or additive reduction at _all_ the primes dividing $m$ then for all primes
$p$,

$\chi(G,E[p^{\infty}])=\chi(\Gamma_{F},E[p^{\infty}]).$

In this case, (8.2) simplifies considerably and becomes

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}=\lambda_{p}(E/F_{\mathrm{cyc}}).$

Now, Theorem 5.1 asserts that

$\operatorname{rank}_{\mathbb{Z}}(E/F_{n})\leq\lambda_{p}(E/F_{\mathrm{cyc}})p^{n}.$

_Case 2:_ When $\ell$ is a prime of _split multiplicative reduction_ we have
the following assertion.

######  Proposition 10.3.

Consider the false Tate curve extension
${\mathcal{F}}_{\infty}={\mathcal{F}}_{\infty}^{(m)}$. Let $E_{/\mathbb{Q}}$
be a fixed elliptic curve with _split multiplicative reduction_ at a prime
dividing $m$. As $p$ varies over all primes of good reduction of $E$, the
$G$-Euler characteristic is _always_ non-trivial.

###### Proof.

Suppose that $\ell$ is a prime divisor of $m$ and $v|\ell$ is a prime of split
multiplicative reduction of $E$. Since $\ell\neq p$, we have that
$v\in{\mathfrak{M}}$. Recall that Lemma 4.6(2) asserts that

$\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1\textrm{ if and only if
}q_{v}\equiv 1\pmod{p}.$

We have $q_{v}=\left\lvert\mathbb{F}_{\ell^{f}}\right\rvert$ where $f=f_{p}$
is the _degree of inertia_ of $\ell$ in $F=\mathbb{Q}(\mu_{p})$. Recall from
our discussion in §8.2 that $f$ the smallest positive integer such that
$\ell^{f}\equiv 1\pmod{p}$. Therefore, we conclude that varying over all
primes $p$, the $G$-Euler characteristic is _always_ non-trivial for an
elliptic curve $E_{/\mathbb{Q}}$ with split multiplicative reduction at
$\ell|m$. ∎

######  Remark 10.4.

The growth of Mordell–Weil ranks inside ${\mathcal{F}}_{\infty}$ has been
studied using Heegner points by H. Darmon and Y. Tian in [DT10] under certain
hypotheses (see also [DL17]). Their results tell us that the Mordell–Weil
ranks are expected to be unbounded inside ${\mathcal{F}}_{\infty}$.

_Case 3:_ If $\ell$ is a prime divisor of $m$ and $v|\ell$ is a prime of _good
reduction_ of $E$, we need to analyse two cases. If $\ell=p$, then
$v\not\in{\mathfrak{M}}$ and there is no contribution to the $G$-Euler
characteristic from the local Euler factor. Otherwise, by definition,

$v\in{\mathfrak{M}}\textrm{ if and only if }E(F_{v})[p^{\infty}]\neq 0.$

We want to evaluate for what proportion of primes $p$ is $E(F_{v})[p]\neq 0$
for a fixed $\ell$. Equivalently (see [Sil09, Proposition VII.3.1]), how often
does $p$ divide $\widetilde{E}(\kappa_{v})$? Recall that

$\\#\widetilde{E}(\kappa_{v})=\ell^{f}+1-a_{v}\equiv 2-a_{v}\pmod{p},$

where $f$ is the smallest positive integer such that $\ell^{f}\equiv
1\pmod{p}$ as in the proof of Proposition 10.3. Thus, the condition
$\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1$ holds precisely when
$a_{v}\equiv 2\pmod{p}$.

######  Proposition 10.5.

Let $E_{/\mathbb{Q}}$ be a fixed elliptic curve with good reduction at
$\ell\geq 5$. As $p$ varies over all primes (distinct from $\ell$), consider
the number field $F=\mathbb{Q}(\mu_{p})$ and the base-change of $E$ to $F$.
Then, for _at least half_ of the primes $p$,

$a_{v}\not\equiv 2\pmod{p}.$

###### Proof.

Since $E_{/\mathbb{Q}}$ and $\ell$ are fixed, the values of $a_{\ell}$ is
determined precisely. Let $\alpha$ and $\beta$ be the roots of
$X^{2}-a_{\ell}X+\ell$ in $\overline{\mathbb{Q}}$ and let $\bar{\alpha}$ and
$\bar{\beta}$ be the roots of $X^{2}-a_{\ell}X+\ell$ in
$\overline{\mathbb{F}_{p}}$. Then $a_{v}=\alpha^{f}+\beta^{f}\equiv 2\mod p$
if and only if $\bar{\alpha}^{f}=1$ since $(\alpha\beta)^{f}=\ell^{f}\equiv
1\mod p$.

When $p\neq\ell$, the constant term of the polynomial $X^{2}-a_{\ell}X+\ell$
is not zero modulo $p$. In particular, this tells us that $\bar{\alpha}\neq
0$. Thus, if $\bar{\alpha}\notin\mathbb{F}_{p}$, then $\bar{\alpha}^{p-1}\neq
1$. As $f|(p-1)$ by definition, we have furthermore $\bar{\alpha}^{f}\neq 1$.

Note that $\bar{\alpha}\notin\mathbb{F}_{p}$ if and only if the polynomial
$X^{2}-a_{\ell}X+\ell$ is irreducible over $\mathbb{F}_{p}$. When $p$ is odd,
this in turn is equivalent to the discriminant $a_{\ell}^{2}-4\ell$ not being
a square modulo $p$. By the Hasse–Weil bound, $a_{\ell}^{2}-4\ell<0$, so it is
not a square in $\mathbb{Q}$. Therefore, Chebotarev density theorem tells us
that for exactly half of the primes $p$, $a_{\ell}^{2}-4\ell$ is not a square
modulo $p$. Thus, the result follows. ∎

######  Proposition 10.6.

Let $m$ be a fixed prime number and consider the false Tate curve extension
${\mathcal{F}}_{\infty}={\mathcal{F}}_{\infty}^{(m)}$. Let $E_{/\mathbb{Q}}$
be a fixed elliptic curve with good reduction at $m$. As $p$ varies over all
primes of good reduction of $E$, for _at least half_ of the primes,

$\chi_{t}(G,E[p^{\infty}])=\chi_{t}(\Gamma_{F},E[p^{\infty}]).$

###### Proof.

The proof is immediate from Theorem 4.9 and Proposition 10.5. ∎

We consider the following special case.

######  Proposition 10.7.

Suppose that $m=\ell$ is a prime number and that $E$ good _supersingular_
reduction at $\ell$ with $a_{\ell}(E)=0$. Let $v$ denote a prime above $\ell$
in $F$. For exactly _two-third_ of the primes $p$, we have $a_{v}\not\equiv
2\pmod{p}$.

###### Proof.

We have (see [Sil09, Exercise 5.15])

$\\#\widetilde{E}(\kappa_{v})=\begin{cases}\ell^{f}+1&\textrm{ if }f\textrm{
is odd}.\\\ \left(\ell^{f/2}-(-1)^{f/2}\right)^{2}&\textrm{ if }f\textrm{ is
even}.\end{cases}$

In particular, $p|\\#\widetilde{E}(\kappa_{v})$ if and only if $f\equiv
2\pmod{4}$. The latter condition is satisfied precisely _one-third_ of the
time by [CM03, Theorem 1.1 with $l=2$]. Hence the result. ∎

Similar to Proposition 10.6, this implies the following.

######  Proposition 10.8.

Let $m$ be a fixed prime number and consider the false Tate curve extension
${\mathcal{F}}_{\infty}={\mathcal{F}}_{\infty}^{(m)}$. Let $E_{/\mathbb{Q}}$
be a fixed elliptic curve with good _supersingular_ reduction at $m$ and
$a_{m}(E)=0$. As $p$ varies over all primes of good reduction of $E$, for
exactly _two-third_ of the primes,

$\chi_{t}(G,E[p^{\infty}])=\chi_{t}(\Gamma_{F},E[p^{\infty}]).$

######  Remark 10.9.

However, when studying the $\mathcal{G}^{(m)}$-Euler characteristic,
${\mathfrak{M}}$ is a (finite) set of rational primes. If $\ell|m$ is a prime
of good reduction,

$\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1\textrm{ if and only if
}\\#\widetilde{E}(\mathbb{F}_{\ell})=q_{v}+1-a_{\ell}=\ell+1-a_{\ell}\equiv
0\pmod{p}.$

As $p$ varies, we count how often $a_{\ell}\equiv\ell+1\pmod{p}$. By an
application of the Hasse-bound $\left\lvert a_{\ell}\right\rvert\leq
2\sqrt{\ell}$. Thus, $\left\lvert L_{v}(E,1)\right\rvert_{p}$ is a $p$-adic
unit for all but finitely many primes $p$.

The above discussion can be summarized as follows.

######  Theorem 10.10.

Let $E_{/\mathbb{Q}}$ be a fixed elliptic curve and $m$ be a fixed positive
integer. Let $F=\mathbb{Q}(\mu_{p})$ and ${\mathcal{F}}_{\infty}$ be the false
Tate curve extension. As $p$ varies over all primes of good reduction of $E$,
the following assertions hold.

  1. (1)

If $E$ has split multiplicative reduction at $\ell|m$, then the $G$-Euler
characteristic is _never_ a $p$-adic unit.

  2. (2)

If $E$ has non-split multiplicative reduction or additive reduction or good
reduction with $E(F_{v})[p^{\infty}]=0$ at all primes $\ell|m$, then

$\chi_{t}(G,E[p^{\infty}])=\chi_{t}(\Gamma_{F},E[p^{\infty}]).$

In particular, if $E$ has Mordell–Weil rank 0 over $\mathbb{Q}$ and $F$ then,
$\chi(G,E[p^{\infty}])=1$ as $p$ varies over all good _ordinary_ primes
outside a set of density 0.

  3. (3)

If $m$ is a fixed _prime number_ , and $E$ has good _supersingular_ reduction
at $m$, then for two-thirds of the primes $p$

$\chi_{t}(G,E[p^{\infty}])=\chi_{t}(\Gamma_{F},E[p^{\infty}]).$

In particular, if $E$ has Mordell–Weil rank 0 over $\mathbb{Q}$ and $F$ then,
$\chi(G,E[p^{\infty}])=1$ for all such primes $p$.

  4. (4)

If $E$ has good reduction reduction at all primes $\ell|m$ then the
$\mathcal{G}$-Euler characteristic is given by

$\chi_{t}(\mathcal{G},E[p^{\infty}])=\chi_{t}(\Gamma_{\mathbb{Q}},E[p^{\infty}]).$

for all but finitely many primes $p$.

