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##  Appendix A $\ell_{p}$ affine embeddings via root leverage score sampling

Following the techniques of [WY23c], we obtain a root leverage score sample
guarantee for $\ell_{p}$ affine embeddings, for both $p<2$ and $p>2$.

Our main result of this section is the following theorem, which shows that
root leverage score sampling yields $\ell_{p}$ affine embeddings.

######  Theorem A.1 (Root leverage score sampling).

Let $\mathbf{A}\in\mathbb{R}^{n\times d}$ and $\mathbf{b}\in\mathbb{R}^{n}$.
Let $1\leq p<\infty$. Let $R\geq\lVert\mathbf{b}\rVert_{p}$. Suppose that

$\frac{\lvert\mathbf{b}(i)\rvert^{p}}{R^{p}}\leq\begin{dcases}\min\left\\{1,n^{p/2-1}\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}\right\\}&p>2\\\
\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}&p<2\end{dcases}$

for every $i\in[n]$. Let $\alpha=\Theta(\varepsilon^{2})/((\log
n)^{3}+\log(1/\delta))$ and let

$q_{i}\geq\begin{dcases}\min\left\\{1,n^{p/2-1}\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}/\alpha\right\\}&p>2\\\
\min\left\\{1,\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}/\alpha\right\\}&p<2\end{dcases}$

Let $\mathbf{S}$ be the $\ell_{p}$ sampling matrix (Definition 1.7) with
sampling probabilities $\\{q_{i}\\}_{i=1}^{n}$. Then, with probability at
least $1-\delta$, for every $\mathbf{x}\in\mathbb{R}^{d}$,

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}=(1\pm\varepsilon)\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm\varepsilon
R^{p}$

###  A.1 Reduction to a small number of scales

Our first task is to reduce the proof of Theorem A.1 to showing a similar
theorem when $\mathbf{A}\mathbf{x}$ is restricted to a certain scale, for a
small number of scales. This is shown in the following lemma:

######  Lemma A.2.

Let $\mathbf{A}\in\mathbb{R}^{n\times d}$ and $\mathbf{b}\in\mathbb{R}^{n}$.
Let $1\leq p<\infty$. Let $R_{0}\geq\left\lVert\mathbf{b}\right\rVert_{p}$ and
$0<\varepsilon<1/2$. Suppose that

$\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}=1}\left\lvert\left\lVert\mathbf{S}\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}\right\rvert\leq\varepsilon$

and that

$\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R_{i}}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert\leq\varepsilon
R_{i}^{p}$

holds for each $R_{i}=2^{i}\cdot R_{0}$, $i\in[I]$, where
$I=O(\log\varepsilon^{-1})$. Then,

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}=(1\pm
2\cdot
4^{p}\varepsilon)\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm
2^{p}\varepsilon R_{0}^{p}$

for every $\mathbf{x}\in\mathbb{R}^{d}$.

###### Proof.

First note that if $\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R_{1}$, then we immediately have

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}=\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm\varepsilon
R_{1}^{p}=\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm
2^{p}\varepsilon R_{0}^{p}.$

Next, suppose that $\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\geq
R_{0}/\varepsilon$. Note that

$\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}=\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\pm\left\lVert\mathbf{b}\right\rVert_{p}=(1\pm\varepsilon)\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}$

and similarly,

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}=\left\lVert\mathbf{S}\mathbf{A}\mathbf{x}\right\rVert_{p}\pm\left\lVert\mathbf{S}\mathbf{b}\right\rVert_{p}=(1\pm
4\varepsilon)\left\lVert\mathbf{S}\mathbf{A}\mathbf{x}\right\rVert_{p}$

Thus,

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}=(1\pm
4\varepsilon)(1\pm\varepsilon)\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}=(1\pm
7\varepsilon)\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}.$

Finally, we handle the intermediate scales between $R_{0}$ and
$R_{0}/\varepsilon$. Consider $\mathbf{x}$ such that
$R_{i}\leq\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}<2\cdot R_{i}$. Note
then that

$\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}\geq\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}-\left\lVert\mathbf{b}\right\rVert_{p}\geq\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}/2\geq
R_{i}/2$

so

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}=\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm\varepsilon\cdot(2R_{i})^{p}=\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm\varepsilon\cdot(4\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p})^{p}.$

This covers all cases. ∎

###  A.2 Reduction to a Gaussian process with flat sensitivities

We now work towards bounding a quantity as the one used in Lemma A.2. The
following lemma is a standard reduction in chaining arguments (see, e.g.,
Lemma D.1 of [WY23c], also [CP15, CD21]):

######  Lemma A.3 (Reduction to Gaussian processes).

Let $\mathbf{A}\in\mathbb{R}^{n\times d}$ and $\mathbf{b}\in\mathbb{R}^{n}$.
Let $1\leq p<\infty$. Let $R\geq\lVert\mathbf{b}\rVert_{p}$. Let $\mathbf{S}$
be a random $\ell_{p}$ sampling matrix (Definition 1.7). Then,

$\operatorname*{\mathbf{E}}_{\mathbf{S}}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert^{l}\leq(2\pi)^{l/2}\operatorname*{\mathbf{E}}_{\mathbf{S}}\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\sum_{i\in
T}\mathbf{g}_{i}\left\lvert[\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})](i)\right\rvert^{p}\right\rvert^{l},$

where $T\subseteq[n]$ is the set of rows with sampling probability $q_{i}<1$.

We will further reduce the problem to a similar problem for an instance with
“flat sensitivities”, following [CP15, WY23c]. We will use the following
trivial generalization of the result of [WY23c, Lemma D.5].

######  Lemma A.4 (Flattening all sensitivities).

Let $1\leq p<\infty$ and $\mathbf{A}\in\mathbb{R}^{n\times d}$ and
$\mathbf{b}\in\mathbb{R}^{n}$. Let $0<\alpha<1$. Then, there exists
$\mathbf{A}^{\prime}\in\mathbb{R}^{m\times d}$ and
$\mathbf{b}^{\prime}\in\mathbb{R}^{m}$ for $m=O(n\alpha^{-1})$ such that

$\boldsymbol{\sigma}_{i}^{q}(\mathbf{A}^{\prime})\leq\alpha,\qquad\boldsymbol{\sigma}_{i}^{q}(\mathbf{b}^{\prime})=\frac{\lvert\mathbf{b}(i)\rvert^{q}}{\left\lVert\mathbf{b}\right\rVert_{q}^{q}}\leq\alpha$

for every $i\in[m]$ and $1\leq q<\infty$. Furthermore, for any $1\leq
q<\infty$ and $\mathbf{x}\in\mathbb{R}^{d}$, we have that
$\left\lVert\mathbf{A}^{\prime}\mathbf{x}+\mathbf{b}^{\prime}\right\rVert_{q}=\Theta(\alpha^{1/p-1/q})\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{q}$.

Using Lemma A.4, we construct the following new instance with bounded
$\ell_{2}$ and $\ell_{p}$ sensitivities:

######  Lemma A.5 (Flattened instance).

Let $\mathbf{A}\in\mathbb{R}^{n\times d}$ and
$\mathbf{b}\in\mathbb{R}^{n\times d}$. Let
$R\geq\left\lVert\mathbf{b}\right\rVert_{p}$. Suppose that

$\frac{\lvert\mathbf{b}(i)\rvert^{p}}{R^{p}}\leq\begin{dcases}\min\left\\{1,n^{p/2-1}\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}\right\\}&p>2\\\
\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}&p<2\end{dcases}$

for every $i\in[n]$. Let $0<\alpha<1$ and let

$q_{i}\geq\begin{dcases}\min\left\\{1,n^{p/2-1}\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}/\alpha\right\\}&p>2\\\
\min\left\\{1,\boldsymbol{\tau}_{i}(\mathbf{A})^{p/2}/\alpha\right\\}&p<2\end{dcases}$

Let $T\subseteq[n]$ be the set of rows $i\in[n]$ with $q_{i}<1$. Let
$\mathbf{S}$ be a diagonal matrix with $\mathbf{S}_{i,i}\leq 1/q_{i}^{1/p}$.
Then, there is $\mathbf{A}^{\prime\prime}\in\mathbb{R}^{m\times d}$ and
$\mathbf{b}^{\prime\prime}\in\mathbb{R}^{m}$ for $m=O(n/\alpha)$ such that

  * •

$\boldsymbol{\tau}_{i}(\mathbf{A}^{\prime\prime})\leq O(\alpha)$ for $p<2$ and
$\boldsymbol{\tau}_{i}(\mathbf{A}^{\prime\prime})\leq O(\alpha)/n^{1-2/p}$ for
$p>2$, for every $i\in[m]$

  * •

$\boldsymbol{\sigma}_{i}^{p}(\mathbf{A}^{\prime\prime})\leq O(\alpha)$ and
$\lvert\mathbf{b}^{\prime\prime}(i)\rvert^{p}/R^{p}\leq O(\alpha)$ for every
$i\in[m]$

  * •

$\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}+\mathbf{b}^{\prime\prime}\right\rVert_{p}^{p}=\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}+\left\lVert\mathbf{S}|_{T}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}$
for every $\mathbf{x}\in\mathbb{R}^{d}$

  * •

$\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{p}^{p}=\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}+\left\lVert\mathbf{S}|_{T}\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}$
for every $\mathbf{x}\in\mathbb{R}^{d}$

###### Proof.

Let $\mathbf{A}^{\prime}\in\mathbb{R}^{m\times d}$ and
$\mathbf{b}^{\prime}\in\mathbb{R}^{m}$ be the flattened instances given by
Lemma A.4, where $m=O(n/\alpha)$. Now let

$\mathbf{A}^{\prime\prime}\coloneqq\begin{pmatrix}\mathbf{A}^{\prime}\\\
\mathbf{S}|_{T}\mathbf{A}\end{pmatrix},\qquad\mathbf{b}^{\prime\prime}\coloneqq\begin{pmatrix}\mathbf{b}^{\prime}\\\
\mathbf{S}|_{T}\mathbf{b}\end{pmatrix}$

be the $(m+n_{\mathbf{S}})\times d$ matrix and
$(m+n_{\mathbf{S}})$-dimensional vector formed by the vertical concatenation
of $\mathbf{A}^{\prime}$ and $\mathbf{b}^{\prime}$ with $\mathbf{S}\mathbf{A}$
and $\mathbf{S}\mathbf{b}$, where $n_{\mathbf{S}}$ is the number of rows
sampled by $\mathbf{S}$.

We now show how to bound the sensitivities of $\mathbf{A}^{\prime\prime}$ and
$\mathbf{b}^{\prime\prime}$.

