The following is the main result of the subsection.

######  Theorem 6.6.

Let $(A,B,M,N,\varphi,\psi)$ be a basic Morita context of Artin algebras,
where $M$ is projective as a left $A$-module. Suppose also that
$\psi:N\otimes_{A}M\longrightarrow B$ is a monomorphism and the algebra
$B/\tau_{B}$ has finite global dimension.

Set $\Lambda=\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$. Then
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ (resp., $\Lambda\mbox{\rm-mod}$ allows for unlimited
iteration of strong tilting) if and only if this is true for
$\mathcal{P}^{<\infty}(A\mbox{\rm-mod})$ (resp., $A\mbox{\rm-mod}$).

In particular, $\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of
strong tilting when $A$ is a Gorenstein algebra or a truncated path algebra or
when $\mathop{\rm l.findim}\nolimits A=0=\mathop{\rm r.findim}\nolimits A$.

###### Proof.

Take $e=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$. The goal is to check that the
pair $(\Lambda\mbox{\rm-mod},e)$ satisfies the blanket hypotheses of Setting
4.1. Bearing in mind that the algebra $e\Lambda e$ is isomorphic to $A$, the
result will then be a direct consequence of Theorem 4.6 and Corollary 5.6.
Note that $B$ and $(1-e)\Lambda(1-e)$ are also isomorphic algebras and hence
the $e\Lambda e$-$(1-e)\Lambda(1-e)$-bimodule $e\Lambda(1-e)$ is isomorphic to
the $A$-$B$-bimodule $M$, with the obvious meaning. Hence $\mathop{\rm
pdim}\nolimits_{e\Lambda e}(e\Lambda(1-e))=0$ and we need to check that
$\mathop{\rm
pdim}\nolimits_{\Lambda}(\frac{\Lambda(1-e)}{J(\Lambda))(1-e)})<\infty$.

We first prove that $\Lambda e\Lambda(1-e)=\begin{pmatrix}0&M\\\
0&\tau_{B}\end{pmatrix}\cong\begin{pmatrix}M\\\ \tau_{B}\end{pmatrix}$ is a
projective left $\Lambda$-module. Indeed the adjunction map
$\epsilon_{\Lambda(1-e)}:\Lambda e\otimes_{e\Lambda
e}e\Lambda(1-e)\longrightarrow\Lambda(1-e)$ gets identified, in the obvious
way, with the map $\begin{pmatrix}A\\\
N\end{pmatrix}\otimes_{A}M\longrightarrow\begin{pmatrix}M\\\ B\end{pmatrix}$,
that can be expressed matricially as $\begin{pmatrix}\mu&0\\\
0&\psi\end{pmatrix}:\begin{pmatrix}A\otimes_{A}M\\\
N\otimes_{A}M\end{pmatrix}\longrightarrow\begin{pmatrix}M\\\ B\end{pmatrix}$.
Here $\mu:A\otimes_{A}M\longrightarrow M$ is the canonical isomorphism given
by multiplication and $\psi:N\otimes_{A}M\longrightarrow B$ is the map in the
Morita context. It follows that $\epsilon_{\Lambda(1-e)}$ is a monomorphism,
and so

$\Lambda e\otimes_{e\Lambda e}e\Lambda(1-e)\cong\Lambda
e\Lambda(1-e)=\text{Im}(\epsilon_{\Lambda(1-e)}).$

But $\Lambda e\otimes_{e\Lambda e}e\Lambda(1-e)\in\text{add}(\Lambda
e\otimes_{e\Lambda e}e\Lambda e)=\text{add}(\Lambda e)$ since
$e\Lambda(1-e)\in e\Lambda e\text{-proj}=\text{add}(_{e\Lambda e}e\Lambda e)$.
Therefore $\Lambda e\Lambda(1-e)$ is a projective left $\Lambda$-module.

Next we should notice that there is an algebra isomorphism
$\frac{\Lambda}{\Lambda e\Lambda}\cong\frac{B}{\tau_{B}}$, so that these
isomorphic algebras have finite global dimension. Moreover, we have an
isomorphism $\frac{\Lambda}{\Lambda e\Lambda}\cong\frac{\Lambda(1-e)}{\Lambda
e\Lambda(1-e)}$ in $\Lambda\mbox{\rm-mod}$, which, by the previous paragraph,
implies that $\frac{\Lambda}{\Lambda e\Lambda}$ has projective dimension $\leq
1$ as a left $\Lambda$-module. Note also that
$\frac{\Lambda(1-e)}{J(\Lambda)(1-e)}$ is canonically a left
$\frac{\Lambda}{\Lambda e\Lambda}$-module. Fix a minimal projective resolution
$0\rightarrow Q_{t}\rightarrow...\rightarrow Q_{1}\rightarrow
Q_{0}\rightarrow\frac{\Lambda(1-e)}{J(\Lambda)(1-e)}\rightarrow 0$ in
$\frac{\Lambda}{\Lambda e\Lambda}\text{-mod}$. This is also an exact sequence
in $\Lambda\mbox{\rm-mod}$, and we have that $\mathop{\rm
pdim}\nolimits_{\Lambda}(Q_{k})\leq 1$ since
$Q_{k}\in\text{add}(\frac{\Lambda}{\Lambda e\Lambda})$, for all $k=0,1,...,t$.
It then follows that $\mathop{\rm
pdim}\nolimits_{\Lambda}(\frac{\Lambda(1-e)}{J(\Lambda)(1-e)})<\infty$, as
desired.

The final statement clearly follows from assertion 3 since in those examples
$A\text{-mod}$ allows for unlimited iteration of strong tilting. ∎

The last theorem and its proof have the following consequence for triangular
matrix algebras.

######  Corollary 6.7.

Let $K$ be a commutative Artinian ring, let $A$ and $B$ be basic Artin
$K$-algebras and let $M$ and $N$ be a finitely generated $A$-$B$-bimodule and
$B$-$A$-bimodule, respectively. Suppose that $B$ has finite global dimension
and $\mathcal{P}^{\infty}(A\mbox{\rm-mod})$ is contravariantly finite in
$A\mbox{\rm-mod}$. The following assertions hold:

  1. (1)

If $\Lambda=\begin{pmatrix}A&0\\\ N&B\end{pmatrix}$, then
$\mathcal{P}^{\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. Moreover if $A\mbox{\rm-mod}$ allows for unlimited
iteration of strong tilting, so does $\Lambda\mbox{\rm-mod}$.

  2. (2)

If $\mathop{\rm pdim}\nolimits(_{A}M)<\infty$, the conclusions of assertion 1
remain true on replacement of $\Lambda$ by $\Gamma=\begin{pmatrix}A&M\\\
0&B\end{pmatrix}$.

###### Proof.

Assertion 1 is a direct consequence of Theorem 6.6. As for assertion 2, note
that if $e=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ then $\mathop{\rm
pdim}\nolimits_{e\Gamma e}(e\Gamma(1-e))<\infty$ since we have $\mathop{\rm
pdim}\nolimits(_{A}M)<\infty$. We will prove that $\mathop{\rm
pdim}\nolimits_{\Gamma}(\Gamma e\Gamma(1-e))<\infty$ and, arguing as in the
last paragraph of the proof of Theorem 6.6, we will conclude that the pair
$(\Gamma\text{-mod},e)$ satisfies the blanket hypotheses of Setting 4.1, and
the result will follow from Theorem 4.6 and Corollary 5.6.

Note that $0=(1-e)\Gamma e$ and so $\Gamma e=e\Gamma e$ is projective as right
$e\Gamma e$-module. Moreover we have a commutative diagram of $K$-modules

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
38.19777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{e\Gamma e\otimes_{e\Gamma
e}e\Gamma(1-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
58.13528pt\raise 6.15pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.32278pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 89.32278pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{e\Gamma(1-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Gamma
e\otimes_{e\Gamma
e}e\Gamma(1-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
86.19777pt\raise-46.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
86.19777pt\raise-46.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\Gamma
e\Gamma(1-e)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where the upper horizontal arrow is the canonical isomorphism and the lower
horizontal one is the multiplication map. It follows that this latter arrow is
an isomorphism, which implies that $\mathop{\rm pdim}\nolimits(_{\Gamma}\Gamma
e\Gamma(1-e))<\infty$ since ${}_{e\Gamma e}e\Gamma(1-e)$ has finite projective
dimension and the functor $\Gamma e\otimes_{e\Gamma e}-:e\Gamma
e\text{-mod}\longrightarrow\Gamma\text{-mod}$ is exact and takes projectives
to projectives. ∎

We end the paper by giving non-triangular examples to which Theorem 6.6
applies. We start with the following elementary observation.

######  Remark 6.8.

Let $A$ and $B$ be basic Artin $K$-algebras and $M$ and $N$ finitely generated
$A$-$B$\- and $B$-$A$-bimodules, respectively. If $N\otimes_{B}M=0$ then any
morphism $\varphi:M\otimes_{B}N\longrightarrow A$ of $A$-$A$-bimodules such
that $\mathop{\rm Im}\nolimits(\varphi)\subseteq J(A)$ gives rise to a basic
Morita context of Artin algebras $(A,B,M,N,\varphi,0)$ since the required
compatibility conditions hold (see, e.g., [20, Exercise IV.35]).

######  Examples 6.9.

Let $A$, $B$ be basic finite dimensional algebras over an algebraically closed
field $K$ and let us fix complete sets of primitive idempontents
$\\{e_{1},....,e_{m}\\}$ and $\\{e_{m+1},....,e_{n}\\}$ in $A$ and $B$,
respectively. Consider either one of the following two situations, where the
unadorned $\otimes$ means $\otimes_{K}$:

  1. (a)

$\mathop{\rm gldim}\nolimits(B)<\infty$, $M$ is any finitely generated
$A$-$B$-bimodule that is projective as a left $A$-module, $N$ is any finitely
generated $B$-$A$-bimodule such that $Ne_{i}=0$ whenever the simple left
$A$-module $Ae_{i}/J(A)e_{i}$ embeds in $\mathop{\rm top}\nolimits(_{A}M)$,
and $\varphi:M\otimes_{B}N\longrightarrow A$ is any morphism of
$A$-$A$-bimodules such that $\text{Im}(\varphi)\subseteq J(A)$. Then
$(A,B,M,N,\varphi,0)$ is a basic Morita context.