In this case as well, we can make some remarks about the $\Lambda(H)$-ranks of
$\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}$. We know from (8.2) that

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}=\lambda_{p}(E/F_{\mathrm{cyc}})+\sum_{\begin{subarray}{c}\ell|N_{E},\
\ell|m\\\ \ell\textrm{ split
multiplicative}\end{subarray}}1+\sum_{\begin{subarray}{c}v|\ell,\ \ell|m,\\\
E(F_{v})[p]\neq 0\end{subarray}}2.$

Therefore, if the fixed elliptic curve has a prime of split multiplicative
reduction at a prime divisor of $m$, then for all primes $p$,

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}>\lambda_{p}(E/F_{\mathrm{cyc}}).$

By the same argument as before, one can also deduce the following. Suppose
that $m$ is a prime number such that $E$ has good reduction at $m$, then for
_at most half_ of the primes $p$,

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}>\lambda_{p}(E/F_{\mathrm{cyc}}).$

### 10.3.

Fix a pair of elliptic curves $(E,E^{\prime})$, both defined over $\mathbb{Q}$
and suppose that $E^{\prime}/\mathbb{Q}$ is a non-CM elliptic curve. Let
$F=\mathbb{Q}(E^{\prime}[p])$ and consider the $p$-adic Lie extension
${\mathcal{F}}_{\infty}=\mathbb{Q}(E^{\prime}[p^{\infty}])$. By the Weil
pairing, we know that $F\supseteq\mathbb{Q}(\mu_{p})$.

Let $\mathcal{G}_{E^{\prime}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}\left(\mathbb{Q}(E^{\prime}[p^{\infty}])/\mathbb{Q}\right)$.
We will study for what proportion of all primes is
$\chi_{t}(\mathcal{G}_{E^{\prime}},E[p^{\infty}])$ a $p$-adic unit. For us,
${\mathfrak{M}}_{E^{\prime}}$ is a set of primes over $\mathbb{Q}$. By
definition, this set contains precisely the primes of potentially
multiplicative reduction of $E^{\prime}$.

Suppose that $v\in{\mathfrak{M}}_{E^{\prime}}$ is a prime of split (resp. non-
split) multiplicative reduction of $E$. By Lemma 4.6(2) (resp. Lemma 4.6(3))
we know that

$\displaystyle\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1$
$\displaystyle\textrm{ if and only if }q_{v}=\ell\equiv 1\pmod{p}$
$\displaystyle(\textrm{resp. }\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1$
$\displaystyle\textrm{ if and only if }q_{v}=\ell\equiv-1\pmod{p}).$

In either case, $\ell$ is a prime which is independent of $p$. As $p$ varies
over all primes, either of the congruence conditions can be satisfied by _at
most_ finitely many primes $p$.

If $v\in{\mathfrak{M}}_{E^{\prime}}$ is a prime of good reduction of $E$, then
Lemma 4.6(1) asserts that

$\left\lvert L_{v}(E,1)\right\rvert_{p}\neq 1\textrm{ if and only if
}\\#\widetilde{E}(\mathbb{F}_{\ell})=\ell+1-a_{\ell}\equiv 0\pmod{p}.$

As before, we see that $\left\lvert L_{v}(E,1)\right\rvert_{p}$ is a $p$-adic
unit for all but finitely many primes $p$.

We have the following theorem.

######  Theorem 10.11.

Let $(E,E^{\prime})$ be a pair of elliptic curves over $\mathbb{Q}$ where
$E^{\prime}$ is a non-CM curve. For all but finitely many primes $p$,

$\chi_{t}\left(\mathcal{G}_{E^{\prime}},E[p^{\infty}]\right)=\chi_{t}\left(\Gamma_{\mathbb{Q}},E[p^{\infty}]\right).$

The proportion of primes for which
$\chi_{t}(\Gamma_{\mathbb{Q}},E[p^{\infty}])=1$ was studied in detail [KR21a,
Section 3]. It was conjectured [KR21a, Conjecture 3.17] that
$\chi_{t}(\Gamma_{\mathbb{Q}},E[p^{\infty}])=1$ should be true for $100\%$ of
the primes of good _ordinary_ reduction.

To study the $G_{E^{\prime}}$-Euler characteristic, we consider the base-
change curve ${E}_{/F}$. Since $F\supseteq\mathbb{Q}(\mu_{p})$ the arguments
of Section 10.2 imply that $q_{v}\equiv 1\pmod{p}$ for all values of $p$.
Indeed, this is because the inertia degree of $\ell$ for the extension
$\mathbb{Q}(\mu_{p})/\mathbb{Q}$ divides the of inertia degree of $\ell$ for
the extension $F/\mathbb{Q}$.

This means, if $\ell$ is a prime of potentially multiplicative reduction of
$E^{\prime}$ which is also a prime of split multiplicative reduction of $E$,
then the $G_{E^{\prime}}$-Euler characteristic of $E$ is _never_ trivial. This
is well-known in the special case $E=E=E^{\prime}$. It was shown in [CH01,
Theorem 1.5] that the $p$-primary Selmer group is infinite dimensional for all
$p\geq 5$.

Suppose that we fix the two distinct elliptic curves $E$ and $E^{\prime}$
(both defined over $\mathbb{Q}$) such that the following properties hold:

  1. (a)

$E^{\prime}$ is a non-CM elliptic curve.

  2. (b)

$E$ has additive reduction or non-split multiplicative reduction at the primes
above potentially multiplicative reduction of $E^{\prime}$.

Then, the contribution of the local Euler factors to the
$G_{E^{\prime}}$-Euler characteristic of $E$ is trivial. Equivalently,

$\chi_{t}\left(G_{E^{\prime}},E[p^{\infty}]\right)=\chi_{t}\left(\Gamma_{F},E[p^{\infty}]\right).$

When $F/\mathbb{Q}$ is a number field and
$\operatorname{rank}_{\mathbb{Z}}E(F)\geq 1$, the analysis of the Euler
characteristic formula is more subtle. The difficulty arises from our lack of
knowledge regarding the normalized $p$-adic regulator and the $p$-part of the
Tate–Shafarevich group over number fields. Also, when
$[F\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$ is ``large'',
computations are expensive and it is hard to obtain meaningful heuristics.
However, it is still possible to make some brief remarks about the
Mordell–Weil rank growth of $E$ in the extension ${\mathcal{F}}_{\infty}/F$.

######  Proposition 10.12.

Suppose $E,E^{\prime}$ are two fixed elliptic curves chosen as described above
with conductors $N_{1},N_{2}$, respectively. Then, for all primes $p\nmid
N_{1}N_{2}$,

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}=\lambda_{p}(E/F_{\mathrm{cyc}}).$

###### Proof.

The hypothesis on $E$ and $E^{\prime}$ guarantee that $E[p^{\infty}]$ is not
rational over $\mathbb{Q}(E^{\prime}[p^{\infty}])$. As $p$ varies over all odd
primes away from the divisors of $N_{1}N_{2}$, the base field
$F=\mathbb{Q}(E^{\prime}[p])$ varies as well. It follows from (5.1) and Lemma
5.5 that

$\operatorname{rank}_{\Lambda(H)}\mathcal{X}(E/{\mathcal{F}}_{\infty})_{f}=\lambda_{p}(E/F_{\mathrm{cyc}}).$

In particular, Theorem 5.1 asserts that

$\operatorname{rank}_{\mathbb{Z}}E(F_{n})\leq\lambda_{p}(E/F_{\mathrm{cyc}})p^{3n}.$

∎

##  Appendix A Classifying conjugacy classes in
$\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$

The results in this appendix are used in the proof of Corollary 8.10. Given a
prime $p$, write $G_{p}=\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$. Given
a matrix $\sigma\in G_{p}$, denote by $\alpha=\alpha(\sigma)$ and
$\beta=\beta(\sigma)$ the two eigenvalues of $\sigma$. Here, $\alpha$ and
$\beta$ are interchangeable, however, assume that each $\sigma$ comes with a
choice of $\alpha$ and $\beta$. As is well known, $\left\lvert
G_{p}\right\rvert=(p^{2}-1)(p^{2}-p)=p(p-1)^{2}(p+1)$. Given an element
$\sigma\in G_{p}$, write $f(\sigma)$ for the smallest integer
$f\in\mathbb{Z}_{\geq 1}$ such that $\det(\sigma)^{f}=1$. Let $H_{p}$ be the
subset of $G_{p}$ consisting of elements $\sigma\in G_{p}$ such that
$\alpha^{f(\sigma)}=\beta^{-f(\sigma)}\neq 1$, where $\alpha$ and $\beta$ are
the eigenvalues of $\sigma$. We divide $G_{p}$ into the following conjugacy
classes. (See [Lan02, Chapter XVIII, Table 12.4] or http://www-
math.mit.edu/~dav/gl2conj.pdf.)

  * •

Let $C_{a,b}$ be the set of diagonalizable matrices with eigenvalues
$a,b\in{\mathbb{F}_{p}}^{\times}$ with $a\neq b$. We have $(p-1)(p-2)/2$
choices of $C_{a,b}$ and for each choice, $\\#C_{a,b}=p(p+1)$.

  * •

Let $\mathcal{C}_{a}$ be the set of non-diagonal matrices with one single
eigenvalue $a\in{\mathbb{F}_{p}}^{\times}$. There are $(p-1)$ choices for
$\mathcal{C}_{a}$ and for each choice, $\\#\mathcal{C}_{a}=p^{2}-1$.

  * •

Let $D_{a}=\left\\{\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}\right\\}$,
$a\in\mathbb{F}_{p}^{\times}$. Then, there are $(p-1)$ choices for $a$ and for
each choice $\\#D_{a}=1$.

  * •

Let $E_{\lambda}$ be the set of matrices whose eigenvalues are $\lambda$ and
$\lambda^{\prime}$, where
$\lambda\in\mathbb{F}_{p^{2}}\setminus\mathbb{F}_{p}$ and $\lambda^{\prime}$
is the conjugate of $\lambda$. There are $p(p-1)/2$ choices for $\lambda$ and
for each choice of $\lambda$, $\\#E_{\lambda}=p^{2}-p$.

Given an element $a\in\bar{\mathbb{F}}_{p}^{\times}$, we write $o(a)$ for the
order of $a$, i.e., the minimal value of $N\in\mathbb{Z}_{\geq 1}$ such that
$a^{N}=1$. Also, let $\varphi$ denote Euler's totient function, where
$\varphi(n)$ is the number of positive integers $m<n$ that are coprime to $n$.
Recall that $\varphi(n)=n\prod_{\ell}\left(1-\frac{1}{\ell}\right)$, where
$\ell$ runs through all prime divisors of $n$ and that
$\sum_{d|n}\varphi(d)=n$.

