\frac{\Gamma}{2}\right).$ (5)

In the derivation of the above expression, the position operator $\hat{X}$ and
momentum operator $\hat{P}$ are written in terms of the annihilation
(creation) operators $\hat{a}$($\hat{a}^{\dagger}$), of radiation field as

$\displaystyle\hat{X}$ $\displaystyle=$
$\displaystyle\sigma(\hat{a}^{\dagger}+\hat{a}),$ (6) $\displaystyle\hat{P}$
$\displaystyle=$ $\displaystyle\frac{i}{2\sigma}(\hat{a}^{\dagger}-\hat{a}),$
(7)

where $\sigma$ is the width of the fundamental Gaussian beam. Here, the
parameter $\Gamma\equiv g/\sigma$, and $D\left(\mu\right)$ is the displacement
operator with complex $\mu$ defined by
$D(\mu)=e^{\mu\hat{a}^{\dagger}-\mu^{\ast}\hat{a}}.$ Note that the coupling
parameter $\Gamma$ represents measurement strength, and the coupling between
the system and pointer is defined as weak (strong) if $\Gamma<1$$(\Gamma>1)$.
We assume throughout this work that the coupling parameter $\Gamma$ covers all
the allowed values both in weak and strong measurement regimes.

The total initial state $|\Phi_{in}\rangle$ evolves by the interaction
Hamiltonian, Eq. (1), as

$\displaystyle|\Psi\rangle$
$\displaystyle=e^{-ig\hat{\sigma}_{x}\otimes\hat{P}}|\psi_{i}\rangle\otimes|\phi\rangle.$
(8)

After taking a postselection with the state
$|\psi_{f}\rangle=|\uparrow_{z}\rangle$ onto $|\Psi\rangle$, the final state
of the pointer and its form can obtained as

$|\Phi\rangle\\!=\\!\frac{\beta}{\sqrt{2}}\left[\left(1\\!\\!+\\!\langle\sigma_{x}\rangle_{w}\right)D\left(\frac{\Gamma}{2}\right)+\left(1\\!-\\!\langle\sigma_{x}\rangle_{w}\right)D\left(\\!-\frac{\Gamma}{2}\right)\right]|\phi\rangle,$
(9)

where

$\displaystyle\beta$
${}^{-2}=1\\!+\\!|\langle\sigma_{x}\rangle_{w}|^{2}\\!+\\!\gamma^{2}e^{-\frac{\Gamma^{2}}{2}}\text{$\times$}$
$\displaystyle\\!\\!Re[(1+\\!\langle\sigma_{x}\rangle_{w})^{\ast}(1\\!-\\!\langle\sigma_{x}\rangle_{w})(\gamma^{-2}\\!-\\!\Gamma^{2}\\!+2iIm[\alpha])e^{2\Gamma
iIm\text{[}\alpha]}]$ (10)

is the normalization coefficient. Here,

$\langle\sigma_{x}\rangle{}_{w}=\frac{\langle\psi_{f}|\hat{\sigma}_{x}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=e^{i\delta}\tan\frac{\varphi}{2}$
(11)

is the weak value of the system observable $\sigma_{x}$, and $Re$ and $Im$
represents the real and imaginary parts of a complex number. From Eq. (11), we
know that when the pre-selected state $|\psi_{i}\rangle$ and the post-selected
state $|\psi_{f}\rangle$ are almost orthogonal, the absolute value of the weak
value can be arbitrarily large. This feature is regarded as a very useful
postselected weak measurement technique which is called weak value
amplification, and bears various applications in many research fields as
mentioned in Sec.II. We have to mention that although the weak value can take
large anomalous value, it is accompanied by low successful postselection
probability
$P_{s}=|\langle\psi_{f}|\psi_{i}\rangle|^{2}=\cos^{2}\frac{\varphi}{2}$. In
next sections we use the normalized final state $|\Phi\rangle$ of SPAC state
after postselection to discuss the related issues.

In order to explain the weak-to-strong measurement transition in the next
section, here we introduce some concepts by following Ref. (Araya-Sossa and
Orszag, 2021). By taking the postselection process into account, a transition
value is introduced to represent the values of the system observable
$\hat{\sigma}_{x}$ in weak and strong measurement regimes which is defined as

$\sigma_{x}^{T}=\langle\Phi|\Psi^{\prime}\rangle,$ (12)

with

$\displaystyle|\Psi^{\prime}\rangle$
$\displaystyle=\langle\psi_{f}|\hat{\sigma}_{x}|\Psi\rangle=\langle\psi_{f}|\psi_{i}\rangle\times$
$\displaystyle\left[\frac{1}{2}\left(1+\langle\sigma_{x}\rangle_{w}\right)D\left(\frac{\Gamma}{2}\right)-\frac{1}{2}\left(1-\langle\sigma_{x}\rangle_{w}\right)D\left(-\frac{\Gamma}{2}\right)\right]|\phi\rangle.$
(13)

