=&\langle
u|\sum_{j,k}\\{\sum_{\alpha}(\hat{x}_{j}(\alpha)-x_{j})b_{j}^{*}(\hat{x}_{k}(\alpha)-x_{k})b_{k}\sqrt{\rho_{x}}M_{\alpha}\sqrt{\rho_{x}}\\\
&-\sum_{\beta}(\hat{x}_{j}(\beta)-x_{j})b_{j}^{*}\sqrt{\rho_{x}}M_{\beta}(\sum_{\alpha}M_{\alpha})\sum_{\gamma}(\hat{x}_{k}(\gamma)-x_{k})b_{k}M_{\gamma}]\sqrt{\rho_{x}}\\}|u\rangle\\\
=&\langle
u|\sum_{\alpha}\\{[\sum_{j}(\hat{x}_{j}(\alpha)-x_{j})b_{j}^{*}\sqrt{\rho_{x}}-\sum_{j}\sum_{\beta}(\hat{x}_{j}(\beta)-x_{j})b_{j}^{*}\sqrt{\rho_{x}}M_{\beta}]M_{\alpha}[\sum_{k}(x_{k}(\alpha)-x_{k})b_{k}\sqrt{\rho_{x}}\\\
&-\sum_{k}\sum_{\gamma}(\hat{x}_{k}(\gamma)-x_{k})b_{k}M_{\gamma}\sqrt{\rho_{x}}]\\}|u\rangle\\\
=&\langle u|\sum_{\alpha}M^{\dagger}(b)M_{\alpha}M(b)]|u\rangle\geq
0,\end{aligned}$ (225)

here
$M(b)=\sum_{k}(x_{k}(\alpha)-x_{k})b_{k}\sqrt{\rho_{x}}-\sum_{k}\sum_{\gamma}(\hat{x}_{k}(\gamma)-x_{k})b_{k}M_{\gamma}\sqrt{\rho_{x}}$.

##  Appendix E Tradeoff relations with RLDs

Let

$S_{u}=\left(\begin{array}[]{cc}A_{u}&B_{u}\\\ B^{\dagger}_{u}&F_{u}\\\
\end{array}\right)\geq 0,$ (226)

with $(A_{u})_{jk}=\langle
u|\sqrt{\rho_{x}}X_{j}X_{k}\sqrt{\rho_{x}}|u\rangle$, $(B_{u})_{jk}=\langle
u|\sqrt{\rho_{x}}X_{j}L_{k}^{R\dagger}\sqrt{\rho_{x}}|u\rangle$,
$(F_{u})_{jk}=\langle
u|\sqrt{\rho_{x}}L_{j}^{R}L_{k}^{R\dagger}\sqrt{\rho_{x}}|u\rangle$, where
$L_{j}^{R}$ is the RLD corresponding to the parameter $x_{j}$.

If we choose a complete basis, $\\{|u_{1}\rangle,\cdots,|u_{d}\rangle\\}$, and
let $S=\sum_{j}S_{u_{j}}=\left(\begin{array}[]{cc}A&B\\\
B^{\dagger}&F^{RLD}\end{array}\right)\geq 0$, where
$(A)_{jk}=Tr(\rho_{x}X_{j}X_{k})$,
$(B)_{jk}=Tr(\rho_{x}X_{j}L_{k}^{R\dagger})=I$,
$(F^{RLD})_{jk}=Tr(\rho_{x}L_{j}^{R}L_{k}^{R\dagger})$, we obtain the RLD
bound

$Cov(\hat{x})\geq A\geq(F^{RLD})^{-1}.$ (227)

This can be equivalently written as

$Cov^{-1}(\hat{x})\leq F^{RLD}=F^{RLD}_{Re}+iF^{RLD}_{Im},$ (228)

with $F^{RLD}_{Re}$ and $F^{RLD}_{Im}$ as the real and imaginary part of
$F^{RLD}$ respectively, $F^{RLD}_{Re}=\frac{1}{2}[F^{RLD}+(F^{RLD})^{T}]$ is
real symmetric and $F^{RLD}_{Im}=\frac{1}{2}[F^{RLD}-(F^{RLD})^{T}]$ is real
skew-symmetric. By taking the transpose, we also have(note $Cov(\hat{x})$ is
symmetric)

$Cov^{-1}(\hat{x})\leq(F^{RLD})^{T}=F^{RLD}_{Re}-iF^{RLD}_{Im}.$ (229)

From which we get

$\displaystyle\begin{aligned}
F_{Q}^{-\frac{1}{2}}Cov^{-1}(\hat{x})F_{Q}^{-\frac{1}{2}}\leq
F_{Q}^{-\frac{1}{2}}F^{RLD}_{Re}F_{Q}^{-\frac{1}{2}}\pm
iF_{Q}^{-\frac{1}{2}}F^{RLD}_{Im}F_{Q}^{-\frac{1}{2}}.\end{aligned}$ (230)

Then for any vector, $|w\rangle$, we have

$\langle
w|F_{Q}^{-\frac{1}{2}}Cov^{-1}(\hat{x})F_{Q}^{-\frac{1}{2}}|w\rangle\leq\langle
w|F_{Q}^{-\frac{1}{2}}F^{RLD}_{Re}F_{Q}^{-\frac{1}{2}}|w\rangle-|\langle
w|F_{Q}^{-\frac{1}{2}}F^{RLD}_{Im}F_{Q}^{-\frac{1}{2}}|w\rangle|.$ (231)

By choosing $|w\rangle$ as all the eigenvectors of
$F_{Q}^{-\frac{1}{2}}F^{RLD}_{Im}F_{Q}^{-\frac{1}{2}}$ and making a summation,
we obtain the tradeoff relation from the standard RLD as

$Tr[F_{Q}^{-1}Cov^{-1}(\hat{x})]\leq
Tr[F_{Q}^{-1}F^{RLD}_{Re}]-\|F_{Q}^{-\frac{1}{2}}F^{RLD}_{Im}F_{Q}^{-\frac{1}{2}}\|_{1}.$
(232)

When there are $\nu$ copies of the state, this gives

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov^{-1}(\hat{x})]\leq
Tr[F_{Q}^{-1}F^{RLD}_{Re}]-\|F_{Q}^{-\frac{1}{2}}F^{RLD}_{Im}F_{Q}^{-\frac{1}{2}}\|_{1}.$
(233)

The bound can be improved by taking transposes on any $S_{u}$. We choose a
complete basis, $\\{|u_{1}\rangle,\cdots,|u_{d}\rangle\\}$, as the orthonormal
eigenvectors of
$\sqrt{\rho_{x}}(L_{j}^{R}L_{k}^{R\dagger}-L_{k}^{R}L_{j}^{R\dagger})\sqrt{\rho_{x}}$.
As mentioned in the main text, for any $|u_{q}\rangle$, $\frac{1}{2i}\langle
u_{q}|\sqrt{\rho_{x}}(L_{j}^{R}L_{k}^{R\dagger}-L_{k}^{R}L_{j}^{R\dagger})\sqrt{\rho_{x}}|u_{q}\rangle$,
which is the imaginary part of $(F_{u_{q}})_{jk}$, is a real number, which we
denote as $t^{q}_{jk}$. We then define

$\displaystyle\bar{S}_{u_{q}}:=\\{\begin{array}[]{cc}S_{u_{q}},\text{when
}t^{q}_{jk}\geq 0,\\\ S^{T}_{u_{q}},\text{when }t^{q}_{jk}<0.\end{array}$
(236)

By summing $\bar{S}_{u_{q}}$ we get

$\displaystyle\bar{S}=\sum_{q}\bar{S}_{u_{q}}=\left(\begin{array}[]{cc}\bar{A}&\bar{B}\\\
\bar{B}^{\dagger}&\bar{F}^{RLD}\end{array}\right),$ (239)

here $\bar{B}=I+i\bar{B}_{Im}$, $\bar{F}^{RLD}=\sum_{q}\bar{F}_{u_{q}}$ with
$\bar{F}_{u_{q}}$ equals to either $F_{u_{q}}$ or $F^{T}_{u_{q}}$ so that the
imaginary part of $(\bar{F}_{u_{q}})_{jk}$ is always positive. The imaginary
part of the $jk$-th entry of $\bar{F}^{RLD}$ is then given by

$(\bar{F}_{Im})_{jk}=\frac{1}{2}\|\sqrt{\rho_{x}}(L_{j}^{R}L_{k}^{R\dagger}-L_{k}^{R}L_{j}^{R\dagger})\sqrt{\rho_{x}}\|_{1},$
(240)

and the real part of $\bar{F}^{RLD}$ remains the same as $F^{RLD}_{Re}$.

By the Schur’s complement we then have
$\bar{F}^{RLD}-\bar{B}^{\dagger}Cov^{-1}(\hat{x})\bar{B}\geq 0$, which can be
equivalently written as

$\displaystyle\bar{F}_{Re}^{RLD}+i\bar{F}_{Im}^{RLD}-[Cov^{-1}(\hat{x})+\bar{B}_{Im}^{T}Cov^{-1}(\hat{x})\bar{B}_{Im}+i(\bar{B}_{Im}^{T}Cov^{-1}(\hat{x})-Cov^{-1}(\hat{x})\bar{B}_{Im})]\geq
0.$ (241)

We first assume $F_{Q}=I$, in this case $Cov^{-1}(\hat{x})\leq F_{Q}=I$. Then
by following the same procedure as previous, we denote
$Cov^{-1}(\hat{x})\bar{B}_{Im}$ as $D$ and get

$\displaystyle\bar{F}_{Re}^{RLD}-Cov^{-1}(\hat{x})-D^{T}D+i(\bar{F}_{Im}^{RLD}+D^{T}-D]\geq
0.$ (242)

By taking a $2\times 2$ principle submatrix we have

$\displaystyle\begin{aligned}
&\left(\begin{array}[]{cc}(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}-(D^{T}D)_{jj}&-Cov^{-1}(\hat{x})_{jk}-(D^{T}D)_{jk}\\\
-Cov^{-1}(\hat{x})_{kj}-(D^{T}D)_{kj}&(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}-(D^{T}D)_{kk}\\\
\end{array}\right)\\\
+&i\left(\begin{array}[]{cc}0&(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}\\\
-(\bar{F}_{Im}^{RLD})_{jk}-D_{kj}+D_{jk}&0\\\ \end{array}\right)\geq
0.\end{aligned}$ (243)

From the positiveness of the determinant, we have

$\displaystyle\begin{aligned}
&[(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}-(D^{T}D)_{jj}][(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}-(D^{T}D)_{kk}]\\\
\geq&[Cov^{-1}(\hat{x})_{jk}+(D^{T}D)_{jk}]^{2}+[(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}]^{2},\end{aligned}$
(244)

from which we can get

$\displaystyle\begin{aligned}
&[(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}-(D^{T}D)_{jj}]+[(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}-(D^{T}D)_{kk}]\\\
&\geq
2\sqrt{[(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}-(D^{T}D)_{jj}][(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}-(D^{T}D)_{kk}]}\\\
&\geq
2\sqrt{[Cov^{-1}(\hat{x})_{jk}+(D^{T}D)_{jk}]^{2}+[(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}]^{2}}\\\
&\geq 2|(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}|,\end{aligned}$ (245)

i.e.,

$\displaystyle(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}+(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}\geq
2|(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}|+(D^{T}D)_{jj}+(D^{T}D)_{kk}.$ (246)