###  A.1. Contributions from $E_{\lambda}$

Let $\sigma\in E_{\lambda}$ for some
$\lambda\in\mathbb{F}_{p^{2}}\setminus\mathbb{F}_{p}$. Since
$\lambda\lambda^{\prime}\in\mathbb{F}_{p}^{\times}$, it follows that
$f(\sigma)=o(\lambda\lambda^{\prime})$ has to be a divisor of $p-1$. However,
the fact that $\lambda\notin\mathbb{F}_{p}^{\times}$ implies that
$\lambda^{p-1}\neq 1$. Thus, $\sigma\in H_{p}$. In particular, this gives that
the number of elements are

$\sum_{\lambda}\\#E_{\lambda}=p^{2}(p-1)^{2}/2.$

###  A.2. Contributions from $D_{a}$ and $\mathcal{C}_{a}$

Let $\sigma\in D_{a}$ or $\mathcal{C}_{a}$ for some
$a\in\mathbb{F}_{p}^{\times}$, then note that $f(\sigma)=o(a^{2})$. As we
shall see, in this case, $\sigma\in H_{p}$ if and only if $o(a)$ is even.
First suppose that $o(a)$ is even, then, we have that $o(a)/2=f(\sigma)$.
Therefore, $a^{f(\sigma)}=-1$, and hence, $\sigma\in H_{p}$, since
$a^{f(\sigma)}\neq 1$. Else, when $o(a)$ is odd, then it is easily checked
that $\sigma\notin H_{p}$.

It remains to determine which $a\in\mathbb{F}_{p}^{\times}$ has $o(a)\in
2\mathbb{Z}$. Suppose that $p-1=2^{e}n$, where $n$ is an odd integer. Then,
$o(a)$ is odd if and only if $o(a)|n$. Therefore, we have in total

$p-1-\sum_{r|n}\varphi(r)=p-1-n=(p-1)(1-2^{-e})$

choices of $a$. It follows that the number of elements in $H_{p}$ is given by

$(1+(p^{2}-1))(p-1)(1-2^{-e})=p^{2}(p-1)(1-2^{-e}).$

######  Remark A.1.

Since $e\geq 1$, on combining the contributions from $E_{\lambda}$, $D_{a}$
and $\mathcal{C}_{a}$, we have

$\frac{\\#H_{p}}{\\#G_{p}}\geq\frac{p^{2}(p-1)^{2}/2+p^{2}(p-1)/2}{p(p-1)^{2}(p+1)}=\frac{p^{2}}{2(p^{2}-1)}.$

###  A.3. Contributions from $C_{a,b}$

Given $a\in\mathbb{F}_{p}^{\times}$, we would like to find $b$ such that
$o(ab)$ is not a multiple of $o(a)$. Recall that for each $d|(p-1)$, there are
$\varphi(d)$ elements of order $d$ in $\mathbb{F}_{p}^{\times}$. Therefore the
number of choices of $ab$ is

$\sum_{o(a)\nmid d|p-1}\varphi(d)=p-1-\sum_{o(a)\left\lvert d\right\rvert
p-1}\varphi(d).$

But $\\{ab\mathrel{\mathop{\mathchar
58\relax}}b\in\mathbb{F}_{p}^{\times}\\}=\mathbb{F}_{p}^{\times}$. Therefore,
we have the same number of choices for $b$. Excluding those choices where
$a=b$ (there are $(p-1)(1-2^{-e})$ such possibilities, where
$e=\operatorname{ord}_{2}(p-1)$) and repetitions (swapping $a$ and $b$), there
are in total

$\displaystyle\frac{1}{2}\left(\sum_{r|p-1}\varphi(r)\left(p-1-\sum_{r\left\lvert
d\right\rvert p-1}\varphi(d)\right)-(p-1)(1-2^{-e})\right)$ $\displaystyle=\ $
$\displaystyle\frac{1}{2}\left((p-1)^{2}-\sum_{r\left\lvert d\right\rvert
p-1}\varphi(r)\varphi(d)-(p-1)(1-2^{-e})\right)$ $\displaystyle=\ $
$\displaystyle\frac{1}{2}\left((p-1)(p-2+2^{-e})-\sum_{d|p-1}d\varphi(d)\right)=\
\frac{1}{2}\left((p-1)(p-2+2^{-e})-\sum_{d|p-1}\varphi(d^{2})\right)$

classes of $C_{a,b}$ belonging to $H_{p}$. In total, the number of elements
are

$\frac{1}{2}\left((p-1)(p-2+2^{-e})-\sum_{d|p-1}\varphi(d^{2})\right)p(p+1).$

######  Remark A.2.

Writing $p-1=\prod_{i}q_{i}^{n_{i}}$, where $q_{i}$ are distinct primes and
$n_{i}\geq 1$, we have

$\displaystyle\sum_{d|p-1}\varphi(d^{2})$
$\displaystyle=\prod_{i}\left(\sum_{m=0}^{n_{i}}\varphi(q_{i}^{2m})\right)=\prod_{i}\left(\sum_{m=0}^{2n_{i}}(-1)^{m}q_{i}^{m}\right)$
$\displaystyle=\prod_{i}q_{i}^{2n_{i}}\frac{1+q_{i}^{-2n_{i}-1}}{1+q_{i}^{-1}}=(p-1)^{2}\prod_{i}\frac{1+q_{i}^{-2n_{i}-1}}{1+q_{i}^{-1}}.$

Let us write $\kappa_{p}=\prod_{i}\frac{1+q_{i}^{-2n_{i}-1}}{1+q_{i}^{-1}}$,
which is a constant strictly smaller than $1$. Then the proportion of elements
in $H_{p}$ coming from $C_{a,b}$ is

$\frac{\frac{1}{2}\left((p-1)(p-2+2^{-e})-(p-1)^{2}\kappa_{p}\right)p(p+1)}{p(p-1)^{2}(p+1)}=\frac{1}{2}\left(\frac{p-2+2^{-e}}{p-1}-\kappa_{p}\right).$

Observe that

$\kappa_{p}\leq\frac{1+2^{-2e-1}}{1+2^{-1}}=\frac{2+2^{-2e}}{3}.$

Therefore, for $p\geq 7$, we have

$\displaystyle\frac{1}{2}\left(\frac{p-2+2^{-e}}{p-1}-\kappa_{p}\right)$
$\displaystyle\geq\frac{1}{2}\left(\frac{p-2+2^{-e}}{p-1}-\frac{2+2^{-2e}}{3}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{p-2}{p-1}-\frac{2}{3}+\frac{2^{-e}}{p-1}-\frac{2^{-2e}}{3}\right)$
$\displaystyle\geq\frac{1}{2}\left(\frac{2p-3}{2(p-1)}-\frac{3}{4}\right),$

which tends to $1/8$ as $p\rightarrow\infty$. The inequality in the second
last line follows from the fact that the real function
$f(x)=\frac{x}{p-1}-\frac{x^{2}}{3}$ attains a minimum at $x=1/2$ in the
interval $(0,1/2]$ when $p\geq 7$.

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††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Heisenberg-limited metrology via weak-value amplification
without using entangled resources

Yosep Kim  Department of Physics, Pohang University of Science and Technology
(POSTECH), Pohang 37673, Korea  Seung-Yeun Yoo  Department of Physics, Pohang
University of Science and Technology (POSTECH), Pohang 37673, Korea  Yoon-Ho
Kim  Department of Physics, Pohang University of Science and Technology
(POSTECH), Pohang 37673, Korea

###### Abstract

Weak-value amplification (WVA) provides a way for amplified detection of a
tiny physical signal at the expense of a lower detection probability. Despite
this trade-off, due to its robustness against certain types of noise, WVA has
advantages over conventional measurements in precision metrology. Moreover, it
has been shown that WVA-based metrology can reach the Heisenberg-limit using
entangled resources, but preparing macroscopic entangled resources remains
challenging. Here we demonstrate a novel WVA scheme based on iterative
interactions, achieving the Heisenberg-limited precision scaling without
resorting to entanglement. This indicates that the perceived advantages of the
entanglement-assisted WVA are in fact due to iterative interactions between
each particle of an entangled system and a meter, rather than coming from the
entanglement itself. Our work opens a practical pathway for achieving the
Heisenberg-limited WVA without using fragile and experimentally-demanding
entangled resources.

Precise measurement of an interaction parameter is essential for studying
various quantum physical effects. The interaction parameter can be estimated
from the state change of a meter after interacting with a system. Since the
system and the meter are generally entangled by the interaction, post-
selection of the system collapses the meter’s state: an elaborate post-
selection scheme enables amplified detection of the interaction parameter. In
the weak interaction regime, the amplification factor corresponds to the so-
called weak value, and thus this metrological protocol is known as weak-value
amplification (WVA) Aharonov88 ; Dressel14 ; Cho10 . WVA has been actively
deployed in precision metrology to amplify tiny physical effects, such as the
optical spin Hall effect Hosten08 , an ultra-small phase shift Strubi13 ; Xu13
, and the nonlinear effect due to a weak beam of light Feizpour11 ; Hallaji17
; Chen18 .

WVA provides enhanced sensitivity over conventional measurements (in the sense
that small signals are physically amplified), but the enhancement does not
necessarily result in a higher signal-to-noise ratio (SNR) due to the reduced
detection probability by post-selection, which in turn causes a greater
statistical error Combes14 ; Ferrie14 ; Knee14 ; Zhang15 . Despite this trade-
off, WVA-based metrology has attracted a lot of attention as it offers
meaningful robustness against certain types of noise, e.g., temporally
correlated noises Feizpour11 ; Sinclair17 , detector jitter Kedem12 ; Jordan14
; Viza15 , and detector saturation Harris17 ; Xu20 . In addition, there are
proposals to achieve the Heisenberg-limited precision scaling in WVA-based
metrology using quantum resources such as squeezing Pang15S  and entanglement
Pang14 ; Pang15E . Recently, the entanglement-assisted WVA has been
demonstrated in photonic systems Chen19 ; Starek20 , but the practical
metrological applications are limited by the difficulty in preparing
macroscopic entangled resources.

Here we demonstrate a novel WVA scheme based on iterative interactions,
achieving the Heisenberg-limit without resorting to entanglement. $N$
iterative interactions enhance the post-selection probability by $N^{2}$ while
keeping the amplification factor unchanged. In a proof-of-principle
experiment, we estimate a path-dependent polarization change using our WVA
scheme, showing a good agreement with the Heisenberg-limited precision
scaling.