After substituting the expressions of $|\Phi\rangle$ and
$|\Psi^{\prime}\rangle$ into Eq. (12), the value of $\sigma_{x}^{T}$ can be
obtained as

$\displaystyle\sigma_{x}^{T}$
$\displaystyle=\frac{1}{2}|\beta|^{2}[4Re[\langle\sigma_{x}\rangle_{w}]-\gamma^{2}(1+\langle\sigma_{x}\rangle_{w})^{\ast}(1-\langle\sigma_{x}\rangle_{w})h(\Gamma)$
$\displaystyle+\gamma^{2}(1-\langle\sigma_{x}\rangle_{w})^{\ast}(1+\langle\sigma_{x}\rangle_{w})h^{\ast}(\Gamma)],$
(14)

with

$h(\Gamma)=e^{-\frac{\Gamma^{2}}{2}}\left(1+(\alpha^{\ast}+\Gamma)(\alpha-\Gamma)\right)e^{2\Gamma
iIm[\alpha]}.$ (15)

It can be seen that if $\Gamma\rightarrow 0$ , the transition value reduced to
the weak value $\langle\sigma_{x}\rangle_{w}$ of the observable
$\hat{\sigma}_{x}$, i.e.,

$\displaystyle\left(\sigma_{x}^{T}\right)_{\Gamma\rightarrow 0}$
$\displaystyle=\langle\sigma_{x}\rangle_{w}.$ (16)

On the other side, if $\Gamma\rightarrow\infty$, then the transition value
gives

$\displaystyle\left(\sigma_{x}^{T}\right)_{\Gamma\rightarrow\infty}$
$\displaystyle=\frac{1}{2}\frac{|1+\langle\sigma_{x}\rangle_{w}|^{2}-|1-\langle\sigma_{x}\rangle_{w}|^{2}}{1+|\langle\sigma_{x}\rangle_{w}|^{2}}$
$\displaystyle=\cos\delta\sin\varphi=\sigma_{x}^{c}.$ (17)

Here, $\sigma_{x}^{c}$ is the conditional expectation value of the system
observable $\hat{\sigma}_{x}$ in strong measurement regime. The value of
$\sigma_{x}^{c}$ can be obtained by Aharonov-Bergmann-Lebowitz rule (Aharonov
_et al._ , 1964) which reads as

$\displaystyle\sigma_{x}^{c}$
$\displaystyle=\sum_{j}a_{j}\frac{|\langle\psi_{f}|a_{j}\rangle\langle
a_{j}|\psi_{i}\rangle|^{2}}{\sum_{i}|\langle\psi_{f}|a_{i}\rangle\langle
a_{i}|\psi_{i}\rangle|^{2}}$
$\displaystyle=\frac{|\langle\psi_{f}|\uparrow_{x}\rangle\langle\uparrow_{x}|\psi_{i}\rangle|^{2}-|\langle\psi_{f}|\downarrow_{x}\rangle\langle\downarrow_{x}|\psi_{i}\rangle|^{2}}{|\langle\psi_{f}|\uparrow_{x}\rangle\langle\uparrow_{x}|\psi_{i}\rangle|^{2}+|\langle\psi_{f}|\downarrow_{x}\rangle\langle\downarrow_{x}|\psi_{i}\rangle|^{2}}$
$\displaystyle=\cos\delta\sin\varphi.$ (18)

As we can see, the two extreme limits of the transition value $\sigma_{x}^{T}$
directly gives the corresponding values of the observable for weak and strong
measurement. Thus, the transition value can be seen as a generalization of
weak value and conditional expectation value with considering all allowed
values of corresponding system observable through weak-to-strong measurements.

##  III Weak-to-strong measurement transition

The average shifts of position $x$ and momentum $p$ variables are defined as

$\delta x=\langle\Phi|\hat{X}|\Phi\rangle-\langle\phi|\hat{X}|\phi\rangle$
(19)

and

$\delta p=\langle\Phi|\hat{P}|\Phi\rangle-\langle\phi|\hat{P}|\phi\rangle,$
(20)

respectively. Here $\hat{X}$ and $\hat{P}$ are position and momentum operators
expressed in Fock space representation in terms of the annihilation (creation)
operator $\hat{a}$ ($\hat{a}^{\dagger}$) as aforementioned in Sec. II. By
using Eq. (9), the explicit expressions of $\delta x$ and $\delta p$ can be
obtained as