Again as $(D^{T}D)_{jj}=\sum_{p}D_{pj}^{2}\geq D_{kj}^{2}$ and
$(D^{T}D)_{kk}=\sum_{p}D_{pk}^{2}\geq D_{jk}^{2}$, we have

$(D^{T}D)_{jj}+(D^{T}D)_{kk}=\sum_{p}(D_{pj}^{2}+D_{pk}^{2})\geq
D_{kj}^{2}+D_{jk}^{2}=\frac{1}{2}(D_{kj}-D_{jk})^{2}+\frac{1}{2}(D_{kj}+D_{jk})^{2}.$
(247)

Thus

$\displaystyle\begin{aligned}
&(\bar{F}_{Re}^{RLD})_{jj}-Cov^{-1}(\hat{x})_{jj}+(\bar{F}_{Re}^{RLD})_{kk}-Cov^{-1}(\hat{x})_{kk}\geq&2|(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}|+(D^{T}D)_{jj}+(D^{T}D)_{kk}.\\\
\geq&2|(\bar{F}_{Im}^{RLD})_{jk}+D_{kj}-D_{jk}|+\frac{1}{2}(D_{kj}-D_{jk})^{2}\\\
\geq&\min\\{\frac{1}{2}|(\bar{F}_{Im})_{jk}|^{2},2\\},\end{aligned}$ (248)

where the last inequality we used the fact that when $|y|\leq 2$,
$2|y+x|+\frac{1}{2}x^{2}\geq\frac{1}{2}y^{2}$ since

$\displaystyle\begin{aligned}
2|y+x|+\frac{1}{2}x^{2}&=2|y+x|+\frac{1}{2}(y+x-y)^{2}\\\
&=2|y+x|+\frac{1}{2}(y+x)^{2}-y(x+y)+\frac{1}{2}y^{2}\\\ &\geq
2|y+x|-|y(x+y)|+\frac{1}{2}y^{2}\\\ &=(2-|y|)|x+y|+\frac{1}{2}y^{2}\\\
&\geq\frac{1}{2}y^{2},\end{aligned}$ (249)

while when $|y|\geq 2$, $2|y+x|+\frac{1}{2}x^{2}\geq 2$ since

$\displaystyle\begin{aligned} 2|y+x|+\frac{1}{2}x^{2}&\geq
2(|y|-|x|)+\frac{1}{2}x^{2}\\\ &=\frac{1}{2}(|x|-2)^{2}-2+2|y|\\\ &\geq
2|y|-2\\\ &\geq 2.\end{aligned}$ (250)

From which we can get

$\displaystyle\begin{aligned}
(\bar{F}^{RLD}_{Re})_{jj}-Cov^{-1}(\hat{x})_{jj}+(\bar{F}^{RLD}_{Re})_{kk}-Cov^{-1}(\hat{x})_{kk}\geq\min\\{\frac{1}{2}|\bar{F}^{RLD}_{Im})_{jk}|^{2},2\\}.\end{aligned}$
(251)

By repeating the procedure for different choices of $j,k$ and make a
summation, we then get the tradeoff relation, under the parametrization that
$F_{Q}=I$, as

$Tr[Cov^{-1}(\hat{x})]\leq
Tr[\bar{F}^{RLD}_{Re}]-\frac{1}{4(n-1)}\|C^{RLD}_{1}\|_{F}^{2},$ (252)

with
$(C^{RLD}_{1})_{jk}=\min\\{\frac{1}{2}\|\sqrt{\rho_{x}}(L_{j}^{R}L_{k}^{R\dagger}-L_{k}^{R}L_{j}^{R\dagger})\sqrt{\rho_{x}}\|_{1},2\\}$.

If we repeat the 1-local measurement on $\nu$ copies of the state, the
tradeoff relation under the 1-local measurement, with the parametrization that
$F_{Q}=I$, is then

$\frac{1}{\nu}Tr[Cov^{-1}(\hat{x})]\leq
Tr[F^{RLD}_{Re}]-\frac{1}{4(n-1)}\|C^{RLD}_{1}\|_{F}^{2}$ (253)

When $F_{Q}\neq I$ initially, we can first make a reparametrization with
$\tilde{x}=F_{Q}^{-\frac{1}{2}}x$. The tradeoff relation in Eq.(253) can then
be expressed in the original parametrization as

$\displaystyle\begin{aligned} &\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\\\
\leq&Tr[F_{Q}^{-1}F^{RLD}_{Re}]-\frac{1}{4(n-1)}\|C^{RLD}_{1}\|_{F}^{2}\end{aligned}$
(254)

with the entries of $C^{RLD}_{1}$ given by

$\displaystyle\begin{aligned}
(C^{RLD}_{1})_{jk}&=\min\\{\frac{1}{2}\|\sqrt{\rho_{x}}(\tilde{L}_{j}^{R}\tilde{L}_{k}^{R\dagger}-\tilde{L}_{k}^{R}\tilde{L}_{j}^{R\dagger})\sqrt{\rho_{x}}\|_{1},2\\}\end{aligned}$
(255)

where $\tilde{L}^{R}_{j}=\sum_{q}(F_{Q}^{-\frac{1}{2}})_{jq}L^{R}_{q}$ and
$\tilde{L}^{R}_{k}=\sum_{q}(F_{Q}^{-\frac{1}{2}})_{kq}L^{R}_{q}$.

For p-local measurements, we can similarly get

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov^{-1}(\hat{x})]\leq
Tr[F_{Q}^{-1}F_{Re}^{RLD}]-\frac{1}{4(n-1)}\|\frac{C_{p}^{RLD}}{p}\|_{F}^{2},$
(256)

where $(C_{p}^{RLD})_{jk}=\min\\{\frac{1}{2}\|\sqrt{\rho_{x}^{\otimes
p}}(\tilde{L}_{jp}^{R}\tilde{L}_{kp}^{R\dagger}-\tilde{L}_{kp}^{R}\tilde{L}_{jp}^{R\dagger})\sqrt{\rho_{x}^{\otimes
p}}\|_{1},2p\\}$.

##  Appendix F Example 2

Here we provided more detailed calculations for example 2.

For mixed states $\rho_{x}=\frac{1}{3}I+\sum_{j}x_{j}G_{j}$ with
$G_{j}=\frac{1}{2}\Lambda_{j}$, where $\\{\Lambda_{j}\\}_{j=1}^{8}$ are the
Gell-Mann matrices,

$\displaystyle\Lambda_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},\Lambda_{2}=\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix},\Lambda_{3}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix},$ (257) $\displaystyle\Lambda_{4}=\begin{pmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},\Lambda_{5}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},$ $\displaystyle\Lambda_{6}=\begin{pmatrix}0&0&0\\\
0&0&1\\\ 0&1&0\end{pmatrix},\Lambda_{7}=\begin{pmatrix}0&0&0\\\ 0&0&-i\\\
0&i&0\end{pmatrix},\Lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&-2\end{pmatrix}.$

If the parameters $x_{j}$ are all close to 0, the SLDs and RLDs are all given
by $L_{j}=3G_{j}$. Thus the tradeoff relations from the SLDs and RLDs will be
the same. The QFI matrix is given as $F_{Q}=F^{RLD}=\frac{3}{2}I$, thus
$\tilde{L}_{j}=\sqrt{\frac{2}{3}}L_{j}=\sqrt{6}G_{j}$. The entries of $C_{1}$
is given by

$(C_{1})_{jk}=\frac{1}{2}\|\sqrt{\rho_{x}}[\tilde{L}_{j},\tilde{L}_{k}]\sqrt{\rho_{x}}\|_{1}=\|[G_{j},G_{k}]\|_{1},$
(258)

from which the matrix form of $C_{1}$ can be computed as

$C_{1}=\begin{pmatrix}0&1&1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0\\\
1&0&1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0\\\
1&1&0&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0&1&\frac{1}{2}&\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&1&0&\frac{1}{2}&\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0&1&\frac{\sqrt{3}}{2}\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&1&0&\frac{\sqrt{3}}{2}\\\
0&0&0&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&0\end{pmatrix},$
(259)

Thus we have

$\displaystyle\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]$ $\displaystyle\leq
n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}$ (260) $\displaystyle=8-\frac{1}{28}\times
2(5\times 1+4\times\frac{3}{4}+16\times\frac{1}{4})=\frac{50}{7}\approx 7.14.$

For $p$-local measurements on $\rho_{x}$, the entries of $C_{p}$ is given by

$(C_{p})_{jk}=\frac{1}{2}\|\sqrt{\rho_{x}^{\otimes
p}}[\tilde{L}_{jp},\tilde{L}_{kp}]\sqrt{\rho_{x}^{\otimes
p}}\|_{1}=\frac{1}{3^{p-1}}\|[G_{jp},G_{kp}]\|_{1},$ (261)

For all $j,k$, the eigenvalues of $[G_{j},G_{k}]$ are
$\\{-\lambda,0,\lambda\\}$, where $\lambda=\frac{1}{2}$ or $\frac{1}{4}$ or
$\frac{\sqrt{3}}{4}$. Suppose that the eigenvectors corresponding to
eigenvalues $\\{-\lambda,0,\lambda\\}$ can be written as
$\\{|\Phi_{l}\rangle\\}_{l\in\\{-\lambda,0,\lambda\\}}$. The eigenvectors of
$[G_{jp},G_{kp}]=\sum_{r=1}^{p}[G_{j}^{(r)},G_{k}^{(r)}]$ are then given by
$\otimes_{r=1}^{p}|\Phi_{l_{r}}\rangle$ with the corresponding eigenvalues
$\sum_{r=1}^{p}l_{r}$, here $l_{r}\in\\{-\lambda,0,\lambda\\}$. , The
recursive relation to obtain the eigenvalues is depicted in Fig. 5(a), where
in Fig. 5(b) a few possible values of $\sum_{r=1}^{p}l_{r}$ have been
listed(note that the $(p+1)$-th row in Fig. 5(b) corresponds to all possible
values of $\sum_{r=1}^{p}l_{r}$). The multiplicity of each eigenvalue can be
obtained as Fig. 5(c), which is just the trinomial triangle that corresponds
to the coefficients of $(1+x+x^{2})^{p}$.

Figure 5: Eigenvalues and multiplicities of
$\sum_{r=1}^{p}[G_{j}^{(r)},G_{k}^{(r)}]$.

Hence the eigenvalues of
$[G_{jp},G_{kp}]=\sum_{r=1}^{p}[G_{j}^{(r)},G_{k}^{(r)}]$ are $\lambda s$ with
multiplicity $\binom{p}{s}_{2}$ for $s=-p,-p+1,\cdots,p$, here
$\binom{p}{s}_{2}=\sum_{i=0}^{p}(-1)^{i}\binom{p}{i}\binom{2p-2i}{p-s-i}$ is
the trinomial coefficient.