Theory.—We first briefly review a typical WVA-based metrology depicted in Fig.
1(a). Given the initial system state as $|\psi\rangle_{\text{s}}$ and the
initial meter state as $|\\!+\\!x\rangle_{\text{m}}$, the weak interaction of
$\hat{U}=e^{i\gamma\hat{A}\otimes\hat{\sigma}_{z}}$ changes the system-meter
state to,

$\hat{U}|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}\approx|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}+i\gamma\hat{A}|\psi\rangle_{\text{s}}|\\!-\\!x\rangle_{\text{m}},$
(1)

where $\hat{A}$ is the system observable, $\hat{\sigma}_{z}$ is the Pauli-$z$,
and $|\\!\pm\\!x\rangle_{\text{m}}$ are the eigenstates of the Pauli-$x$.
Assuming the initial states and the system observable are known, the
interaction parameter $\gamma$ is estimated from the change of the meter
state. Moreover, the state change can be significantly amplified by projecting
the system onto $|\phi\rangle_{\text{s}}$ and post-selecting the outcome,

${}_{\text{s}}\langle\phi|\hat{U}|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}\approx\
\\!_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}(|\\!+\\!x\rangle_{\text{m}}+i\gamma\langle\hat{A}\rangle_{\text{w}}|\\!-\\!x\rangle_{\text{m}}).$
(2)

The post-selection probability is
$P\approx|{}_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}|^{2}$, and the
amplification factor corresponds to the weak value Aharonov88 ,

$\langle\hat{A}\rangle_{\text{w}}=\frac{{}_{\text{s}}\langle\phi|\hat{A}|\psi\rangle_{\text{s}}}{{}_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}}.$
(3)

The weak value can be outside the eigenvalue spectrum of $\hat{A}$ and is
generically a complex number. These intriguing features have been extensively
studied in the context of amplified detection Hosten08 ; Strubi13 ; Xu13 ;
Feizpour11 ; Hallaji17 ; Chen18  and, more recently, in the context of quantum
information science, e.g., quantum tomography Lundeen11 ; Kim18 , quantum
foundation Kocsis11 ; Mahler16 ; Cho19 ; Ramos20 ; Kim21 , and decoherence
management Kim12 .

Figure 1: Three weak-value amplification (WVA) scenarios for estimating an
interaction strength $\gamma$ with a total of $N$ interactions between a
system and a meter. (a) The meter state is measured independently $N$ times
after a single interaction with a system particle and the post-selection of
the system. (b) The meter state is measured once after $N$ consecutive
interactions with each particle of an $N$-partite entangled system and the
post-selection of the system. (c) The meter state is measured once after $N$
iterative interactions with a system particle and the post-selection of the
system.

When repeating $N$ independent WVA measurements of the meter state in Eq. (2),
the minimum achievable uncertainty of an unbiased estimator $\gamma$ is given
by the quantum Cramér-Rao bound Braunstein94 ; Braunstein96 ,

$\Delta\gamma_{\text{ind}}^{(N)}\geq\frac{1}{\sqrt{NPI(\gamma)}},$ (4)

where $P$ is the post-selection probability and $I(\gamma)\approx
4|\langle\hat{A}\rangle_{\text{w}}|^{2}$ is the quantum Fisher information
contained in the meter state under the weak interaction condition of
$|\gamma|\ll|1/\langle\hat{A}\rangle_{\text{w}}|$, see Supplemental Material
for the details. The result in Eq. (4) shows that the measurement uncertainty
of a typical WVA-based metrology scales with $1/\sqrt{N}$ according to the
standard quantum limit Giovannetti04 . To achieve a large amplification factor
in Eq. (3) and large quantum Fisher information, the post-selection state
$|\phi\rangle_{\text{s}}$ has to be set to reduce a post-selection probability
$P\approx|{}_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}|^{2}$. Considering
the uncertainty bound in Eq. (4), the perceived enhancement from the amplified
detection is, therefore, inevitably compromised by the small detection
probability. However, note again that the amplified detection enables us to
suppress certain types of noise Feizpour11 ; Sinclair17 ; Kedem12 ; Jordan14 ;
Viza15 ; Harris17 ; Xu20 .

We can leverage an entangled system to overcome the standard quantum limit in
Eq. (4) Pang14 ; Pang15E ; Chen19 ; Starek20 . The entanglement-assisted WVA
scheme is depicted in Fig. 1(b). Each particle of the $N$-partite entangled
system in $|\Psi\rangle_{\text{s}}$ interacts consecutively with the meter in
$|\\!+\\!x\rangle_{\text{m}}$, and then the system is post-selected by
$|\Phi\rangle_{\text{s}}$ for amplified detection of the interaction parameter
$\gamma$. The interaction is written as $\hat{U}^{\otimes
N}=e^{i\gamma\sum_{k=1}^{N}\hat{A}_{k}\otimes\hat{\sigma}_{z}}$, where
$\hat{A}_{k}=\hat{\mathbb{I}}\otimes\dots\otimes\hat{A}\otimes\dots\otimes\hat{\mathbb{I}}$
denotes the observable $\hat{A}$ for the $k$-th system particle. According to
the original proposal Pang14 , the entangled system is initially prepared as a
maximally entangled state and is post-selected as follows:

$\displaystyle|\Psi\rangle_{\text{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\lambda_{\text{max}}\rangle_{\text{s}}^{\otimes
N}+|\lambda_{\text{min}}\rangle_{\text{s}}^{\otimes N}),$ (5)
$\displaystyle|\Phi\rangle_{\text{s}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{2}|\langle\hat{A}\rangle_{\text{w}}|}\bigg{[}(\langle\hat{A}\rangle_{\text{w}}^{*}-N\lambda_{\text{min}})|\lambda_{\text{max}}\rangle_{\text{s}}^{\otimes
N}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(\langle\hat{A}\rangle_{\text{w}}^{*}-N\lambda_{\text{max}})|\lambda_{\text{min}}\rangle_{\text{s}}^{\otimes
N}\bigg{]}.$

Here $|\lambda_{\text{max}(\text{min})}\rangle_{\text{s}}$ is the eigenstate
of $\hat{A}$ and has the maximum (minimum) eigenvalue of
$\lambda_{\text{max}(\text{min})}$. Then, under the assumption of a large
amplification factor and a small interaction strength,

$|N\lambda_{\text{max(min)}}|\ll|\langle\hat{A}\rangle_{\text{w}}|\ll|\gamma|^{-1},$
(6)

we can estimate the interaction parameter $\gamma$ with the enhanced post-
selection probability,

$P_{\text{ent}}^{(N)}\approx|{}_{\text{s}}\langle\Phi|\Psi\rangle_{\text{s}}|^{2}\approx\frac{N^{2}(\lambda_{\text{max}}-\lambda_{\text{min}})^{2}}{4|\langle\hat{A}\rangle_{\text{w}}|^{2}},$
(7)

while keeping the amplification factor in the meter state the same with Eq.
(2),

${}_{\text{s}}\langle\Phi|\hat{U}^{\otimes
N}|\Psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}\approx\
\\!_{\text{s}}\langle\Phi|\Psi\rangle_{\text{s}}(|\\!+\\!x\rangle_{\text{m}}+i\gamma\langle\hat{A}\rangle_{\text{w}}|\\!-\\!x\rangle_{\text{m}}),$
(8)

For a fair comparison, the number of interactions in the entanglement-assisted
WVA in Fig. 1(b) needs to be the same with $N$ independent measurements of the
typical WVA in Fig. 1(a). To do so, we estimate the quantum Cramér-Rao bound
for a single entanglement-assisted WVA measurement to be,

$\Delta\gamma^{(N)}_{\text{ent}}\geq\frac{1}{\sqrt{P^{(N)}_{\text{ent}}I(\gamma)}}=\frac{1}{N|\lambda_{\text{max}}-\lambda_{\text{min}}|},$
(9)

where the meter state in Eq. (8) gives $I(\gamma)\approx
4|\langle\hat{A}\rangle_{\text{w}}|^{2}$. The uncertainty bound clearly shows
that the use of an $N$-partite entangled system enables the Heisenberg-limited
precision scaling, achieving the precision improvement of $\sqrt{N}$ over the
case of $N$ independent measurements in Eq. (4).

Figure 2: Experimental schematic to estimate a path-dependent polarization
change via WVA based on iterative interactions. The polarization mode and the
path mode of a single photon are employed as the system and the meter,
respectively, and they are initialized using the first beam displacer (BD) and
half-wave plate (HWP). The path-dependent polarization change of $\gamma$ is
implemented by rotating the next HWP in each path in opposite directions, and
the $N$ iterative interactions are introduced by setting the HWP angles at
$N\gamma/2$ and $-N\gamma/2$, respectively. The system (polarization mode) is
then projected onto $|\phi\rangle_{\text{s}}$ by post-selecting the photon
coming from the lower path of the second BD, and the post-selection state is
determined by the HWP angle $\phi$. The operation of the second BD converts
the path mode state into the polarization mode state, so the meter readout is
carried out using the polarizing beam splitter (PBS).

We now present a WVA scheme which can achieve the Heisenberg-limit without
using entangled resources. As shown in Fig. 1(c), we consider $N$ iterative
interactions, $\hat{U}^{N}=e^{iN\gamma\hat{A}\otimes\hat{\sigma}_{z}}$,
between the system and the meter prior to the detection. Interestingly, if the
initial and post-selection system states are set to have the same relative
amplitudes as those in Eq. (5),

$\displaystyle|\psi\rangle_{\text{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\lambda_{\text{max}}\rangle_{\text{s}}+|\lambda_{\text{min}}\rangle_{\text{s}}),$
(10) $\displaystyle|\phi\rangle_{\text{s}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{2}|\langle\hat{A}\rangle_{\text{w}}|}\bigg{[}(\langle\hat{A}\rangle_{\text{w}}^{*}-N\lambda_{\text{min}})|\lambda_{\text{max}}\rangle_{\text{s}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(\langle\hat{A}\rangle_{\text{w}}^{*}-N\lambda_{\text{max}})|\lambda_{\text{min}}\rangle_{\text{s}}\bigg{]},$

under the condition of Eq. (6), the amplification factor is fixed to
$\langle\hat{A}\rangle_{\text{w}}$ independently of $N$,

${}_{\text{s}}\langle\phi|\hat{U}^{N}|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}\approx\
\\!_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}(|\\!+\\!x\rangle_{\text{m}}+i\gamma\langle\hat{A}\rangle_{\text{w}}|\\!-\\!x\rangle_{\text{m}}),$
(11)

and the system post-selection probability becomes identical to Eq. (7),

$P_{\text{iter}}^{(N)}\approx|_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}|^{2}\approx\frac{N^{2}(\lambda_{\text{max}}-\lambda_{\text{min}})^{2}}{4|\langle\hat{A}\rangle_{\text{w}}|^{2}}.$
(12)

Consequently, the quantum Cramér-Rao bound for a single WVA measurement with
$N$ iterative interactions also follows the Heisenberg-limited scaling,

$\Delta\gamma^{(N)}_{\text{iter}}\geq\frac{1}{\sqrt{P^{(N)}_{\text{iter}}I(\gamma)}}=\frac{1}{N|\lambda_{\text{max}}-\lambda_{\text{min}}|}.$
(13)

This result indicates that the perceived advantages of the entanglement-
assisted WVA Pang14 ; Pang15E ; Chen19 ; Starek20  are in fact due to
iterative local interactions between each particle of the entangled system and
the meter, rather than coming from the entanglement itself.