  2. (b)

Suppose that the quiver $Q_{B}$ of $B$ has no oriented cycles, let
$\mathop{\rm Soc}\nolimits(B)$ be the socle of $B$ as a $B$-$B$-bimodule and
fix any index $k\in\\{m+1,...,n\\}$ such that $W:=\mathop{\rm
Soc}\nolimits(B)e_{k}\neq 0$. Note that $W$ is a subbimodule of ${}_{B}B_{B}$
isomorphic to $Y\otimes\frac{e_{k}B}{e_{k}J(B)}$, for some semisimple left
$B$-module $Y$ that, due to the absence of oriented cycles in $Q_{B}$,
satisfies $e_{k}Y=0$. Take $M=Ae_{i}\otimes\frac{e_{k}B}{e_{k}J(B)}$ and
$N=Y\otimes\frac{e_{i}A}{e_{i}J(A)}$, for some $i=1,...,m$. We then have
$M\otimes_{B}N=0$ and an isomorphism of $B$-$B$-bimodules

$N\otimes_{A}M\cong
Y\otimes\frac{e_{i}Ae_{i}}{e_{i}J(A)e_{i}}\otimes\frac{e_{k}B}{e_{k}J(B)}\cong
Y\otimes\frac{e_{k}B}{e_{k}J(B)}\cong W.$

Taking as $\psi:N\otimes_{A}M\longrightarrow B$ the composition of this latter
isomorphism followed by the inclusion $W\hookrightarrow B$, we get a basic
Morita context $(A,B,M,N,0,\psi)$ (cf. the left-right symmetric version of
Remark 6.8).

In the situations (a) and (b), the associated algebra
$\Lambda=\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$ satisfies that
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ $($resp., $\Lambda\mbox{\rm-mod}$ allows for unlimited
iteration of strong tilting$)$ if, and only if, so does the algebra $A$.

###### Proof.

In situation (a), we have an isomorphism of left $A$-modules
$M\cong\oplus_{i\in I}Ae_{i}^{t_{i}}$, where $I$ is the set of
$i\in\\{1,...,m\\}$ such that the simple left $A$-module $Ae_{i}/J(A)e_{i}$
embeds in $\text{top}(_{A}M)$ and $t_{i}>0$ for all $i\in I$. It follows that
$N\otimes_{A}M=0$ and the existence of the mentioned Morita context follows by
Remark 6.8. Hence in both situations $M$ is projective as a left $A$-module
and the map $\psi:N\otimes_{A}M\longrightarrow B$ is a monomorphism. In (a) we
have that $\tau_{B}=0$ and in (b) the quiver of $B/\tau_{B}$ has no oriented
cycles. Therefore $\text{gl.dim}(B/\tau_{B})<\infty$ in both cases and
assertions 1 and 2 are direct consequences of Theorem 6.6. ∎

The following example is a combinatorial version of Example 6.9(a).

######  Example 6.10.

Let $A:=KQ_{e}/\langle R_{e}\rangle$ and $B:=KQ_{1-e}/\langle R_{1-e}\rangle$
be finite dimensional algebras given as quotients of path algebras modulo
relations, on which we fix bases $B_{e}$ and $B_{1-e}$ consisting of paths in
$Q_{e}$ and $Q_{1-e}$, respectively. Consider the algebra $\Lambda=KQ/\langle
R\rangle$, also given by quiver and relations, where:

  1. (1)

The quiver $Q$ is obtained from the disjoint union quiver $Q_{e}\bigsqcup
Q_{1-e}$ by adding two finite sets of arrows (some possibly empty)
$\\{\alpha_{1},...,\alpha_{s}\\}$ and $\\{\beta_{1},...,\beta_{t}\\}$, where
the $\alpha_{i}$ go from vertices in $Q_{1-e}$ to vertices in $Q_{e}$ and the
$\beta_{j}$ from vertices in $Q_{e}$ to vertices in $Q_{1-e}$, with the only
restriction that $t(\alpha_{k})\neq o(\beta_{l})$ for all $k=1,...,s$ and
$l=1,...,t$ (here $o(\gamma)$ and $t(\gamma)$ denote, respectively, the origin
and terminus of any arrow $\gamma$);

  2. (2)

The set of relations is $R=R_{e}\cup R_{1-e}\cup R_{1-e,e}\cup R_{e,e}$, where

$R_{1-e,e}=\\{\beta_{l}q\text{: }l\in\\{1,...,t\\}\text{ and
}q\in\bigcup_{k=1}^{s}e_{o(\beta_{l})}B_{e}e_{t(\alpha_{k})}\\}$, and

$R_{e,e}=\\{\alpha_{k}p\beta_{l}\text{: }l\in\\{1,...,t\\}\text{,
}k\in\\{1,...,s\\}\text{ and }p\in
e_{o(\alpha_{k})}B_{1-e}e_{t(\beta_{l})}\\}$.

When $\mathop{\rm gldim}\nolimits(B)<\infty$, the subcategory
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ (resp. $\Lambda\mbox{\rm-mod}$ allows for arbitrary
iteration of strong tilting) if, and only if, the corresponding property is
true for the algebra $A$.

###### Proof.

Let $\\{e_{1},...,e_{m},e_{m+1},...,e_{n}\\}$ be the set of primitive
idempotents of $\Lambda$ corresponding to the vertices of $Q$, where we assume
that $i$ is a vertex of $Q_{e}$ if and only if $i\leq m$. We have the
canonical Morita context associated to $e:=e_{1}+...+e_{m}$, so that
$\Lambda\cong\begin{pmatrix}e\Lambda e&e\Lambda(1-e)\\\ (1-e)\Lambda
e&(1-e)\Lambda(1-e)\end{pmatrix}$. The relations $R_{1-e,e}$ and $R_{e,e}$
guarantee that the nonzero paths in $e\Lambda e$ and $(1-e)\Lambda(1-e)$
consist exclusively of arrows of $Q_{e}$ and $Q_{1-e}$, respectively. That is,
we have $e\Lambda e\cong A$ and $(1-e)\Lambda(1-e)\cong B$.

On the other hand, one can check that the chosen set of relations implies that
$e\Lambda(1-e)=\sum_{k=1}^{s}\sum_{p\in
e_{o(\alpha_{k})}B_{1-e}}e\Lambda\alpha_{k}p$, that this sum is direct and
that the map $e\Lambda e_{t(\alpha_{k})}\longrightarrow e\Lambda\alpha_{k}p$
($x\rightarrow x\alpha_{k}p$) is an isomorphism, for all $k=1,...,s$.
Therefore $e\Lambda(1-e)$ is a projective left $e\Lambda e$-module. Moreover
the simple left $e\Lambda e$-module ${e\Lambda e_{i}}/{eJe_{i}}$ embeds in
$\mathop{\rm top}\nolimits(_{e\Lambda e}e\Lambda(1-e))$ if and only if
$i=t(\alpha_{k})$ for some $k=1,...,s$. But the relations in $R_{1-e,e}$ imply
that $(1-e)\Lambda e_{t(\alpha_{k})}=0$, for all $k=1,...,s$.

We are thus in the situation of Example 6.9(a), and the conclusions follow
from that example. ∎

###  6.3. A specific path algebra modulo relations

The final example is a non-monomial path algebra modulo relations whose
category of left modules allows for unlimited iteration of strong tilting. Our
reduction technique renders verification of this fact significantly less
labor-intensive.

######  Example 6.11.

Let $\Lambda=KQ/I$ be a specimen of the following class of finite dimensional
algebras over a field $K$, which depends on $4$ parameters
$c_{1},\dots,c_{4}\in K^{*}$. The quiver $Q$ is

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\delta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\nu}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$

and $I\subseteq KQ$ is the ideal generated by $\gamma\alpha-c_{1}\delta\beta$,
$\gamma\beta-c_{2}\delta\alpha$, $\alpha\rho-c_{3}\beta\sigma$, and
$\alpha\rho-c_{4}\tau\nu$, next to monomial relations which are apparent from
the graphs of the indecomposable projective left $\Lambda$-modules:

|
---|---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\nu}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\delta}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\gamma}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{4}$$\textstyle{3}$$\textstyle{5}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\delta}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\nu}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\nu}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\nu}$$\textstyle{4}$$\textstyle{4}$$\textstyle{2}$$\textstyle{2}$$\textstyle{5}$

Then $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite
in $\Lambda\mbox{\rm-mod}$, and $\Lambda\mbox{\rm-mod}$ allows for unlimited
iteration of strong tilting.

###### Proof.

First one checks that the simple left $\Lambda$-modules corresponding to the
vertices $e_{4}$ and $e_{5}$ have finite projective dimension, namely
$\mathop{\rm pdim}\nolimits_{\Lambda}S_{4}=1$ and $\mathop{\rm
pdim}\nolimits_{\Lambda}S_{5}=3$. Choose $e:=e_{1}+e_{2}+e_{3}$. To check that
$e\Lambda(1-e)$ is projective in $e\Lambda e\mbox{\rm-mod}$, observe that
$\Lambda^{\prime}:=e\Lambda e$ is the algebra determined by the quiver
$Q^{\prime}$ and the indecomposable projective left $\Lambda^{\prime}$-modules
shown below.

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\alpha=c_{1}\delta\beta}$$\scriptstyle{\gamma\beta=c_{2}\delta\alpha}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$ |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\alpha}$$\scriptstyle{\gamma\beta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\alpha}$$\textstyle{3}$$\textstyle{3}$$\textstyle{2}$$\textstyle{2}$$\textstyle{3}$

We now read off that $e\Lambda e_{4}\cong(\Lambda^{\prime}e_{3})^{2}$ and
$e\Lambda e_{5}\cong\Lambda^{\prime}e_{3}$ in $\Lambda^{\prime}\mbox{\rm-
mod}$, to find that the left $\Lambda^{\prime}$-module $e\Lambda(1-e)$ is
indeed projective. To see that
$\mathcal{P}^{<\infty}(\Lambda^{\prime}\mbox{\rm-mod})$ is contravariantly
finite in $\Lambda^{\prime}\mbox{\rm-mod}$, we check that $\mathop{\rm
l.findim}\nolimits\Lambda^{\prime}=0$ by applying Bass’s criterion [4]:
Indeed, the graphs of the indecomposable projective right
$\Lambda^{\prime}$-modules,

---
$\textstyle{1\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$$\scriptstyle{\mu}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\alpha}$$\scriptstyle{\gamma\beta}$$\textstyle{3}$$\textstyle{3}$$\textstyle{1}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{2}$$\textstyle{1}$

show that all simple right $\Lambda^{\prime}$-modules embed into the right
socle of $\Lambda^{\prime}$. Consequently, Theorem 4.6 yields contravariant
finiteness of $\mathcal{P}^{<\infty}({\Lambda\mbox{\rm-mod}})$ in
$\Lambda\mbox{\rm-mod}$.