$\displaystyle\delta x$
$\displaystyle=\langle\Phi|\hat{X}|\Phi\rangle-\langle\phi|\hat{X}|\phi\rangle$
$\displaystyle=2\sigma Re\left[\langle\Phi|\hat{a}|\Phi\rangle\right]-2\sigma
Re\left[\langle\phi|\hat{a}|\phi\rangle\right]$
$\displaystyle=\sigma|\beta|^{2}\gamma^{2}\\{|1+\langle\sigma_{x}\rangle_{w}|^{2}(\Gamma\gamma^{-2}+4Re[\alpha]+2Re[\alpha]|\alpha|^{2})$
$\displaystyle+|1-\langle\sigma_{x}\rangle_{w}|^{2}(-\Gamma\gamma^{-2}+4Re[\alpha]+2Re[\alpha]|\alpha|^{2})$
$\displaystyle+Re[(1+\langle\sigma_{x}\rangle_{w})^{\ast}(1-\langle\sigma_{x}\rangle_{w})f(-\Gamma)]$
$\displaystyle+Re[(1-\langle\sigma_{x}\rangle_{w})^{\ast}(1+\langle\sigma_{x}\rangle_{w})f(\Gamma)]\\}$
$\displaystyle-2\sigma\gamma^{2}(2+|\alpha|^{2})Re[\alpha],$ (21)

and

$\displaystyle\delta p$
$\displaystyle=\langle\Phi|\hat{P}|\Phi\rangle-\langle\phi|\hat{P}|\phi\rangle$
$\displaystyle=\frac{\hbar}{\sigma}Im\left[\langle\Phi|a|\Phi\rangle\right]-\frac{\hbar}{\sigma}Im\left[\langle\phi|\hat{a}|\phi\rangle\right]$
$\displaystyle=\frac{\hbar}{2\sigma}|\beta|^{2}\gamma^{2}\\{|1+\langle\sigma_{x}\rangle_{w}|^{2}(\Gamma\gamma^{-2}+4Im[\alpha]+2Im[\alpha]|\alpha|^{2})$
$\displaystyle+|1-\langle\sigma_{x}\rangle_{w}|^{2}(-\Gamma\gamma^{-2}+4Im[\alpha]+2Im[\alpha]|\alpha|^{2})$
$\displaystyle+Im[(1+\langle\sigma_{x}\rangle_{w})^{\ast}(1-\langle\sigma_{x}\rangle_{w})f(-\Gamma)]$
$\displaystyle+Im[(1-\langle\sigma_{x}\rangle_{w})^{\ast}(1+\langle\sigma_{x}\rangle_{w})f(\Gamma)]\\}$
$\displaystyle-\frac{\hbar}{\sigma}\gamma^{2}(2+|\alpha|^{2})Im[\alpha],$ (22)

respectively, with

$\displaystyle f(\Gamma)$ $\displaystyle=e^{-2\Gamma iIm[\alpha]}\times$
$\displaystyle\left[2\alpha(2+|\alpha|^{2})+3\Gamma\gamma^{-2}-2\alpha^{2}\Gamma+\Gamma^{2}(\alpha^{\ast}-3\alpha)\right]e^{-\frac{\Gamma^{2}}{2}}.$
(23)

These average shifts are valid for all measurement regimes, and their
magnitudes can be controlled by weak value $\langle\sigma_{x}\rangle_{w}$ and
coupling parameter $\Gamma$. As aforementioned, the weak-to-strong measurement
transition can occur in continuously by adjusting the magnitude of coupling
parameter $\Gamma$. If we assume the coupling parameter is very small, then
the above position and momentum shifts are reduced to the corresponding
position and momentum shifts in postselected weak measurement regime, i.e.,

$\displaystyle\left(\delta x\right)_{\Gamma\rightarrow 0}$
$\displaystyle=W_{x}=gRe\left[\langle\sigma_{x}\rangle_{w}\right]-g\frac{\partial
Var(X)_{|\phi\rangle}}{2\sigma^{2}\partial\theta}Im\left[\langle\sigma_{x}\rangle_{w}\right]$
(24)

and

$\left(\delta p\right)_{\Gamma\rightarrow
0}=W_{p}=\frac{2g}{\hbar}Var\left(P\right)_{|\phi\rangle}Im\left[\langle\sigma_{x}\rangle_{w}\right],$
(25)

respectively. Here,

$\displaystyle Var\left(X\right)_{|\phi\rangle}$
$\displaystyle=\sigma^{2}\gamma^{4}\left(3+4|\alpha|^{2}\sin^{2}\theta+|\alpha|^{4}\right)$
(26)

and

$Var(P)_{|\phi\rangle}=\frac{\hbar^{2}}{4\sigma^{2}}\gamma^{4}\left[3+4|\alpha|^{2}\cos^{2}\theta+|\alpha|^{4}\right]$
(27)

are the variances of position and momentum variables under the initial SPAC
state $|\phi\rangle$, respectively.