Experiment.—To demonstrate our WVA scheme based on iterative interactions, we
experimentally estimate a small path-dependent polarization change. The
experimental schematic is depicted in Fig. 2. We employ the polarization mode
and the path mode of a single-photon as the system and the meter,
respectively. The system state can be written as a linear combination of the
right circular $|\\!+\\!y\rangle_{\text{s}}$ and the left circular
$|\\!-\\!y\rangle_{\text{s}}$ polarization states, and the upper and lower
path states are denoted by $|\\!+\\!z\rangle_{\text{m}}$ and
$|\\!-\\!z\rangle_{\text{m}}$, respectively. Here, $|\\!\pm\\!k\rangle$ have
the eigenvalues of $\pm 1$ for the Pauli-$k$ operator, $\hat{\sigma}_{k}$. To
initialize the system-meter state to,

$|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}=\frac{1}{\sqrt{2}}(|\\!+\\!y\rangle_{\text{s}}+|\\!-\\!y\rangle_{\text{s}})|\\!+\\!x\rangle_{\text{m}},$
(14)

we prepare a horizontally polarized single-photon in the path superposition,
$|\\!+\\!x\rangle_{\text{m}}=(|\\!+\\!z\rangle_{\text{m}}+|\\!-\\!z\rangle_{\text{m}})/\sqrt{2}$,
by using a beam displacer (BD) and a half-wave plate (HWP). The details of the
single-photon source are presented in Supplemental Material.

As shown in Fig. 2, the weak interaction that causes a small path-dependent
polarization change of $\gamma$ is implemented by rotating the angles of HWPs
conditioned on the paths. The $N$ iterative interactions are introduced by
setting the HWP angles at $N\gamma/2$ for $|\\!+\\!z\rangle_{\text{m}}$ and
$-N\gamma/2$ for $|\\!-\\!z\rangle_{\text{m}}$. With an additional HWP set at
$0^{\circ}$, the conditional polarization operations are given as $e^{\pm
iN\gamma\hat{\sigma}_{y}}$, and thus the iterative weak interactions are
written as,

$\displaystyle\hat{U}^{N}$ $\displaystyle=$ $\displaystyle
e^{iN\gamma\hat{\sigma}_{y}}\otimes{}_{\text{m}}|\\!+\\!z\rangle\langle+z|_{\text{m}}+e^{-i\gamma
N\hat{\sigma}_{y}}\otimes{}_{\text{m}}|\\!-\\!z\rangle\langle-z|_{\text{m}}$
(15) $\displaystyle=$ $\displaystyle
e^{iN\gamma\hat{\sigma}_{y}\otimes\hat{\sigma}_{z}}.$

For the amplified detection of $\gamma$, we project the system (the
polarization mode) onto,

$|\phi\rangle_{\text{s}}=\frac{1}{\sqrt{2}}(|\\!+\\!y\rangle_{\text{s}}+e^{4i\phi}|\\!-\\!y\rangle_{\text{s}}),$
(16)

using the two HWPs at $\phi$ and $45^{\circ}$ and the second BD. Then, the
meter (the path mode) state is found to be,

${}_{\text{s}}\langle\phi|\hat{U}^{N}|\psi\rangle_{\text{s}}|\\!+\\!x\rangle_{\text{m}}\approx\
\\!_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}(|\\!+\\!x\rangle_{\text{m}}+i\gamma\langle
N\hat{\sigma}_{y}\rangle_{\text{w}}|\\!-\\!x\rangle_{\text{m}}),$ (17)

where the weak value of $\langle
N\hat{\sigma}_{y}\rangle_{\textrm{w}}={}_{\text{s}}\langle\phi|N\hat{\sigma}_{y}|\psi\rangle_{\text{s}}/{}_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}$
can be large by choosing $\phi$ so that
${}_{\text{s}}\langle\phi|\psi\rangle_{\text{s}}\approx 0$ at the limit of
$|\gamma|\ll|1/\langle N\hat{\sigma}_{y}\rangle_{\text{w}}|$. The interaction
parameter $\gamma$ in the meter state can be obtained from the Pauli-$z$
expectation of the meter state:

$\langle\hat{\sigma}_{z}\rangle_{\text{m}}\approx 2\gamma\
\\!\text{Im}[\langle N\hat{\sigma}_{y}\rangle_{\text{w}}].$ (18)

This shows that, in the meter readout, the interaction parameter $\gamma$ is
amplified by the imaginary weak value $\text{Im}[\langle
N\hat{\sigma}_{y}\rangle_{\text{w}}]$. As the operation of the second BD in
Fig. 2 converts the path mode state into the polarization mode state, the
meter readout is carried out using a polarizing beam splitter (PBS) and is
estimated from
$\langle\hat{\sigma}_{z}\rangle_{\text{m}}=(n_{1}-n_{2})/(n_{1}+n_{2})$, where
$n_{1}$ and $n_{2}$ are the photon counts detected at D1 and D2, repectively.

Figure 3: The amplified meter readouts
$\langle\hat{\sigma}_{z}\rangle_{\text{m}}\approx 2\gamma\
\\!\text{Im}[\langle N\hat{\sigma}_{y}\rangle_{\text{w}}]$ and the detection
probabilities $P^{(N)}_{\text{iter}}$ for the WVA measurements of
$\gamma=1^{\circ}$ with (a) a single interaction and (b) four iterative
interactions. As the post-selection state approaches the completely orthogonal
state of the initial system state, $\phi=45^{\circ}$, the amplification effect
on the meter expectation value $\langle\hat{\sigma}_{z}\rangle_{\text{m}}$ is
enhanced while the detection probability decreases. The solid lines show the
exact theoretical curves, and the error bars represent one standard deviation
of Poissonian counting statistics. Figure 4: The detection probabilities
$P^{(N)}_{\text{iter}}$ of WVA based on $N$ iterative interactions. At each
$N$, we adjust the post-selection state $|\phi\rangle_{\text{s}}$ according to
Eq. (10) in order to fix the amplification factor to $|\langle
N\hat{\sigma}_{y}\rangle_{\text{w}}|=11.5$. For $N\\!=\\!1,2,3,4$, the
amplified meter expectation values for $\gamma=1^{\circ}$ are, respectively,
measured as $\langle\hat{\sigma}_{z}\rangle_{\text{m}}=0.41\pm 0.16$, $0.52\pm
0.08$, $0.37\pm 0.06$, $0.35\pm 0.04$ with
$\phi=47.5^{\circ},50.0^{\circ},52.3^{\circ},54.6^{\circ}$. The red bars
represent the experimental data, and the dashed bars represent the ideally
expected $P^{(N)}_{\text{iter}}$. The corresponding Heisenberg-limited scaling
of $P^{(N)}_{\text{H}}\propto N^{2}$ is shown in the beige bars. The detection
probabilities $P^{(N)}_{\text{iter}}$ have slight deviations from
$P^{(N)}_{\text{H}}$ since the condition of Eq. (6), $N\ll|\langle
N\hat{\sigma}_{y}\rangle_{\text{w}}|=11.5$, does not hold as $N$ increases.
This deviation can be elimilated by increasing the amplification factor. The
error bars represent one standard deviation of Poissonian counting statistics.

Experimental results.—First, by varying the post-selection angle $\phi$ in Eq.
(16), we study the trade-off relation between the meter expectation of
$\langle\hat{\sigma}_{z}\rangle_{\text{m}}$ and the detection probability of
$P^{(N)}_{\text{iter}}$ with $\gamma=1^{\circ}$ at $N=1$ and $4$. Figure 3
shows that the meter expectation is more amplified as the detection
probability decreases. In addition, as expected from Eq. (18), we observe that
the WVA with $N=4$ iterative interactions provides nearly four times greater
meter expectation values $\langle\hat{\sigma}_{z}\rangle_{\text{m}}$ compared
to the WVA with $N=1$ while keeping the detection probability
$P^{(N)}_{\text{iter}}$ almost independent of the number of interactions. The
amplification factor is not exactly four times greater as the approximation
condition, $|\gamma|\ll|1/\langle N\hat{\sigma}_{y}\rangle_{\text{w}}|$, does
not satisfy well with a large amplification factor.

Next, we measure the enhanced detection probabilities from $N=1$ to 4 with a
fixed amplification factor $|\langle N\hat{\sigma}_{y}\rangle_{\text{w}}|$.
For the purpose of this demonstration, we set the amplification factor to 11.5
by varying the post-selection state $|\phi\rangle_{\text{s}}$ at each $N$
according to Eq. (10). The experimental results in Fig. 4 show almost
quadratic enhancement in the detection probability $P^{(N)}_{\text{iter}}$
with increased $N$. Given the same amplification factor in the meter state, as
the minimum achievable uncertainty bound in Eq. (13) is inversely proportional
to the square root of the detection probability $P^{(N)}_{\text{iter}}$, the
quadratic enhancement of the detection probability results in the Heisenberg-
limited precision scaling for $\Delta\gamma^{(N)}_{\text{iter}}$ by reducing
statistical errors. Since the quadratic scaling of the detection probability
is achieved under the condition in Eq. (6), $N\ll|\langle
N\hat{\sigma}_{y}\rangle_{\text{w}}|=11.5$, the experimental data in Fig. 4
have slight deviations from the Heisenberg-limited scaling as $N$ increases.
However, this deviation can be eliminated by increasing the amplification
factor.

Discussion.—We have experimentally demonstrated that the Heisenberg-limited
precision scaling can be achieved using a novel WVA scheme based on iterative
interactions. This result indicates that the perceived advantages of the
entanglement-assisted WVA Pang14 ; Pang15E ; Chen19 ; Starek20  are in fact
due to iterative local interactions between each particle of an entangled
system and a meter, rather than coming from the entanglement itself. Although
quantum entanglement enables simultaneous interactions between all system
particles and a meter, the fast interaction time is usually compromised by
large decoherence which increases linearly with the size of the system Dur04 .
Therefore, the entanglement-assisted WVA does not offer a meaningful advantage
over our scheme under decoherence Boixo12 ; Demkowicz14 . Rather, since a
large-scale entangled state is difficult to produce with high fidelity, our
iterative interaction approach would provide practical applications in
achieving the Heisenberg-limited precision scaling for weak-value
amplification.

This work was supported in part by the National Research Foundation of Korea
(Grant No. 2019R1A2C3004812) and the ITRC support program
(IITP-2021-2020-0-01606).