For any row $i$ corresponding to a row of $\mathbf{A}^{\prime}$, the
$\ell_{2}$ sensitivities are already bounded by $\alpha$, and furthermore,
$\ell_{2}$ sensitivities can clearly only decrease with row additions. For any
row $i$ corresponding to a row of $\mathbf{S}\mathbf{A}$ that is sampled with
probability $q_{i}<1$, we have that

$\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{2}}{\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{2}^{2}}\leq\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{2}}{\left\lVert\mathbf{A}^{\prime}\mathbf{x}\right\rVert_{2}^{2}}=\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{2}}{\Theta(\alpha^{2/p-1})\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{2}^{2}}\leq\frac{1}{q_{i}^{2/p}}\frac{\left\lvert[\mathbf{A}\mathbf{x}](i)\right\rvert^{2}}{\Theta(\alpha^{2/p-1})\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{2}^{2}}\leq\frac{\boldsymbol{\tau}_{i}(\mathbf{A})}{\Theta(\alpha^{2/p-1})q_{i}^{2/p}}=O(\alpha).$

In fact, for $p>2$, we have the stronger bound of

$\frac{\boldsymbol{\tau}_{i}(\mathbf{A})}{\Theta(\alpha^{2/p-1})q_{i}^{2/p}}\leq\frac{O(\alpha)}{n^{1-2/p}}.$

Thus, we have that
$\boldsymbol{\tau}_{i}(\mathbf{A}^{\prime\prime})=\boldsymbol{\sigma}_{i}^{2}(\mathbf{A}^{\prime\prime})\leq
O(\alpha)$ for every row $i$ of $\mathbf{A}^{\prime\prime}$.

For $p<2$, the max sensitivity is bounded by $O(\alpha)$ by the monotonicity
of max sensitivities. For $p>2$, we have by reverse monotonicity of max
sensitivities that

$\boldsymbol{\sigma}_{i}^{p}(\mathbf{A})\leq
n^{p/2-1}\boldsymbol{\tau}_{i}(\mathbf{A})$

so for any row $i$ corresponding to a row of $\mathbf{S}\mathbf{A}$ sampled
with probability $q_{i}<1$, we have that

$\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{p}}{\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{p}^{p}}\leq\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{p}}{\left\lVert\mathbf{A}^{\prime}\mathbf{x}\right\rVert_{p}^{p}}=\frac{\left\lvert[\mathbf{S}\mathbf{A}\mathbf{x}](i)\right\rvert^{p}}{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}}\leq\frac{1}{q_{i}}\frac{\left\lvert[\mathbf{A}\mathbf{x}](i)\right\rvert^{p}}{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}}\leq\frac{\boldsymbol{\sigma}_{i}^{p}(\mathbf{A})}{q_{i}}\leq
O(\alpha).$

By similar reasoning, we have that

$\frac{\lvert[\mathbf{S}\mathbf{b}](i)\rvert^{p}}{R^{p}}\leq\frac{1}{q_{i}}\frac{\lvert\mathbf{b}(i)\rvert^{p}}{R^{p}}\leq
O(\alpha).\qed$

The next lemma shows that in order to bound the Gaussian process in Lemma A.3,
it suffices to bound a similar Gaussian process for
$\mathbf{A}^{\prime\prime}$ and $\mathbf{b}^{\prime\prime}$.

######  Lemma A.6 (Reduction to flattened instance).

Let $\mathbf{S}$, $\mathbf{A},\mathbf{b}$, and
$\mathbf{A}^{\prime\prime},\mathbf{b}^{\prime\prime}$ be as given in Lemma
A.5. Let $\delta,\varepsilon,l$ be such that $\delta\varepsilon^{l}\leq 1/2$.
Furthermore, suppose that

$\operatorname*{\mathbf{E}}_{\mathbf{S}}\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\sum_{i\in
T}\mathbf{g}_{i}\left\lvert[\mathbf{A}^{\prime\prime}\mathbf{x}+\mathbf{b}](i)\right\rvert^{p}\right\rvert^{l}\leq\delta\varepsilon^{l}(R^{p})^{l}$

for every $R\geq\lVert\mathbf{b}\rVert_{p}$. Then,

$\operatorname*{\mathbf{E}}_{\mathbf{S}}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert^{l}\leq
2\delta(2^{3p}\varepsilon R^{p})^{l}$

for every $R\geq\left\lVert\mathbf{b}\right\rVert_{p}$.

###### Proof.

Fix an outcome of $\mathbf{S}$ and let

$F_{\mathbf{S},R}=\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert.$

Note then that for any $\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R$, we have

$\displaystyle\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{p}$
$\displaystyle\leq\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}+\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}+\left\lVert\mathbf{S}\mathbf{b}\right\rVert_{p}$
$\displaystyle\leq
R+\left\lparen\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}+\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert\right\rparen^{1/p}+\left\lparen\left\lVert\mathbf{b}\right\rVert_{p}^{p}+\left\lvert\left\lVert\mathbf{S}\mathbf{b}\right\rVert_{p}^{p}-\left\lVert\mathbf{b}\right\rVert_{p}^{p}\right\rvert\right\rparen^{1/p}$
$\displaystyle\leq
R+\left\lparen\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}+F_{\mathbf{S},R}\right\rparen^{1/p}+\left\lparen\left\lVert\mathbf{b}\right\rVert_{p}^{p}+F_{\mathbf{S},R}\right\rparen^{1/p}$
$\displaystyle\leq
R+\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}+F_{\mathbf{S},R}^{1/p}+\left\lVert\mathbf{b}\right\rVert_{p}+F_{\mathbf{S},R}^{1/p}$
$\displaystyle\leq 4R+2F_{\mathbf{S},R}^{1/p}.$

Thus,

$\displaystyle\operatorname*{\mathbf{E}}_{\mathbf{S}}F_{\mathbf{S},R}^{l}$
$\displaystyle\leq\operatorname*{\mathbf{E}}_{\mathbf{S}}\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\sum_{i\in
T}\mathbf{g}_{i}\left\lvert[\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})](i)\right\rvert^{p}\right\rvert^{l}$
Lemma A.3
$\displaystyle\leq\operatorname*{\mathbf{E}}_{\mathbf{S}}\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\sum_{i}\mathbf{g}_{i}\left\lvert[\mathbf{A}^{\prime\prime}\mathbf{x}+\mathbf{b}^{\prime\prime}](i)\right\rvert^{p}\right\rvert^{l}$
$\displaystyle\leq\operatorname*{\mathbf{E}}_{\mathbf{S}}\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}^{\prime\prime}\mathbf{x}\right\rVert_{p}\leq
4R+2F_{\mathbf{S},R}^{1/p}}\left\lvert\sum_{i}\mathbf{g}_{i}\left\lvert[\mathbf{A}^{\prime\prime}\mathbf{x}+\mathbf{b}^{\prime\prime}](i)\right\rvert^{p}\right\rvert^{l}$
$\displaystyle\leq\operatorname*{\mathbf{E}}_{\mathbf{S}}\delta\varepsilon^{l}((4R+2F_{\mathbf{S},R}^{1/p})^{p})^{l}$
by hypothesis
$\displaystyle\leq\operatorname*{\mathbf{E}}_{\mathbf{S}}\delta(2^{2p}\varepsilon)^{l}(((2R)^{p})^{l}+F_{\mathbf{S},R}^{l})$
$\displaystyle=\delta(2^{2p}\varepsilon)^{l}\left[((2R)^{p})^{l}+\operatorname*{\mathbf{E}}_{\mathbf{S}}F_{\mathbf{S},R}^{l}\right]$

so rearranging gives

$\frac{\operatorname*{\mathbf{E}}_{\mathbf{S}}F_{\mathbf{S},R}^{l}}{(2^{p}R^{p})^{l}+\operatorname*{\mathbf{E}}_{\mathbf{S}}F_{\mathbf{S},R}^{l}}\leq\delta(2^{2p}\varepsilon)^{l}.$

In turn, this implies that

$\operatorname*{\mathbf{E}}_{\mathbf{S}}F_{\mathbf{S},R}^{l}\leq\frac{\delta(2^{2p}\varepsilon)^{l}(2^{p}R^{p})^{l}}{1-\delta(2^{2p}\varepsilon)^{l}}\leq
2\delta(2^{3p}\varepsilon R^{p})^{l}.\qed$

###  A.3 Bounds on the Gaussian process

In this section, we present results from [WY23c] (which in turn are based on
[BLM89, LT91]) which will allow us to bound a Gaussian process of the form of
Lemma A.6.

The following is a straightforward generalization of Lemma C.3 of [WY23c].

######  Lemma A.7.

Let $1\leq p<\infty$ and let $\mathbf{A}\in\mathbb{R}^{n\times d}$ and
$\mathbf{b}\in\mathbb{R}^{n}$. Let
$R\geq\left\lVert\mathbf{b}\right\rVert_{p}$. Define the pseudo-metric

$d_{X}(\mathbf{y},\mathbf{y}^{\prime})\coloneqq\left\lparen\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\left\lvert\sum_{i=1}^{n}\mathbf{g}_{i}\lvert\mathbf{y}(i)\rvert^{p}-\sum_{i=1}^{n}\mathbf{g}_{i}\lvert\mathbf{y}^{\prime}(i)\rvert^{p}\right\rvert^{2}\right\rparen^{1/2}$

Let $\sigma\geq\max_{i\in
S}^{n}\boldsymbol{\sigma}_{i}^{p}(\mathbf{A})+\lvert\mathbf{b}(i)\rvert^{p}/R^{p}$.
Then, for any $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{b}$ and
$\mathbf{y}^{\prime}=\mathbf{A}\mathbf{x}^{\prime}+\mathbf{b}$ with
$\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p},\left\lVert\mathbf{A}\mathbf{x}^{\prime}\right\rVert_{p}\leq
R$,

$d_{X}(\mathbf{y},\mathbf{y}^{\prime})\leq\begin{dcases}O(1)\left\lVert\mathbf{A}(\mathbf{x}-\mathbf{x}^{\prime})\right\rVert_{\infty}^{p/2}R^{p/2}&p<2\\\
O(1)\sigma^{1/2-1/p}\cdot\left\lVert\mathbf{A}(\mathbf{x}-\mathbf{x}^{\prime})\right\rVert_{\infty}R^{p-1}&p>2\end{dcases}$

With Lemma A.7 in hand, the following result follows immediately from the
proofs in [WY23c]:

######  Theorem A.8.

Let $1\leq p<\infty$ with $p\neq 2$ be fixed and let
$\mathbf{A}\in\mathbb{R}^{n\times d}$ and $\mathbf{b}\in\mathbb{R}^{n}$. Let
$R\geq\lVert\mathbf{b}\rVert_{p}$. Let
$\tau\geq\max_{i=1}^{n}\boldsymbol{\tau}_{i}(\mathbf{A})$ and let
$\sigma\geq\max_{i=1}^{n}\boldsymbol{\sigma}_{i}^{p}(\mathbf{A})$. Define

$\mathcal{E}\coloneqq\begin{dcases}\tau^{1/2}(\sigma n)^{1/2-1/p}\cdot(\log
n)^{3/2}&p>2\\\ \tau^{1/2}\cdot\left\lparen\log n\right\rparen^{3/2}&p<2\\\
\end{dcases}.$

Then,

$\operatorname*{\mathbf{E}}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I}_{n})}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R}\left\lvert\sum_{i=1}^{n}\mathbf{g}_{i}\lvert[\mathbf{A}\mathbf{x}+\mathbf{b}](i)\rvert^{p}\right\rvert^{l}\leq\left[(2\mathcal{E})^{l}(\mathcal{E}/\sigma^{1/2})+O(\sqrt{l}\sigma^{1/2})^{l}\right](R^{p})^{l}$

###  A.4 Proof of main sampling theorems

We now prove Theorem A.1 by combining the previous results of this section.