At the other extreme, the $\delta x$ and $\delta p$ give the values
corresponding to the strong measurement regime as

$\displaystyle\left(\delta x\right)_{\Gamma\rightarrow\infty}$
$\displaystyle=\frac{2gRe\left[\langle\sigma_{x}\rangle_{w}\right]}{1+|\langle\sigma_{x}\rangle|^{2}}=g\cos\delta\sin\varphi=g\sigma_{x}^{c},$
(28)

and

$\left(\delta p\right)_{\Gamma\rightarrow\infty}=0,$ (29)

respectively. In Fig. 1, we show the weak-to-strong measurement transition of
pointer’s shifts, Eq. (21) and Eq. (22), as a function weak value
characterized by $\varphi$. As shown in Fig. 1, the measurement transition
from weak-to-strong regime occurred in continuously with the increasing of the
coupling parameter $\Gamma$, and two extreme limits ($\Gamma\rightarrow 0$ and
$\Gamma\rightarrow\infty$) also match well with our theoretical results. In
the current work, the beam width $\sigma$ of SPAC state is assumed to be fixed
so that the coupling parameter $\Gamma$ is controlled by only adjusting the
coupling strength $g$ between the pointer and measured system, which is
assumed to be an impulsive interaction independent of time. In general, the
coupling strength $g$ can be the function of time so that the measurement
transition can be implemented experimentally in three ways, each corresponding
to tuning one of the three parameters $g$, $t$ and $\sigma$. Yet, tuning of
the coupling duration time $t$ is the most straightforward approach to
implement the weak-to-strong transition (Pan  _et al._ , 2020).

Figure 1: (Color online) Measurement transition of the pointer’s shifts in the
position (a) and the momentum (b) for SPAC state after taking postselection as
a function of weak value quantified by $\varphi$ for different coupling
parameters $\Gamma$. Here, the other parameters are set as
$r=2,\theta=\frac{\pi}{6}$ and $\delta=\frac{\pi}{6}$.

##  IV Signal-to-Noise Ratio

In remaining parts of this work, we discuss the advantages of SPAC state in
precision measurement processes. Firstly, we investigate the advantages of
SPAC state in signal amplification process after postselection measurement. In
general, the signal amplification of position shift can be characterized by
SNR (Agarwal, 2013). Thus, in order to show the advantages of postselected
measurement over non-postselected one based on SPAC state for its position
shift, we investigate the ratio of SNRs between postselected and
nonpostselected measurments, i.e.,

$\chi=\frac{\mathcal{R}_{p}}{\mathcal{R}_{n}}.$ (30)

Here, $\mathcal{R}_{p}$ represents the SNR of postselected von Neumann
measurement defined as

$\mathcal{R}_{p}=\frac{\sqrt{NP_{s}}\delta x}{\triangle x},$ (31)

with the variance of position operator

$\triangle
x=\sqrt{\langle\Phi|\hat{X}^{2}|\Phi\rangle-\langle\Phi|\hat{X}|\Phi\rangle^{2}}$
(32)

and the average shift of the pointer variable $x$ after postselected
measurement

$\displaystyle\delta x$
$\displaystyle=\langle\Phi|\hat{X}|\Phi\rangle-\langle\phi|\hat{X}|\phi\rangle,$
(33)

respectively. Here, $\hat{X}=\sigma\left(\hat{a}+\hat{a}^{\dagger}\right)$ is
the position operator, $N$ is the total number of measurements, $P_{s}$ is the
success probability of postselection, and $|\Phi\rangle$ denotes the
normalized state of SPAC state after postselection, i.e., Eq. (9).

The $\mathcal{R}_{n}$ for non-postselected measurement is defined as (Agarwal,
2013)

$\mathcal{R}_{n}=\frac{\sqrt{N}\delta x^{\prime}}{\triangle x^{\prime}},\ $
(34)

with

$\triangle
x^{\prime}=\sqrt{\langle\Psi|\hat{X}^{2}|\Psi\rangle-\langle\Psi|\hat{X}|\Psi\rangle^{2}}$
(35)

and

$\displaystyle\delta x^{\prime}$
$\displaystyle=\frac{\langle\Psi|\hat{X}|\Psi\rangle}{\langle\Psi|\Psi\rangle}-\langle\phi|\hat{X}|\phi\rangle=g\sin\varphi\cos\delta,$
(36)

respectively. Here, $|\Psi\rangle$ is the unnormalized final state of the
total system without postselection which is given in Eq. (8). The explicit
expression of $\delta x$ is given in Eq. (21) and other quantities also can be
calculated with some algebra easily. In Fig. 2 we show the ratio $\chi$ of
SNRs between postselected and nonpostselected von Neumann measurement for
different system parameters. As presented in Fig. 2 (a), the ratio $\chi$
increases with the decreasing of the weak value in the weak measurement regime
where the coupling parameters $\Gamma$ $\in(0.3,0.5)$. The ratio $\chi$ can
even be much larger than unity near $\Gamma=0.3$. Furthermore, in Fig. 2 (b)
we plot the ratio $\chi$ as a function of the state parameter $r$ with
different weak values for the coupling parameter $\Gamma=0.3$ corresponding to
the weak measurement regime. From the Fig. 2(b) we can see that in the weak
measurement regime the ratio $\chi$ of SNRs exhibits a slightly damping
periodic oscillation with the increasing of system parameter $r$ which
characterizes the nonclassicality of the initial SPAC state (Agarwal and Tara,
1991). Interestingly, we noticed that for SPAC state based postselected von
Neumann measurement the $\chi$ is decreased with very large weak values, which
is contrary to the signal amplification feature as verified in previous
studies(Pang and Brun, 2015; Turek  _et al._ , 2015b; Turek, 2020). In a word,
from the above discussions we can conclude that the SPAC state can be utilized
to improve the SNR in postselected weak measurement rather than the
nonpostselected measurement.