Denote $\mathcal{N}_{p}=\sum_{s=0}^{p}s\binom{p}{s}_{2}$, we then have

$(C_{p})_{jk}=\frac{1}{3^{p-1}}\|[G_{jp},G_{kp}]\|_{1}=(C_{1})_{jk}\frac{\mathcal{N}_{p}}{3^{p-1}}.$
(262)

The Frobenius norm of $C_{p}$ is then given by

$\displaystyle\|C_{p}\|_{F}$
$\displaystyle=\sqrt{\sum_{jk}(C_{p})_{jk}^{2}}=\sqrt{\sum_{jk}\left((C_{1})_{jk}\frac{1}{3^{p-1}}\mathcal{N}_{p}\right)^{2}}$
(263)
$\displaystyle=\frac{1}{3^{p-1}}\mathcal{N}_{p}\sqrt{\sum_{jk}\left((C_{1})_{jk}\right)^{2}}=\frac{1}{3^{p-1}}\mathcal{N}_{p}\|C_{1}\|_{F}.$

Using the tradeoff relations for $p$-local measurements, we have

$\displaystyle\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]$ $\displaystyle\leq
n-\frac{1}{4(n-1)}\|\frac{C_{p}}{p}\|_{F}^{2}$ (264)
$\displaystyle=n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$
$\displaystyle=8-\frac{6}{7}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$

Specifically, for 2-local measurements,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{2}}{2}\|_{F}^{2}=8-\frac{6}{7}\times\frac{1}{4}\times\frac{16}{9}=\frac{160}{21}\approx
7.62$ (265)

and for 3-local measurements,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{3}}{3}\|_{F}^{2}=8-\frac{6}{7}\times\frac{1}{9}\times\frac{25}{9}=\frac{1462}{189}\approx
7.74$ (266)

If we choose the basis $\\{|u\rangle_{q}\\}$ as computational basis
$|u\rangle_{0}=|0\rangle$, $|u\rangle_{1}=|1\rangle$,
$|u\rangle_{2}=|2\rangle$, the matrices $F_{u_{q}}$ are given as

$F_{u_{0}}=\frac{1}{2}\begin{pmatrix}1&i&0&0&0&0&0&0\\\ -i&1&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&\frac{\sqrt{3}}{3}\\\ 0&0&0&1&i&0&0&0\\\ 0&0&0&-i&1&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&\frac{\sqrt{3}}{3}&0&0&0&0&\frac{1}{3}\end{pmatrix},F_{u_{1}}=\frac{1}{2}\begin{pmatrix}1&-i&0&0&0&0&0&0\\\
i&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&-\frac{\sqrt{3}}{3}\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&i&0\\\ 0&0&0&0&0&-i&1&0\\\
0&0&-\frac{\sqrt{3}}{3}&0&0&0&0&\frac{1}{3}\end{pmatrix}$ (267)
$F_{u_{2}}=\frac{1}{2}\begin{pmatrix}0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&1&-i&0&0&0\\\ 0&0&0&i&1&0&0&0\\\ 0&0&0&0&0&1&-i&0\\\
0&0&0&0&0&i&1&0\\\
0&0&0&0&0&0&0&\frac{4}{3}\end{pmatrix},\bar{F}_{Im}=\begin{pmatrix}0&1&0&0&0&0&0&0\\\
-1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\end{pmatrix}$ (268)

Let $\bar{F}=F_{u_{0}}+F_{u_{1}}^{T}+F_{u_{2}}^{T}$, this gives a bound as

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{(n-2)}{(n-1)^{2}}\|\bar{F}_{Im}\|_{F}^{2}=8-\frac{6}{49}\times
4\approx 7.51,$ (269)

If we only estimate $\\{x_{1},x_{2},x_{4},x_{5}\\}$, the associated matrices
are given by the $4\times 4$ submatrices of the original ones,

$\displaystyle C_{1}=\begin{pmatrix}0&1&\frac{1}{2}&\frac{1}{2}\\\
1&0&\frac{1}{2}&\frac{1}{2}\\\ \frac{1}{2}&\frac{1}{2}&0&1\\\
\frac{1}{2}&\frac{1}{2}&1&0\end{pmatrix},\bar{F}_{Im}=\begin{pmatrix}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{pmatrix},$ (270)

which further gives $\|C_{1}\|_{F}=\sqrt{6}$, $\|\bar{F}_{Im}\|_{F}=2$. Then
we have

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}=4-\frac{1}{2}=\frac{7}{2}=3.5,$ (271)
$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{(n-2)}{(n-1)^{2}}\|\bar{F}_{Im}\|_{F}^{2}=4-\frac{8}{9}=\frac{28}{9}\approx
3.11.$ (272)

For $p$-local measurements, by following the same derivation as the previous
case, we have

$\displaystyle\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]$ $\displaystyle\leq
n-\frac{1}{4(n-1)}\|\frac{C_{p}}{p}\|_{F}^{2}$ (273)
$\displaystyle=n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$
$\displaystyle=4-\frac{1}{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$

Specifically, for p=2 we have

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{2}}{2}\|_{F}^{2}=4-\frac{1}{2}\times\frac{1}{4}\times\frac{16}{9}=\frac{34}{9}\approx
3.78,$ (274)

and for p=3,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{3}}{3}\|_{F}^{2}=4-\frac{1}{2}\times\frac{1}{9}\times\frac{25}{9}=\frac{623}{162}\approx
3.85.$ (275)

If we choose the basis $\\{|u_{q}\rangle\\}$ as the computational basis
$|u_{0}\rangle=|00\rangle$, $|u_{1}\rangle=|01\rangle$,
$|u_{2}\rangle=|02\rangle$, $|u_{3}\rangle=|10\rangle$,…,
$|u_{8}\rangle=|22\rangle$, the imaginary part of the matrices $F_{u_{q}}$ are
given as

$\displaystyle F_{u_{0}Im}=\begin{pmatrix}0&\frac{1}{3}&0&0\\\
-\frac{1}{3}&0&0&0\\\ 0&0&0&\frac{1}{3}\\\
0&0&-\frac{1}{3}&0\end{pmatrix},F_{u_{4}Im}=\begin{pmatrix}0&-\frac{1}{3}&0&0\\\
\frac{1}{3}&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix},F_{u_{8}Im}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&-\frac{1}{3}\\\ 0&0&\frac{1}{3}&0\end{pmatrix},$ (276) $\displaystyle
F_{u_{1}Im}=F_{u_{3}Im}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\frac{1}{3}\\\
0&0&-\frac{1}{3}&0\end{pmatrix},F_{u_{2}Im}=F_{u_{6}Im}=\frac{1}{2}\begin{pmatrix}0&\frac{1}{3}&0&0\\\
-\frac{1}{3}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix},$ $\displaystyle
F_{u_{5}Im}=F_{u_{7}Im}=\frac{1}{2}\begin{pmatrix}0&-\frac{1}{3}&0&0\\\
\frac{1}{3}&0&0&0\\\ 0&0&0&-\frac{1}{3}\\\ 0&0&\frac{1}{3}&0\end{pmatrix}.$

The optimal $\bar{F}_{Im2}$ is then given by
$\bar{F}_{Im2}=F_{u_{0}Im}+F_{u_{4}Im}^{T}+F_{u_{8}Im}^{T}+(F_{u_{1}Im}+F_{u_{3}Im})+(F_{u_{2}Im}+F_{u_{6}Im})+(F_{u_{5}Im}+F_{u_{7}Im})^{T}$,
i.e.,

$\bar{F}_{Im2}=\begin{pmatrix}0&\frac{4}{3}&0&0\\\ -\frac{4}{3}&0&0&0\\\
0&0&0&\frac{4}{3}\\\ 0&0&-\frac{4}{3}&0\end{pmatrix},$ (277)

which gives a tighter bound as

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{(n-2)}{(n-1)^{2}}\|\frac{\bar{F}_{Im2}}{2}\|_{F}^{2}=4-\frac{2}{9}\times\frac{1}{4}\times
4\times\frac{16}{9}=4-\frac{32}{81}\approx 3.60.$ (278)

If we only estimate $\\{x_{1},x_{2},x_{5}\\}$, the associated matrices are
given by the $3\times 3$ submatrices of the original ones,

$\displaystyle C_{1}=\begin{pmatrix}0&1&\frac{1}{2}\\\ 1&0&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}&0\end{pmatrix},\bar{F}_{Im}=\begin{pmatrix}0&1&0\\\
-1&0&0\\\ 0&0&0\end{pmatrix},$ (279)

which further gives $\|C_{1}\|_{F}=\sqrt{3}$, $\|\bar{F}_{Im}\|_{F}=\sqrt{2}$.
Then we have the tradeoff relations

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}=3-\frac{3}{8}=\frac{21}{8}=2.625,$ (280)
$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{(n-2)}{(n-1)^{2}}\|\bar{F}_{Im}\|_{F}^{2}=3-\frac{1}{2}=\frac{5}{2}=2.5.$
(281)

For $p$-local measurements, we have

$\displaystyle\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]$ $\displaystyle\leq
n-\frac{1}{4(n-1)}\|\frac{C_{p}}{p}\|_{F}^{2}$ (282)
$\displaystyle=n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$
$\displaystyle=3-\frac{3}{8}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$

Specifically, for p=2,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{2}}{2}\|_{F}^{2}=3-\frac{3}{8}\times\frac{1}{4}\times\frac{16}{9}=\frac{17}{6}\approx
2.83,$ (283)

and for p=3,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{3}}{3}\|_{F}^{2}=3-\frac{3}{8}\times\frac{1}{9}\times\frac{25}{9}=\frac{623}{216}\approx
2.88.$ (284)

For 2-local measurements, if we choose the basis $\\{|u_{q}\rangle\\}$ as the
computational basis $|u_{0}\rangle=|00\rangle$, $|u_{1}\rangle=|01\rangle$,
$|u_{2}\rangle=|02\rangle$, $|u_{3}\rangle=|10\rangle$,…,
$|u_{8}\rangle=|22\rangle$, the imaginary part of the matrices $F_{u_{q}}$ are
given as

$\displaystyle F_{u_{0}Im}=\begin{pmatrix}0&\frac{1}{3}&0\\\
-\frac{1}{3}&0&0\\\
0&0&0\end{pmatrix},F_{u_{4}Im}=\begin{pmatrix}0&-\frac{1}{3}&0\\\
\frac{1}{3}&0&0\\\
0&0&0\end{pmatrix},F_{u_{1}Im}=F_{u_{3}Im}=F_{u_{8}Im}=\mathbf{0},$ (285)
$\displaystyle
F_{u_{2}Im}=F_{u_{6}Im}=\frac{1}{2}\begin{pmatrix}0&\frac{1}{3}&0\\\
-\frac{1}{3}&0&0\\\
0&0&0\end{pmatrix},F_{u_{5}Im}=F_{u_{7}Im}=\frac{1}{2}\begin{pmatrix}0&-\frac{1}{3}&0\\\
\frac{1}{3}&0&0\\\ 0&0&0\end{pmatrix}.$

The optimal $\bar{F}_{Im2}$ is then given by
$\bar{F}_{Im2}=F_{u_{0}Im}+F_{u_{4}Im}^{T}+(F_{u_{2}Im}+F_{u_{6}Im})+(F_{u_{5}Im}+F_{u_{7}Im})^{T}$,
i.e.,

$\bar{F}_{Im2}=\begin{pmatrix}0&\frac{4}{3}&0\\\ -\frac{4}{3}&0&0\\\
0&0&0\end{pmatrix},$ (286)

which gives

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{(n-2)}{(n-1)^{2}}\|\frac{\bar{F}_{Im2}}{2}\|_{F}^{2}=3-\frac{1}{4}\times\frac{1}{4}\times
2\times\frac{16}{9}=3-\frac{2}{9}\approx 2.78.$ (287)

This is tighter than the bound given by $C_{2}$.