###### Proof of Theorem A.1.

By Lemma A.2, it suffices to show that

$\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R_{i}}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert\leq\varepsilon
R_{i}^{p}$ (18)

for $R_{i}=2^{i}R$ for $i\in[I]$, $I=O(\log\varepsilon^{-1})$. The
corresponding statement for bounding

$\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}=1}\left\lvert\left\lVert\mathbf{S}\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}^{p}\right\rvert\leq\varepsilon$

will follow from the exact same analysis by setting $\mathbf{b}=0$ and $R=0$.

In order to obtain (18) for a single scale with high probability, we will
bound the $l$th moment for a large even power $l$. We will bound this quantity
by passing to a Gaussian process bound in Lemma A.6, which we can in turn
bound using Theorem A.8. Note that by our construction of the flattened
instance $\mathbf{A}^{\prime\prime}$ and $\mathbf{b}^{\prime\prime}$ in Lemmas
A.5 and A.6, we have $\tau=O(\alpha)$ and $\sigma=O(\alpha)$ for $p<2$ and
$\tau=O(\alpha)/n^{1-2/p}$ and $\sigma=O(\alpha)$ for $p>2$, so we can bound
the $\mathcal{E}$ parameter in Theorem A.8 by

$\mathcal{E}\leq\begin{cases}O(\alpha^{1-1/p})(\log n)^{3/2}&p>2\\\
O(\alpha^{1/2})(\log n)^{3/2}&p<2\end{cases}$

In turn, the bound on the Gaussian process in Theorem A.8 is
$\delta\varepsilon^{l}(R_{i}^{p})^{l}/(I+1)$ by our choice of $\alpha$, for

$l=O\left\lparen\log\log
n+\log\log\frac{1}{\varepsilon}+\log\frac{1}{\delta}\right\rparen.$

That is, we have shown that

$\operatorname*{\mathbf{E}}_{\mathbf{S}}\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R_{i}}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert^{l}\leq\frac{\delta}{I+1}\varepsilon^{l}(R_{i}^{p})^{l}.$

Then by Markov’s inequality, we have that

$\operatorname*{\mathbf{Pr}}_{\mathbf{S}}\left\\{\sup_{\left\lVert\mathbf{A}\mathbf{x}\right\rVert_{p}\leq
R_{i}}\left\lvert\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}-\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\right\rvert\leq\varepsilon
R_{i}^{p}\right\\}\geq 1-\frac{\delta}{I+1}.$

Now by a union bound, this is simultaneously true for all $i\in[I]$ as well as
for $\mathbf{b}=0$ and $R=0$ by a union bound, all with probability at least
$1-\delta$. In turn, we have that

$\left\lVert\mathbf{S}(\mathbf{A}\mathbf{x}+\mathbf{b})\right\rVert_{p}^{p}=(1\pm
2\cdot
4^{p}\varepsilon)\left\lVert\mathbf{A}\mathbf{x}+\mathbf{b}\right\rVert_{p}^{p}\pm
2^{p}\varepsilon R_{0}^{p}$

by Lemma A.2. We conclude with the desired conclusion by rescaling
$\varepsilon$ and $\delta$ up to constant factors. ∎

##  Appendix B Sharper bounds for representative subspaces

We provide sharper bounds for the result of [SW18, Theorem 10].

###  B.1 Sharper scalar inequalities

The following result simplifies and sharpens [SW18, Claim 2].

######  Lemma B.1.

Let $u,v,w\geq 0$ satisfy $u^{2}=v^{2}-w^{2}$. Then,

$u^{p}\leq\begin{cases}\min\\{\varepsilon
v^{p},2^{p-1}\varepsilon^{1-2/p}(v^{p}-w^{p})\\}&1\leq p\leq 2\\\
v^{p}-w^{p}&2\leq p<\infty\end{cases}$

###### Proof.

The second inequality follows from the subadditivity of $(\cdot)^{p/2}$ [SW18]
so it remains to show the first. We may assume that $v=1$ by scaling. We also
reparameterize $w=1-x$ for some $0\leq x\leq 1$. Then,

$u^{p}=(1-(1-x)^{2})^{p/2}\leq(2x)^{p/2}$

and

$\frac{u^{p}}{v^{p}-w^{p}}=\frac{u^{p}}{1-(1-x)^{p}}\leq\frac{(2x)^{p/2}}{x}=2^{p/2}x^{p/2-1}$

Thus, if $x\leq\varepsilon^{2/p}/2$, then $u^{p}\leq\varepsilon$, and if
$x\geq\varepsilon^{2/p}/2$, then $u^{p}\leq
2^{p-1}\varepsilon^{1-2/p}(v^{p}-w^{p})$. ∎

The following result sharpens [SW18, Claim 5].

######  Lemma B.2.

Let $u,v\geq 0$ and $1\leq p<\infty$. Then,

$(u+v)^{p}\leq(1+\varepsilon)u^{p}+\frac{(2p)^{p}}{\varepsilon^{p-1}}v^{p}$

###### Proof.

We may assume that $u=1$ by scaling. If $v\geq 1$, then $(1+v)^{p}\leq
2^{p}v^{p}$ so assume that $v\leq 1$. Then, $(1+v)^{p}\leq 1+2pv$ so if
$2pv\leq\varepsilon$, then $(1+v)^{p}\leq\varepsilon$, while if
$2pv\geq\varepsilon$, then

$(1+v)^{p}\leq 1+2pv=1+\frac{2p}{v^{p-1}}v^{p}\leq
1+\frac{(2p)^{p}}{\varepsilon^{p-1}}v^{p}.$

∎

The following result generalizes [SW18, Lemma 4] to $p>1$.

######  Lemma B.3.

Let $a,b,f,g\geq 0$ and $1\leq p<\infty$. Then,

$\lvert(a^{2}+b^{2})^{p/2}-(f^{2}+g^{2})^{p/2}\rvert\leq\frac{(4p)^{p}}{2\varepsilon^{p-1}}\left\lparen\lvert
a-f\rvert^{p}+\lvert
b-g\rvert^{p}\right\rparen+\varepsilon\left\lparen(a^{2}+b^{2})^{p/2}+(f^{2}+g^{2})^{p/2}\right\rparen.$

###### Proof.

By Lemma B.2, we have that

$\lVert(a,b)\rVert_{2}^{p}\leq(\lVert(a-f,b-g)\rVert_{2}+\lVert(f,g)\rVert_{2})^{p}\leq(1+\varepsilon)\lVert(f,g)\rVert_{2}^{p}+\frac{(2p)^{p}}{\varepsilon^{p-1}}\lVert(a-f,b-g)\rVert_{2}^{p}$

and similarly

$\lVert(f,g)\rVert_{2}^{p}\leq(\lVert(a-f,b-g)\rVert_{2}+\lVert(a,b)\rVert_{2})^{p}\leq(1+\varepsilon)\lVert(a,b)\rVert_{2}^{p}+\frac{(2p)^{p}}{\varepsilon^{p-1}}\lVert(a-f,b-g)\rVert_{2}^{p}.$

Thus,

$\lvert\lVert(a,b)\rVert_{2}^{p}-\lVert(f,g)\rVert_{2}^{p}\rvert\leq\frac{(2p)^{p}}{\varepsilon^{p-1}}\lVert(a-f,b-g)\rVert_{2}^{p}+\varepsilon(\lVert(a,b)\rVert_{2}^{p}+\lVert(f,g)\rVert_{2}^{p}).$

Finally, we bound

$\lVert(a-f,b-g)\rVert_{2}^{p}\leq\lVert(a-f,b-g)\rVert_{1}^{p}\leq
2^{p-1}(\lvert a-f\rvert^{p}+\lvert b-g\rvert^{p}).$

∎

###  B.2 Representative subspace theorem

The first lemma shows that if
$\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup F})\rVert_{2}$ is
small, then the projection of a vector $\mathbf{a}$ onto $S\cup F$ is close to
its projection onto $S$, and the projection of
$\mathbf{a}^{\top}\mathbf{P}_{S\cup F}$ onto $F$ is close to the projection of
$\mathbf{a}^{\top}\mathbf{P}_{S}$ onto $F$.

######  Lemma B.4.

Let $S,F\subseteq\mathbb{R}^{d}$ be subspaces and let
$\mathbf{a}\in\mathbb{R}^{d}$ be a vector. Then,

  * •

$\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{2}=\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}\pm\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}$

  * •

$\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{F})\rVert_{2}=\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}\pm\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}$

###### Proof.

These are proven in [SW18]. We reproduce a proof for the reader’s convenience.
The first inequality is just the triangle inequality, so it remains to show
the latter. One direction of the inequality follows by

$\displaystyle\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{F})\rVert_{2}$ $\displaystyle=\min_{\mathbf{x}\in
F}\lVert\mathbf{a}^{\top}\mathbf{P}_{S\cup F}-\mathbf{x}\rVert_{2}$
$\displaystyle\leq\lVert\mathbf{a}^{\top}\mathbf{P}_{S\cup
F}-\mathbf{a}^{\top}\mathbf{P}_{S}\mathbf{P}_{F}\rVert_{2}\leq\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{S})\rVert_{2}+\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}$

and the other by

$\displaystyle\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}$
$\displaystyle=\min_{\mathbf{x}\in
F}\lVert\mathbf{a}^{\top}\mathbf{P}_{S}-\mathbf{x}\rVert_{2}$
$\displaystyle\leq\lVert\mathbf{a}^{\top}\mathbf{P}_{S}-\mathbf{a}^{\top}\mathbf{P}_{F}\rVert_{2}\leq\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}+\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{F})\rVert_{2}$

∎

We may combine Lemma B.4 with Lemma B.3 to show the following, which states
that the projection cost of $\mathbf{a}$ onto $F$ is approximately the sum of
the cost of projecting onto $S$, and then projecting onto $F$.