Figure 2:  (Color online) The ratio $\chi$ of SNRs between postselected and
nonpostselected measurement for SPAC state. Here we take
$\delta=\frac{5\pi}{12},\theta=\frac{\pi}{2}$. (a) $\chi$ is plotted as a
function of coupling parameter $\Gamma$ for different weak values for fixed
$r=5$. (b) $\chi$ is plotted as a function of initial SPAC state parameter $r$
for different weak values for the fixed coupling parameter $\Gamma=0.3$.

##  V Fisher Information

In this section, we study the usefulness of SPAC state after postselected weak
measurement on parameter estimation process over initial input state. Fisher
information is the maximum amount of information about the unknown parameter
that we can extract from the system. For a pure quantum state
$|\Phi_{\Gamma}\rangle$, the quantum Fisher information for estimating the
unknown parameter $\Gamma$ is

$F=4[\langle\dot{\Phi}_{\Gamma}|\dot{\Phi}_{\Gamma}\rangle-|\langle\text{$\Phi$}|\dot{\Phi}_{\Gamma}\rangle|^{2}],$
(37)

where state $|\text{$\Phi_{\Gamma}$}\rangle$ represents the final pointer
states of the system which is given in Eq. (9),
$|\dot{\Phi}_{\Gamma}\rangle=\partial_{\Gamma}|\Phi\rangle$, and $\Gamma\equiv
g/\sigma$ is the measurement coupling parameter which is directly related to
coupling constant $g$ in our Hamiltonian, Eq. (1).

In the information theory, the Cramer-Rao bound (CRB) is the fundamental limit
in the minimum uncertainty for parameter estimation, and it equals to the
inverse of the Fisher information, i.e.

$\Delta\Gamma\geq\frac{1}{NF^{(Q)}}.$ (38)

Here, $N$ the total number of measurement trials, and $F^{(Q)}=P_{s}F$ is the
fisher information of our scheme after taking the successful postselection
probability $P_{s}=\cos^{2}\frac{\varphi}{2}$ into account. As indicated in
Eq. (38), the larger the Fisher information is, the better estimation of
parameter $\Gamma$ achieves for the fixed measurement trials.

Figure 3: (Color online) The quantum Fisher information $F^{(Q)}$ of SPAC
state after postselected measurement as a function of coupling parameter
$\Gamma$ and weak value quantified by $\varphi$. Here, we take $N=1$, and
other parameters are the same as Fig. 1. Figure 4: (Color online) The quantum
Fisher information $F^{(Q)}$ of SPAC state after postselected measurement as a
state parameter $r$ for different coupling parameter $\Gamma$. Here, we take
$N=1$, and other parameters are the same as Fig. 1.

In order to evaluate the variance $\triangle\Gamma$, we have to calculate the
Fisher information $F^{(Q)}$ corresponding to the state $|\Phi\rangle$. The
analytical results for the Fisher information are presented in Fig. 3 and Fig.
4, respectively. In Fig. 3, we plot the Fisher information $F^{(Q)}$ as a
function of the coupling parameter $\Gamma$ and weak value quantified by
$\varphi$ for the coherent state parameter $r=2$. As shown in Fig. 3, the
Fisher information $F^{(Q)}$ is larger around $\Gamma=1$ with small weak
values compared to the weak measurement regime ($\Gamma\ll 1$) and
nonpostselected case ($\Gamma=0$). To further verify our claims, we plot the
$F^{(Q)}$ as a function of the coherent state parameter $r$ for different
coupling parameters $\Gamma$ with small weak value,
$\langle\sigma_{x}\rangle_{w}\approx 0.234+0.135i$, corresponding to
$\delta=\varphi=\frac{\pi}{6}$ in Eq. (11). It is very clear from the Fig. (4)
that after postselected measurement the Fisher information of SPAC state is
increased significantly for moderate coupling parameters $\Gamma$ as
increasing the state parameter $r$.

From the discussions of weak-to-strong measurement transition in Section. III,
we can know that with increasing the coupling parameter $\Gamma$, the weak
value transformed to the conditional expectation value. Thus, one can deduce
that in strong measurement regime there is no any amplification effect caused
by the “weak value”, and it can be verified in numerical results presented in
Figs. 2-4, respectively. Furthermore, the above results also indicate that the
postselection process can improve the parameter estimation rather than the
nonpostselection one (Arvidsson-Shukur  _et al._ , 2020).