To confirm that $\Lambda\mbox{\rm-mod}$ even allows for unlimited iteration of
strong tilting, we show that this is true for $\Lambda^{\prime}$-mod, whence
Corollary 5.6 will yield our claim. Since $\mathop{\rm
l.findim}\nolimits\Lambda^{\prime}=0$, the basic strong tilting object in
$\Lambda^{\prime}\mbox{\rm-mod}$ is
${{}_{\Lambda^{\prime}}T^{\prime}}={{}_{\Lambda^{\prime}}\Lambda^{\prime}}$,
which yields $\widetilde{\Lambda^{\prime}}\cong\Lambda^{\prime}$.

It is not difficult to directly ascertain that
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda^{\prime})$ is in turn
contravariantly finite in $\mbox{\rm mod-}\Lambda^{\prime}$, but another
application of Theorem 4.6 cuts this task short: Setting
$e^{\prime}=e^{\prime}_{2}+e^{\prime}_{3}$, where the $e^{\prime}_{i}$ are the
primitive idempotents of $\Lambda^{\prime}$ corresponding to the vertices of
$Q^{\prime}$, it is effortless to check that $e^{\prime}$ satisfies the
conditions of Setting 4.1 relative to $\mbox{\rm mod-}\Lambda^{\prime}$, and
that $\mathop{\rm r.findim}\nolimits e^{\prime}\Lambda^{\prime}e^{\prime}=0$.
Hence $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}e^{\prime}\Lambda^{\prime}e^{\prime})$ is contravariantly finite in
$\mbox{\rm mod-}e^{\prime}\Lambda^{\prime}e^{\prime}$, and consequently so is
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda^{\prime})$ in $\mbox{\rm
mod-}\Lambda^{\prime}$. By Theorem 3.1, we thus conclude that
$\Lambda^{\prime}\mbox{\rm-mod}$ allows for unlimited iteration of strong
tilting. This completes the argument. ∎

## References

  * [1] M. AUSLANDER and I. REITEN, Applications of contravariantly finite subcategories. Adv. Maths. 86(1) (1991), 111-152.
  * [2] M. AUSLANDER and S.O. SMALØ, Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61-122.
  * [3] M. AUSLANDER and S.O. SMALØ, Almost split sequences in subcategories, J. Algebra 69 (1981), 426-454.
  * [4] H. BASS, Finitistic dimension and a homological generalization of semi-primary rings. Trans. Am. Math. Soc. 95(3) (1960), 466-488.
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  * [12] P.J. HILTON and U. STAMMBACH, A Course in Homological Algebra, 2nd edition. Springer-Verlag, 1997.
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  * [14] B. HUISGEN-ZIMMERMANN and M. SAORÍN, Dualities for modules of finite projective dimension. Contemp. Maths. 761 (2021), 61-101.
  * [15] B. HUISGEN-ZIMMERMANN and S.O. SMALØ, A homological bridge between finite and infinite dimensional representations, Alg. & Rep. Theory 1 (1998), 169-188.
  * [16] K. IGUSA, S.O. SMALØ and G. TODOROV, Finite projectivity and contravariant finiteness. Proc. Amer. Math. Soc., 109(4) (1990), 937-941.
  * [17] J.P. JANS, Some aspects of torsion. Pacific J. Math. 15 (1965), 1249–1259.
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correlated noise, encoded by the phases $\varphi_{0}(t)$ and $\varphi_{1}(t)$.
More generally, we can ask which matrix elements
$\langle\vec{\alpha}|\rho(t)|\vec{\beta}\rangle$ in the $z$ basis are
“quantum-noise-insensitive” (QNI). The answer depends upon the symmetries of
the noise. For general non-collective dephasing, there are no further
simplifications to Eqs. (22)-(24). If we want both $\varphi_{0}(t)$ and
$\varphi_{1}(t)$ to vanish, it is easy to see that either
$\\{\alpha_{n}=\beta_{n},\,\forall\,n\\}$ or
$\\{\alpha_{n}=-\beta_{n},\,\forall\,n\\}$. Thus, only $2^{N+1}$ of the
$2^{2N}$ matrix elements, the diagonal and anti-diagonal ones, are QNI. For
collective and even-odd noise, the higher degree of symmetry results in a
greater number of QNI states. Let us work out the math of each of them:

Collective noise. With reference to Sec. 3.3, the collective dynamic
coefficients are index-independent, with the non-vanishing decay and phase
being given by

$\displaystyle\gamma(t)^{\text{coll}}=\kappa(t)(m-m^{\prime})^{2},\quad\varphi_{0}(t)^{\text{coll}}=\xi(t)(m^{\prime
2}-m^{2}).$ (83)

Here, $2m=\sum_{n=1}^{N}\alpha_{n},2m^{\prime}=\sum_{n=1}^{N}\beta_{n}$ and
$\kappa(t),\xi(t)$ are obtained by replacing $B_{n}(t)$ with $B(t)$ in Eqs.
(25)-(26) and $m,m^{\prime}$ represent the $z$ angular momentum component of
$|\vec{\alpha}\rangle$, $|\vec{\beta}\rangle$. Note that permutation
invariance forces $\varphi_{1}(t)^{\text{coll}}=0$. Thanks to its higher
degree of symmetry, the reduced matrix evolution can be described exclusively
in terms of the permutation-invariant Dicke states
$\\{|J,-J\rangle,|J,-J+1\rangle,\ldots,|J,m\rangle,\ldots,|J,J\rangle\\}$,
where $J=N/2$ is the total angular momentum [36]. It is clear from Eq. (83)
that the QNI matrix elements must have $m=\pm m^{\prime}$, and are thus of the
form $\langle J,m|\rho_{0}|J,\pm m\rangle$, for $-J\leq m\leq J$. There are
$2(2J+1)$ such elements in the Dicke basis or, noting that each $|J,m\rangle$
consists of the equally weighted sum of all ${2J\choose m+J}$ kets
$|\vec{\alpha}\rangle$ such that $\sum_{n=0}^{N}\alpha_{n}=2m$, $2{4J\choose
2J}$ such elements in the $z$ basis.

Even-odd noise. As in the main text, half of the $N=2J$ qubits (the even ones)
couple to the environment via operator $B_{\text{e}}(t)$, and the other half
(the odd ones) couple to the environment via $B_{\text{o}}(t)\neq
B_{\text{e}}(t)$. Permutation invariance is then retained only among qubits
belonging to the same cluster, and we can represent ZMGSD even-odd dynamics in
terms the basis states $|J/2,m_{\text{e}};J/2,m_{\text{o}}\rangle$, with
$-J/2\leq m_{\text{e}},m_{\text{o}}\leq J/2$, for a total of $(J+1)^{2}$
elements. The even-odd decay and phase
$\gamma(t)^{\text{eo}},\varphi_{0}(t)^{\text{eo}}$ are now given by:

$\displaystyle\gamma(t)^{\text{eo}}$
$\displaystyle=\Big{\\{}\kappa_{\text{ee}}(t)\left(m_{\text{e}}-m^{\prime}_{\text{e}}\right){}^{2}+\kappa_{\text{oo}}(t)\left(m_{\text{o}}-m^{\prime}_{\text{o}}\right){}^{2}+2\kappa_{\text{eo}}(t)\left(m_{\text{e}}-m_{\text{o}}\right)\left(m^{\prime}_{\text{e}}-m^{\prime}_{\text{o}}\right)\Big{\\}},$
$\displaystyle\varphi_{0}(t)^{\text{eo}}$
$\displaystyle=\Big{\\{}\xi_{\text{ee}}(t)(m_{\text{e}}^{2}-m_{\text{e}}^{{}^{\prime}2})+\xi_{\text{oo}}(t)(m_{\text{o}}^{{}^{\prime}2}-m_{\text{o}}^{{}^{\prime}2})+2\xi_{\text{eo}}(t)\left(m_{\text{e}}m_{\text{o}}-m^{\prime}_{\text{e}}m^{\prime}_{\text{o}}\right)\Big{\\}},$
$\displaystyle\varphi_{1}(t)^{\text{eo}}$
$\displaystyle=\Big{\\{}2\vartheta_{\text{eo}}(t)\left(m_{\text{e}}m^{\prime}_{\text{o}}-m^{\prime}_{\text{e}}m_{\text{o}}\right)\Big{\\}},$
(84)

It follows that in order for $\varphi_{0}(t)^{\text{eo}}$ and
$\varphi_{1}(t)^{\text{eo}}$ to be zero, we must have
$\\{m_{{\text{e}}}=m^{\prime}_{{\text{e}}},m_{{\text{o}}}=m^{\prime}_{\text{o}}\\}$
or
$\\{m_{\text{e}}=-m^{\prime}_{\text{e}},m_{\text{o}}=-m^{\prime}_{\text{o}}\\}$.
Let us consider the former case first. If $m_{\text{e}}+J/2=k$ spins in the
even cluster are up $(0\leq k\leq J)$, we have ${J\choose k}$ combinations of
spin up/down states giving $m_{\text{e}}$, and ${J\choose k}^{2}$ combinations
making $m_{\text{e}}=m^{\prime}_{\text{e}}$. For each of these combinations,
we must also have that $m_{\text{o}}=m^{\prime}_{\text{o}}$. If
$m_{\text{o}}+J/2=q$ spins in the odd cluster are up ($0\leq q\leq J$), this
gives ${J\choose q}^{2}$ possibilities. Thus, for fixed $m_{\text{e}}$ and
$m_{\text{o}}$, there are ${J\choose k}^{2}{J\choose q}^{2}$ matrix elements
fulfilling
$\\{m_{\text{e}}=m^{\prime}_{\text{e}},m_{\text{o}}=m^{\prime}_{\text{o}}\\}$.
Summing over all possible values of $k,q$ and multiplying by two to account
for the
$\\{m_{\text{e}}=-m^{\prime}_{\text{e}},m_{\text{o}}=-m^{\prime}_{\text{o}}\\}$
case, we get:

$2\bigg{(}\sum_{k=0}^{J}{J\choose
k}^{2}\bigg{)}\bigg{(}\sum_{q=0}^{J}{J\choose q}^{2}\bigg{)}=2{2J\choose
J}^{2}.$

Thus, there are $2{2J\choose J}^{2}$ matrix elements that are QNI, and
${2J\choose J}^{2}$ that altogether feel no noise. We can further extend this
reasoning to $L$ clusters of $J/L$ qubits, resulting in a number of
$2{4J/L\choose 2J/L}^{L}$ QNI matrix elements.