######  Lemma B.5.

Let $S,F\subseteq\mathbb{R}^{d}$ be subspaces and let
$\mathbf{a}\in\mathbb{R}^{d}$ be a vector. Then,

$\displaystyle\left\lvert\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{p}-(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{2})^{p/2}\right\rvert$
$\displaystyle\leq\left\lparen\frac{(4p)^{p}}{\varepsilon^{p-1}}+2^{p-1}\varepsilon\right\rparen\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}^{p}$ $\displaystyle\hskip
50.00008pt+(2^{p-1}+1)\varepsilon\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{p}$

###### Proof.

Note that by orthogonality,

$\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{2}=\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{F})\rVert_{2}^{2}.$

Then, we apply Lemma B.3 with
$a=\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S\cup F})\rVert_{2}$,
$b=\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup F}-\mathbf{P}_{F})\rVert_{2}$,
$f=\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}$, and
$g=\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}$
as well as the bound

$\lvert a-b\rvert,\lvert
f-g\rvert\leq\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}$

from Lemma B.4 to see that

$\displaystyle\lvert\lVert(a,b)\rVert_{2}^{2}-\lVert(f,g)\rVert_{2}^{2}\rvert^{p/2}$
$\displaystyle=\left\lvert\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{p}-(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{2})^{p/2}\right\rvert$
$\displaystyle\leq\frac{(4p)^{p}}{2\varepsilon^{p-1}}(\lvert
a-f\rvert^{p}+\lvert
b-g\rvert^{p})+\varepsilon(\lVert(a,b)\rVert_{2}^{p}+\lVert(f,g)\rVert_{2}^{p})$
$\displaystyle\leq\frac{(4p)^{p}}{\varepsilon^{p-1}}\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}^{p}$ $\displaystyle\hskip
50.00008pt+\varepsilon(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{p}+(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{2})^{p/2}).$

Note that

$\displaystyle(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{2})^{p/2}$
$\displaystyle\leq
2^{p-1}(\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{2}^{2}+\lVert\mathbf{a}^{\top}(\mathbf{P}_{S\cup
F}-\mathbf{P}_{F})\rVert_{2}^{2})^{p/2}$ $\displaystyle\hskip
50.00008pt+2^{p-1}\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}^{p}$
$\displaystyle=2^{p-1}\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{F})\rVert_{2}^{p}+2^{p-1}\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}^{p}$

so combining the bounds gives the claimed result. ∎

It remains to construct a subspace $S$ such that
$\lVert\mathbf{A}(\mathbf{P}_{S}-\mathbf{P}_{S\cup F})\rVert_{p,2}^{p}$ is
small for every $k$-dimensional subspace $F$.

######  Lemma B.6.

Let $1\leq p<\infty$ and $k\in\mathbb{N}$. There is an $s$-dimensional
subspace $S$ where $s=O(k/\varepsilon^{\max\\{2,p\\}})$ such that for every
$k$-dimensional subspace $F$,

$\lVert\mathbf{A}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p}\leq\varepsilon^{p}\mathsf{OPT}$

where
$\mathsf{OPT}=\min_{F\in\mathcal{F}_{k}}\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{F})\rVert_{p,2}^{p}$.

###### Proof.

The proof largely follows [SW18] combined with our improved inequalities
proved earlier. We reproduce a proof for the reader’s convenience.

Lemma 6 in [SW18] shows that there is an $s$-dimensional subspace $S$ such
that

$\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S})\rVert_{p,2}^{p}-\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p}\leq\varepsilon^{\max\\{2,p\\}}\mathsf{OPT}$ (19)

for every $k$-dimensional subspace $F\in\mathcal{F}_{k}$. We now use the fact
that for any vector $\mathbf{a}\in\mathbb{R}^{d}$,

$\lVert\mathbf{a}^{\top}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{2}^{2}=\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}^{2}-\lVert\mathbf{a}^{\top}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{2}^{2}$

by orthogonality and Lemma B.1 (with $\varepsilon^{\prime}=\varepsilon^{p}$)
to show that

$\lVert\mathbf{A}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p}\leq\begin{cases}\varepsilon^{p}\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S})\rVert_{p,2}^{p}+2\varepsilon^{p-2}(\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S})\rVert_{p,2}^{p}-\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p})&1\leq p<2\\\
\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S})\rVert_{p,2}^{p}-\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p}&2\leq p<\infty\end{cases}$

by summing up the inequalities over vectors $\mathbf{a}_{i}$ for $i\in[n]$. By
(19), we have that

$\lVert\mathbf{A}(\mathbf{P}_{S}-\mathbf{P}_{S\cup F})\rVert_{p,2}^{p}\leq
3\varepsilon^{p}\mathsf{OPT}$

in any case. Rescaling $\varepsilon$ by constant factors yields the statement
of the theorem. ∎

Finally, we combine this bound with Lemma B.5 to conclude Theorem 3.1.

##  Appendix C Efficient constructions of nearly linear representative
subspaces

In this section, we sharpen a result of [FKW21] and show that representative
subspaces of dimension $\tilde{O}(k)\varepsilon^{-\Theta(p)}$ can be
constructed in polynomial time. This implies that strong coresets of size
$\tilde{O}(k^{\max\\{1,p/2\\}})\operatorname{poly}(\varepsilon^{-1})$ _with an
added dimension_ can be constructed in time
$\tilde{O}(\operatorname{\mathsf{nnz}}(\mathbf{A}))+d\operatorname{poly}(k/\varepsilon)$.
Indeed, note that the work of [HV20, WY23a] shows that strong coresets of size
$\operatorname{poly}(k/\varepsilon)$ can be constructed in input sparsity time
via sensitivity sampling, so we may assume that $n$ and $d$ are at most
$\operatorname{poly}(k/\varepsilon)$. Thus, it suffices to design a polynomial
time algorithm for constructing representative subspaces. For this, we use the
observation of [FKW21] that we only need an efficient bicriteria algorithm
that outputs a subspace which has objective value at most $(1+\varepsilon)$
times the rank $k$ optimum. The work of [FKW21] uses an algorithm with
bicriteria rank $\operatorname{poly}(k/\varepsilon)$. We will show instead how
to improve this to an algorithm with bicriteria rank
$\tilde{O}(k/\varepsilon)$.

We first start with a sketching result which sharpens [CW15, Theorem 32].

######  Lemma C.1.

Let $\mathbf{A}\in\mathbb{R}^{n\times d}$. Let $\mathbf{G}$ be an $d\times r$
random Gaussian matrix for $r=\tilde{O}((k+\log(n/\delta))/\varepsilon)$.
Then,

$\min_{\operatorname{rank}(\mathbf{X})\leq
k}\lVert\mathbf{A}\mathbf{G}\mathbf{X}-\mathbf{A}\rVert_{p,2}^{p}\leq(1+\varepsilon)\min_{\operatorname{rank}(F)\leq
k}\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{F})\rVert_{p,2}^{p}.$

###### Proof.

Let $\mathbf{V}^{*}$ be an orthonormal basis for the optimal rank $k$
projection $\mathbf{P}^{*}$, so that

$\mathsf{OPT}=\lVert\mathbf{A}\mathbf{V}^{*}(\mathbf{V}^{*})^{\top}-\mathbf{A}\rVert_{p,2}^{p}=\min_{\mathbf{X}\in\mathbb{R}^{n\times
k}}\lVert\mathbf{X}(\mathbf{V}^{*})^{\top}-\mathbf{A}\rVert_{p,2}^{p}.$

Note that the latter problem can be viewed as $n$ different $\ell_{2}$ linear
regression problems. Then by sketching with $\mathbf{G}$ [Woo14], for each
$i\in[n]$, we have that with probability at least $1-\delta/n$ that
$\hat{\mathbf{x}}=\mathbf{e}_{i}^{\top}\mathbf{A}\mathbf{G}((\mathbf{V}^{*})^{\top}\mathbf{G})^{-}$
satisfies

$\lVert\hat{\mathbf{x}}(\mathbf{V}^{*})^{\top}-\mathbf{e}_{i}^{\top}\mathbf{A}\rVert_{2}\leq(1+\varepsilon)\min_{\mathbf{x}\in\mathbb{R}^{k}}\lVert\mathbf{x}(\mathbf{V}^{*})^{\top}-\mathbf{e}_{i}^{\top}\mathbf{A}\rVert_{2}.$

By a union bound over the $n$ rows, this is simultaneously true for all
$i\in[n]$ with probability at least $1-\delta$. Then by taking $p$-th powers
and summing, we have

$\lVert\mathbf{A}\mathbf{G}((\mathbf{V}^{*})^{\top}\mathbf{G})^{-}(\mathbf{V}^{*})^{\top}-\mathbf{A}\rVert_{p,2}^{p}\leq(1+O(\varepsilon))\min_{\mathbf{X}\in\mathbb{R}^{n\times
k}}\lVert\mathbf{X}(\mathbf{V}^{*})^{\top}-\mathbf{A}\rVert_{p,2}^{p}=(1+\varepsilon)\mathsf{OPT}.\qed$

The next lemma shows how Lemma C.1 implies an efficient bicriteria algorithm
of rank $\tilde{O}(k/\varepsilon)$.