##  VI conclusion and outlook

In summary, we investigated the weak-to-strong measurement transition and some
precision measurement processes based on SPAC state after postselected von
Neumann measurement. To achieve this aim, we take the polarization and spatial
degrees of freedom of initial SPAC state as measured system and pointer,
respectively. After the standard process of postselected weak measurement we
obtained the final state of the pointer, and discussed the related problems.
Firstly, we studied the weak-to-strong measurement transition for SPAC state
via a coupling parameter $\Gamma$ that involves the system and pointer
coupling. We found that the weak-to-strong measurement transition
characterized by pointer’s shifts can occur in continuously by controlling the
coupling parameter $\Gamma$. We also investigated the advantages of SPAC state
afer postselected measurement on some precision measurement processes such as
system’s signal amplification and parameter estimation. We found that SPAC
state can improve the SNR after taking postselection in weak measurement
regime for small weak values compared to nonpostselection process. We also
noticed that the ratio of SNRs between postselected and nonpostselected von
Neumann measurement can be much larger than unity and oscillates periodically
with a slight damping in the weak measurement regime. In parameter estimation
process, we calculated the Fisher information for the final state of SPAC
state after postselected von Neumann measurement for estimating the variance
of unknown coupling parameter $\Gamma$ which is quantified by CRB. We found
that SPAC state after postselected measurement can increase the estimation of
unknown coupling parameter in moderate coupling strength regimes with small
weak value. Last but not the least, contrary to the previous studies, the
large anomalous weak values can’t improve the efficiency of related precision
measurement processes based on SPAC state.

The single photon state and the coherent state correspond to the two limit
cases (for $|\alpha|$→0 or $|\alpha|$$\gg 1$) of the SPAC state. Therefore,
our current postselected von Nuemman measurement proposal based on SPAC
pointer state not only covers both of the two extreme cases, but may also
provide an alternate method to improve the related quantum information
processing and quantum metrology (Zavatta  _et al._ , 2011) based on the SPAC
pointer state. Another interesting point is that, contrary to the other
possible pointer states including coherent state, Fock state and squeezed
state which are widely used in weak measurement studies, the SPAC state
possesses both the key features normally associated to quantum states: the
negativity of the Wigner function and the reduced fluctuations along one
quadrature. As previous works investigated, the squeezed nature of SPAC state
potentially offers an advantage in metrology such as parameter estimation
(Filippov  _et al._ , 2013) and quantum sensing (Braun  _et al._ , 2014; Gard
_et al._ , 2016; Schnabel, 2017) compared to the coherent state and Fock
state, and helps to develop security protocols in quantum key distribution
(Assche, 2006; Loepp and Wootters, 2006; Barnett, 2009; Barnett  _et al._ ,
2018). Furthermore, in our recent work, we studied the effects of postselected
von Neumann measurement on the nonclassicality of SPAC state including
squeezing and photon statistics (Turek, 2021) and found that the postselected
von Neumann measurement characterized by weak value really has positive
effects to optimize the inherent properties of SPAC state. However, the
comparison of advantages of different pointer states in postselected von
Nuemann measurement based quantum metrology is not the main goal of our
present work, and it is an open problem worth to study. Work along this line
is in progress, and results will be presented in near future.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No. 11865017, No. 11864042), the Natural Science Foundation of
Xinjiang Uyghur Autonomous Region of China (Grant No. 2020D01A72), and the
Introduction Program of High Level Talents of Xinjiang Ministry of Science.