###  A.3 Classicality of the noisy reduced dynamics

The possibility to exactly represent, or “simulate” purely dephasing quantum
dynamics in terms of stochastically fluctuating classical fields has been long
investigated as part of an effort to obtain a deeper understanding of
nonclassicality notions – resulting in the concept of random unitary (RU)
dynamics [46]. As an application of the explicit form of the reduced density
operator that the above cumulant-approach enables, we now establish a
sufficient condition on the noise for the reduced dynamics to be RU.

Suppose that the evolution of a quantum system up to time $t>0$ is described
by a completely-positive, trace-preserving map,

$\rho_{0}\mapsto\rho(t)=\mathcal{E}_{t}[\rho_{0}]=\sum_{i}K_{i}(t)\rho_{0}K_{i}^{\dagger}(t),\qquad\sum_{i}K_{i}^{\dagger}(t)K_{i}(t)={\mathbb{I}}.$

Recall that the dynamics are RU if the evolution may be expressed as a convex
sum of unitary maps,

$\mathcal{E}_{t}[\rho_{0}]=\sum_{i\in\mathcal{I}}p_{i}\,U_{i}(t)\,\rho_{0}\,U_{i}^{\dagger}(t),\quad
U_{i}(t)U_{i}^{\dagger}(t)={\mathbb{I}},\qquad\sum_{i\in\mathcal{I}}p_{i}=1,p_{i}>0,$

or, more generally, as an integral
$\mathcal{E}_{t}[\rho_{0}]=\int_{\Lambda}\,\mathbb{P}[d\lambda]\,U_{\lambda}(t)\,\rho_{0}\,U^{\dagger}_{\lambda}(t)$,
for an appropriate probability measure ${\mathbb{P}}$. Note that, by
construction, RU dynamics is necessarily doubly-stochastic (unital), since the
identity operator is preserved. Thus,
$\sum_{i}K_{i}(t)K_{i}^{\dagger}(t)={\mathbb{I}}$ as well. Unlike in the
classical dynamical setting, however, not every unital quantum map
$\mathcal{E}_{t}$ arises as a mixture of unitaries, which makes the problem of
determining whether or not a given map on a system of arbitrary dimension is
RU highly challenging [62] and computationally NP-hard in general [63]. We
prove the following result for ZMGSD:

Claim. The reduced dynamics resulting from a ZMGSD model with
$\varphi_{1}(t)=0$ is random unitary.

Proof. The proof is constructive. We start by noting that the action of the
system Hamiltonian, $H_{S}$, and the contribution of the phase
$\varphi_{0}(t)$ [Eq. (23) in the main text] can be factored out in terms of
unitaries $U_{0}(t)=\exp\left\\{\\!-ibt\sum_{n=1}^{N}\sigma_{n}^{z}\right\\}$
and
$U_{Q}(t)=\exp\left\\{\\!-i4\sum_{n,m=1}^{N}\xi_{nm}(t)\sigma_{n}^{z}\sigma_{m}^{z}\right\\}$,
respectively. It follows that the reduced density matrix at time $t$ can be
expressed as
$\rho(t)=U_{0}(t)U_{Q}(t)\mathcal{E}_{t}\left[\rho_{0}\right]U_{Q}^{\dagger}(t)U_{0}^{\dagger}(t),$
with

$\mathcal{E}_{t}\left[\rho_{0}\right]=\sum_{\vec{\alpha},\vec{\beta}}e^{-\sum_{n,m=1}^{N}\kappa_{nm}(t)(\alpha_{n}-\beta_{n})(\alpha_{m}-\beta_{m})}\langle\vec{\alpha}|\rho_{0}|\vec{\beta}\rangle|\vec{\alpha}\rangle\langle\vec{\beta}|,$

representing the decay of coherence. The key step in our construction consists
on explicitly expressing $\mathcal{E}_{t}\left[\rho_{0}\right]$ as an RU map
(albeit in terms of explicitly time-dependent Hamiltonians). To that end, we
define the zero-mean, Gaussian set of stochastic processes
$\vec{\lambda}(t)\equiv\\{\lambda_{n}(t)\\},\,n=1,\ldots,N$, such that

$\langle\lambda_{n}(t)\rangle_{\text{c}}=0,\forall
n,\qquad\frac{1}{2}\int_{0}^{t}\\!ds\\!\int_{0}^{t}\\!ds^{\prime}\,\langle\lambda_{n}(s)\lambda_{m}(s^{\prime})\rangle_{\text{c}}=\kappa_{nm}(t),$

with $\langle
f(\vec{\lambda})\rangle_{\text{c}}=\int_{\Lambda}d\vec{\lambda}P(\vec{\lambda})f(\vec{\lambda})$
denoting a classical ensemble average and $f(\vec{\lambda})$ a function of any
subset of $\\{\lambda_{1},\ldots,\lambda_{N}\\}$. Let now
$U(\vec{\lambda},t)\equiv\exp\left(\\!-i\sum_{n}\sigma_{n}^{z}\int_{0}^{t}ds\,\lambda_{n}(s)\right)$.
Using Gaussianity, we can evaluate

$\left\langle
U(\vec{\lambda},t)\,\rho_{0}\,U^{\dagger}(\vec{\lambda},t)\right\rangle_{\text{c}}=\sum_{\vec{\alpha},\vec{\beta}}e^{-\sum_{n,m=1}^{N}\kappa_{nm}(t)(\alpha_{n}-\beta_{n})(\alpha_{m}-\beta_{m})}\langle\vec{\alpha}|\rho_{0}|\vec{\beta}\rangle|\vec{\alpha}\rangle\langle\vec{\beta}|=\mathcal{E}_{t}[\rho_{0}],$

making $\mathcal{E}_{t}[\rho_{0}]$ an RU channel, as claimed. As the
composition of $\mathcal{E}_{t}[\rho_{0}]$ with $U_{0}(t)$ and $U_{Q}(t)$ is
still manifestly RU, so is the resulting dynamics. $\blacksquare$

According to the above claim, ZMGSD noise can always be mimicked by a
classical phase damping process in the collective regime, consistent with the
fact that, as noted in Sec. 3.3, the bath operators
$\\{U_{\vec{\alpha}}(t)\\}$ commute with one another. One may wonder whether
the condition $\varphi_{1}(t)=0$ is also _necessary_ for a non-collective
dephasing model to be RU. A rigorous answer is very nontrivial and not
available to the best of our knowledge.

It is natural to think that, for ZMGS non-Markovian dephasing as we consider,
the phase $\varphi_{1}(t)$ may play a role similar to what the Hamiltonian
obstruction does in the simpler setting of Markovian dephasing models [77]. To
make the connection with the Markovian limit clearer, we may recast the
evolution of the qubit state under ZMGSD noise in terms of an exact non-
Markovian master equation, by differentiating the expression for the reduced
density matrix elements given in Eq. (21). We find:

$\displaystyle\frac{d}{dt}\rho(t)=-i[H_{\text{eff}}(t),\rho(t)]+\sum_{n,m=1}^{N}c_{nm}(t)\Big{(}\sigma_{n}^{z}\,\rho(t)\,\sigma_{m}^{z}-\frac{1}{2}\\{\rho(t),\sigma_{n}^{z}\sigma_{m}^{z}\\}\Big{)},$
(85) $\displaystyle
H_{\text{eff}}(t)=\frac{b}{2}\sum_{n=1}^{N}\sigma_{n}^{z}+\sum_{n,m=1}^{N}\dot{\xi}_{nm}(t)\sigma_{n}^{z}\sigma_{m}^{z},\quad
c_{nm}(t)=2\Big{(}\dot{\kappa}_{nm}(t)+i\dot{\vartheta}_{nm}(t)\Big{)}.$ (86)

Explicitly, the time derivatives of the dynamic coefficients are given by:

$\displaystyle\dot{\xi}_{nm}(t)=\frac{1}{16}\int_{-t}^{t}ds\,\text{sign}(s)\,\langle[B_{n}(s),B_{m}(0)]\rangle_{\mathrm{B}},$
$\displaystyle\dot{\kappa}_{nm}(t)=\frac{1}{16}\int_{-t}^{t}ds\,\langle\\{B_{n}(s),B_{m}(0)\\}\rangle_{\mathrm{B}},\quad\dot{\vartheta}_{nm}(t)=\frac{1}{16}\int_{-t}^{t}ds\,\langle[B_{n}(s),B_{m}(0)]\rangle_{\mathrm{B}},$

with sign$(s)$ the sign function that is $1$ for $s>0$, $0$ for $s=0$ and $-1$
for $s<0$. Thus, one sees explicitly that the bath-induced phase
$\varphi_{0}(t)$, which depend upon $\xi_{nm}(t)$, corresponds to a bath-
mediated, entangling Ising Hamiltonian, which enters the dynamics unitarily,
akin to a (non-Markovian) Lamb shift. In the Markovian limit,
$\partial_{t}c_{nm}(t)=0$, resulting in a constant relaxation matrix
$C\equiv\\{c_{nm}\\}$; Eq. (85) then reduces to the well-known Gorini-
Kossakowski-Sudarshan-Lindblad form. As noted in Sec. 3.3, the parameters
$\vartheta_{nm}(t)$ (hence $\varphi_{1}(t)$) can still be non-zero as long as
the noise spatial correlation matrix obeys Im$\,C\neq 0$. The latter condition
immediately implies a non-vanishing Hamiltonian obstruction (see Eq. (28) in
[77]). In such a setting, vanishing of the obstruction is both sufficient and
necessary for an RU representation in terms of smooth, diffusive (Wiener)
processes to exist; however, a Markovian dephasing model may still be RU with
a non-zero obstruction, as long as that arises only from discontinuous Poisson
noise.