If we only estimate $\\{x_{1},x_{2}\\}$, the associated matrices are given by
the $2\times 2$ submatrices of the original ones,

$\displaystyle C_{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (288)

which further gives $\|C_{1}\|_{F}=\sqrt{2}$. Then we have the tradeoff
relation

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}=2-\frac{1}{2}=\frac{3}{2}.$ (289)

For $p$-local measurements, we have

$\displaystyle\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]$ $\displaystyle\leq
n-\frac{1}{4(n-1)}\|\frac{C_{p}}{p}\|_{F}^{2}$ (290)
$\displaystyle=n-\frac{1}{4(n-1)}\|C_{1}\|_{F}^{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$
$\displaystyle=2-\frac{1}{2}\left(\frac{1}{p3^{p-1}}\mathcal{N}_{p}\right)^{2}$

Specifically, for p=2,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{2}}{2}\|_{F}^{2}=2-\frac{1}{2}\times\frac{1}{4}\times\frac{16}{9}=\frac{16}{9}\approx
1.78,$ (291)

and for p=3,

$\frac{1}{\nu}Tr[F_{Q}^{-1}Cov(\hat{x})^{-1}]\leq
n-\frac{1}{4(n-1)}\|\frac{C_{3}}{3}\|_{F}^{2}=2-\frac{1}{2}\times\frac{1}{9}\times\frac{25}{9}=\frac{299}{162}\approx
1.85.$ (292)

We plot the bound with different $p$ in Fig.6. It can be seen that the Holevo
bound, which equals to the QCRB since the weak commutative condition holds in
this case, is only achievable when $p\rightarrow\infty$.

Figure 6: Upper bound on $\Gamma_{p}$ and the QCRB/Holevo bound with the
number of parameters equal to $8,4,3,2$ respectively.


# Missing for 20 years: MeerKAT re-detects the elusive binary pulsar M30B

Vishnu Balakrishnan Max-Planck-Institut fur Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany  Paulo C. C. Freire Max-Planck-Institut fur
Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany  S. M. Ransom
National Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville, VA
22903, USA  Alessandro Ridolfi INAF – Osservatorio Astronomico di Cagliari,
Via della Scienza 5, I-09047 Selargius (CA), Italy  Max-Planck-Institut fur
Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany  E. D. Barr Max-
Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
W. Chen Max-Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121
Bonn, Germany  Vivek Venkatraman Krishnan Max-Planck-Institut fur
Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany  D. Champion Max-
Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
M. Kramer Max-Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121
Bonn, Germany  Jodrell Bank Centre for Astrophysics, Department of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, UK  T. Gautam
Max-Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,
Germany  Prajwal V. Padmanabh Max Planck Institute for Gravitational Physics
(Albert Einstein Institute), D-30167 Hannover, Germany  Leibniz Universität
Hannover, D-30167 Hannover, Germany  Max-Planck-Institut fur Radioastronomie,
Auf dem Hügel 69, D-53121 Bonn, Germany  Yunpeng Men Max-Planck-Institut fur
Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany  F. Abbate Max-
Planck-Institut fur Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
B. W. Stappers Jodrell Bank Centre for Astrophysics, Department of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, UK  I. Stairs
Department of Physics and Astronomy, University of British Columbia, 6224
Agricultural Road, Vancouver, BC V6T 1Z1, Canada  E. Keane 6 School of
Physics, Trinity College Dublin, University of Dublin, College Green, Dublin
2, D02 PN40, Ireland  A. Possenti INAF – Osservatorio Astronomico di Cagliari,
Via della Scienza 5, I-09047 Selargius (CA), Italy

###### Abstract

PSR J2140$-$2311B is a 13-ms pulsar discovered in 2001 in a 7.8-hour Green
Bank Telescope (GBT) observation of the core-collapsed globular cluster M30
and predicted to be in a highly eccentric binary orbit. This pulsar has eluded
detection since then, therefore its precise orbital parameters have remained a
mystery until now. In this work, we present the confirmation of this pulsar
using observations taken with the UHF receivers of the MeerKAT telescope as
part of the TRAPUM Large Survey Project. Taking advantage of the beamforming
capability of our backends, we have localized it, placing it
$1.2(1)\text{\,}\mathrm{\SIUnitSymbolArcminute}$ from the cluster centre. Our
observations have enabled the determination of its orbit: it is highly
eccentric ($e=0.879$) with an orbital period of $6.2$ days. We also measured
the rate of periastron advance, $\dot{\omega}=0.078\pm 0.002\,\rm
deg\,yr^{-1}$. Assuming that this effect is fully relativistic, general
relativity provides an estimate of the total mass of the system, $M_{\rm
TOT}=2.53\pm 0.08$ M⊙, consistent with the lightest double neutron star
systems known. Combining this with the mass function of the system gives the
pulsar and companion masses of $m_{p}<1.43\,\rm M_{\odot}$ and
$m_{c}>1.10\,\rm M_{\odot}$ respectively. The massive, undetected companion
could either be a massive WD or a NS. M30B likely formed as a result of a
secondary exchange encounter. Future timing observations will allow the
determination of a phase-coherent timing solution, vastly improving our
uncertainty in $\dot{\omega}$ and likely enabling the detection of additional
relativistic effects which will determine $m_{p}$ and $m_{c}$.

Star:neutron (1108) — Globular Clusters:individual: M30 (656) — Pulsars:
individual PSR J2140$-$2311B (1306)

tablenum SIXtablenum

##  1 Introduction

Globular clusters (GCs) are dense spheroidal arrangement of stars held
together by their own gravity. Due to their Their high stellar densities
(10${}^{3-6}\,\rm pc^{-3}$) and old age ($\sim 10\,\rm Gyr$) GCs are prime
targets to discover old, ‘dead’ neutron stars (NS) recycled by the accretion
of matter from a companion star (1987Natur.328..399L). This is because the
high densities within these clusters aid dynamical interactionsenable
dynamical interactions, where some of these old NSs — which would have
remained undetectable if located in the Galactic disk — can gain a stellar
mass companion, for example, a main-sequence (MS) star (Verbunt1987). As the
companion evolves and becomes a red giant , it fills its Roche lobe and either
in binary - single star encounters, or direct NS - MS encounters (Verbunt1987;
1995ApJS...99..609S; 1995MNRAS.276..876D; 1998MNRAS.301...15D).

In the resulting binaries, a MS or giant star transfers mass and angular
momentum to the NS. During this accretion process, commonly referred to as
‘recycling’, thermal X-ray emission is produced due to frictional heating from
the in-falling matter, making these systems detectable as low, intermediate,
or high-mass X-ray binaries depending on the mass of the donor star. In GCs,
only low-mass stars are still on the main sequence, therefore GCs are
populated exclusively by any X-ray binaries containing NSs are low-mass X-ray
binaries (LMXBs; Clark1975).

Because low-mass MS companions evolve slowly, LMXBs are very long-lived. At
the end of this process, a fully recycled radio pulsar (a ‘millisecond
pulsar’, or MSP) emerges rotating several hundred times per second , with
$P<20$ ms) emerges (e.g. 1982Natur.300..728A). These systems are quite mostly
resemble the MSPs in the Galactic disk, which have a wide variety of
companions (black widows, redbacks, white dwarfs and some isolated MSPs as
well); all of them produced by unperturbed stellar evolution; the majority
have orbits with low ($e<10^{-3}$) eccentricities
(2005AJ....129.1993M)111https://www.atnf.csiro.au/research/pulsar/psrcat/.

These dynamical formation channels for LMXBs explain why LMXBs and MSPs are so
abundant in GCs : at the relative to the Galaxy. At the time of writing (2022
November), 265 December), 272 radio pulsars in 36 38 GCs are known (see ‘GC
Pulsar Catalog‘222See http://www.naic.edu/~pfreire/GCpsr.html for an up-to
date count.of which 239 ) of which 245 ($\sim 90\%$) are MSPs($P\leq\rm
20\,ms$). Per unit stellar mass, GCs are estimated to have three orders of
magnitude more LMXBs and MSPs than the Galactic disk (Clark1975;
vandenBerg2020). (Clark1975; vandenBerg2020). Some of these channels, like
direct collisions of NSs with MS stars (1992ApJ...401..246D), can also explain
some of the exotic objects found in GCs, like ultra-compact X-ray binaries
(2005ApJ...621L.109I), see 2022ApJ...931...84Y for a recent review.
Furthermore, some dynamical interactions, like perturbations from nearby
stars, explain why a significant percentage of the MSPs - white dwarf systems
in GCs have mildly eccentric orbits (1992RSPTA.341...39P; 2005ASPC..328..147C;
2008IAUS..246..291R).

The stability of MSP rotation However, the extreme stellar densities at the
cores of the GCs with the highest interaction rates per binary, $\gamma$
(2014A&A...561A..11V) \- especially the core-collapsed GCs - imply that stars
are likely to go through repeated gravitational interactions with other stars
and binaries in the core. This leads to the formation, in these GCs, of binary
systems where already fully recycled MSPs acquire massive companions in
additional exchange encounters (1991ApJ...374L..41P; 2004ApJ...606L..53F;
2012ApJ...745..109L; 2015ApJ...807L..23D; 2021MNRAS.504.1407R;
2022A&A...664A..27R; 2022ApJ...934L...1K). If these massive companions are
degenerate, so these orbits retain the high eccentricity of the systems after
the exchange encounters. Such systems could even include MSP - black hole
binaries (e.g. 2019ApJ...877..122Y).