## References

  * von Neumann J (1955) von Neumann J,  _Mathematical Foundations of Quantum Mechanics_ (Princeton University Press, Princeton, NJ, 1955).
  * Li  _et al._ (2019) Y. Li, H.-S. Zhong, Y.-H. Luo, L.-C. Peng, C.-Y. Lu, N.-L. Liu, J. Zhang, L. Li, and J.-W. Pan, Optica 6, 1199 (2019).
  * Brodutch and Cohen (2016) A. Brodutch and E. Cohen, Phys. Rev. Lett. 116, 070404 (2016).
  * Zurek (2003) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
  * Wheeler and Zurek (1983) J. A. Wheeler and W. H. Zurek,  _Quantum Theory and Measurement_ (Princeton University Press, Princeton, 1983).
  * Braginsky and Khalili (1992) V. B. Braginsky and F. Y. Khalili,  _Quantum Measurement_ (Cambridge University Press, Cambridge, England, 1992).
  * Jacobs (2014) K. Jacobs,  _Quantum Measurement Theory and its Applications_ (Cambridge University Press, Cambridge, England, 2014).
  * Debnath  _et al._ (2016) S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, Nature 536, 63 (2016).
  * Barends  _et al._ (2014) R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O¡¯Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Nature 508, 500 (2014).
  * Ofek  _et al._ (2016) N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Nature 536, 441 (2016).
  * Kielpinski  _et al._ (2002) D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002).
  * Biamonte  _et al._ (2017) J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017).
  * LeCun  _et al._ (2015) Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
  * Monz  _et al._ (2016) T. Monz, D. Nigg, E. A. Martinez, M. F. Brandl, P. Schindler, R. Rines, S. X. Wang, I. L. Chuang, and R. Blatt, Science 351, 1068 (2016).
  * Gyongyosi and Imre (2019) L. Gyongyosi and S. Imre, Scientific Reports 9, 6755 (2019).
  * Xu  _et al._ (2020) L. Xu, Z. Liu, A. Datta, G. C. Knee, J. S. Lundeen, Y.-q. Lu, and L. Zhang, Phys. Rev. Lett. 125, 080501 (2020).
  * Chen  _et al._ (2018a) G. Chen, N. Aharon, Y.-N. Sun, Z.-H. Zhang, W.-H. Zhang, D.-Y. He, J.-S. Tang, X.-Y. Xu, Y. Kedem, C.-F. Li, and G.-C. Guo, Nat. Commun 9, 93 (2018a).
  * Brunner and Simon (2010) N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
  * Xu  _et al._ (2013) X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 111, 033604 (2013).
  * Yu  _et al._ (2019) S. Yu, F. AlbarrÃ¡n-Arriagada, J. C. Retamal, Y.-T. Wang, W. Liu, Z.-J. Ke, Y. Meng, Z.-P. Li, J.-S. Tang, E. Solano, L. Lamata, C.-F. Li, and G.-C. Guo, Adv. Quantum. Technol 2, 1800074 (2019).
  * Huang  _et al._ (2011) J. S. Huang, L. F. Wei, and C. H. Oh, Phys. Rev. A 83, 032110 (2011).
  * Gaikwad  _et al._ (2021) A. Gaikwad, Arvind, and K. Dorai, Quant. Infor. Proc 20, 19 (2021).
  * Řeháček  _et al._ (2001) J. Řeháček, Z. Hradil, and M. Ježek, Phys. Rev. A 63, 040303 (2001).
  * Zurek (1981) W. H. Zurek, Phys. Rev. D 24, 1516 (1981).
  * Aharonov  _et al._ (1988) Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
  * Tollaksen  _et al._ (2010) J. Tollaksen, Y. Aharonov, A. Casher, T. Kaufherr, and S. Nussinov, New. J. Phys 12, 013023 (2010).
  * Dressel  _et al._ (2014) J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014).
  * Turek  _et al._ (2015a) Y. Turek, H. Kobayashi, T. Akutsu, C.-P. Sun, and Y. Shikano, New Journal of Physics 17, 083029 (2015a).
  * Nakamura  _et al._ (2012) K. Nakamura, A. Nishizawa, and M.-K. Fujimoto, Phys. Rev. A 85, 012113 (2012).
  * Zhu  _et al._ (2011) X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, Phys. Rev. A 84, 052111 (2011).
  * Pan  _et al._ (2020) Y. Pan, J. Zhang, E. Cohen, C.-w. Wu, P.-X. Chen, and N. Davidson, Nat. Phys 16, 1206 (2020).
  * Turek  _et al._ (2015b) Y. Turek, W. Maimaiti, Y. Shikano, C.-P. Sun, and M. Al-Amri, Phys. Rev. A 92, 022109 (2015b).
  * Araya-Sossa and Orszag (2021) K. Araya-Sossa and M. Orszag, Phys. Rev. A 103, 052215 (2021).
  * Jozsa (2007) R. Jozsa, Phys. Rev. A 76, 044103 (2007).
  * Agarwal and Tara (1991) G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
  * Zavatta  _et al._ (2004) A. Zavatta, S. Viciani, and M. Bellini, Science 306, p.660 (2004).
  * Zavatta  _et al._ (2005) A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72, 023820 (2005).
  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011).