##  Appendix B Additional technical details for OATS dynamics

###  B.1 OATS expectation values $\langle J_{y}(\vec{r},t)\rangle$, $\langle
J_{y}^{2}(\vec{r},t)$ for arbitrary qubit positions

We first evaluate $\langle J_{y}(\vec{r},t)\rangle$ for an OATS with qubits in
positions $\vec{r}\equiv\\{\vec{r}_{1},\ldots,\vec{r}_{n}\\}$. Let us write
the OATS as

$\displaystyle\rho_{0}=|S\rangle\langle S|,\qquad|S\rangle=\exp(-i\beta
J_{x})\exp(-i\theta J_{z}^{2})|+\rangle^{\otimes N},$ (87)

with $|+\rangle$ the positive eigenstate of $\sigma_{x}$. By expressing the
angular momentum components in terms of single qubit matrices,
$J_{\alpha}=\sum_{n=1}^{N}\sigma_{n}^{\alpha}/2$, with $\alpha\in\\{x,y,z\\}$,
we may identify a contribution involving qubit $n$ and a second contribution
which does not, that is, $J_{\alpha}=\sigma_{n}^{\alpha}/2+J^{\alpha}_{\neq
n}$. We can then rewrite the OATS as

$\displaystyle e^{i\theta/8}\exp(-i\beta J^{x}_{\neq
n})\exp[-i\theta(J^{z}_{\neq
n})^{2}/2]\exp(-i\beta\sigma_{n}^{x}/2)\exp(-i\theta J^{z}_{\neq
n}\sigma_{n}^{z}/4)|+\rangle_{n}|+\rangle_{\neq n},$ (88)

where $|+\rangle_{\neq n}=\bigotimes_{\ell\neq n}|+\rangle_{\ell}$ is a CSS
excluding qubit $n$. We also write the effective propagator corresponding to
qubit $n$, Eq. (14) in the main text, as

$\displaystyle e^{i\Phi_{n}(t)}=\exp\\{-i[\varphi(t)+\psi_{\neq
n}(t)]\sigma_{n}^{z}\\},\quad\psi_{\neq n}(t)=\sum_{\ell\neq
n}\Psi_{n\ell}(t)\sigma_{\ell}^{z}.$ (89)

We then substitute Eqs. (87) and (88) into Eq. (14), and evaluate all
expectation values of operators for qubit $n$ with respect to $|+\rangle_{n}$
exactly. Summing over all qubits, this results in

$\displaystyle\langle J_{y}(\vec{r},t)\rangle$
$\displaystyle=\sum_{n}\frac{e^{\chi_{nn}(t)/2}}{4}\Big{[}\frac{e^{i\varphi(t)}}{2}\sin(\beta)(\mathcal{E}^{\neq
n}_{--+}-\mathcal{E}^{\neq n}_{+--})$ (90)
$\displaystyle+ie^{-i\varphi(t)}\sin(\beta/2)^{2}\mathcal{E}^{\neq
n}_{+-+}-ie^{i\varphi(t)}\cos(\beta/2)^{2}\mathcal{E}^{\neq
n}_{+++}+\text{c.c.}\Big{]},$

where $\mathcal{E}^{\neq n}_{s_{1}s_{2}s_{3}}$ represents the remaining
position-dependent expectation value with respect to all qubits $\ell\neq n$:

$\displaystyle\mathcal{E}^{\neq n}_{s_{1}s_{2}s_{3}}(\vec{r},t)=\langle
S_{\neq n}|\exp(is_{1}\theta J^{z}_{\neq n}/2)\exp[is_{2}\tilde{\psi}_{\neq
n}(\vec{r},t)]\exp(is_{3}\theta J^{z}_{\neq n}/2)|S_{\neq n}\rangle,$ (91)

with $s_{1},s_{2},s_{3}\in\\{+1,-1\\}$, and

$\displaystyle\tilde{\psi}_{\neq n}(\vec{r},t)=\exp(i\beta J^{x}_{\neq
n})\psi_{\neq n}(\vec{r},t)\exp(-i\beta J^{x}_{\neq n}),\quad|S_{\neq
n}\rangle=\exp[-i\theta(J^{z}_{\neq n})^{2}]|+\rangle_{\neq n}.$ (92)

So far, no approximation has been made. Note that Eq. (90) no longer involves
operators involving qubit $n$. The mean values $\mathcal{E}^{\neq
n}_{s_{1}s_{2}s_{3}}(\vec{r},t)$ can be approximately evaluated by performing
a generalized cumulant expansion in the sense of Kubo [78]555 Note that,
unlike the cumulant expansion carried out in A.1, which was a way to re-
express an _exact_ average taken _over the bath degrees of freedom_ , the
generalized cumulant operation considered here is a tool to _approximately_
evaluate mean values _over the system degrees of freedom_ (excluding qubit
$n$). . Truncating this expansion to the second order and expressing cumulants
in terms of moments then gives

$\displaystyle\langle
J(\vec{r},t)\rangle\simeq\exp\Big{\\{}i(s_{1}+s_{3})\frac{\theta}{2}\langle
J^{z}_{\neq n}\rangle+is_{2}\langle\tilde{\psi}_{\neq
n}\rangle-\frac{\theta^{2}}{4}(1+s_{2}s_{3})\left(\langle(J^{z}_{\neq
n})^{2}\rangle-\langle J^{z}_{\neq n}\rangle^{2}\right)$ (93)
$\displaystyle-\frac{1}{2}[\langle(\tilde{\psi}_{\neq
n})^{2}\rangle-\langle\tilde{\psi}_{\neq
n}\rangle^{2}]\\!-\frac{\theta}{2}\left[s_{1}s_{2}\langle J^{z}_{\neq
n}\tilde{\psi}_{\neq n}\rangle+s_{2}s_{3}\langle\tilde{\psi}_{\neq
n}J^{z}_{\neq n}\rangle-(s_{1}s_{2}+s_{2}s_{3})\langle\tilde{\psi}_{\neq
n}\rangle\langle J^{z}_{\neq n}\rangle\right]\Big{\\}},$

where all expectation values are taken with respect to $|S_{\neq n}\rangle$.
Evaluating these expectations and substituting in Eq. (93) then yields an
approximate expression for $\langle J_{y}(\vec{r},t)\rangle$ for arbitrary
spatial correlations:

$\displaystyle\langle J_{y}(\vec{r},t)\rangle$
$\displaystyle=\sum_{n=1}^{N}e^{-\chi_{nn}(t)}\exp[-\langle\tilde{\psi}_{\neq
n}\rangle/2]\bigg{\\{}e^{-\theta^{2}(N-1)/8}\Big{[}\cos\left(\frac{\beta}{2}\right)^{2}e^{-\theta\langle
R\rangle}+\sin\left(\frac{\beta}{2}\right)^{2}e^{\theta\langle
R\rangle}\Big{]}$ (94)
$\displaystyle+\sin(\beta)\sin\Big{[}\theta\sin(\beta)\cos\left(\frac{\theta}{2}\right)\sum_{\ell\neq
n}\Psi_{n\ell}(t)\Big{]}\bigg{\\}}\sin(\varphi(t)),$

where

$\displaystyle\langle\tilde{\psi}_{\neq n}\rangle$
$\displaystyle=\sum_{\ell}^{\prime}\Psi_{n\ell}(t)^{2}+\sum_{\ell_{1},\ell_{2}\neq\ell_{1}}^{\prime}\Psi_{n\ell_{1}}(t)\Psi_{n\ell_{2}}(t)\bigg{[}\frac{1}{2}\sin\left(\frac{\beta}{2}\right)^{2}(1-\cos(\theta)^{N-3})$
(95)
$\displaystyle+\sin(2\beta)\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)^{N-3}\bigg{]},$
$\displaystyle\langle R\rangle$
$\displaystyle=\sum_{\ell}^{\prime}\Psi_{n\ell}(t)^{2}\cos(\beta)+\sum_{\ell_{1},\ell_{2}\neq\ell_{1}}^{\prime}\Psi_{n\ell_{2}}(t)\sin(\beta)\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)^{N-3},$
(96)

are the position-dependent relevant mean values, and $\sum^{\prime}$ stands
for a summation excluding the index $n$.

Evaluation of $\langle J_{y}^{2}(\vec{r},t)\rangle$ proceeds along similar
steps as above. To treat qubits $n,m$ separately from qubits $\ell\neq n,m$ we
introduce a new collective spin operator

$\displaystyle J^{\alpha}_{\neq n}\equiv\sum_{\ell\neq
nm}\frac{\sigma_{\ell}^{\alpha}}{2},\quad\alpha\in\\{x,y,z\\}.$ (97)

Using $J^{\alpha}=J^{\alpha}_{\neq
n}+(\sigma^{\alpha}_{n}+\sigma^{\alpha}_{m})/2$, we then write the initial
OATS in Eq. (87) as

$\displaystyle|S\rangle$ $\displaystyle=e^{i\theta/4}\exp[-i\beta J^{x}_{\neq
nm}]\exp[-i\theta(J^{z}_{\neq
nm})^{2}/2]\exp[-i\beta(\sigma_{n}^{x}+\sigma_{m}^{x})/2]$ (98)
$\displaystyle\qquad\quad\exp[-i\theta J^{z}_{\neq
nm}(\sigma_{n}^{x}+\sigma_{m}^{x})]\exp[-i\theta\sigma_{n}^{z}\sigma_{m}^{z}/4]|+\rangle_{n}|+\rangle_{m}|+\rangle_{\neq
nm},$

where $|+\rangle_{\neq nm}=\bigotimes_{\ell\neq nm}|+\rangle_{\ell}$. We also
write the effective propagator corresponding to qubits $n$ and $m$, Eq. (15)
in the main text, as

$\displaystyle
e^{-i\Phi_{nm}(t)}=\exp[-i\varphi(t)(\sigma^{z}_{n}+\sigma^{z}_{m})-\chi_{nm}(t)\sigma_{n}\sigma_{m}-i\psi_{\neq
nm}(t)\sigma_{n}^{z}-i\psi_{\neq mn}(t)\sigma_{m}^{z}],$ (99)

where $\psi_{\neq nm}(t)\equiv\sum_{\ell\neq n,m}\Psi_{n\ell}(t)$. We then
substitute Eqs. (98) and (99) into Eq. (15) for $\langle
J^{2}_{y}(\vec{r},t)\rangle$ and evaluate all expectation values of operators
for qubits $n$ and $m$ with respect to $|+\rangle_{n}|+\rangle_{m}$ exactly:

$\displaystyle\langle J^{2}_{y}(\vec{r},t)\rangle$
$\displaystyle\simeq\frac{N}{4}+\frac{1}{2}\tilde{\sum_{\ell_{1},\ell_{2}\neq\ell_{1}}}e^{-\chi_{nn}(t)-\chi_{mm}(t)}\bigg{[}\,e^{\chi_{nm}(t)+\chi_{mn}(t)}\,\langle\Theta_{+-}|\Theta_{-+}\rangle$
(100)
$\displaystyle-\cos(2\varphi(t))\,e^{-\chi_{nm}(t)-\chi_{mn}(t)}\,\langle\Theta_{++}|\Theta_{--}\rangle\bigg{]},$

where $\tilde{\sum}$ excludes indexes $n,m$ from the summation, and
$\langle\Theta_{+-}|\Theta_{-+}\rangle$,
$\langle\Theta_{++}|\Theta_{--}\rangle$ are expectation values of operators
acting on all qubits $\neq n,m$, taken with respect to the state
$|S\rangle_{\neq nm}\equiv\exp[-i\theta(J^{z}_{\neq nm})^{2}/2]|+\rangle_{\neq
nm}$. As before, these can be approximately evaluated by performing a cumulant
expansion and truncating to the second order, yielding the following
expressions:

$\displaystyle\langle\Theta_{+-}|\Theta_{-+}\rangle\simeq\bigg{\\{}\frac{1}{8}\sin\left(\beta\right)^{2}\Big{[}\cos(2\theta\langle
I_{-}\rangle)+\cosh(2\theta\langle R_{-}\rangle)e^{-2\theta^{2}\langle
F_{x}^{2}\rangle}\Big{]}-\frac{1}{2}\sin(2\beta)$
$\displaystyle\bigg{[}e^{-\theta\langle
R_{-}\rangle}\sin\\!\bigg{(}\frac{\theta}{2}(1-2\langle
I_{-}\rangle)\bigg{)}+e^{\theta\langle
R_{-}\rangle}\sin\\!\left(\frac{\theta}{2}(1+2\langle
I_{-}\rangle)\right)\bigg{]}e^{-\theta^{2}\langle
F_{x}^{2}\rangle/2}+\frac{1}{4}\cos(\beta)^{2}\bigg{\\}}e^{-\langle
g_{-}^{2}\rangle/2},$
$\displaystyle\langle\Theta_{++}|\Theta_{--}\rangle\simeq\bigg{\\{}\frac{1}{4}\bigg{[}\cos\\!\left(\frac{\beta}{2}\right)^{4}e^{-2\langle
R_{+}\rangle}+\sin\\!\left(\frac{\beta}{2}\right)^{4}e^{2\langle
R_{+}\rangle}\bigg{]}e^{-2\theta^{2}\langle
F_{x}^{2}\rangle}-\frac{1}{2}\cos\\!\left(\frac{\beta}{2}\right)^{2}\sin\\!\left(\frac{\beta}{2}\right)^{2}$
$\displaystyle\quad\cos(2\theta\langle
I_{+}\rangle)+\frac{1}{2}\sin(\beta)\bigg{[}\cos\\!\left(\frac{\beta}{2}\right)^{2}\sin\\!(\theta(1-2\langle
I_{+}\rangle))e^{-\theta\langle
R_{+}\rangle}-\sin\\!\left(\frac{\beta}{2}\right)\sin\\!(\theta(1+2\langle
I_{+}\rangle))$ $\displaystyle\quad e^{\theta\langle
R_{+}\rangle}\bigg{]}e^{-\theta^{2}\langle
F_{x}^{2}\rangle/2}+\frac{1}{4}\sin\\!\left(\beta\right)^{2}\bigg{\\}}e^{-\langle
g^{2}_{+}\rangle/2}.$

Finally, introducing the notation
$\Delta\Psi_{\ell}(t)^{\pm}\equiv\Psi_{n\ell}(t)\pm\Psi_{m\ell}(t)$, the
position-dependent moments and covariances entering the above equation can be
expressed as:

$\displaystyle\langle R_{\pm}\rangle$
$\displaystyle=\frac{\cos(\beta)}{8}\tilde{\sum}_{\ell}\Delta\Psi_{\ell}^{\pm}+\frac{1}{8}\sin(\beta)\sin\\!\left(\frac{\theta}{2}\right)\cos\\!\left(\frac{\theta}{4}\right)^{\\!N-4}\\!\\!\tilde{\sum_{\ell_{1},\ell_{2}\neq\ell_{1}}}\Delta\Psi_{\ell_{2}}^{\pm},$
$\displaystyle\langle I_{\pm}\rangle$
$\displaystyle=-\frac{\sin(\beta)}{8}\cos\left(\frac{\theta}{2}\right)^{\\!N-3}\tilde{\sum_{\ell}}\Delta\Psi_{\ell}^{\pm},$
$\displaystyle\langle g_{\pm}^{2}\rangle$
$\displaystyle=\frac{1}{16}\tilde{\sum_{\ell}}(\Delta\Psi_{\ell}^{\pm})^{2}+\frac{1}{16}\tilde{\sum_{\ell_{1},\ell_{2}\neq\ell_{1}}}\Delta\Psi_{\ell_{1}}^{\pm}\Delta\Psi_{\ell_{2}}^{\pm}\bigg{[}\frac{1}{2}\sin(\beta)^{2}(1-\cos(\theta)^{N-4})+$
$\displaystyle\quad\;\sin(2\beta)\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)^{N-4}\bigg{]},$
$\displaystyle\langle F_{x}^{2}\rangle$ $\displaystyle=(N-2)/4.$

Putting things together, we obtain the desired expression for
$\Delta\hat{b}(t)$ by replacing Eqs. (94) and (100) into the error propagation
formula of Eq. (6).

###  B.2 Optimality of noiseless OATS angles

Given an initial OATS subject to collective or even-odd spin-boson  dephasing
noise, we claimed in the main text that the best possible short-time
uncertainty scaling is achieved by the squeezing and rotation angles
$(\theta_{\text{opt}},\beta_{\text{opt}})$ of Eq. (48) that minimize the
uncertainty along the $y$ axis in the noiseless case. To justify the claim,
consider $N$ qubit probes prepared in an OATS as in Eq. (87), subject to the
Ramsey interferometry protocol. After measuring along the $y$ axis, we can
approximate $\Delta\hat{b}(t)$ around $t=0$ as follows:

$\Delta\hat{b}(t)\approx\Delta\hat{b}_{\text{app}}(t)=\frac{\sqrt{a_{0}(\theta,\beta,N)+a_{2}(\theta,\beta,N)(\omega_{c}t)^{2}}}{N/2\,\sqrt{T\;t}\;h_{0}(\theta,\beta,N)},$
(101)

where the $a_{0}(\theta,\beta,N),a_{2}(\theta,\beta,N)$ and
$h_{0}(\theta,\beta,N)$ coefficients come from Taylor expanding $\Delta
J^{2}_{y}(t)$ and $\langle J_{y}(t)\rangle$ with respect to time, to second
and zero order, respectively. Their specific functional form depends on the
nature of the noise. Remarkably, the agreement between the collective and
even-odd uncertainties with their short-time approximation (101) is excellent
for arbitrary squeezing and rotation angles. The simple time dependence of
$\Delta\hat{b}_{\text{app}}(t)$ allows us to easily minimize it with respect
to $t$. We find that:

$\omega_{c}\tau_{\text{opt}}=\frac{\sqrt{a_{0}(\theta,\beta,N)}}{\sqrt{a_{2}(\theta,\beta,N)}},\qquad\Delta\hat{b}_{\text{opt}}=\sqrt{\frac{\omega_{c}}{T}}\frac{\sqrt{2}\left[a_{0}(\theta,\beta,N)\;a_{2}(\theta,\beta,N)\right]^{1/4}}{h_{0}(\theta,\beta,N)\,N/2}.$
(102)

Let us now tackle each case separately.

Collective noise. We find, using Mathematica:

$\displaystyle a_{0}(\theta,\beta,N)$
$\displaystyle=N^{2}\cos(\beta)e^{-N\theta^{2}/2}\left[2+\cos(\beta)\left(e^{N\theta^{2}/2}-1\right)-4\sin(\beta)\sin\left(\theta/2\right)e^{N\frac{3}{8}\theta^{2}}\right],$
$\displaystyle a_{2}(\theta,\beta,N)$
$\displaystyle=\frac{1}{16}N^{2}\text{$\chi_{0}$}^{2}e^{-N\theta^{2}/2}\left(3+\cos(2\beta)+2\sin(\beta)^{2}e^{N\theta^{2}/2}+4\sin(2\beta)\sin\left(\theta/2\right)e^{N\frac{3}{8}\theta^{2}}\right),$
$\displaystyle h_{0}(\theta,\beta,N)$
$\displaystyle=e^{-N\frac{1}{8}\theta^{2}},$

where we allow for the squeezing and rotation angles to depend on $N$,
$\\{\theta,\beta\\}\rightarrow\\{\theta(N),\beta(N)\\}$. Note that, unless
$\theta$ scales like $\theta(N)\propto N^{-\alpha}$ with $\alpha>1/2$, then
$e^{-N\theta^{2}/2}\ll 1$. Having $e^{-N\theta^{2}/2}\ll 1$ leads, in turn, to
the following limiting expressions:

$\displaystyle a_{0}(\theta,\beta,N)$
$\displaystyle=N\Big{(}1+\frac{N}{2}\,\cos(\beta)^{2}\Big{)},\;\;\;a_{2}(\theta,\beta,N)=\frac{1}{8}N^{2}\chi_{0}^{2}\sin(\beta)^{2},\;\;\;h_{0}=e^{-\frac{1}{8}N\theta^{2}},$

and replacing in Eq. (102) the optimal uncertainty becomes

$\Delta\hat{b}_{\text{opt}}=\sqrt{\frac{\omega_{c}}{T}}\;e^{\frac{1}{8}N\theta^{2}}\;\frac{\left[\chi_{0}^{2}N^{3}/8\left(1+N/2\;\cos(\beta)^{2}\right)\sin(\beta)^{2}\right]^{1/4}}{N/2},$

which grows exponentially with $N$. Thus, we require $\theta(N)$ to scale like
$N^{-\alpha}$ with $\alpha>1/2$. We then expand
$a_{0}(\theta,\beta,N),a_{2}(\theta,\beta,N)$ in this limit, keeping the
angles $N$ dependence implicit for now. We would like to see whether there is
some set of angles $\\{\theta(N),\beta(N)\\}$ that reduces the scaling order
of either of those quantities. Replacing $e^{\pm\theta N^{2}/4}\approx 1$ and
$\sin\left(\frac{\theta}{2}\right)\propto\frac{1}{2}N^{-\alpha}\approx 0$ in
the above expression for $a_{2}(\theta,\beta,N)$, we get