These systems are especially interesting because the rotational stability of
MSPs, which rivals atomic clocks on long timescales (e.g. 2020MNRAS.491.5951H)
. This makes them extremely useful for a diverse array of applications.
Pulsars located in GCs : their orbital eccentricities and large companion
masses enables precise mass measurements for the MSPs and their companions
(2012ApJ...745..109L; 2019MNRAS.490.3860R) and, at least in one case so far,
tests of gravity theories (2006ApJ...644L.113J). Even if they are not in
eccentric binaries, MSPs in globular clusters can be used to probe their
gravitational potentials (2017MNRAS.471..857F; 2017ApJ...845..148P;
2017MNRAS.468.2114P; 2018MNRAS.481..627A); pulsars located in binary systems
can be used to explore the gravitational fields of their companions, acting in
many cases as outstanding probes of strong-field gravity (e.g.
2021PhRvX..11d1050K).

M30 (NGC 7099) is a GC located at a distance of 8.1 kpc from the sun, at
Galactic coordinates $l=27.18^{\circ}$, $b=-46.84^{\circ}$
(1996yCat.7195....0H, 2010 revision), and has an estimated age of 12.9 Gyr
(2010MNRAS.404.1203F). This GC is of particular interest for pulsar searching
as it is a core-collapsed cluster and has shown significant evidence of mass
segregation (Howell_2000). Its core has a radius of
$3.6\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ and its half-light radius is
$61.8\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ (Harris2010).

The extreme stellar densities at the cores of such clusters imply that stars
are likely to go through repeated gravitational interactions with other stars
and binaries in the core; i.e., they have a high interaction rate per binary,
$\gamma$ (2014A&A...561A..11V). This leads to the formation, in these GCs, of
exotic binary systems where MSPs have massive companions and highly eccentric
orbits (1991ApJ...374L..41P; 2004ApJ...606L..53F; 2012ApJ...745..109L;
2015ApJ...807L..23D; 2021MNRAS.504.1407R; 2022A&A...664A..27R). This formation
mechanism could potentially form a MSP-black hole (BH) system. The high
orbital eccentricities and large companion masses of these systems can enable
precise MSP mass measurements (2012ApJ...745..109L; 2019MNRAS.490.3860R) and,
at least in one case so far, tests of gravity theories (2006ApJ...644L.113J).

Previous searches for pulsars in M30 using observations taken at the 100-m
Green Bank Telescope (GBT), yielded the discovery of two radio pulsars: PSR
J2140$-$2310A (M30A), a 11.01-ms eclipsing pulsar in a 4.17-h orbit; and PSR
J2140$-$2311B (M30B) (2004ApJ...604..328R), a 13.0-ms binary pulsar. Based on
the spin frequency evolution seen in the 7.8 h discovery observation in 2001,
2004ApJ...604..328R could infer that this pulsar must be in a highly eccentric
($e\geq 0.45$), relativistic orbit. However, a precise characterisation of
this orbit was not possible because the pulsar was not detected in any other
observations made with the GBT with the total time spent on source adding up
to $30\text{\,}\mathrm{h}$. We believe that the The reason for this is that it
was discovered while its flux was being amplified by diffractive
scintillation. , which produces a strong modulation in the flux densities of
the pulsars in this cluster (see Fig. 3 of 2004ApJ...604..328R, which shows
the flux density variations observed for M30A). Because of this, the basic
characteristics of this system have remained a mystery for the last 20 years.

Aided by sensitivity gains offered by the MeerKAT telescope, we present the
first set of detections of this pulsar since the discovery observation in
2001, which have revealed the nature of this system.

##  2 Observations and data reduction

We observed M30 using MeerKAT, with at least 56 antennas per observation, on 9
occasions between 2020 December and 2022 September (see table 1 for details).
Our initial campaign consisted of four observations, each lasting 60 minutes,
taken as part of the TRansients And PUlsars with Meerkat
(TRAPUM333http://www.trapum.org; Stappers_Kramer2016) Large Survey Project
(LSP) between December 2020 and January 2021.

Table 1: List of MeerKAT observations of M30B recorded and analysed for this
work. $t_{\rm samp}:$ sampling time, $N_{\rm pol}:$ Number of polarizations
recorded, $f_{\rm c}:$ central frequency, $\Delta f:$ bandwidth, $N_{\rm
ant:}$ number of MeerKAT antennas used, $N_{\rm beam}:$ number of tied-array
beams recorded. All APSUSE observations were incoherently dedispersed to DM =
$25.06\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$ and downsampled to 256
channel filterbank files. All PTUSE observations were recorded with full
stokes information, coherently dedispersed to a DM of
$25.06\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$ and downsampled to 256
channel psrfits files to save disk space.

Obs. id | Start Time | Start Time | Length | Backend | $t_{\rm samp}$ | $N_{\rm pol}$ | $f_{\rm c}$ | $\Delta f$ | $N_{\rm chan}\phantom{*}$ | $N_{\rm ant}$ | $N_{\rm beam}$
---|---|---|---|---|---|---|---|---|---|---|---
| (Date) | (MJD) | (s) |  |  ($\mu$s) |  | (MHz) | (MHz) |  |  |
01L | 2021 Dec 17 | 59200.512 | 3600 | APSUSE | 76.56 | 1 | 1284 | 856 | 4096 | 56 | 287
02L | 2021 Dec 29 | 59212.572 | 3600 | APSUSE | 76.56 | 1 | 1284 | 856 | 4096 | 56 | 287
03U | 2021 Jan 22 | 59236.453 | 3600 | APSUSE | 60.24 | 1 | 816 | 544 | 4096 | 56 | 287
04U | 2021 Jan 30 | 59244.488 | 3600 | APSUSE | 60.24 | 1 | 816 | 544 | 4096 | 56 | 275
05Ua | 2022 Jun 29 | 59759.837 | 3600 | PTUSE | 9.41 | 4 | 816 | 544 | 1024 | 60 | 1
06Ua | 2022 Jun 30 | 59760.179 | 3600 | PTUSE | 9.41 | 4 | 816 | 544 | 1024 | 60 | 1
07Ua | 2022 Jun 30 | 59760.880 | 3600 | PTUSE | 9.41 | 4 | 816 | 544 | 1024 | 60 | 1
08Ua | 2022 Jul 02 | 59762.877 | 3600 | PTUSE | 9.41 | 4 | 816 | 544 | 1024 | 60 | 1
09U | 2022 Sep 09 | 59831.785 | 16200 | PTUSE | 9.41 | 4 | 816 | 544 | 1024 | 64 | 1

  * a

These observations are phase connected in our timing solution provided in
table 2. The remaining observations have been fitted with arbitrary time-
offsets in the form of ‘JUMP’ statements.

Of these, the first two observations were recorded with the L-band receivers,
with a central frequency of $1284\text{\,}\mathrm{MHz}$ and a bandwidth of
$856\text{\,}\mathrm{MHz}$ split into 4096 frequency channels and sampled
every $76.56\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The next two observations
were recorded with the Ultra High Frequency (UHF) receivers, centered at the
frequency of $816\text{\,}\mathrm{MHz}$ with a bandwidth of
$544\text{\,}\mathrm{MHz}$, also split into 4096 channels, and sampled every
$60.24\text{\,}\mathrm{\SIUnitSymbolMicro s}$. We used the Filterbanking
Beamformer User Supplied Equipment (FBFUSE, Barr2018) as the backend to form
between 275-287 synthesized beams on the sky with an overlap fraction444The
boundary of the synthesized beams in the tiling overlap each other with the
power level equal to this ratio. of 0.8, enabling arcsecond localization of
pulsars after initial detection. The beam tiling pattern corresponding to each
pointing was estimated based on an optimal hexagonal packing approach of
elliptical beams. This was done using the
Mosaic555https://github.com/wchenastro/Mosaic software (2021JAI....1050013C).
The beams were processed on-line by the Accelerated Pulsar Search User
Supplied Equipment (APSUSE) computing cluster, where data from each beam are
converted into an 8-bit, Stokes-I pulsar-search mode file, based on the
sigproc filterbank format (2011ascl.soft07016L). We then incoherently
dedispersed the observations off-line, using a dispersion measure (DM) of
$25.06\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$ (which corresponds to
the DM of the other known pulsar, M30A), and then downsampled the data in
frequency by a factor of 16 (bringing the total number of frequency channels
to 256) so as to reduce the data volume.

Based on the results obtained from the TRAPUM observations, we also carried
out a follow-up orbital campaign, with a pseudo-log cadence, between 2022 June
and July . Each of these observations was 60 minutes, with the exception of a
280-minute (4.5 h) periastron-passage observation in September 2022. These
follow-up observations were made using at least 60 antennas, the UHF receivers
and the Pulsar Timing User Supplied Equipment (PTUSE) backend
(2020PASA...37...28B). PTUSE recorded the data from a single tied-array beam
placed at the nominal centre of M30.666Additionally, we also recorded TRAPUM
search-mode observations in parallel to enable further pulsar and transient
searches of the cluster. The full bandwidth of $544\text{\,}\mathrm{MHz}$ was
split into 1024 frequency channels, coherently dedispersed to a DM of
$25.06\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$, downsampled to 256
frequency channels and saved in PSRFITS format (2004PASA...21..302H). PTUSE
observations were also recorded in search-mode since at the time we did not
yet have an accurate orbital ephemeris, and to allow further pulsar searches.
These observations were recorded in full-Stokes mode to enable polarimetric
measurements, and sampled every $15.06\text{\,}\mathrm{\SIUnitSymbolMicro s}$
for high-resolution pulsar timing.

All the observations used the Inter-Quartile Range Mitigation algorithm
(2022MNRAS.510.1393M) as a first-pass to filter out bright Radio-frequency
interference (RFI) signals. The data were then shipped on hard drives to
Garching, Germany, where the primary search analysis was conducted using the
Max Planck Computing and Data Facility (MPCDF)
Hercules777https://docs.mpcdf.mpg.de/doc/computing/clusters/systems/Radioastronomy.html
cluster.

###  2.1 Search Analysis

Our search analysis consists of two pipelines implementing different search
algorithms. First is
pulsar_miner888https://github.com/alex88ridolfi/PULSAR_MINER, a user-friendly
wrapper of presto999https://github.com/scottransom/presto which is a fourier-
domain acceleration search pipeline sensitive to binary pulsars in constant
acceleration ($P\mathrm{{}_{orb}}~{}\gtrsim$ 10 $T\mathrm{{}_{obs}}$;
2002AJ....124.1788R; 2003ApJ...589..911R). Our second pipeline called
3d/5d_peasoup101010https://github.com/vishnubk/5D_Peasoup uses the template-
bank algorithm to search for pulsars in compact circular orbit binaries by
coherently searching across three Keplerian parameters in the time-domain
(2022MNRAS.511.1265B).

Before commencing searching, we cleaned all of the observations again using
presto’s rfifind program, which masks frequency channels and sub-integrations
contaminated by radio frequency interference (RFI) using appropriate user-
defined thresholds. Additionally, we removed Fourier frequencies identified in
a topocentric zero-DM timeseries as they are almost certainly caused by RFI.
We then dedispersed the data between 23-28
$\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$ and transformed our
observation frame of reference to the solar-system barycenter by adding or
subtracting appropriate delays from the dedispersed timeseries using the DE421
JPL Ephemerides (2009IPNPR.178C...1F). We searched for both isolated and
binary pulsars using the accelsearch routine in presto. For our searches, we
used the GPU version111111https://github.com/jintaoluo/presto_on_gpu of presto
with a zmax = 1200 and performed incoherent harmonic summing of up-to 16
harmonics to be sensitive to narrow-duty cycle pulsars.