######  Lemma C.2.

There is a $\operatorname{poly}(nd)$ time algorithm which, given an input
matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$, outputs a rank
$\tilde{O}(k/\varepsilon)$ subspace $\hat{F}$ such that

$\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{\hat{F}})\rVert_{p,2}^{p}\leq(1+\varepsilon)\min_{\operatorname{rank}(F)\leq
k}\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{F})\rVert_{p,2}^{p}.$

###### Proof.

By Lemma C.1, we have that

$\min_{\operatorname{rank}(\mathbf{X})\leq
k}\lVert\mathbf{A}\mathbf{G}\mathbf{X}-\mathbf{A}\rVert_{p,2}^{p}\leq(1+\varepsilon)\min_{\operatorname{rank}(F)\leq
k}\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{F})\rVert_{p,2}^{p}.$

Then, we can simply drop the rank $k$ constraint on $\mathbf{X}$ to solve for
a matrix $\mathbf{Y}$ of rank $\tilde{O}(k/\varepsilon)$ such that
$\lVert\mathbf{A}\mathbf{Y}-\mathbf{A}\rVert_{p,2}^{p}\leq(1+\varepsilon)\mathsf{OPT}$.
Note that $\mathbf{Y}$ can be computed efficiently by, e.g., applying
Dvoretzky’s theorem to approximate the $(p,2)$-norm by a $(p,p)$-norm and then
solving an instance of $\ell_{p}$ regression $d$ times. Finally, we can take
$\hat{F}=\operatorname{rowspan}(\mathbf{Y})$ to produce a rank
$\tilde{O}(k/\varepsilon)$ subspace such that

$\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{\hat{F}})\rVert_{p,2}^{p}\leq(1+\varepsilon)\mathsf{OPT}.\qed$

Finally the above bicriteria algorithm can be combined with the iteration
algorithm idea of [SW18], as done in [FKW21], to prove the following theorem:

######  Theorem C.3.

There is an algorithm which, given an input matrix
$\mathbf{A}\in\mathbb{R}^{n\times d}$, runs in time $\operatorname{poly}(nd)$
and outputs a rank $\tilde{O}(k/\varepsilon)$ subspace $S$ such that

$\lVert\mathbf{A}(\mathbf{I}-\mathbf{P}_{F})\rVert_{p,2}^{p}=(1\pm\varepsilon)\lVert[\mathbf{A}\mathbf{P}_{S}(\mathbf{I}-\mathbf{P}_{F}),\mathbf{b}_{S}]\rVert_{p,2}^{p}$

for every $F\in\mathcal{F}_{k}$, where
$\mathbf{b}_{S}(i)=\lVert\mathbf{a}_{i}^{\top}(\mathbf{I}-\mathbf{P}_{S})\rVert_{2}$.

###### Proof.

To construct a linear size representative subspace in polynomial time, we will
use the work of [FKW21], which observes that $(1+\varepsilon)$ approximation
algorithms in the algorithm of [SW18] can be replaced by $(1+\varepsilon)$
bicriteria approximation algorithms, which can in turn be implemented
efficiently. The proof of this theorem now simply replaces the bicriteria
algorithm of [FKW21] with our bicriteria algorithm of Lemma C.2 and our
sharpened analysis of [SW18] in Appendix B. Note that to find a space $S$ such
that

$\lVert\mathbf{A}(\mathbf{P}_{S}-\mathbf{P}_{S\cup
F})\rVert_{p,2}^{p}\leq\varepsilon\mathsf{OPT}$

the algorithm will add a rank $\tilde{O}(k/\varepsilon)$ bicriteria space for
$\varepsilon^{-\max\\{2/p,1\\}}$ iterations, so the resulting subspace has
rank at most $\tilde{O}(k)\varepsilon^{-\max\\{2/p+1,2\\}}$. We need to apply
this with $\varepsilon$ replaced by $\varepsilon^{p}$, so the result bound on
the rank of the representative subspace is
$\tilde{O}(k)\varepsilon^{-\max\\{2+p,2p\\}}$. ∎


# Acute entanglement and Photon/Phonons statistics in a balanced/unbalanced
$PT$-symmetry systems

Muhammad Abid  Areeda Ayoub  Javed Akram<EMAIL_ADDRESS>Department
of Physics, COMSATS University Islamabad, Pakistan

###### Abstract

We study the significance of Photon/Phonons bunching and antibunching on the
dynamics of the quantum entanglement in the presence of coupled $PT$-symmetry
systems with balanced/unbalanced gain and loss. We suggest a hybrid
electromechanical system to realize a strong and tunable coupling between a
Coplanar-Waveguide (CPW) microwave cavity and a nanomechanical resonator
(NAMR) via a superconducting Transmon qubit. The hybrid electromechanical
system consists of a non-hermitian Hamiltonian with balanced/unbalanced gain
and loss. The interplay between the quantum entanglement and the $PT$-symmetry
systems is also thoroughly investigated. We frame a connection between Number
operators, Photon/Phonons antibunching, and entanglement. It has been observed
that the relative Photon/Phonons numbers play a key role in the quantum
entanglement dynamics. Furthermore, we study that quantum entanglement can be
characterized by defining a Photon/Phonons antibunching. The Photon/Phonons
antibunching is strongly dependent on the initial squeezed state and the rate
of balanced/unbalanced gain and loss of the system.

###### pacs:

42.50.Nn, 42.50.Dv, 03.65.−w, 03.70.+k

##  I Introduction

Recently, the most promising idea pursued by the researchers is to combine two
or more systems to form hybrid quantum systems, intending to harness the
strengths and advantages of different physical systems in order to explore new
physical phenomena and potentially give rise to novel quantum technologies
Lukin (2003); Hanson and Awschalom (2008); Ashhab  _et al._ (2008); Cheng  _et
al._ (2015); Tian (2005); Chen  _et al._ (2007); Xue  _et al._ (2007a); Akram
and Saif (2008); Xiang  _et al._ (2013); Xue  _et al._ (2007b); Cohen and Di
Ventra (2013); Schmidt  _et al._ (2010); Zhang  _et al._ (2005); Xia and Evers
(2009); Zhang  _et al._ (2009); Liu  _et al._ (2010); Pirkkalainen  _et al._
(2015); Hussain  _et al._ (2020). They provide new tools and platforms to
investigate deeper into unexplored quantum regimes. Among these systems, the
nanomechanical resonators (NAMRs) embedded in qubits (including artificial
atoms Lukin (2003), spins Hanson and Awschalom (2008), superconducting qubits
Ashhab  _et al._ (2008); Cheng  _et al._ (2015), and so on) are considered as
a good candidate for exploring different quantum phenomena. For example,
quantum entanglement Tian (2005); Chen  _et al._ (2007); Ayoub and Akram
(2021), controllable coupling Xue  _et al._ (2007a), hybrid quantum circuits
Xiang  _et al._ (2013), quantum squeezing Cohen and Di Ventra (2013); Xue  _et
al._ (2007b), quantum detection in nanomechanical systems Schmidt  _et al._
(2010), ground-state cooling Zhang  _et al._ (2005); Xia and Evers (2009);
Zhang  _et al._ (2009) and phonon blockade Liu  _et al._ (2010) have already
been proposed. Moreover, a controllable coupling between NAMR and Coplanar-
Waveguide (CPW) microwave cavity has been realized through a qubit
Pirkkalainen  _et al._ (2015).

On the other hand, the $PT$-symmetry has been studied in different quantum
systems Rüter  _et al._ (2010); Cartarius and Wunner (2012); Jing  _et al._
(2014); Xu  _et al._ (2015); Hang  _et al._ (2013); Haag  _et al._ (2014);
Quijandría  _et al._ (2018); Akram  _et al._ (2021). A wide class of non-
hermitian Hamiltonians respecting $PT$-symmetry exhibit entirely real spectrum
of eigenvalues. The most interesting feature of such Hamiltonians is $PT$
phase transition, in which the eigenvalues spectrum switches from being
entirely real to being completely imaginary, which is marked by the presence
of exceptional point (EP), where two or more eigenvalues and their
corresponding eigenvectors coalesce and become degenerate Rüter  _et al._
(2010). Therefore, many interesting physical properties can also be studied
with the $PT$ symmetric device the dynamics of which is governed by an
intrinsic quantum-mechanical law. For example, PT-symmetric Bose-Einstein
condensate in a $\delta$-function double-well potential Cartarius and Wunner
(2012); Hussain  _et al._ (2019), optomechanical devices as a phonon laser
Jing  _et al._ (2014); Xu  _et al._ (2015), non-linear dynamics in cold atoms
Haag  _et al._ (2014), and $PT$\- symmetric circuit QED systems Quijandría
_et al._ (2018) have been proposed. Many intrinsic quantum properties have
been studied with the help of $PT$ devices such as decoherence dynamics Gardas
_et al._ (2016), information retrieval and criticality Kawabata  _et al._
(2017), entanglement Chen  _et al._ (2014), and chiral population transfer Xu
_et al._ (2016); Doppler  _et al._ (2016). Moreover, due to the considerable
progress in theories and experiments for the generation of mechanical gain and
cavity loss Zhang  _et al._ (2018); Li  _et al._ (2018), more interesting
properties have been studied. However, the true quantumness of $PT$ system is
still questionable because to preserve the proper commutation relation,
quantum noises associated with the gain (amplifying) and loss (dampening) are
often ignored, which shows a drastic difference than usual predictions Agarwal
and Qu (2012); Kepesidis  _et al._ (2016); Zhang  _et al._ (2019).

Entanglement, being an inherent form of quantum correlation, has become an
invaluable resource for quantum computing and quantum information processing.
As it is well known that quantum entanglement is very fragile under the
influence of environmental noises Yu and Eberly (2004); Huang and Zhu (2008).
Notably, due to the presence of quantum noises, the continuous variable (CV)
entanglement generated in a system of two coupled waveguides is badly affected
Vashahri-Ghamsari  _et al._ (2017). Therefore, experimentalists are mostly
concerned to preserve the entanglement for a long time, as unavoidable
interaction with an external environment is significantly detrimental to the
generation of entanglement. A wide variety of decoherence entangled pairs such
as photons Laurat  _et al._ (2007), atoms Almeida  _et al._ (2007), continuous
Gaussian states Paz and Roncaglia (2008), and spin chains Cormick and Paz
(2008) have been analyzed both experimentally and theoretically for the
environment-induced sudden death of entanglement. The presence of quantum
noise enables us to understand their effects in broken $PT$ -symmetric systems
more deeply.

In this article, we present a general framework for two paired quantum systems
that adhere to $PT$-symmetry. Relevant linear quantum systems shared
balanced/unbalanced gain and loss with their neighbors. By defining unbalanced
gain and loss, we try to understand the effects of quantum noises on
entanglement. In this work, we describe the dynamical behavior of entanglement
at exceptional points in the presence of balanced/unbalanced gain and loss. We
study the effect of $PT$-symmetry exceptional point on the evolution of the
number of photons and phonons with balanced/unbalanced gain and loss. We
investigate the dynamics of a quantum correlation function in the different
regimes across the exceptional points balanced/unbalanced gain and loss and
try to build a relation between photon/phonon numbers and phase coherence. The
rest of the paper is organized as follows. In Sec. II we present the
description of the hybrid electromechanical system with the non-hermitian
Hamiltonian. A hybrid electromechanical system to realize a strong and tunable
coupling between a CPW microwave cavity and a NAMR via a superconducting
Transmon qubit. The direct coupling between NAMR and CPW microwave cavities
would be weak and uncontrollable as well due to their size mismatch. In Sec.
III, we determine the effective Hamiltonian between the CPW microcavity and
the NAMR by using the Fröhlich-Nakajima transformation. In Sec. IV, we
illustrate the evolution of entanglement, photon/phonon numbers, and a kind of
quantum correlation function in different regimes across the exceptional
points. In the same section, we describe the effect of quantum noise on the
dynamics of entanglement and the growth of photon/phonon numbers.
Additionally, we also study the influence of squeezing on the dynamics of
entanglement. In Sec. V, we discuss in detail how to do altering of the
entanglement dynamics by changing the initial squeezing parameter for balanced
and unbalanced scenarios. Finally, we conclude our findings in Sec. VI.