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# Some properties of Cayley signed graphs on finite abelian groups

Mohammad A. Iranmanesh∗ and Nasrin Moghaddami

###### Abstract.

Let $\Sigma=(\Gamma,\sigma)$ is a signed graph(or sigraph in short), where
$\Gamma$ is a underlying graph of $\Sigma$ and
$\sigma:E\longrightarrow\\{+,-\\}$ is a function. Consider
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$, where all $p_{1},p_{2},\ldots,p_{k}$ are distinct
prime factors and
$\Phi=\varphi_{p_{1}}\times\varphi_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}}$. For any positive integer $n$,
$\varphi_{n}=\\{\ell|1\leq\ell<n,\gcd(\ell,n)=1\\}$. Motivated by [13], we
will investigate balancing in $\Sigma$ and $L(\Sigma)$, clusterability and
sign-compatibility of $\Sigma$.

MSC(2010): Primary: 20E99.
Keywords: Abelian group, balancing, Cayley graph, clustering, signed graph,
sign-compatible.
$*$Corresponding author

##  1\. Introduction

Let $(G,\cdot)$ be a group and $S=S^{-1}$ be a non empty subset of $G$ not
containing the identity element e of $G$. The Cayley graph
${\mathrm{Cay}}(G,S)$ is the simple graph having vertex set $G$ and edge set
$\\{\\{v,vs\\}|v\in G,s\in S\\}$. To find enough information about Cayley
graphs, refer to books Biggs [5] and Godsil, Royle [7].

A signed graph(or sigraph in short) is an ordered pair
$\Sigma=(\Gamma,\sigma)$ where $\Gamma=(V,E)$ is a graph called the underlying
graph of $\Sigma$ and $\sigma:E\longrightarrow\\{+,-\\}$ is a function.
$\Sigma$ is all-positive (all-negative) if all its edges are positive
(negative). Moreover, it is said to be homogeneous if it is either all-
positive or all-negative and heterogeneous otherwise. $d^{-}(v)$ ($d^{+}(v)$)
represents the number of negative (positive) edges incident at $v$ in
$\Sigma$. A marked sigraph is an ordered pair $\Sigma_{\mu}=(\Sigma,\mu)$,
where $\Sigma=(\Gamma,\sigma)$ is a sigraph and
$\mu:V(\Sigma)\longrightarrow\\{+,-\\}$ is a function, called a marking of
$\Sigma$ (see [15, 16, 17]).

Behzad and Chartrand [3] defined line sigraph $L(\Sigma)$ as the sigraph in
which the edges of $\Sigma$ are represented as vertices, two of these vertices
are defined adjacent whenever the corresponding edges in $\Sigma$ have a
vertex in common, any such edge ef is defined to be negative whenever both $e$
and $f$ are negative edges in $\Sigma$. A positive cycle in $\Sigma$ is a
cycle which contains even number of negative edges. A sigraph $\Sigma$ is a
balanced sigraph if every cycle in $\Sigma$ is positive. The first
classification of balanced graphs was done by Harary in 1953 [8].

A clusterable sigraph is a sigraph in which its vertex set is partitioned into
pairwise disjoint subsets in such a way that whenever there are positive edges
they lie in the same subset and negative edges lie in the different subsets.
These subsets are called clusters (see [6]). A sigraph is called sign-
compatible [11] if it is possible to mark the vertices of the sigraph in such
a manner that for every negative edge there are negatively marked vertices on
its both ends and for any positive edge its ends are not assigned negative,
sign-incompatible otherwise.

Let $\Sigma=(\Gamma,\sigma)$ is a sigraph whose underlying graph is
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$ where $p_{1},p_{2},\ldots,p_{k}$ are distinct prime
factors and
$\Phi=\varphi_{p_{1}}\times\varphi_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}}$ and also for an edge $ab$ of $\Sigma$, the function
$\Sigma$ is defined as follows [13].

$\sigma(ab)=\left\\{\begin{array}[]{ll}+&~{}~{}~{}~{}~{}if~{}~{}a\in\Phi~{}~{}or~{}~{}b\in\Phi,\vspace{0.5cm}\\\
-&~{}~{}~{}~{}~{}otherwise.\end{array}\right.$

By $\mathbb{Z}_{n}$ we denote the abelian group of order $n$. For $p_{1}=2$,
the signed graph $\Sigma$, is a disconnected sigraph with two connected
components, say $\Sigma_{1}$ and $\Sigma_{2}$, where
$V(\Sigma_{1})=\\{(1,v)|v~{}is~{}odd\\}\cup\\{(0,v)|v~{}is~{}even\\}$ and
$V(\Sigma_{2})=\\{(0,v)|v~{}is~{}odd\\}\cup\\{(1,v)|v~{}is~{}even\\}$. Since
every Cayley graph ${\mathrm{Cay}}(G,S)$ is $|S|$-regular (see for example
[7]), we find that $\Sigma$ is $|\Phi|$-regular.

In this article some properties of the signed graph $\Sigma$, such as
balancing, clusterability and sign-compatibility, are investigated.

##  2\. balance in sigraph $\Sigma$ and line sigraph $\Sigma$

In this chapter, we examined the number of positive and negative edges for
$\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$ and one of the prime factors say $p_{i}$ is $2$ and
$\alpha_{j}\geq 1$ for any $j=1,2,\ldots,k$. Moreover we find the number of
positive and negative edges of
$\Gamma_{1}={\mathrm{Cay}}(\mathbb{Z}_{p}\times\mathbb{Z}_{p^{\alpha}},\Phi)$,
where $\alpha\geq 1$, $p\geq 3$, and
$\Gamma_{2}={\mathrm{Cay}}(\mathbb{Z}_{p}\times\mathbb{Z}_{pq},\Phi)$, where
$p,q\geq 3$ are prime numbers. Next we study the balanced property of $\Sigma$
and $L(\Sigma)$.

When $\Gamma={\mathrm{Cay}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2},\Phi)$, sigraph
$\Sigma$ has two connected components. One of the components has a positive
edge and the other component has a negative edge. (According to the
explanation given in the last paragraph of the introduction,
$V(\Sigma_{1})=\\{(0,0),(1,1)\\}$ and $V(\Sigma_{2})=\\{(0,1),(1,0)\\}$. Since
$(1,1)\in\Phi$ and $(0,0)$ is adjacent to $(1,1)$, so, by definition of
$\sigma$, the edge $(0,0)(1,1)$ is positive. Also, since the vertices of
$V(\Sigma_{2})$ are not in $\Phi$ and $(0,1)$ is adjacent to $(1,0)$, hence
$(0,1)(1,0)$ is negative edge). Note that the tree has no cycle and using
Theorem 1.6 [18], we can conclude that each tree is balanced. Since $\Sigma$
is the union of two trees, so it is balanced. Also $L(\Sigma)$ has only two
vertices so we may consider it is a balansed graph. Therefore in the proof of
Theorems 2.6, 2.8, this case is not considered.

######  Proposition 2.1.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$ and one of the prime factors is $2$. Then $\Sigma$
has $|\Phi|^{2}$ positive edges.

###### Proof.

Since one of $p_{i}$ is $2$, so without loss of generality we may assume that
$p_{1}=2$. In this case $\Sigma$ is a disconnected sigraph with exactly two
connected components $\Sigma_{1}=(\Gamma_{1},\sigma)$ and
$\Sigma_{2}=(\Gamma_{2},\sigma)$, where
$V(\Sigma_{1})=\\{(1,v)|v~{}is~{}odd\\}\cup\\{(0,v)|v~{}is~{}even\\}$ and
$V(\Sigma_{2})=\\{(0,v)|v~{}is~{}odd\\}\cup\\{(1,v)|v~{}is~{}even\\}$. Note
that $\Sigma$ is $|\Phi|$-regular hence
$|E(\Sigma_{1})|=|E(\Sigma_{2})|=\dfrac{1}{4}.|\Phi|.p_{1}^{\alpha_{1}+1}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}$.

Suppose $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$ is an arbitrary edge of
$\Sigma_{1}$, where $(u,v)\in\\{(1,v)|v~{}is~{}odd\\}$ and
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(0,v)|v~{}is~{}even\\}$. Since
$\Phi\subseteq\\{(1,v)|v~{}is~{}odd\\}$ and by the definition of $\sigma$ the
edge that has at least one end in $\Phi$ is positive, so a number of edges in
$\Sigma_{1}$ are positive. Also each of the members of $\Phi$ is adjacent to
the $|\Phi|$ vertices of $\\{(0,v)|v~{}is~{}even\\}$. Hence we have
$|\Phi|^{2}$ positive edges. Note that if
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2^{\alpha_{1}}},\Phi)$
then $\Phi=\\{(1,v)|v~{}is~{}odd\\}$. Therefore $\Sigma_{1}$ is all-positive
component of sigraph $\Sigma$ (since the edges of $\Sigma_{1}$ have one end in
$\\{(1,v)|v~{}is~{}odd\\}$ and the other in $\\{(0,v)|v~{}is~{}even\\}$, so by
definition of $\sigma$ are all positive). Also
$|E(\Sigma_{1})|=\dfrac{1}{4}.|\Phi|.p_{1}^{\alpha_{1}+1}=|\Phi|.2^{\alpha_{1}-1}=|\Phi|^{2}$,
since the number of odd numbers in $\\{1,2,3,\ldots,2^{\alpha_{1}}\\}$ is
equal to $2^{\alpha_{1}-1}=|\Phi|$.

Now suppose that $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})\in E(\Sigma_{2})$,
where $(u,v)\in\\{(0,v)|v~{}is~{}odd\\}$ and
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(1,v)|v~{}is~{}even\\}$. Clearly $u$
and $v^{{}^{\prime}}$ are multiples of $2$. So $(u,v)$ and
$(u^{{}^{\prime}},v^{{}^{\prime}})$ are not in $\Phi$. This implies that
$\Sigma_{2}$ is the all-negative component of $\Sigma$ (since the vertices of
$\Sigma_{2}$ are not in $\Phi$, so all the edges in $\Sigma_{2}$ are
negative).