Additionally, in order to be sensitive to binary pulsars in more compact
orbits, we performed the search not only on the full 60-minute observation,
but also split the data into 15- and 30-minute chunks and searched each
segment with the same zmax value of 1200. This gives us extra sensitivity
towards pulsars in compact orbits (i.e. P${}_{\rm orb}\gtrsim$ $150\,\rm min$
for the 15-min segments) at the cost of raising our minimum flux density limit
as sensitivity improves with $T\mathrm{{}_{obs}}^{1/2}$. These searches also
improve our chances for finding a pulsar which shows significant diffractive
scintillation.

Finally, in order to take advantage of the sensitivity offered by the full
60-minute observation, but increase our sensitivity towards compact orbits
(i.e not be limited to $P\mathrm{{}_{orb}}~{}\gtrsim$ 10 $T\mathrm{{}_{obs}}$)
we carried out a template-bank search (2009PhRvD..79j4017M;
2011PhDT.......293K; 2013ApJ...774...93K; Allen_2013) where we expand from a
one-dimensional acceleration search to a three-dimensional Keplerian parameter
search (orbital period, projected semi major axis and initial orbital phase)
assuming a circular orbit binary. Here we searched for orbits between 4 and 10
h in the 60-minute observation and between 2 and 5 h in the 30-minute chunks,
with initial orbital phase between 0 and 2 $\pi$, mismatch of 10 % and a
coverage of 90 % assuming a minimum pulsar spin-period of 2 ms, minimum pulsar
mass of 1.4 M⊙ and a maximum companion mass of 8 M⊙. We refer the interested
readers to §3 of 2021MNRAS.504.1407R and §2 of 2022MNRAS.511.1265B for a more
in-depth review of the pulsar_miner and the 3d/5d_peasoup pipelines
respectively. On average, we folded approximately 350 pulsar candidates per
beam for all our acceleration searches and 1000 pulsar candidates per beam for
our template-bank searches121212Larger numbers of candidates are expected for
higher-order template-bank searches since our computational trials scale up
due to the addition of more binary parameters.. Multiplying this number by the
total number of synthesized beams times the four initial observations taken
under the TRAPUM project, approximately 1.5 million pulsar candidates were
produced which was infeasible to inspect manually. Therefore we used machine-
learning based pipelines to extract the most interesting pulsar candidates. We
used the Pulsar Image Classification system (PICS; 2014ApJ...781..117Z) and
Pulsar Candidate Identification Using Semi-Supervised Generative Adversarial
Networks (SGAN; 2021MNRAS.505.1180B) and manually inspected candidates which
scored above a threshold of 0.5 using either of these algorithms. This reduced
our candidate viewing load from 1.5 million to approximately 5000 ($\sim
0.3\%$). No new pulsars have been discovered in our analysis. However, our
pipelines blindly re-detected and confirmed the elusive binary pulsar PSR
J2140$-$2311B described in the next section.

##  3 Results

Figure 1: Bottom Panel: High S/N polarization pulse profile of M30B obtained
by summing phase-connected observations between 05U-08U recorded in the UHF
band. Black line represents the total intensity, red and blue lines display
the linear and circular polarization pulse profile respectively. Top Panel:
Measurements of Position Angle (PA) in degrees above four sigma of the linear
polarization.

###  3.1 Confirmation of PSR J2140$-$2311B (M30B)

We redetected M30B in two out of four of our initial set of TRAPUM
observations. Both of our detections were from data recorded in the UHF band
taken during 2021 January 22 and 30 (obs id: 03U and 04U) with a folded
significance of 12.4 and 14.0 sigma respectively. The measured barycentric
spin-period of 12.9896559(7) ms with a detected DM =
$25.07\text{\,}\mathrm{pc}\text{\,}{\mathrm{cm}}^{-3}$ in the Jan 22
observation immediately confirmed that this was indeed M30B initially reported
in 2004ApJ...604..328R. The absence of any clear detection from searches in
the L-band is most likely due to the pulsar being a steep spectrum source.
While the original GBT detection of consistent with the result of most GBT
observations.

M30B was made at L-band, we think that is most likely due to an extreme
diffractive scintillation event. M30B has been consistently detected in our
follow-up orbital campaigns between 2022 June-September, which were recorded
in the UHF band; this is likely caused by a combination of factors: the steep
spectrum of the pulsar, the fact that scintles are narrower at these lower
frequencies (thus leading to more of them being averaged within a given band),
higher sensitivity of MeerKAT (Gain G = 2.63 K/Jy, assuming 60 antennas used)
compared to GBT (G = 2.0 K/Jy) and the wider frequency bandwidth of the
MeerKAT UHF receivers (Bandwidth: 544 MHz) compared to the GBT 820 MHz
receiver (Bandwidth: 200 MHz). Using PTUSE full-Stokes observations taken
between 2022 Jun - July (obs id: 05U-08U), we were able to obtain a high-S/N
polarization pulse profile of this pulsar using standard routines in the
PSRCHIVE131313https://psrchive.sourceforge.net/ (2012AR&T....9..237V) software
suite. We detected a rotation measure (RM) = 8 $\pm$ 6 rad m-2 which was used
to de-Faraday the folded archive to produce a linear and circular polarization
pulse profile. This is shown in Figure 1.

Bottom Panel: High S/N polarization pulse profile of M30B obtained by summing
phase-connected observations between 05U-08U recorded in the UHF band. Black
line represents the total intensity, red and blue lines display the linear and
circular polarization pulse profile respectively. Top Panel: Measurements of
Position Angle (PA) in degrees above four sigma of the linear polarization.

###  3.2 Localization

The full width at half maximum (FWHM) of the GBT beam at 20 cm is about
$9\text{\,}\mathrm{\SIUnitSymbolArcminute}$. When M30B was initially
discovered, its position within the GBT telescope beam and by extension within
the cluster was not yet known. The position of a pulsar with respect to the
cluster centre gives us important clues regarding the evolutionary history of
the system. As mentioned in §2, MeerKAT observations of this cluster made with
the TRAPUM backend typically consist of over 270 synthesized beams (hereafter
referred to as coherent beams) on the sky. These beams are much smaller than
the typical sky area observed by a single-dish radio telescope at the same
frequency. For example, the size141414Defined here with an elliptical fit at
50-percent of the power level of the semi-major and semi-minor axes of the
coherent beam during the beginning of the January 22nd UHF band were 27.3 and
21.9 arcseconds respectively151515The size of the coherent beam depends on the
configuration of the antennas used, on the central frequency of the
observation and on the source elevation. Therefore, a detection in one of the
beams already tells us that the pulsar is within or very near that beam. This
position measurement can be improved further if detections in multiple beams
are available which was the case with M30B where we detected the pulsar in
five neighbouring beams in the 2021 January 22nd observation. We used the
SeeKAT multibeam localiser
software161616https://github.com/BezuidenhoutMC/SeeKAT (Bezuidenhout et al.
submitted.) to improve our estimate for the position of this pulsar. SeeKAT
takes in the position of the coherent beam, the detected S/N within it and the
beam point-spread function (PSF) calculated using the software Mosaic and
performs a maximum-likelihood analysis to get a better estimate of the
pulsar’s position. In Figure 2, we show the known radio timing position of
M30A and the recently localized M30B’s position within the cluster along with
the TRAPUM beam tiling pattern for the beginning of the observation taken on
2021 January 22. M30B is located
$1.2(1)\text{\,}\mathrm{\SIUnitSymbolArcminute}$ away from the cluster centre
and just outside the half-light radius. The consequences of this in terms of
the evolutionary history of M30B is discussed in §3.5.

Figure 2: Beam tiling pattern at the start of the TRAPUM observation of M30
recorded in UHF band on 2021 January 22. Boundary of the ellipse plotted here
corresponds to 50 % of the maximum power level. The beam pattern was generated
using Mosaic. The boundaries of the core and half-light radius have been
obtained from the Harris2010 catalog which are marked as dashed and dash-
dotted lines respectively. Overlaid on the plots in red are the two known
pulsars within this cluster both of which were reported in
2004ApJ...604..328R. While M30A was discovered and regularly monitored since
then, M30B remained undetected since the discovery detection which has now
finally been confirmed, localized and solved using TRAPUM. The inset shows a
zoomed out version of the same plot.

###  3.3 Preliminary Orbital analysis

Given our new detections from TRAPUM and MeerTIME observations, the first step
was to obtain an accurate orbital solution for M30B which requires the
identification of its orbital period $P_{b}$. In order to do this, we
extracted the barycentric epoch and spin-period corresponding to each
detection. We then used a modified
version171717https://github.com/mcbernadich/CandyCracker of the
2008MNRAS.387..273B roughness algorithm to get an initial estimate of the
orbital period of this system. Roughness $\phi$ is calculated by folding the
data with a number of trial $P_{b}$ values. For each iteration, we then obtain
data for the observed spin-period versus orbital phase. From this, we compute
the summation of the squared differences of the observed spin period between
adjacent values of $\phi$. The idea here is that the roughness should be
minimum for the correct value of $P_{b}$. Building on the original algorithm,
we added a variance measure to obtain a more robust measurement of $P_{b}$.
Using this method, we estimated the $P_{\rm b}$ of M30B to be 6.215 days. This
estimate was then used as an initial guess to build a first-pass orbital
ephemeris using the program
fitorbit181818https://github.com/vivekvenkris/fitorbit.

###  3.4 Pulsar timing

The next step was to improve this orbital ephemeris through a process known as
pulsar timing. We started by folding all our observations using the
dspsr191919https://dspsr.sourceforge.net/ software package
(2011PASA...28....1V) modulo the predicted spin-period from the orbital
ephemeris obtained from fitorbit. Then we formed a stable integrated pulse
profile by summing the data in frequency, time, polarization. We then cross-
correlated this high-S/N pulse profile with an analytic template to extract
times of arrival (TOAs) at our telescope site for a particular rotational
phase of the pulsar. These preprocessing steps were done using standard
routines from the PSRCHIVE202020https://psrchive.sourceforge.net/
(2012AR&T....9..237V) software suite.

Using TOAs derived from our MeerKAT detections, we were able to determine the
orbital parameters of M30B using the software
tempo212121https://tempo.sourceforge.net/ and the theory-independent “DD”
model (1986AIHS...44..263D). To get a better estimate of DM, we extracted TOAs
per frequency channel by summing each observation in time and polarization and
averaging our data across frequency from 256 to 4 frequency channels. These
TOAs were then used to fit for DM by keeping all other orbital and spin
parameters fixed using tempo. Our ephemeris was precise enough to fold and
detect the pulsar in the 2021 and 2022 UHF observations (obs id: 03U - 08U).
However, since this ephemeris is not a phase-connected solution, a priori we
do not yet know the rotation count between groups of TOAs of different
observations. This is estimated by adding an arbitrary time-offset for each
observation in the form of ‘JUMP’ statements between each set of locally
connected TOAs, and then attempting to establish the rotation counts between
close sets of observations, as described in detail in §3 of
2018MNRAS.476.4794F. Given the general sparsity of the detections, especially
between 2001 and 2022, we could not connect all observations this way, so we
used Dracula (described in §4 of 2018MNRAS.476.4794F) to automatically
identify the unknown number of rotations between observations. However, even
with this algorithm, we cannot yet determine a unique phase-coherent timing
solution for this all the data of the pulsar, as the number of observations is
still too small for that.