##  II Hamiltonian of the hybrid system

We consider a hybrid electromechanical system, which consists of a
superconducting CPW microwave cavity, a superconducting Transmon qubit, and a
NAMR, as depicted in Fig. (1). The Hamiltonian of CPW microwave cavity can be
expressed as $H_{c}=\hbar\omega_{c}c^{{\dagger}}c$, where $c(c^{{\dagger}})$
denotes the bosonic annihilation (creation) operator of the cavity mode with
resonant frequency $\omega_{c}=\sqrt{1/L_{r}C_{r}}$. Here, $L_{r}$ and
$C_{r}$, respectively, represent the total inductance and capacitance of the
cavity. The Hamiltonian of the NAMR is defined as
$H_{m}=\hbar\omega_{m}b^{{\dagger}}b$, with $b(b^{{\dagger}})$ being the
bosonic annihilation (creation) operator of the mechanical mode and
$\omega_{m}$ is the resonant frequency of the NAMR. The Hamiltonian of
superconducting Transmon qubit is given as
$H_{q}=4E_{C}(n-n_{g})^{2}-E_{J}\cos{\phi}$ Koch  _et al._ (2007), where $n$
denotes the number of Cooper pairs, $n_{g}$ describes the effective offset
charge due to environmental sources and $E_{C}=e^{2}/2C_{eq}$ defines the
charging energy of the capacitor, in which $C_{eq}=C_{NR}(x)+C_{t}+C_{q}$ is
the total capacitance. $C_{NR}(x)=C_{NR}(1-x/d)$ describes the capacitive
coupling between the Transmon qubit and the NAMR as a function of resonator
displacement $x$. Normally the length $d$ is large as compared to the
oscillation displacement $x$. $C_{t}$ illustrates the coupling capacitance
between the Transmon qubit and the CPW microwave cavity and $C_{q}$ defines
the total capacitance of the Transmon qubit. The effective Josephson coupling
energy defines the Transmon qubit is given as
$E_{J}=E_{J_{1}}+E_{J_{2}}=E_{J_{0}}\cos{(\pi\Phi_{ext}/\Phi_{0}})$, where
$E_{J_{0}}$ is the maximum Josephson energy, $\Phi_{ext}$ is the externally
applied magnetic flux, and $\Phi_{0}$ is the superconducting magnetic flux
quantum. In the rotating wave approximation, the effective Hamiltonian of the
coupled system reads ($\hbar=1$)Abdi  _et al._ (2015)

Figure 1: (a) Schematic description of the hybrid electromechanical quantum
system, which comprises a CPW microwave cavity, a superconducting qubit, and a
NAMR. The coupling between the Transmon qubit and the NAMR can be achieved by
applying external dc voltage in the NAMR. (b) Equivalent circuit of the hybrid
quantum system.

$H=\omega_{c}c^{{\dagger}}c+\omega_{m}b^{{\dagger}}b+\frac{\omega_{q}}{2}\sigma_{z}+g\Big{(}c^{{\dagger}}\sigma_{-}+c\sigma_{+}\Big{)}+\lambda\Big{(}b^{{\dagger}}\sigma_{-}+b\sigma_{+}\Big{)},$
(1)

where $\omega_{q}$ and $\sigma_{z}=\ket{e}\bra{e}-\ket{g}\bra{g}$ denote,
respectively, the transition frequency and the Pauli operator for the Transmon
qubit. $\sigma_{+}=\ket{e}\bra{g}(\sigma_{-}=\ket{g}\bra{e})$ is the raising
(lowering) operator, where $\ket{e}(\ket{g})$ describes the excited (ground)
state of the Transmon qubit. The coupling strength between the CPW microwave
cavity and the Transmon qubit is defined as
$g=-2{V_{rms}^{0}}C_{t}/C_{eq}\hbar$ , where
$V_{rms}^{0}=\sqrt{\hbar\omega_{c}/2C_{r}}$ is the rms of vacuum voltage
fluctuations of the CPW microwave cavity.
$\lambda=x_{zpf}(4E_{c}V_{NR}/e\hbar)dC_{NR}/dx$ is the coupling strength
between the NAMR and the qubit, where $x_{zpf}$ represents the zero-point
fluctuation of the NAMR oscillation displacement Irish and Schwab (2003);
Aspelmeyer  _et al._ (2014), and $V_{NR}$ is the applied external dc voltage
to the NAMR. Thus, the physical properties of the hybrid quantum system can be
adjusted by changing the external voltage, which modulate the coupling
$\lambda$ between the qubit and the NAMR.

##  III  Effective Coupling between The CPW microwave cavity and the NAMR

To decouple the Transmon qubit from hybrid quantum system, we use Fröhlich-
Nakajima transformation Fröhlich (1950) to Hamiltonian

$H_{eff}=e^{-S}He^{S}=H+[H,S]+[[H,S],S]/2+...,$ (2)

where

$S=\frac{g}{\Delta_{c}}\Big{(}c^{{\dagger}}\sigma_{-}+c\sigma_{+}\Big{)}+\frac{\lambda}{\Delta_{m}}\Big{(}b^{{\dagger}}\sigma_{-}+b\sigma_{+}\Big{)}.$
(3)

We have considered the hybrid quantum system in dispersive regime where the
detuning frequencies of the Transmon qubit from the CPW microwave cavity
($\Delta_{c}\equiv\omega_{q}-\omega_{c}>0$) and NAMR
($\Delta_{m}\equiv\omega_{q}-\omega_{cm}>0$), respectively, are much larger
than the coupling strengths $g$ and $\lambda$ i.e., $g/\Delta_{c}<<1$ and
$\lambda/\Delta_{m}<<1$. Since, both the coefficients $g/\Delta_{c}$ and
$g/\Delta_{m}$ are very small, the higher-order terms can be neglected and
only the second-order term needs to be kept in Eq. (2). Therefore, the
effective Hamiltonian can be rewritten as

$H_{eff}\approx\omega_{c0}c^{{\dagger}}c+\omega_{m0}b^{{\dagger}}b-G(c^{{\dagger}}b+cb^{{\dagger}}),$
(4)

where the effective microwave cavity frequency $\omega_{c0}$, the effective
NAMR frequency $\omega_{m0}$, and the effective coupling strength $G$ between
the microwave cavity and the NAMR are defined as

$\omega_{c0}=\omega_{c}-\frac{g^{2}}{\Delta_{c}},\hskip
14.22636pt\omega_{m0}=\omega_{m}-\frac{\lambda^{2}}{\Delta_{m}},$
$G=g\lambda\Big{(}\frac{\Delta_{c}+\Delta_{m}}{2\Delta_{c}\Delta_{m}}\Big{)}.$
(5)

While deriving Eq. (4), we have assumed that the Transmon qubit is at the
ground state ($\sigma_{+}\sigma_{-}=0$). In this way, an effective coupling
between the CPW microwave cavity and NAMR can be achieved by adiabatically
eliminating the degrees of freedom of the qubit. From Eq. (5), we see that the
frequencies of both the cavity mode and the mechanical mode are shifted due to
their large detunings with the qubit, and the effective electromechanical
coupling strength $G$ depends upon the coupling strengths, $g$ and $\lambda$,
and the frequency detunings, $\Delta_{c}$ and $\Delta_{m}$. By modulating the
NAMR-qubit coupling strength $\lambda$ with the external voltage, the
effective electromechanical coupling $G$ can be controlled.

##  IV Entanglement in PT-Symmetric electromechanical system

By considering the effect of the thermal environment, the quantum Heisenberg-
Langevin equations for the system are written as

$\displaystyle\dot{c}=-\Big{(}\dot{\iota}\omega_{c0}+\frac{\kappa}{2}\Big{)}c+\dot{\iota}Gb+\sqrt{\kappa}c^{in},$
(6a)
$\displaystyle\dot{b}=-\Big{(}\dot{\iota}\omega_{m0}-\frac{\gamma}{2}\Big{)}b+\dot{\iota}Gc+\sqrt{\gamma}b^{in},$
(6b)

where $\kappa$ defines the cavity loss and $\gamma$ describes the mechanical
gain. The vacuum input noise operators are defined as $c^{in}$ and $b^{in}$,
which satisfying following correlation functions:

$\displaystyle\langle c^{in{\dagger}}(t){c^{in}}(t^{\prime})\rangle=0,$ (7a)
$\displaystyle\langle
c^{in}(t){c^{in{\dagger}}}(t^{\prime})\rangle=\delta(t-t^{\prime}),$ (7b)
$\displaystyle\langle
b^{in{\dagger}}(t){b^{in}}(t^{\prime})\rangle=n_{th}\delta(t-t^{\prime}),$
(7c) $\displaystyle\langle
b^{in}(t){b^{in{\dagger}}}(t^{\prime})\rangle=(n_{th}+1)\delta(t-t^{\prime}),$
(7d)

where $n_{th}=[exp(\hbar\omega_{m0}/k_{B}T)-1]^{-1}$ illustrates the mean
thermal photon numbers of the mechanical resonator at temperature $T$ and
$k_{B}$ describes the Boltzmann constant. Under the Markovian assumption, the
noise operators $c^{in}$ and $b^{in}$ have zero mean values. Next, by
introducing two slowly varying operators,
$\tilde{c}=ce^{\dot{\iota}\omega_{c0}t}$ and
$\tilde{b}=be^{\dot{\iota}\omega_{m0}t}$, Eq. (6) can be rewritten as

$\displaystyle\dot{\tilde{c}}=-\frac{\kappa}{2}\tilde{c}+\dot{\iota}G\tilde{b}e^{\dot{\iota}(\omega_{c0}-\omega_{m0})t}+\sqrt{\kappa}\tilde{c}^{in},$
(8a)
$\displaystyle\dot{\tilde{b}}=\frac{\gamma}{2}\tilde{b}+\dot{\iota}G\tilde{c}e^{-\dot{\iota}(\omega_{c0}-\omega_{m0})t}+\sqrt{\gamma}\tilde{b}^{in}.$
(8b)