Next assume that $p_{1}\geq 3$. According to the definition of a Cayley graph,
vertex $(0,0)$ is adjacent to all vertices of $\Phi$. Also each vertex $\Phi$
is adjacent to $|\Phi|-1$ of other vertices of $\Sigma$. By definition of
$\sigma$, all edges between these vertices are positive and their number is
equal to $|\Phi|+|\Phi|(|\Phi|-1|)=|\Phi|^{2}$. We represent the set of all
these vertices with $V_{1}$ such that
$V_{1}=\\{(0,v)|v~{}is~{}even\\}\cup\\{(u,v)|u\in\\{1,2,\ldots,p_{1}-1\\}~{}and~{}v~{}is~{}not~{}odd$-$multiple~{}of~{}prime~{}factors\\}$.
Consider the vertices of $V_{2}$, where

$V_{2}=\\{(0,v)|v~{}is~{}odd\\}\cup\\{(u,v)|u\in\\{1,2,\ldots,p_{1}-1\\}~{}and~{}v~{}is~{}odd$-$multiple~{}of~{}prime~{}factors\\}$.
Thus $V=V_{1}\cup V_{2}$. Assume that $(u,v)$ and
$(u^{{}^{\prime}},v^{{}^{\prime}})$ are arbitrary vertices of $V_{2}$. Since
$v$ and $v^{{}^{\prime}}$ are both odd so $v-v^{{}^{\prime}}$ is even, hence
$(u,v)-(u^{{}^{\prime}},v^{{}^{\prime}})\notin\Phi$. Therefore none of the
vertices $V_{2}$ are adjacent to each other. Let’s assume now $(u,v)\in V_{2}$
is adjacent to $(u^{{}^{\prime}},v^{{}^{\prime}})\in V_{1}$. Because $v$ is an
odd, so $v^{{}^{\prime}}$ should be even. Thus
$(u^{{}^{\prime}},v^{{}^{\prime}})\notin\Phi$. Moreover it is easy to see that
the vertices of $V_{2}$ are not in $\Phi$. This implies that all edges
incident with the vertices of $V_{2}$ are negative and their number is
$|V_{2}|\times|\Phi|=\dfrac{n|\Phi|}{2}-|\Phi|^{2};~{}n=p_{1}^{\alpha_{1}+1}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}$. ∎

######  Example 2.2.

Let $\Sigma=({\mathrm{Cay}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2\times
3},\Phi),\sigma)$ and
$\Sigma^{{}^{\prime}}=({\mathrm{Cay}}(\mathbb{Z}_{3}\times\mathbb{Z}_{2\times
3},\Phi),\sigma)$ which are shown in Figures 1 and 2, respectively. Positive
edges with straight line and negative edges are shown with dash.

Figure 1.  Two connected components of $\Sigma$, left $\Sigma_{1}$, right
$\Sigma_{2}$ Figure 2.  sigraph $\Sigma^{{}^{\prime}}$

######  Lemma 2.3.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p}\times\mathbb{Z}_{p^{\alpha}},\Phi)$,
$\alpha\geq 1$ and $p\geq 3$ is prime number. Then the number of negative
edges is $p^{\alpha-1}|\Phi|$.

###### Proof.

Assume that $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$ is an arbitrary edge of
$\Sigma$. By the definition of $\sigma$, we have negative edge only when
$u,v^{{}^{\prime}}$ or $u^{{}^{\prime}},v$ are multiples of $p$. Without loss
of generality we suppose that $u,v^{{}^{\prime}}$ are multiples of $p$. Since
in $\mathbb{Z}_{p}$ only zero is multiple of $p$, hence
$u=0,v\in\varphi_{p^{\alpha}}$ and also $u^{{}^{\prime}}\in\varphi_{p}$ and
$v^{{}^{\prime}}$ is multiple of $p$ less than $p^{\alpha}$. We know that
$\varphi_{p^{\alpha}}=p^{\alpha-1}(p-1)$ so $(u,v)$ can have
$p^{\alpha-1}(p-1)$ cases. Moreover $u^{{}^{\prime}}\in\\{1,2,\ldots,p-1\\}$
and we can consider $p^{\alpha-1}$ cases for $v^{{}^{\prime}}$, because the
number of multiples of $p$ less than $p^{\alpha}$ is equal to $p^{\alpha-1}$.
Therefore $(u^{{}^{\prime}},v^{{}^{\prime}})$ can have $p^{\alpha-1}(p-1)$
cases. This implies that $\Sigma$ has
$(p^{\alpha-1}(p-1))^{2}=p^{\alpha-1}.|\varphi_{p}|.|\varphi_{p^{\alpha}}|=p^{\alpha-1}|\Phi|$
negative edges. ∎

######  Example 2.4.

Let $\Sigma=({\mathrm{Cay}}(\mathbb{Z}_{3}\times\mathbb{Z}_{3},\Phi),\sigma)$.
This sigraph, which is shown in figure 3, has $4$ negative edges.

Figure 3.  sigraph $\Sigma$

######  Remark 1.

Suppose $\lambda$ is the length of the longest sequence of consecutive
integers in $\mathbb{Z}_{n}$, each of which shares a prime factor with $n$.
Assume that $n=pq$, where $p,q$ are prime numbers and $p,q\geq 3$. In
$\mathbb{Z}_{pq}$ we have $\lambda=2$. Now we prove that $\lambda$ appears
exactly in two places.

Let $kp$ and $k^{\prime}q$ be the first consecutive integers in
$\mathbb{Z}_{pq}$, which are multiples of $p$ and $q$, respectively. It is
clear that $pq-k^{\prime}q$ and $pq-kp$ are the final consecutive integers in
$\mathbb{Z}_{pq}$. Suppose that $fp$ and $tq$ are other consecutive integers
in $\mathbb{Z}_{pq}$, where $f>k$ and $t>k^{\prime}$. Then $(fp-
tq)-(k^{\prime}q-kp)=0$ and therefore $(f+k)p=(t+k^{\prime})q$. Hence there
exists an integer $r$ with $f+k=rq$ and $t+k^{\prime}=rp$. Since $1\leq
k<f\leq q-1$ and $1\leq k^{\prime}<t\leq p-1$, we have $r=1$. So $f=q-k$ and
$t=p-k^{\prime}$. Therefore $fp$ and $tq$ are the final consecutive integers
in $\mathbb{Z}_{pq}$.

######  Lemma 2.5.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p}\times\mathbb{Z}_{pq},\Phi)$ and $p,q\geq
3$. Then $\Sigma$ has $|\Phi|(2p+q-3)$ negative edges.

###### Proof.

Let’s consider the arbitrary edge $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$ of
$\Sigma$. Only when we have a negative edge that the vertices $(u,v)$ and
$(u^{{}^{\prime}},v^{{}^{\prime}})$ are not in $\Phi$. Also this is possible
if at each of the vertices $(u,v),(u^{{}^{\prime}},v^{{}^{\prime}})$ at least
one of theire elements multiples of one of the prime factors $p$ or $q$ .The
following cases are the only possible cases for the formation of negative
edges. Note that we show nonzero multiples of $p$ and $q$ by $\alpha p$ and
$\beta q$, respectively.

Case(i) $(u,v)\in\\{(0,v)|v\in\varphi_{pq}\\}$,
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(u^{{}^{\prime}},v^{{}^{\prime}})|u^{{}^{\prime}}\in\\{1,2,\ldots,p-1\\},v^{{}^{\prime}}=0~{}or~{}v^{{}^{\prime}}=\alpha
p,v-v^{{}^{\prime}}\neq\beta q~{}or~{}v^{{}^{\prime}}=\beta
q,v-v^{{}^{\prime}}\neq\alpha p\\}$.

Clearly there are $|\varphi_{pq}|$ cases for $(u,v)$. For
$(u^{{}^{\prime}},v^{{}^{\prime}})$ if $v^{{}^{\prime}}=0$ then we have $p-1$
cases and if $v^{{}^{\prime}}=\alpha p,v-v^{{}^{\prime}}\neq\beta q$ then by
Remark 1, we have $(p-1)(q-2)$ cases and also if $v^{{}^{\prime}}=\beta
q,v-v^{{}^{\prime}}\neq\alpha p$ then we have $(p-1)(p-2)$ cases for
$(u^{{}^{\prime}},v^{{}^{\prime}})$. Therefore in this case , the number of
negative edges is $|\varphi_{pq}|((p-1)+(p-1)(q-2)+(p-1)(p-2))=|\Phi|(p+q-3)$.

Case(ii) $(u,v)\in\\{(0,v)|v=\alpha p\\}$,
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(u^{{}^{\prime}},v^{{}^{\prime}})|u^{{}^{\prime}}\in\\{1,2,\ldots,p-1\\},v^{{}^{\prime}}=\beta
q\\}$.

In this case we have $q-1$ cases for $(u,v)$ and $(p-1)(p-1)$ cases for
$(u^{{}^{\prime}},v^{{}^{\prime}})$. Hence there are $(q-1)(p-1)(p-1)=|\Phi|$
cases for $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$.

Case(iii) $(u,v)\in\\{(0,v)|v=\beta q\\}$,
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(u^{{}^{\prime}},v^{{}^{\prime}})|u^{{}^{\prime}}\in\\{1,2,\ldots,p-1\\},v^{{}^{\prime}}=\alpha
p\\}$.

Similar to Case(ii), there are $(p-1)(p-1)(q-1)=|\Phi|$ cases for
$(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$.

Case(iv) $(u,v)\in\\{(u,v)|u\in\\{1,2,\ldots,p-1\\},v=\alpha p\\}$,
$(u^{{}^{\prime}},v^{{}^{\prime}})\in\\{(u^{{}^{\prime}},v^{{}^{\prime}})|u^{{}^{\prime}}\in\\{1,2,\ldots,p-1\\},u^{{}^{\prime}}\neq
u,v^{{}^{\prime}}=\beta q\\}$. There are $p-1,~{}q-1,~{}p-2$ and $p-1$ cases
for $u,~{}v,~{}u^{{}^{\prime}}$ and $v^{{}^{\prime}}$, respectively. Therefore
we have $(p-1)(q-1)(p-2)(p-1)=|\Phi|(p-2)$ cases for
$(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$.

By considering all the above cases, the number of negative edges in $\Sigma$
is equal to:

$|\Phi|(p+q-3)+|\Phi|+|\Phi|+|\Phi|(p-2)=|\Phi|(2p+q-3)$. ∎

######  Theorem 2.6.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$. Then $\Sigma$ is balanced if and only if one of
the prime factors is $2$.

###### Proof.

Necessity: Let $\Sigma$ is balanced. On contrary, assume that $p_{i}\geq 3$
for any $i=1,2,\ldots,k$. Sinse $p_{i}\geq 3$ hence both numbers $1$ and $2$
belong to $\varphi_{p_{1}}$ and
$\varphi_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots p_{k}^{\alpha_{k}}}$.
This implies that $\Sigma$ contains the cycle $C=((0,1),(1,2),(2,0),(0,1))$.
Note that $(1,2)\in\Phi$ hence $(0,1)(1,2)$ and $(1,2)(2,0)$ are positive
edges. Moreover $(0,1)$ and $(2,0)$ are not in $\Phi$, so $(2,0)(0,1)$ is
negative edge. We conclude that $C$ is a negative cycle. Therefore, by
definition of balanced sigraph, $\Sigma$ is unbalanced. Which is in
contradiction with the assumption.