$a_{2}(\theta,\beta,N)\approx\frac{1}{16}N^{2}\text{$\chi_{0}$}^{2}\left[3+\cos(2\beta)+2\sin(\beta)^{2}\right]=\frac{1}{4}N^{2}\chi_{0}^{2},$

which is clearly always positive, making $a_{2}(\theta,\beta,N)\propto N^{2}$
to leading order, regardless of our choice of squeezing and rotation angles.
However, noting that $a_{0}(\theta,\beta,N)$ is the value of $\Delta
J_{y}^{2}(t)$ at $t=0$, we know that for the optimal squeezing and rotation
angles, $a_{0}(\theta_{\text{opt}},\beta_{\text{opt}},N)\propto N^{1/3}$, and
this scaling is best possible666Strictly speaking, $a_{0}(\theta,\beta,N)$ is
the approximation to variance $\Delta J_{y}^{2}(0)$ of Eq. (46) coming from a
cumulant expansion with respect to the qubit operators, see B.1. It can be
checked, however, that it matches the exact formula if $N\gg 1$.. Replacing
$a_{0}(\theta_{\text{opt}},\beta_{\text{opt}},N)\propto
N^{1/3},a_{2}(\beta_{\text{opt}},\theta_{\text{opt}},N)\propto N^{2}$ in Eq.
(102) leads to the $N^{-5/12}$ scaling of the uncertainty. As the short-time
expansion in Eq. (101) reproduces the uncertainty very accurately, we can then
conclude it is also best possible for $\Delta\hat{b}(t)$.

Even-odd noise. Proving optimality of
$\\{\theta_{\text{opt}},\beta_{\text{opt}}\\}$ in the even-odd setting can be
shown analogously. Performing the short-time expansion in this case, we have
the same values for $a_{0}(\theta,\beta,N)$ and $h_{0}(\theta,\beta,N)$ as in
the collective case analyzed above, whereas $a_{2}(\theta,\beta,N)$ in the
$N\gg 1$ limit is given by

$\displaystyle a_{2}(\theta,\beta,N,x)$
$\displaystyle=\frac{N}{16}\Big{\\{}-N\chi_{0}^{2}\left(\cos(2\beta)\left(e^{\theta^{2}N/2}-1\right)-4\sin(2\beta)\sin\left(\theta/2\right)e^{\frac{3}{4}\theta^{2}N/2}\right)$
$\displaystyle-\chi_{0}^{2}\left(e^{\theta^{2}N/2}-2N+3\right)+N\chi_{1}^{2}(x)\left(e^{\theta^{2}N/2}+1\right)\Big{\\}}e^{\theta^{2}(-N/2)}.$

Again, we must require $\theta(N)\propto N^{-\alpha}$ with $\alpha>1/2$ for
the uncertainty not to grow exponentially in the asymptotic $N\gg 1$ regime.
In this limit $e^{\pm\theta N^{2}}\approx 1$ and
$\sin\left(\frac{\theta}{2}\right)\propto\frac{1}{2}N^{-\alpha}\approx 0$,
leading to an $a_{2}(\theta,\beta,N,x)$ that scales like $N^{2}$ regardless of
the $\\{\theta,\beta\\}$ values:

$a_{2}(\theta,\beta,N,x)\approx\frac{N^{2}}{2}(\chi_{0}^{2}+\chi_{1}^{2}(x))\geq
0,\;\;\;\forall x.$

This implies that the optimal choice for rotation and squeezing angles must
once again minimize $a_{0}(\theta,\beta,N)$, and the result follows. For both
of these noise regimes, a numerical optimization of $\Delta\hat{b}(t)$ over
$\theta,\beta$ for fixed $N$ confirmed the analytic results we just derived.

##  Appendix C Additional technical details for randomized coupling dynamics

As stated in the main text, we carry out the calculations for $D=1$, assuming
a spectral density with a Gaussian cutoff, $K(\omega,\omega_{c})\equiv
e^{-\omega^{2}/\omega_{c}^{2}}$ and in the short time regime $\omega_{c}t\ll
1$.

###  C.1 First order cumulants under spatial averaging

The means
$\mathbb{E}\\{\gamma(\vec{r},t)\\},\mathbb{E}\\{\varphi_{0}(\vec{r},t)\\}$ are
easily computed. For $n\neq m$ the spatial average in Eqs. (62)-(63) in the
main text may be evaluated by using Eq. (58), which makes it apparent why it
was useful to express the dynamic coefficients in Eqs. (62)-(63) in terms of
discrete sums. It becomes then possible to express the mean of the dynamic
coefficients as frequency integrals, namely:

$\displaystyle\mathbb{E}\\{\kappa_{nn}(t)\\}$ $\displaystyle\equiv$
$\displaystyle\bar{\kappa}_{0}(t)=\frac{1}{2}\int_{0}^{\infty}J(\omega)\frac{1-\cos\omega
t}{\omega^{2}}d\omega,$ (103)
$\displaystyle\mathbb{E}\\{\kappa_{nm}(\vec{r},t)\\}$ $\displaystyle\equiv$
$\displaystyle\bar{\kappa}_{1}(t)=\frac{1}{2}\int_{0}^{\infty}J(\omega)\frac{1-\cos\omega
t}{\omega^{2}}e^{-\omega^{2}\epsilon^{2}/v^{2}}d\omega,\quad n\neq m,$ (104)
$\displaystyle\mathbb{E}\\{\xi_{nm}(\vec{r},t)\\}$ $\displaystyle\equiv$
$\displaystyle\bar{\xi}(t)=\frac{1}{2}\int_{0}^{\infty}J(\omega)\frac{\omega
t-\sin\omega t}{\omega^{2}}e^{-\omega^{2}\epsilon^{2}/v^{2}}d\omega,\quad
n\neq m.$ (105)

Note that $\bar{\kappa}_{0}(t)$, which corresponds to the $n=m$ case, is
unaffected by the spatial average. The integrals can be given an approximate
analytic expression in the short time regime $1\ll\omega_{c}t$,

$\displaystyle\bar{\kappa}_{0}(t)$
$\displaystyle\approx\frac{1}{8}\Gamma\left(\frac{s+1}{2}\right)\left(\omega_{c}t\right)^{2}\equiv\bar{\kappa}_{0}^{2}\left(\omega_{c}t\right)^{2},$
(106) $\displaystyle\bar{\kappa}_{1}(t)$
$\displaystyle\approx\eta^{s+1}\frac{1}{8}\Gamma\left(\frac{s+1}{2}\right)\left(\omega_{c}t\right)^{2}\equiv\eta^{s+1}\bar{\kappa}_{0}^{2}\left(\omega_{c}t\right)^{2},$
(107) $\displaystyle\bar{\xi}(t)$
$\displaystyle\approx\eta^{s+2}\frac{1}{48}s\;\Gamma\left(\frac{s}{2}\right)(\text{$\omega$}_{c}t)^{3}\equiv\eta^{s+2}\bar{\xi}_{0}^{\,3}\,(\text{$\omega$}_{c}t)^{3}.$
(108)

Therefore, we see that if $\eta\ll 1$, both $\bar{\kappa}_{1}(t)$ and
$\bar{\xi}(t)$ are vanishingly small, whereas the “qubit-local”
$\bar{\kappa}_{0}(t)$ term is unaffected by the procedure, consistent with
intuition.

Introducing the state-dependent angle
$N\cos\left(\theta_{\alpha\beta}\right)\equiv\sum_{i=1}^{N}\alpha_{n}\beta_{n}$,
we can write
$\mathbb{E}\\{\gamma(\vec{r},t)\\},\mathbb{E}\\{\varphi_{0}(\vec{r},t)\\}$ in
terms of $\theta_{\alpha\beta}$ and the $J_{z}$ angular momentum components of
$|\vec{\alpha}\rangle$ and $|\vec{\beta}\rangle$, $m$ and $m^{\prime}$:

$\displaystyle\bar{\gamma}(t)$
$\displaystyle=4\Big{[}N\sin\left(\theta_{\alpha\beta}/2\right)^{2}\bar{\kappa}_{0}(t)+\Big{(}(m-m^{\prime})^{2}-N\sin\left(\theta_{\alpha\beta}/2\right)^{2}\Big{)}\bar{\kappa}_{1}(t)\Big{]},$
$\displaystyle\bar{\varphi}_{0}(t)$ $\displaystyle=4(m^{2}-m^{\prime
2})\bar{\xi}(t).$

###  C.2 Second order cumulants under spatial averaging

The calculation of the phase and decay variances and covariance is more
involved, but follows similar lines. The resulting expressions are still
products of a state-dependent structure, captured by functions of
$m,m^{\prime}$ and the angle $\theta_{\alpha\beta}$ introduced above, and a
time-dependent contribution, represented by coefficients
$F_{\ell}(t),G_{\ell}(t),FG_{2}(t)$, as given below:

$\displaystyle\Delta\bar{\gamma}(t)^{2}$
$\displaystyle=32\Big{\\{}\Big{[}N\sin\left(\theta_{\alpha\beta}/2\right)^{2}(N\sin\left(\theta_{\alpha\beta}/2\right)^{2}-1)\Big{]}F_{1}(t)$
$\displaystyle+2\Big{[}(m-m^{\prime})^{2}(N\sin\left(\theta_{\alpha\beta}/2\right)^{2}-2)-N\sin\left(\theta_{\alpha\beta}/2\right)^{2}(N\sin\left(\theta_{\alpha\beta}\right)^{2}-1)\Big{]}F_{2}(t)\Big{\\}},$
$\displaystyle\Delta\bar{\varphi}_{0}(t)^{2}$
$\displaystyle=4N\Big{\\{}N\sin\left(\theta_{\alpha\beta}\right)^{2}G_{1}(t)$
$\displaystyle+2\Big{[}(m^{2}-2mm^{\prime}\cos\left(\theta_{\alpha\beta}\right)+m^{\prime
2})-N\sin\left(\theta_{\alpha\beta}\right)^{2}\Big{]}G_{2}(t)\Big{\\}},$
$\displaystyle\text{Cov}(\gamma(\vec{r},t),$
$\displaystyle\varphi_{0}(\vec{r},t))=32\Big{[}(m^{2}-m^{\prime
2})\left(N\sin\left(\theta_{\alpha\beta}\right)^{2}-1\right)\Big{]}FG_{2}(t).$
(111)