  * Vitelli  _et al._ (2010) C. Vitelli, N. Spagnolo, L. Toffoli, F. Sciarrino, and F. De Martini, Phys. Rev. Lett. 105, 113602 (2010).
  * Joo  _et al._ (2011) J. Joo, W. J. Munro, and T. P. Spiller, Phys. Rev. Lett. 107, 083601 (2011).
  * Joo  _et al._ (2012) J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, Phys. Rev. A 86, 043828 (2012).
  * Hu  _et al._ (2014) L.-Y. Hu, C.-P. Wei, J.-H. Huang, and C.-J. Liu, Opt. Commun 323, 68 (2014).
  * Magnitskii  _et al._ (2017) S. A. Magnitskii, D. N. Frolovtsev, D. P. Agapov, A. V. Demin, V. N. Krutikov, and G. G. Levin, Meas. Tech 60, 235 (2017).
  * Huang  _et al._ (2015) J. Huang, X. Qin, H. Zhong, Y. Ke, and C. Lee, Sci. Rep. 5, 17894 (2015).
  * Liu  _et al._ (2016) J. Liu, X.-M. Lu, Z. Sun, and X. Wang, J. Phys. A: Math. Theor. 49, 115302 (2016).
  * Huang  _et al._ (2018) J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee, Phys. Rev. A 98, 012129 (2018).
  * Chen  _et al._ (2018b) G. Chen, L. Zhang, W.-H. Zhang, X.-X. Peng, L. Xu, Z.-D. Liu, X.-Y. Xu, J.-S. Tang, Y.-N. Sun, D.-Y. He, J.-S. Xu, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 121, 060506 (2018b).
  * Maleki  _et al._ (2021) Y. Maleki, M. O. Scully, and A. M. Zheltikov, Phys. Rev. A 104, 053712 (2021).
  * Maleki (2021) Y. Maleki, Eur. Phys. J. Plus 136, 1028 (2021).
  * Solntsev  _et al._ (2021) A. S. Solntsev, G. S. Agarwal, and Y. S. Kivshar, Nat. Photonics 15, 327 (2021).
  * Dai and Jing (2008) Q. Dai and H. Jing, Inter. J. Theor. Phys 47, 2716 (2008).
  * Pinheiro and Ramos (2013) P. V. P. Pinheiro and R. V. Ramos, Quant. Infor. Proc 12, 537 (2013).
  * Wang  _et al._ (2017) Y. Wang, W.-S. Bao, H.-Z. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, Phys. Lett. A 381, 1393 (2017).
  * Miranda and Mundarain (2017) M. Miranda and D. Mundarain, Quant. Infor. Proc 16, 298 (2017).
  * Srikara  _et al._ (2020) S. Srikara, K. Thapliyal, and A. Pathak, Quant. Infor. Proc 19, 371 (2020).
  * Zhu  _et al._ (2018) J.-R. Zhu, C.-Y. Wang, K. Liu, C.-M. Zhang, and Q. Wang, Quant. Infor. Proc 17, 294 (2018).
  * Wang  _et al._ (2014) D. Wang, M. Li, F. Zhu, Z.-Q. Yin, W. Chen, Z.-F. Han, G.-C. Guo, and Q. Wang, Phys. Rev. A 90, 062315 (2014).
  * Filippov  _et al._ (2013) S. N. Filippov, V. I. Man'ko, A. S. Coelho, A. Zavatta, and M. Bellini, Phys. Scripta T153, 014025 (2013).
  * Gard  _et al._ (2016) B. T. Gard, D. Li, C. You, K. P. Seshadreesan, R. Birrittella, J. Luine, S. M. H. Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, B. E. Koltenbah, C. G. Parazzoli, B. A. Capron, R. W. Boyd, C. C. Gerry, H. Lee, and J. P. Dowling, “Photon added coherent states: Nondeterministic, noiseless amplification in quantum metrology,” (2016), arXiv:1606.09598 [quant-ph] .
  * Braun  _et al._ (2014) D. Braun, P. Jian, O. Pinel, and N. Treps, Phys. Rev. A 90, 013821 (2014).
  * Schnabel (2017) R. Schnabel, Phys. Rep. 684, 1 (2017).
  * Chen  _et al._ (2020) J.-J. Chen, C.-H. Zhang, J.-M. Chen, C.-M. Zhang, and Q. Wang, Quant. Infor. Proc 19, 198 (2020).
  * Aharonov  _et al._ (1964) Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964).
  * Agarwal (2013) G. Agarwal,  _Quantum Optics_ (Cambridge University Press, Cambridge, England, 2013).
  * Pang and Brun (2015) S. Pang and T. A. Brun, Phys. Rev. Lett. 115, 120401 (2015).
  * Turek (2020) Y. Turek, Chin. Phys. B 29, 090302 (2020).
  * Arvidsson-Shukur  _et al._ (2020) D. R. M. Arvidsson-Shukur, N. Yunger Halpern, H. V. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd, Nat. Commun 11, 3775 (2020).
  * Zavatta  _et al._ (2011) A. Zavatta, J. Fiurasek, and M. Bellini, Nat. Photonics 5, 52 (2011).
  * Assche (2006) G. V. Assche,  _Quantum Cryptography and Secret-Key Distillation_ (Cambridge University Press, Cambridge, England, 2006).
  * Loepp and Wootters (2006) S. Loepp and W. K. Wootters,  _Protecting Information_ (Cambridge University Press, Cambridge, England, 2006).
  * Barnett (2009) S. M. Barnett,  _Quantum Information_ (Oxford University, Oxford, 2009).
  * Barnett  _et al._ (2018) S. M. Barnett, G. Ferenczi, C. R. Gilson, and F. C. Speirits, Phys. Rev. A 98, 013809 (2018).
  * Turek (2021) Y. Turek, Eur. Phys. J. Plus 136, 221 (2021).