However, we obtained a good orbital solution (see Table 2, $\chi^{2}=1.36$)
that can fold all the observations, including the early 2001 observation, and
another observation taken during periastron on 2022 Sept (obs id: 09U).
Additionally, all the TOAs from the 4-day orbital campaign (obs id: 05U-08U)
made in 2022 are phase connected; this provides an important contribution to
the precision of our solution. The post-fit residuals for this model can be
found in Figure 3 with the arbitrary time offsets subtracted. The flat
residuals indicate that the model provides, within its current limitations, a
good description of the timing. We were also able to recover the pulsar in one
of our MeerKAT L-band observations (obs id: 02L) which was undetected by our
blind searches. However, even with the aid of this ephemeris, we still could
not recover the signal in any of the old GBT observations besides the
discovery observation, nor in two 30-minute observations of M30 at a frequency
of 400 MHz taken with the Giant Metrewave Radio Telescope (GMRT) in India as
part of the GC survey presented by 2022A&A...664A..54G.

These non-detections can be explained because we do not yet have a phase
coherent timing solution, but also because of the difference of sensitivity of
the MeerKAT, GBT and GMRT observations. The survey sensitivity of TRAPUM GC
observations for NGC 1851 have been reported previously in §2.1 of
2022A&A...664A..27R using the modified radiometer equation
(1985ApJ...294L..25D). Adjusting these numbers for the values reported in
table 1 for TRAPUM observations of M30, we get a minimum detectable flux
density of $14.1\,\rm\mu Jy$ at L-BAND and $19.9\,\rm\mu Jy$ at the UHF band
compared to $109.3\,\rm\mu Jy$ for the 400 MHz GMRT observation of M30
reported in 2022A&A...664A..54G. 2004ApJ...604..328R reported that the GBT
observations of M30 were sensitive to normal millisecond pulsars in the range
of $\sim 50-100\,\rm\mu Jy$.

Table 2: Timing parameters for PSR J2140$-$2311B (M30B) presented in Barycentric Dynamical Time (TDB). For right ascension and declination, we report the values that correspond to the maximum likelihood reported by SeeKAT along with their one-sigma uncertainty. Both the position and DM have been kept fixed for our timing analysis given the sparsity of our detections. Pulsar |  J2140$-$2311B
---|---
Fitting program | TEMPO
Time Units | TDB
Terrestrial Time Standard | UTC(NIST)
Solar System Ephemeris | DE440
Right Ascension, $\alpha$ (J2000) | 21:40:25.2(3)
Declination, $\delta$ (J2000) |  $-$23:11:45(7)
Spin Frequency, $f$ ($\rm Hz$) | 76.9833055(6)
1st Spin Frequency derivative, $\dot{f}$ ($\rm Hz\,s^{-1}$) |  3(15)$\rm\times 10^{-15}$
Reference Epoch (MJD) | 59763.520924
Start of Timing Data (MJD) | 52161.993
End of Timing Data (MJD) | 59831.942
Dispersion Measure, DM (pc cm-3) | 25.063(3)
Number of TOAs | 73
Residuals RMS ($\mu$s) | 39.42
Binary Parameters
Binary Model | DD
Projected Semi-major Axis, $x_{\rm p}$ (lt-s) | 19.5222(7)
Orbital Eccentricity, $e$ | 0.87938(2)
Longitude of periastron, $\omega$ (deg) | 160.8007(4)
Epoch of periastron Passage, $T_{0}$ (MJD) | 59763.520649(6)
Rate of periastron advance, $\dot{\omega}$ (deg/yr) | 0.078(2)
Orbital Period, $P_{b}$ (days) | 6.21565400(6)
Derived Parameters
Galactic longitude, $l$ (∘) | 27.161(1)
Galactic latitude, $b$ (∘) |  $-$46.851(2)
Total system Mass, $M_{\rm TOT}$ (M⊙) | 2.53(8)
Companion mass, $M_{\rm c}$ (${\rm M}_{\odot}$) |  $\geq$ 1.10
Pulsar mass, $M_{\rm p}$ (${\rm M}_{\odot}$) |  $\leq$ 1.43
Spin Period, $P$ (s) | 0.01298982933(2)
1st Spin Period derivative, $\dot{P}$ (s s-1) |  $-$6(25)$\times 10^{-19}$
Mass Function, $f(M_{\rm p})$ (${\rm M}_{\odot}$) | 0.2067
Total offset from GC center, $\theta_{\perp}$ (arcmin) | 1.2(1)
Figure 3: Postfit timing residuals of PSR J2140$-$2311B as a function of time
and orbital phase using the DD Binary model. Colors indicate different
telescopes and backends used to obtain the TOAs.

An important parameter in this orbital solution is the system’s rate of
advance of periastron, which we measure to be $\dot{\omega}=0.078\pm
0.002\,\rm deg\,yr^{-1}$. The precision of this measurement greatly benefits
from the inclusion of the 2001 TOAs, even with the fit of an arbitrary time
offset. Using the DDGR model (1987grg..conf..209T; 1989ApJ...345..434T), which
assumes that general relativity (GR) accounts for the relativistic effects
observed in the timing, the total mass of the system derived from
$\dot{\omega}$ is $M_{\rm tot}=2.53\pm 0.08$ M⊙. The mass-mass diagram
assuming GR along with our measurement of $\dot{\omega}$ is shown in Figure 4.
In this Figure, we can see that combining the total mass measurement with the
constraint from the mass function, we can estimate a minimum companion mass of
1.10 M⊙ and a maximum pulsar mass of 1.43 M⊙.

Note that, although our measurement of the first-spin frequency derivative
$\dot{f}$ is not significant, it is important to fit for this parameter in
order to obtain a realistic uncertainty for $\dot{\omega}$. Indeed, if we
don’t fit for $\dot{f}$, the uncertainty of $\dot{\omega}$ will be one order
of magnitude smaller. Assuming such a the small $\dot{f}$ typical of a MSP
MSPs is not warranted because it could have a significant contribution from
the system’s acceleration in the cluster.

Further timing observations are necessary in order to fully connect all the
MeerKAT observations. A significant extension of the timing baseline will
result in much more precise measurements of $\dot{f}$ and all orbital
parameters, especially $P_{b}$, and $\dot{\omega}$; this might eventually
allow a full phase connection back to 2001 and the detection of additional
relativistic effects. We plan to continue timing M30B as part of the MeerTime
GC pulsar timing programme.

Figure 4: Mass-mass diagram of M30B assuming GR obtained from pulsar timing.
In the left plot, we show the mass of the companion $M_{\rm c}$ versus the
cosine of the inclination angle. The gray region here is excluded by the
requirement that $M_{\rm p}>0$. On the right we present $M_{\rm c}$ versus the
mass of the pulsar $M_{\rm p}$. The gray region is excluded by the mass
function and orbital geometry (i.e $\sin{i}\leq 1$). The black triplet of
lines shows the regions consistent with the measurement of the advance of
periastron $\dot{\omega}$ along with their 1 $\sigma$ uncertainty. As we can
see, from these measurements we can derive an upper limit for $M_{\rm p}$ and
a lower limit for $M_{\rm c}$.

###  3.5 System origin

Most binary MSPs in our Galaxy are in highly circularised orbits with a low-
mass He-WD or a non-degenerate or semi-degenerate ultra-light companion (Mc
$\leq$ 0.08 M⊙, 2012MNRAS.425.1601T and references within). These systems are
highly recycled ($P\leq$10\text{\,}\mathrm{ms}$$) due to the long mass
transfer phase which transfers matter and angular momentum from the companion
to the pulsar, spinning it up to very rapid rates and reducing the magnetic
field strengths of the NS (1991PhR...203....1B; 2002JApA...23...67B;
2006csxs.book..623T). Intermediate spin-period pulsars
($$10\text{\,}\leq$P\leq$20\text{\,}\mathrm{ms}$$$$10\text{\,}\leq$\;P\leq$20\text{\,}\mathrm{ms}$$,
2001ApJ...548L.187C) tend to have massive CO or ONeMg WD companions (e.g. PSR
J1802$-$2104, 2010ApJ...711..764F), but their orbits still have very low
eccentricities.

If the companion star is massive enough to undergo its own supernova (SN)
explosion and if the binary orbit survives, then a double neutron star system
(DNS, e.g. PSR J1757$-$1854; 2022arXiv220315995Csee 2017ApJ...846..170T for a
review) will form. These systems tend to be mildly recycled
($P\geq$17\text{\,}\mathrm{ms}$$, see 2018ApJ...854L..22S) as their massive
companions do not live for long, halting the recycling process earlier. Unlike
systems with WD companions, they have highly eccentric orbits because of the
kick and mass loss from the second SN (1995MNRAS.274..461B;
2017ApJ...846..170T).

Given its high orbital eccentricity, M30B could in principle have formed like
the DNSs in the Galactic disk. The total mass is similar to that of the
lightest known DNS systems (like PSR PSRs J1411+2551 and J1946+2052,
2018ApJ...854L..22S2017ApJ...851L..29M; 2018ApJ...854L..22S). However, this
scenario is unlikely given the spin period of M30B —
$P\sim$12.98\text{\,}\mathrm{ms}$$, faster than for any pulsars in a DNS seen
to date in the Galaxy. Furthermore, there are currently no massive stars in
GCs that would provide a second SN; the last time these existed in GCs was
more than 10 Gyr ago. Thus, if M30B was such a primordial DNS, its
characteristic age ($\tau_{\rm c}$) would have to be at least $\sim 10\,\rm
Gyr$. We note, though, that given the large $\gamma$ of the host cluster it is
very unlikely that the system would still resemble its original configuration.

The spin period of M30B is more compatible with a relatively massive CO or
ONeMg WD companion. In the Galactic disk, these systems have low orbital
eccentricities, but given the large number of stellar encounters in GCs, the
orbital eccentricity could have been greatly increased (1992RSPTA.341...39P).
Thus, it is possible that the companion is a massive WD star whose progenitor
recycled the pulsar. As in the case of a primordial DNS, we should in this
case ; in this case we should also expect $\tau_{\rm c}\sim 10\,\rm Gyr$.