Here, we define two noise operators
$\tilde{c}^{in}=ce^{\dot{\iota}\omega_{c0}t}$ and
$\tilde{b}^{in}=be^{\dot{\iota}\omega_{m0}t}$, which possess the correlation
functions as described in Eq. (7). We assume that the CPW microwave cavity is
resonant with NAMR i.e., $\omega_{c0}=\omega_{m0}$ to obtain

$\displaystyle\dot{\tilde{c}}=-\frac{\kappa}{2}\tilde{c}+\dot{\iota}G\tilde{b}+\sqrt{\kappa}\tilde{c}^{in},$
(9a)
$\displaystyle\dot{\tilde{b}}=\frac{\gamma}{2}\tilde{b}+\dot{\iota}G\tilde{c}+\sqrt{\gamma}\tilde{b}^{in}.$
(9b)

Next, we ignore quantum noises to recast Eq. (9) as
$\dot{u}(t)=-\dot{\iota}\tilde{H}u(t)$. Here,
$u^{T}(t)=(q_{1}(t),p_{1}(t),q_{2}(t),p_{2}(t))$ is the state vector, which
can be written in terms of dimensionless CV quadrature as
$q_{1}\equiv(c+c^{{\dagger}})/\sqrt{2}$,
$p_{1}\equiv(c-c^{{\dagger}})/\dot{\iota}\sqrt{2}$,
$q_{2}\equiv(b+b^{{\dagger}})/\sqrt{2}$ and
$p_{2}\equiv(b-b^{{\dagger}})/\dot{\iota}\sqrt{2}$. The non-Hermitian
Hamiltonian of our system can be defined as

$\tilde{H}=\dot{\iota}\begin{pmatrix}\frac{\gamma}{2}&0&0&-G\\\ \\\
0&\frac{\gamma}{2}&G&0\\\ \\\ 0&-G&-\frac{\kappa}{2}&0\\\ \\\
G&0&0&-\frac{\kappa}{2}\end{pmatrix}.$ (10)

It can be easily verified that the Hamiltonian $\tilde{H}$ remains invariant
under the simultaneous $PT$ operation i.e., $[PT,\tilde{H}]=0$. To study the
$PT$ phase transition, we first diagonalize the Hamiltonian and find the eigen
frequencies of two supermodes

$\omega_{\pm}=\frac{\dot{\iota}(\gamma-\kappa)}{4}\pm\sqrt{G^{2}-\left(\frac{\gamma+\kappa}{4}\right)^{2}}$
(11)

where the real and imaginary parts correspond to the effective frequency and
the dissipation in the system, respectively. From Eq. (11), we find that the
eigenfrequencies $\omega_{\pm}$ of the two modes are highly dependent on the
coupling strength $G$ and there is a critical point $G_{c}=(\gamma+\kappa)/4$.
Next, we consider the general situation where the mechanical gain and the
cavity loss are defined with respect to each other i.e., $\gamma=s\kappa$,
where $s$ defines the ratio of gain and loss. By defining gain in terms of
loss, we can rewrite critical coupling strength as $G_{c}=\kappa(s+1)/4$.

Figure 2: For the case of balanced gain-loss i.e., $\gamma=\kappa$ (a) the
real and (b) the imaginary parts the eigen frequencies $\omega_{\pm}$ are
plotted as a function of dimensionless coupling strength $G/\kappa$.

###  IV.1 Balanced Gain $\&$ Loss ($s=1$)

In this subsection, we first study the dynamics of the hybrid
electromechanical system for the case when the cavity loss and the mechanical
gain are strictly balanced i.e., $\gamma=\kappa(s=1)$. In this situation, the
value of the critical coupling strength becomes $|G_{c}|=\kappa/2$. For the
case of balanced gain and loss, the real and the imaginary parts of the
eigenfrequencies $\omega_{\pm}$ are plotted as a function of dimensionless
coupling strength $G/\kappa$, as shown in Fig. 2(a, b). It can be seen in Fig.
2(a, b), for $|G|>\kappa/2$ and $|G|<\kappa/2$, respectively, there exist two
distinct phases: one that includes all the real eigenvalues named as the $PT$
symmetric phase and the other which possesses purely imaginary eigenvalues
named as broken $PT$ symmetric phase. Both of these phases are separated by
critical coupling strength $|G_{c}|=\kappa/2$, also known as an exceptional
point (EP). Physically, the EP refers to the situation where the mechanical
coupling rate $G$ is balanced with the respective mechanical heating and
cooling rate. Then, for $|G|>{\kappa}/{2}$, we get a coherent exchange of
energy between the CPW microwave cavity and the NAMR which corresponds to the
$PT$ symmetric phase and for $|G|<{\kappa}/{2}$ i.e., the coupling strength
becomes weak enough to support the energy exchange, there is a localization of
energy in the hybrid system which in turn leads to the broken $PT$ symmetric
phase.

Next, by taking quantum noises into account Eq. (9) can be written into more
compact form as $\dot{u}(t)=Au(t)+n(t)$. Here, $u(t)$ is the same $CV$ state
vector, $A=-\dot{\iota}\tilde{H}$ is the drift matrix and
$n^{T}(t)=\sqrt{\kappa}X^{in},\sqrt{\kappa}Y^{in},\sqrt{\gamma}Q^{in},\sqrt{\gamma}P^{in}$
is the matrix of corresponding noises. The input noise quadratures used in
$n^{T}(t)$ are defined as:
$X^{in}\equiv(\tilde{c}^{in}+{\tilde{c}^{in^{\dagger}}})/\sqrt{2}$,
$Y^{in}\equiv(\tilde{c}^{in}-{\tilde{c}^{in^{\dagger}}})/\dot{\iota}\sqrt{2}$,
$Q^{in}\equiv(\tilde{b}^{in}+{\tilde{b}^{in^{\dagger}}})/\sqrt{2}$, and
$P^{in}\equiv(\tilde{b}^{in}-{\tilde{b}^{in^{\dagger}}})/\dot{\iota}\sqrt{2}$.
The solution of this Langevin equation is given as
$u(t)=e^{At}u(0)+\int_{0}^{t}dse^{A(t-s)}n(s)$. The possible solutions of the
system can be found for negative eigenvalues of $A$. It follows that only when
the system remains in the $PT$ symmetric phase, we can find such solutions.

We also observe that the system preserves its Gaussian properties due to the
above-linearized dynamics and zero-mean Gaussian nature of the quantum noises.
To completely describe the system, one can use the usual covariance matrix
(CM) approach Braunstein and van Loock (2005). Let $W_{ij}(t)$ be the CM with
each element defined as

$W_{ij}(t)=\frac{\bra{z}{u_{i}(t)u_{j}(t)+u_{j}(t)u_{i}(t)}\ket{z}}{2}.$ (12)

Here, we consider a squeezed state as input state, i.e.,
$\ket{z}=e^{r(c^{\dagger}b^{\dagger}-cb)}\ket{0,0}$, which is a two-mode
squeezed vacuum state, with $r$ being the squeezing parameter. For realization
of such squeezed states in a hybrid system one may look into other striking
proposals in Refs. Børkje  _et al._ (2011); Riedinger (2018); Ockeloen-Korppi
_et al._ (2018); Marinković  _et al._ (2018). The equation of motion as
satisfied by the CM is given by:

$\dot{W}(t)=AW(t)+W(t)A^{T}+Z.$ (13)

Here, $Z$ defines the matrix of noising correlation which is equal to
$Z=[\frac{\gamma}{2}+\kappa(n_{th}+\frac{1}{2})]$ can be obtained by using the
Markovian assumption and
$({n_{i}(t)n_{j}(t)+n_{j}(t)n_{i}(t)})/{2}=\delta(t-t^{{}^{\prime}})Z_{ij}$.
Eq. (13) is a first-order non-homogeneous differential equation that can be
solved analytically and numerically by using proper initial conditions (Please
see the appendix for an analytical solution). Here, our main concern is to
investigate quantum entanglement. The formal solution of the Eq. (13) can be
written as

$W=\left(\begin{array}[]{cccc}W_{A}&W_{AB}\\\ W_{AB}^{T}&W_{B}\\\
\end{array}\right),$ (14)

where $W_{A}$, $W_{AB}$, $W_{B}$ and $W_{AB}^{T}$ are the $2\times 2$ sub-
matrices, respectively, corresponding to the local covariance matrices of the
CPW microwave cavity and the NAMR and the non-local correlation between them.
One can calculate the so-called logarithmic negativity $E_{N}$ to determine
the degree of quantum entanglement, such as

$E_{N}=max[0,-ln(2W^{-})],$ (15)

where $W^{-}\equiv\frac{1}{\sqrt{2}}{\left[\sum(W)-\sqrt{\sum{(W)}^{2}-4\det
W}\right]}^{\frac{1}{2}}$ is the smallest simplistic eigenvalue of the partial
transpose of the $W$ with
$\sum(W)\equiv\det(W_{A})+\det(W_{B})-2\det(W_{AB})$. It is well established
that the relative phase of the input fields plays important role in
entanglement dynamics. Therefore, we also investigate nonclassicality feature
and try to build a correlation between the entanglement and the antibunching.
For the cavity and mechanical resonator modes $c$ and $b$, respectively, the
condition for inter-mode antibunching is defined as follows Naikoo  _et al._
(2019)

$\mathcal{A}(b,c)=\frac{\langle b^{{\dagger}}c^{{\dagger}}bc\rangle-\langle
b^{{\dagger}}b\rangle\langle c^{{\dagger}}c\rangle}{\langle
c^{{\dagger}}c\rangle\langle b^{{\dagger}}b\rangle}.$ (16)

Figure 3: For $s=1$ (a) Entanglement evolution (line-circles) between the gain
and loss resonators with antibunching (line-squares) for $G=1.5$ (b) for
$G=0.7$. Dynamics of photon numbers (line circles) and phonons (line-squares)
numbers for $G=1.5$ (c) and $G=0.7$ (d).

Here, the first term on the right-hand side represents the simultaneous
detection in the output of two coupled microwave cavities and the NAMR, while
the second term defines the product of individual photon and phonons numbers.
The time evolution of our system for the balanced gain-loss case is shown in
Fig. (3). We have observed that when the system is in $PT$-symmetric phase the
entanglement, antibunching, number of photons $n_{p}$, and number of phonons
$n_{s}$ oscillate periodically.