Sufficiency: Suppose one of the prime factors is $2$. Assume first that
$p_{1}=2$. Then by proof of Proposition 2.1, $\Sigma$ has exactly two
connected components $\Sigma_{1}$ and $\Sigma_{2}$ where $\Sigma_{2}$ is all-
negative. We show that every cycle in $\Sigma_{2}$ is positive. Suppose
$C=(x_{1},x_{2},\ldots,x_{m},x_{1})$ is an arbitrary cycle in $\Sigma_{2}$.
Without loss of generality, let $x_{1}\in\\{(0,v)|v~{}is~{}odd\\}$. Since
neither of the vertices of $\\{(0,v)|v~{}is~{}odd\\}$ is not adjacent to each
other, so $x_{2},x_{m}\in\\{(1,v)|v~{}is~{}even\\}$. By continuing this
process, it is easy to see that $C$ contains an even number of negative edges.
Which implies that $\Sigma_{2}$ is balanced.

Now we prove that $\Sigma_{1}$ is also balanced. Let
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2^{\alpha}},\Phi)$.
Then $\Phi$ is consists of all the members of $\\{(1,v)|v~{}is~{}odd\\}$ and
since $\\{(1,v)|v~{}is~{}odd\\}\subseteq V(\Sigma_{1})$ Thus $\Sigma_{1}$ is
all-positive. Clearly in this case $\Sigma_{1}$ is balanced. Assume that
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$. Consider the arbitrary cycle
$C^{{}^{\prime}}=(x^{{}^{\prime}}_{1},x^{{}^{\prime}}_{2},\ldots,x^{{}^{\prime}}_{n},x^{{}^{\prime}}_{1})$.
The edges in $\Sigma_{1}$ are such that each edge has a vertex in
$\\{(0,v)|v~{}is~{}even\\}$ and a vertex in $\\{(1,v)|v~{}is~{}odd\\}$. Now
without loss of generality assume that
$x^{{}^{\prime}}_{1}\in\\{(0,v)|v~{}is~{}even\\}$ hence
$x^{{}^{\prime}}_{2},\in\\{(1,v)|v~{}is~{}odd\\}$. Hence we have two following
cases. If $x^{{}^{\prime}}_{2}\in\Phi$ then all edges joint to
$x^{{}^{\prime}}_{2}$ will be positive. So
$x^{{}^{\prime}}_{1}x^{{}^{\prime}}_{2}$ and
$x^{{}^{\prime}}_{2}x^{{}^{\prime}}_{3}$ are positive. But if
$x^{{}^{\prime}}_{2}\notin\Phi$, because $x^{{}^{\prime}}_{1}\notin\Phi$, then
$x^{{}^{\prime}}_{1}x^{{}^{\prime}}_{2}$ is negative. Since
$x^{{}^{\prime}}_{3}\in\\{(0,v)|v~{}is~{}even\\}$ hence
$x^{{}^{\prime}}_{2}x^{{}^{\prime}}_{3}$ is also negative. This implies that
after each negative edge in $C^{{}^{\prime}}$ we will have another negative
edge. Therefore, the number of negative edges in $C^{{}^{\prime}}$ is even.
Thus $\Sigma_{1}$ is balanced.

Next consider the case where $p_{1}\geq 3$. In this case $\Sigma$ is connected
sigraph. Suppose $C^{{}^{\prime\prime}}$ is an arbitrary cycle in $\Sigma$. If
all the edges in $C^{{}^{\prime\prime}}$ are positive then the cycle is
positive. Now assume that $C^{{}^{\prime\prime}}$ contains a negative edge
$(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$. Using the proof of Proposition 2.1
and Figure 2 of Example 2.2, $(u,v)(u^{{}^{\prime}},v^{{}^{\prime}})$ has a
vertex in $V_{1}$ and other vertex in $V_{2}$. Without loss of generality
suppose that $(u,v)\in V_{1}$ and $(u^{{}^{\prime}},v^{{}^{\prime}})\in
V_{2}$. Because neither of the vertices of $V_{2}$ is adjacent to each other,
so $(u^{{}^{\prime}},v^{{}^{\prime}})$ must be connected to one of the
vertices of $V_{1}$, for example
$(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})$. Since all the edges that have
a vertex in $V_{2}$ are negative, thus
$(u^{{}^{\prime}},v^{{}^{\prime}})(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})$
is negative edge. This implies that, in order to form cycle
$C^{{}^{\prime\prime}}$, for each negative edge in $C^{{}^{\prime\prime}}$,
another negative edge is required. We conclude that the number of negative
edges in $C^{{}^{\prime\prime}}$ is even. Therefore $\Sigma$ is balanced. ∎

######  Theorem 2.7.

[1] For a sigraph $S$, its line sigraph $L(S)$ is balanced if and only if the
following conditions hold:

for any cycle $Z$ in $S$,
(a) if $Z$ is all-negative, then $Z$ has even length;
(b) if $Z$ is heterogeneous, then $Z$ has even number of negative sections
with even length;

for $v\in S$, if $d(v)>2$, then there is at most one negative edge incident at
$v$ in $S$.

######  Theorem 2.8.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$. Then $L(\Sigma)$ is balanced if and only if one of
the prime factors is $2$.

###### Proof.

Necessity: Let $L(\Sigma)$ is balanced. Assume that the conclusion is false.
Suppose $p_{i}\geq 3$ for any $i=1,2,\ldots,k$. It is easy to see that vertex
$(0,1)$ is adjacent to vertices $(1,0),(2,0),\ldots,(p_{1}-1,0)$. By
definition $\Phi$, the vertices $(0,1),(1,0),(2,0),\ldots,(p_{1}-1,0)$ are not
in $\Phi$. Hence $d^{-}((0,1))\geq 2$. By the Theorem 2.7, condition (ii) does
not hold. Therefore $L(\Sigma)$ is unbalansed. Which is in contradiction with
the assumption. Thus one of the prime factors is $2$.

Sufficiency: Suppose one of the prime factors is $2$. By Theorem 2.6, $\Sigma$
is balanced so, according to Lemma 1.5[18] and Theorem 1.6[18], $\Sigma$ is
switching equivalent to a graph whose all edges are positive. Hence $\Sigma$
can be considered all-positive. Therefore $L(\Sigma)$ is balanced. ∎

##  3\. Clusterability and sign-compatibility of $\Sigma$

In this chapter two properties namely clusterability and sign-compatibility
for $\Sigma$ are investigated.

######  Theorem 3.1.

[6] Let $S$ be any signed graph. Then $S$ has a clustering if and only if $S$
contains no cycle having exactly one negative line.

######  Theorem 3.2.

[11] A sigraph $S$ is sign-compatible if and only if $S$ does not contain a
subsigraph that is isomorphic to either of the following two sigraphs formed
from the path graph $P4=(x,u,v,y)$; $S1$ with both edges $xu$ and $vy$
negative and the edge $uv$ positive and $S2$ obtained from $S1$ by identifying
the vertices $x$ and $y$ (see Figure 4 ).

Figure 4.  Forbidden subsigraphs for a sign-compatible sigraph.

######  Theorem 3.3.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$. Then $\Sigma$ is clusterable if and only if
$\Sigma$ is balanced.

###### Proof.

Nesssity: Suppose $\Sigma$ is clusterable. Assume that the conclusion is
false. Suppose $\Sigma$ is unbalanced. Using Theorem 2.6, $p_{i}\geq 3$ for
any $i=1,2,\ldots,k$. Because $p_{i}\geq 3$ so $(2,2)\in\Phi$ and also it is
adjacent to vertices $(0,1)$ and $(1,0)$. We now consider the cycle
$C=((0,1),(2,2),(1,0),(0,1))$ in $\Sigma$. By the definition of $\sigma$, we
have $\sigma((0,1)(2,2))=\sigma((2,2)(1,0))=+$ and $\sigma((1,0)(0,1))=-$.
Hence $C$ is a cycle with exactly one negative edge. Therefore, according to
Theorem 3.1, $\Sigma$ can not be clusterable, a contradiction to the
hypothesis. Hence one of the prime factors is $2$ so $\Sigma$ is balanced.

Sufficiency: Suppose $\Sigma$ is balanced. Thus all cycles of $\Sigma$ are
positive. So the number of negative edges in them is even. Hence it can not
include the cycle with a single negative edge. Therefore, according to Theorem
3.1, $\Sigma$ is clusterable. ∎

######  Theorem 3.4.

Let $\Sigma=(\Gamma,\sigma)$, where
$\Gamma={\mathrm{Cay}}(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{k}^{\alpha_{k}}},\Phi)$ and $p_{i}\geq 2$ for any $i=1,2,\ldots,k$. Then
$\Sigma$ is sign-compatible.

###### Proof.

Assume $\Sigma$ is sign-incompatible. Then by Theorem 3.2, $\Sigma$ contains a
subsigraph isomorphic to either $S1$ or $S2$. Let
$P=((u,v),(u^{{}^{\prime}},v^{{}^{\prime}}),(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}}),(u^{{}^{\prime\prime\prime}},v^{{}^{\prime\prime\prime}}))$
is a path in $\Sigma$ such that
$\sigma((u,v)(u^{{}^{\prime}},v^{{}^{\prime}}))=\sigma((u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})(u^{{}^{\prime\prime\prime}},v^{{}^{\prime\prime\prime}}))=-$
and
$\sigma((u^{{}^{\prime}},v^{{}^{\prime}})(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}}))=+$.
According to the definition of $\sigma$, since
$(u^{{}^{\prime}},v^{{}^{\prime}})(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})$
is positive edge hence two cases may occure.

If $(u^{{}^{\prime}},v^{{}^{\prime}})$ and
$(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})$ belong to $\Phi$ then all the
edges on $P$ are positive, a contradiction to hypothesis. Now if one of the
vertices $(u^{{}^{\prime}},v^{{}^{\prime}})$ and
$(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})$ belong to $\Phi$. Without loss
of generality assume that $(u^{{}^{\prime}},v^{{}^{\prime}})\in\Phi$ and
$(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})\notin\Phi$. Then we have
$\sigma((u,v)(u^{{}^{\prime}},v^{{}^{\prime}}))=((u^{{}^{\prime}},v^{{}^{\prime}})(u^{{}^{\prime\prime}},v^{{}^{\prime\prime}}))=+$
and
$\sigma((u^{{}^{\prime\prime}},v^{{}^{\prime\prime}})(u^{{}^{\prime\prime\prime}},v^{{}^{\prime\prime\prime}}))=-$.
Which is contradiction the hyposesis. Thus $\Sigma$ does not contain a
subsigraph isomorfic to $S1$.

Now let $\Sigma$ contains a subsigraph isomorphic to $S2$. As proved for $S1$,
according to the definition of $\sigma$, no positive edge can be placed
between two negative edges. Hence $\Sigma$ does not contain a subsigraph
isomorfic to $S2$. Therefore $\Sigma$ is sign-compatible. ∎

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