A more likely hypothesis is that M30B is a result of an exchange encounter,
where the lighter star that recycled the pulsar was ejected during the
binary’s chaotic encounter with a more massive degenerate star, which is the
current companion. Given the random dynamics of such an encounter, we cannot
decide on the nature of the companion - either a massive WD or a NS — based on
arguments from stellar evolution . (although massive white dwarfs are
generally more likely given their larger abundance in the cluster, see e.g.,
2019ApJ...877..122Y; 2021ApJ...917...28K).

Several secondary exchange products have eccentric MSP binaries with massive
companions recently been found in core-collapsed GCs (which have the highest
$\gamma$ values: PSR B2127+11C in M15 1991ApJ...374L..41P;
2006ApJ...644L.113J, PSR J1807$-$2500B in NGC 6544 2012ApJ...745..109L, PSR
J1835$-$3259A in NGC 6652 2015ApJ...807L..23D and PSR J1823$-$3021G in NGC
6624 2021MNRAS.504.1407R). These are trought to be exchange products because
their spin periods are very small compared to binary pulsars in the Galactic
disk with similarly massive companions. In all core-collapsed GCs, these
systems represent about approximately 1/3 of the known population of binary
radio pulsars. For this calculation, we have adopted the definition of core-
collapsed GC from the Harris catalog (1996yCat.7195....0H, 2010 revision) and
only considered binary pulsars with well measured orbits. We then assume that
pulsars in an eccentric orbit ($e>0.1$) orbiting a massive companion
($m_{c}>0.38\;\rm M_{\odot}$) are likely to be the result of exchange
products. Using this definition and the updated numbers from the ‘GC Pulsar
catalog’, we find that 5 out of the 14 ($\sim$ 35.7 %) binary pulsars in core-
collapsed GCs are likely to be exchange products; in non core-collapsed GCs
only 10 out of 109 ($\sim$ 9.2%) binary pulsars fulfill these criteria. The
location of M30B in a core-collapsed GC is thus an indication that it likely
originated in an exchange encounter. Another clue is its high eccentricity:
Among secondary exchange products, M30B has the third-highest orbital
eccentricity after PSR J1835$-$3259A ($e\sim 0.96$, 2015ApJ...807L..23D) and
PSR J0514$-$4002A ($e\sim 0.88$, 2007ApJ...662.1177F; 2019MNRAS.490.3860R).

Another datum that points towards an exchange encounter is that M30B is
located $1.2(1)\text{\,}\mathrm{\SIUnitSymbolArcminute}$ from the cluster
centre and just outside its half-light radius. Normally, mass segregation
causes the pulsar population in dense GCs to be very centrally condensed, with
most pulsars within, or near their cores (e.g., 2017MNRAS.471..857F;
2017ApJ...845..148P; 2018MNRAS.481..627A). This is the case for M30A, which is
very near the centre of the cluster. The large distance of M30B from the
centre is likely the result of the recoil caused by the ejection of the
previous light companion. This situation is very similar to that of PSR
B2127+11C, which is $0.94\text{\,}\mathrm{\SIUnitSymbolArcminute}$ from the
centre of M15 (1991ApJ...374L..41P; 2006ApJ...644L.113J). Overall, pulsars
with such large distances from the centre (in core radii) occur more often in
high-$\gamma$ GCs (2014A&A...561A..11V).

When exchange encounters are so frequentthat they happen even very frequent,
they might even happen during the LMXB phase. In this case, the recycling
process is in many cases truncated, resulting in a relatively slow and young
MSP. This partially recycled pulsar that still has a relatively high magnetic
field and will therefore appear young. Unlike MSPs, such mildly recycled
pulsars spin down fast, which means that their LMXB disruption must be recent.
This explains why slow, apparently young pulsars in GCs are overwhelmingly
found in high-$\gamma$ GCs and lie below the pulsar spin-up line
(2014A&A...561A..11V; 2022MNRAS.513.2292A), although in this regard we must
keep in mind that there are alternative explanations for the formation of
apparently young pulsars in GCs (e.g. 2008MNRAS.386..553I).

The recent disruption of a LMXB by a massive degenerate intruder that then
becomes the pulsar’s companion is a likely explanation for the parameters of
PSR B2127+11C ($P=30.5\,\rm ms$ and $\tau_{\rm c}\sim 0.1\,\rm Gyr$, ), a
binary pulsar located in the core-collapsed GC M15. Although this system
superficially resembles a Galactic DNS (see 2019ApJ...880L...8A), its
characteristic age - less than 1% of the age of the GC , 1991ApJ...374L..41P),
and explains why high-$\gamma$ GCs have many more slow, mildly recycled
pulsars (2014A&A...561A..11V; 2022MNRAS.513.2292A). If this \- is too small
for it to be a DNS formed from the primordial population of massive stars of
M15 (1991ApJ...374L..41P). If the recent disruption of a LMXB is the
explanation for the relatively slow spin of M30B, then we might also expect
the pulsar to be much younger than the have a characteristic age much smaller
than the age of the M30 GC itself. As in the case

One clue that suggests that PSR B2127+11C was recently involved in an exchange
encounter is its large distance
($0.94\text{\,}\mathrm{\SIUnitSymbolArcminute}$) from the centre of M15
(1991ApJ...374L..41P; 2006ApJ...644L.113J). Normally, mass segregation causes
the pulsar population in dense GCs to be very centrally condensed, with most
pulsars within, or near their cores (e.g., 2017MNRAS.471..857F;
2017ApJ...845..148P; 2018MNRAS.481..627A). This is the case for all other
pulsars in M15 (1993PhDT.........2A) and for M30A (2004ApJ...604..328R). The
large distance of PSR B2127+11C , the from the center of M15 is possibly the
result of the recoil caused by the ejection of the previous light companion
that partially recycled the pulsar; the time elapsed since the recoil is
clearly still ($\sim 0.1\rm Gyr$, if it is the same event that disrupted the
LMXB) is presumably too short for the system to have migrated back to the
centre of the GC via dynamical friction. Interestingly, M30B is located
$1.2(1)\text{\,}\mathrm{\SIUnitSymbolArcminute}$ from the cluster centre and
just outside its half-light radius, which again suggests a recent exchange
interaction. Overall, pulsars with such large distances from the centre (in
core radii) occur more often in high-$\gamma$ GCs (2014A&A...561A..11V).

Further timing

###  3.6 Prospects

We plan to continue timing M30B as part of the MeerTime GC pulsar timing
programme. These observations are necessary to in order to fully connect all
the MeerKAT observations (possibly all the way back to 2001) and shed light
into the nature of the M30B system. The first step is to obtain a A phase-
coherent timing solution , which will yield much improved astrometric, spin
and orbital parameters. The spin and orbital period derivatives will be
extremely important, because their measurement allows will allow the
determination of $\tau_{c}$, which is will be crucial for distinguishing
between a primordial binary with a massive companion or an a more recent
exchange product. This will also greatly improve all orbital parameters,
especially $P_{b}$ and $\dot{\omega}$.

The following step would will be to determine the individual masses via the
detection of additional relativistic effects. Dense orbital campaigns in the
near future might lead to the detection of the Shapiro delay
(1964PhRvL..13..789S). If this is not possible (the timing precision might be
insufficient or the orbital inclination might be too low ): for an inclination
angle of $60^{\circ}$, the $h_{3}$ parameter (2010MNRAS.409..199F) would be
$1.2\mu s$. This can be detected with 2-$\sigma$ significance with 4000 ToAs
with the current timing precision. The parameter will be larger (and therefore
detected with higher significance) for higher inclinations, another
possibility is to somehow improve the timing precision. If the orbital
inclination is low and/or we’re unable to improve the timing precision, then
we might will have to wait to detect the Einstein delay. However, the
longitude of periastron $\omega$ of M30B is not optimal for this goal (see
detailed discussion in 2019MNRAS.490.3860R), so if the Shapiro delay is not
detectable, measuring the component masses will take a long timewith the
Einstein delay will take several decades.

##  4 Conclusion

In this letter, we presented the confirmation of M30B with the first set of
new detections of this pulsar since its discovery in 2001. We found that the
pulsar can be reliably detected with the MeerKAT UHF receivers; this has
finally allowed, 20 years after the discovery, a detailed characterization of
this system: it is located $1.2(1)\text{\,}\mathrm{\SIUnitSymbolArcminute}$
from the cluster centre and the pulsar is in a highly eccentric ($e=0.879$)
orbit around a companion that could either be a massive WD or a NS. We also
measured the rate of periastron advance, which indicates a total system mass
consistent with that of the lightest known DNSs in our Galaxy
(2017ApJ...851L..29M; 2018ApJ...854L..22S)222222For a list of NS mass
measurements, see https://www3.mpifr-bonn.mpg.de/staff/pfreire/NS_masses.html.
M30B was likely formed as the result of a secondary exchange encounter,
similar systems have been observed in other GCs with very dense cores. Further
timing observations are necessary to obtain a phase-connected timing solution
for this pulsar, which would yield much improved astrometric, spin and orbital
parameters. Continued timing might result in the detection of additional
relativistic effects and the determination of the individual masses of the
components. The characterization of M30B is a demonstration of the unrivalled
sensitivity of MeerKAT for radio sources in the Southern celestial hemisphere.

The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory, which is a facility of the National Research Foundation, an
agency of the Department of Science and Innovation. SARAO acknowledges the
ongoing advice and calibration of GPS systems by the National Metrology
Institute of South Africa (NMISA) and the time space reference systems
department of the Paris Observatory. PTUSE was developed with support from the
Australian SKA Office and Swinburne University of Technology. MeerTime data is
housed on the OzSTAR supercomputer at Swinburne University of Technology. The
OzSTAR program receives funding in part from the Astronomy National
Collaborative Research Infrastructure Strategy (NCRIS) allocation provided by
the Australian Government. The authors also acknowledge MPIfR funding to
contribute to MeerTime infrastructure. TRAPUM observations used the FBFUSE and
APSUSE computing clusters for data acquisition, storage and analysis. These
clusters were funded and installed by the Max-Planck-Institut für
Radioastronomie and the Max-Planck-Gesellschaft. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. The Green Bank
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. VB, AR, EDB, FA, DJC,
WC, PCCF, TG, MK, PVP and VVK acknowledge continuing valuable support from the
Max-Planck Society. VB gratefully acknowledges support from ERC Synergy Grant
‘BlackHoleCam’ Grant Agreement Number 610058. SMR is a CIFAR Fellow and is
supported by the NSF Physics Frontiers Center awards 1430284 and 2020265. This
work is supported by the Max-Planck Society as part of the ”LEGACY”
collaboration on low-frequency gravitational wave astronomy. AR and AP
gratefully acknowledge financial support by the research grant “iPeska” (P.I.
Andrea Possenti) funded under the INAF national call Prin-SKA/CTA approved
with the Presidential Decree 70/2016. AR, MBu and AP also acknowledge support
from the Ministero degli Affari Esteri e della Cooperazione Internazionale -
Direzione Generale per la Promozione del Sistema Paese - Progetto di Grande
Rilevanza ZA18GR02. BWS acknowledges funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 694745). Pulsar research at UBC is supported by
an NSERC Discovery Grant and by the Canadian Institute for Advanced Research.