This oscillation could be ascribed by the nature of the eigenvalues
$\pm\sqrt{G^{2}-G_{c}^{2}}$ of $A$, as obtained for $G>G_{c}$. In Fig. 3(a),
for $G=1.5$, one can see that the entanglement oscillates rapidly with a
smaller period and antibunching which is a non-classical feature oscillates
briskly and have a positive value for maximum entanglement. However, when we
approach near to the EP i.e., $G=0.7$, we observe lesser oscillation with a
smaller amplitude and longer period. So, in the close vicinity of the EP, the
entanglement almost freezes out, which means that a longer period is required
to complete one oscillation, as shown in Fig. 3(b). We investigate the acute
entanglement happens only when the number of photons $n_{p}$ and phonons
$n_{s}$ becomes equal during their oscillations as predicted in Fig. 3(c,d).
One can observe that at $G=1.5$, i.e., away from the EP, $n_{p}(t)$ and
$n_{s}(t)$ oscillate very rapidly with very smaller time period. Whereas, in
the close vicinity of the EP, $n_{p}$ and $n_{s}$ oscillate slowly with a
longer period. The oscillatory behavior of the curves is attributed to the
fact that the elements of the $W$ matrix are sinusoidal. We also noticed that
the entanglement becomes minimum when the difference of $n_{p}(t)$ and
$n_{s}(t)$ becomes maximum and the entanglement becomes maximum when change
between $n_{p}(t)$ and $n_{s}(t)$ becomes minimum. These results show that a
direct connection between the number of photon/phonons, antibunching, and
entanglement.

Figure 4:  For the case of unbalanced gain-loss i.e., $\gamma\neq\kappa$ (a)
the real and (b) the imaginary parts the eigenfrequencies $\omega_{\pm}$ are
plotted as a function of dimensionless coupling strength $G/\kappa$, here the
value of $s=2$. Figure 5: For $s=2$ (a) Entanglement (line-circles) and
antibunching (line-squares) evolution for the hybrid system for $G=2.3$ (b)
for $G=1.3$. In (a) and (b), inlet is plotted to emphasize the positive values
of antibunching. Dynamics of photon numbers (line-circles) and phonons numbers
(line-squares) for $G=2.3$ (c) and $G=1.3$ (d).

In this subsection, we consider the general situation when the mechanical gain
and the cavity loss are not balanced, i.e., $s\neq 1$, with $s$ being the
gain-loss ratio. For the gain-loss ratio $s=2$, i.e., $\gamma=2\kappa$ means
that in our system mechanical gain is larger as compared to the cavity-loss.
In Fig. 4(a, b), we plot the eigenfrequencies of the system, which looks
identical to Fig. (2), however, the imaginary part of the eigenfrequency is
shifted at $0.25$ from zero, this shifted eigenfrequency responsible for the
gain in the overall system. In another way, we can say that in an unbalanced
gain-loss case eigenfrequencies $\omega_{\pm}$ are not purely real and the
system will always be in broken $PT$ symmetric phase, as shown in Fig. 4(a,
b). As compared to the balanced gain-loss rate, we observe that with the
larger gain, the critical point $G_{c}$ has been shifted to the right and a
stronger coupling strength $G$ is required in this situation to enter into a
so-called exceptional point. Moreover, we also study the dynamics of our
system for the unbalanced gain-loss case. In Fig. 5(a), we show the
entanglement evolution for $s=2$. One can see that for unbalanced gain-loss
cases, oscillations decrease with time and eventually decays to zero a typical
entanglement sudden death-like behavior, which means that unbalanced gain-loss
rate adds noise into our system due to which entanglement ceases to oscillate
and eventually dies out. However, a notable feature is the delayed
entanglement death which can be achieved by pushing the system away from the
EP i.e., $G=0.75$, as predicted in Fig. 5(b). We also investigate the
antibunching phenomenon for this special scenario which shows a positive
magnitude for the maximum entanglement. We note that the non-classical feature
i.e., antibunching phenomenon can not be measured for weak entanglement. In
the sub-Fig. 5(a,b) we plot the zoom out of the antibunching phenomenon, to
show the positive magnitude of the antibunching phenomena. In this case, we
also note that when the difference of $n_{p}(t)$ and $n_{s}(t)$ becomes
maximum, the entanglement grows and vise versa. We realize that for the
unbalanced gain-loss case both $n_{p}$ and $n_{s}$ grow together, waning the
distinction between gain and loss hybrid system. Since there is more gain in
the system, therefore the $n_{p}$ and $n_{s}$ increase and it is due to
external pumping into the system. We also find out that the number of photons
and phonons grows exponentially which leads to a classical system therefore
the entanglement dies out even though the difference between $n_{p}(t)$ and
$n_{s}(t)$ becomes zero.

Figure 6: Entanglement (line-circles) and antibunching (line-squares)
evolution of hybrid electromechanical system for $G=1.5$ (a) and for $G=0.7$
(b) for $s=1$ and $r=2$. In (a) and (b), inlet is plotted to emphasize the
positive values of antibunching. Dynamics of number of photons (line-circles)
and phonons (line-squares) for $G=1.5$ (c) and $G=0.7$ (d).

##  V Altering the entanglement by squeezing parameter

In this section, we investigate the effect of squeezing parameter $r$ on
quantum engagement dynamics for balanced and unbalanced hybrid $PT$
electromechanical systems. For the balanced $s=1$ cavity loss and mechanical
gain, and for the squeezing parameter $r=2$, the logarithmic negativity
oscillates between $[0,4]$ as shown in Fig. 6(a-b). The initial average number
of photons is increased ten times as predicted in Fig. 6(c-d), in comparison
to the scenario when squeezing parameter $r=1$ as represented in Fig. 2(c-d).
We note that the change in the squeezing parameter has no big impact on the
oscillation of entanglement, antibunching, and the number of photon/phonons
other than their magnitude as depicted in Fig. 2(a-d) and 6(a-d). For the
unbalanced $s=2$ cavity loss and mechanical gain, and for the squeezing
parameter $r=2$, we note that the so-called sudden death of the entanglement
time delay is increased predicted in Fig. 7(a-b) as compared to the Fig.
5(a-b). It means that the strong squeezing prolongs the sudden death of the
entanglement and reduces the influence of quantum noise on entanglement. The
strong squeezing also increases the number of photons and phonons as depicted
in Fig. 7(c-d). For the weak squeezing $r=0.1$, we note that the dynamics of
the entanglement oscillate periodically. Additionally, we also find out that
the amplitude of the entanglement dynamics increases near the exceptional
point for the balanced gain and loss system as shown in Fig. 8(a). For the
unbalanced gain and loss system, we note that the significant delay in
entanglement sudden death can be realized by pushing the electromechanical
quantum system towards an exceptional point as illustrated in Fig. 8(b).

Figure 7: Entanglement (line-circles) and antibunching (line-squares)
evolution of hybrid electromechanical system for $G=2.3$ (a) and for $G=1.3$
(b) for $s=2$ and $r=2$. In (a) and (b), inlet is plotted to emphasize the
positive values of antibunching. Dynamics of number of photons (line-circles)
and phonons (line-squares) for $G=2.3$ (c) and $G=1.3$ (d).

##  VI Conclusion and Summary

In this study, we studied a gain-loss hybrid electromechanical system and
emphasized the conditions necessary for exhibiting parity-time ($PT$)
invariance. To realize a strong and tunable coupling between a Coplanar-
Waveguide (CPW) microwave cavity and a nanomechanical resonator (NAMR) we
purposed coupling of this hybrid system through a superconducting Transmon
qubit. We examine the hybrid electromechanical system as a non-hermitian
Hamiltonian with balanced/unbalanced gain and loss. The gain in our system is
introduced in the NAMR and the loss of the system is defined for the CPW when
gain and loss are balanced in the hybrid system the energy spectrum gets a
real spectrum after the exceptional point. However, when the gain and loss do
not equal, then the energy spectrum has some constant imaginary value that
defines the gain in the system. With this setting, we examined the
entanglement, antibunching, average photon, and phonon number with the initial
squeezed state. We found out that the time of entanglement sudden death can be
delayed near the exceptional points.

Figure 8: Entanglement evolution of hybrid electromechanical system for
balanced $s=1$ (a) and for unbalance $s=2$ (b) system, when the squeezing
parameter is defined as $r=0.1$

We also noticed that the entanglement gets maximum when the difference between
the photons and phonons numbers becomes minimum. We further examined a non-
classical feature like antibunching for different balanced and unbalanced
scenarios. We found out that the entanglement gets maximum when the
antibunching gets positive. Later, we showed that for unbalanced gain-loss
cases, oscillations decrease with time and eventually decay to zero a typical
entanglement sudden death-like behavior, which means that an unbalanced gain-
loss rate adds noise into our system due to which entanglement ceases to
oscillate and eventually dies out. In the end, we investigated the dependence
of the entanglement dynamics on the initial squeezing parameter, we analyzed
that the initial squeezing parameter is not affecting the dynamics of the
entanglement for the balanced gain and loss scenario. However, for unbalanced
gain and loss, the sudden death of entanglement can be delayed by increasing
the initial squeezing parameter. For the weak initial squeezing parameter, the
significant delay in entanglement sudden death can be achieved even in a noisy
environment. The present study helps to understand the role of $PT$-symmetry
in the dynamics of the entanglement and in facilitating the preservation of
the entanglement for a longer period. With this study, we build a relation
between entanglement, antibunching, and the photons and phonons numbers in the
physical systems, which are very much relevant in the field of quantum optics
and quantum information processing.

Finally, we give a brief description of the experimental prospect of our
proposed system. In our system, an external dc voltage source was applied to
the NAMR, which established the coupling between the NAMR and the qubit. In a
recent experiment Pirkkalainen  _et al._ (2015), the coupling between the NAMR
and the qubit can be varied between 80 MHz to 160 MHz, by adjusting the
external voltage (${V_{NR}=5-10V}$). Thus, in our system, a strong and tunable
coupling between the NAMR and the CPW microwave cavity mediated via Transmon
qubit can be achieved. With this setup, the transfer of quantum information
between the NAMR and CPW microwave cavity can be accomplished. On the other
hand, it has been reported in many experimental studies that the mechanical
gain can be achieved via directly driving the mechanical modes including
Josephson Phase qubit O’Connell  _et al._ (2010), piezoelectric pump Xiong
_et al._ (2013), microwave electrical driving Bochmann  _et al._ (2013), and
by phonon lasing method Bahl  _et al._ (2012). Hence, by considering both the
cavity gain and the mechanical loss, the transition from the broken
$\bm{PT}$-symmetry phase to the unbroken $\bm{PT}$-symmetry phase can be
obtained in our system.

##  VII Acknowledgment

Jameel Hussain gratefully acknowledges support from the COMSATS University
Islamabad for providing him a workspace.

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