Here, $F_{\ell}(t),G_{\ell}(t),FG_{2}(t)$, $\ell=\\{1,2\\}$, are the variance
and covariance of the relevant dynamic coefficients:

$\displaystyle F_{1}(t)=$
$\displaystyle\mathbb{E}\\{\chi_{nm}(\vec{r},t)^{2}\\}-\mathbb{E}\\{\chi_{nm}(\vec{r},t)\\}^{2},\quad
F_{2}(t)=\mathbb{E}\\{\chi_{nm}(\vec{r},t)\chi_{np}(\vec{r},t)\\}-\mathbb{E}\\{\chi_{nm}(\vec{r},t)\\}^{2},$
$\displaystyle G_{1}(t)=$
$\displaystyle\mathbb{E}\\{\xi_{nm}(\vec{r},t)^{2}\\}-\mathbb{E}\\{\xi_{nm}(\vec{r},t)\\}^{2},\quad\;G_{2}(t)=\mathbb{E}\\{\xi_{nm}(\vec{r},t)\xi_{np}(\vec{r},t)\\}-\mathbb{E}\\{\xi_{nm}(\vec{r},t)\\}^{2},$
$\displaystyle FG_{2}(t)=$
$\displaystyle\mathbb{E}\\{\chi_{nm}(\vec{r},t)\xi_{np}(\vec{r},t)\\}-\mathbb{E}\\{\chi_{nm}(\vec{r},t)\\}\;\mathbb{E}\\{\xi_{np}(\vec{r},t)\\},$

with $n\neq m\neq p$. For $D=1$, in particular, they can be written as double
integrals in the frequency domain:

$\displaystyle F_{\ell}(t)$
$\displaystyle=\frac{1}{4}\int_{0}^{\infty}\\!\\!\int_{0}^{\infty}\\!\\!J(\omega)J(\omega^{\prime})e^{-(\omega^{2}+\omega^{\prime
2})\epsilon^{2}}f(\omega,t)f(\omega^{\prime},t)h(\omega,\omega^{\prime}),$
(112) $\displaystyle G_{\ell}(t)$
$\displaystyle=\frac{1}{4}\int_{0}^{\infty}\\!\\!\int_{0}^{\infty}\\!\\!J(\omega)J(\omega^{\prime})e^{-(\omega^{2}+\omega^{\prime
2})\epsilon^{2}}g(\omega,t)g(\omega^{\prime},t)h(\omega,\omega^{\prime}),$
(113) $\displaystyle FG_{2}(t)$
$\displaystyle=\frac{1}{4}\int_{0}^{\infty}\\!\\!\int_{0}^{\infty}\\!\\!J(\omega)J(\omega^{\prime})e^{-(\omega^{2}+\omega^{\prime
2})\epsilon^{2}}f(\omega,t)g(\omega^{\prime},t)[\cosh(\omega\omega^{\prime}\epsilon^{2}/v^{2})-1]\,d\omega
d\omega^{\prime},$ (114)

where $\ell\in\\{1,2\\}$ and the functions $f(\omega,t),g(\omega,t)$, and
$h(\omega,\omega^{\prime})$ are given by

$f(\omega,t)={[1-\cos(\omega t)]}/{\omega^{2}},\quad g(\omega,t)=[{\omega
t-\sin(\omega t)]}/{\omega^{2}},\quad
h(\omega,\omega^{\prime})=\cosh[(3-\ell)\omega\omega^{\prime}\epsilon^{2}/v^{2}]-1.$

In the short-time regime, we have $f(\omega,t)\approx\frac{1}{2}t^{2}$,
$g(\omega,t)\approx\frac{1}{6}\omega t^{3}$. Introducing these approximations
into Eqs. (112)-(114) the dynamic coefficients variances and covariance can
also be evaluated, resulting in the following expressions:

$\displaystyle F_{1}(t)$ $\displaystyle\approx\eta\;\pi^{3/2}\frac{\sec(\pi
s)}{2^{s+\frac{13}{2}}\Gamma\left(\frac{1}{2}-s\right)}\left(\omega_{c}t\right)^{4},$
(115) $\displaystyle G_{1}(t)$
$\displaystyle\approx-\eta\;\pi^{3/2}\frac{1}{36}\frac{\sec(\pi
s)}{\;2^{s+\frac{11}{2}}\Gamma\left(-s-\frac{1}{2}\right)}\left(\omega_{c}t\right)^{6},$
(116) $\displaystyle F_{2}(t)$
$\displaystyle\approx\eta^{2(s+1)}\;\frac{1}{64}\Gamma\left(\frac{s+1}{2}\right)^{2}\left[\,{}_{2}F_{1}\left(\frac{s+1}{2},\frac{s+1}{2};\frac{1}{2};\frac{1}{4}\right)-1\right]\left(\omega_{c}t\right)^{4},$
(117) $\displaystyle G_{2}(t)$
$\displaystyle\approx\eta^{2(s+2)}\;\frac{1}{576}\Gamma\left(\frac{s}{2}+1\right)^{2}\left[\,{}_{2}F_{1}\left(\frac{s+2}{2},\frac{s+2}{2};\frac{1}{2};\frac{1}{4}\right)-1\right]\left(\omega_{c}t\right)^{6},$
(118) $\displaystyle FG_{2}(t)$
$\displaystyle\approx\eta^{2s+3}\;\frac{1}{192}\;\Gamma\left(\frac{s}{2}+1\right)\Gamma\left(\frac{s+1}{2}\right)\left[2^{s}\left(1+3^{-(s+1)}\right)-1\right]\left(\omega_{c}t\right)^{5},$
(119)

where ${}_{2}F_{1}(a;b;z)$ denotes the confluent hypergeometric function of
the second kind.

We now outline the explicit calculations to obtain Eqs. (C.2)-(111). Consider
$\Delta\bar{\gamma}(t)^{2}=\mathbb{E}\\{\gamma(\vec{r},t)^{2}\\}-\mathbb{E}\\{\gamma(\vec{r},t)\\}^{2}$.
The position-dependent decay rate $\gamma(\vec{r},t)$ in Eq. (22) has dynamic
coefficients $\kappa_{nm}(\vec{r},t)$ given by Eq. (62). The desired variance
is then:

$\displaystyle\Delta\bar{\gamma}(t)^{2}=\\!\\!\\!\\!\sum_{n,m,n^{\prime},m^{\prime}=1}^{N}\\!\\!\\!\\!(\alpha_{n}-\beta_{n})(\alpha_{m}-\beta_{m})(\alpha_{n^{\prime}}-\beta_{n^{\prime}})\,\text{Cov}\left(\kappa_{nm}(\vec{r},t)\,\kappa_{n^{\prime}m^{\prime},}(\vec{r},t)\right)$
(120)

where we recall that the covariance of arbitrary functions $f(\vec{r},t)$ and
$g(\vec{r},t)$ is given by
$\text{Cov}\left(f(\vec{r},t)\,g(\vec{r},t)\right)=\mathbb{E}\\{f(\vec{r},t)\;g(\vec{r},t)\\}-\mathbb{E}\\{f(\vec{r},t)\\}\mathbb{E}\\{g(\vec{r},t)\\}$.
Using Eq. (62) can then write:

$\displaystyle\text{Cov}\left(\kappa_{nm}(\vec{r},t)\,\kappa_{n^{\prime}m^{\prime}}(\vec{r},t)\right)=\frac{1}{16}\sum_{\vec{k},\vec{k^{\prime}}}|g_{k}|^{2}|g_{k^{\prime}}|^{2}\frac{1-\cos\Omega_{k}t}{\Omega_{k}^{2}}\coth\left(\beta\,\Omega_{k}/2\right)$
$\displaystyle\qquad\frac{1-\cos\Omega_{k^{\prime}}t}{\Omega_{k^{\prime}}^{2}}\coth\left(\beta\,\Omega_{k^{\prime}}/2\right)\,\text{Cov}\left(\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\cos(\vec{k}\cdot\left(\vec{r}_{n^{\prime}}-\vec{r}_{m^{\prime}})\right)\right)$
(121)

The spatial dependence is then fully captured by
$\text{Cov}\left(\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\cos(\vec{k}\cdot\left(\vec{r}_{n^{\prime}}-\vec{r}_{m^{\prime}})\right)\right)$.
It is straightforward to see that this quantity vanishes unless the indexes
obey:

  1. 1.

$\\{n=n^{\prime},m=m^{\prime}\\}$ or $\\{n=m^{\prime},m=n^{\prime}\\}$, with
$n\neq m$. Let us call this subset of the indexes $\mathcal{S}_{1}$. We then
have for the first case:

$\text{Cov}\left(\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\right)=e^{-\epsilon^{2}(k^{2}+k^{\prime
2})}\prod_{i=1}^{D}\cosh(2\epsilon^{2}k_{i}k_{i}^{\prime})-1],$

with the $\\{n=m^{\prime},m=n^{\prime}\\}$ case yielding the same result by
symmetry.

  2. 2.

Three different indexes out of the four ones, with the matching pair being
either $\\{n=n^{\prime}\\}$, $\\{n=m^{\prime}\\}$, $\\{m=n^{\prime}\\}$ or
$\\{m=m^{\prime}\\}$. We call this subset of the indexes $\mathcal{S}_{2}$.
Let us consider the case where $\\{n=n^{\prime}\\}$:

$\text{Cov}\left(\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m^{\prime}})\right)\right)=e^{-\epsilon^{2}(k^{2}+k^{\prime
2})}\prod_{i=1}^{D}\cosh(\epsilon^{2}k_{i}k_{i}^{\prime})-1],$

with the other three cases yielding the same result by symmetry.

Replacing the spatial average in Eq. (121), we obtain an expression for the
second order time-dependent coefficients,

$\displaystyle F_{\ell}(t)=$
$\displaystyle\frac{1}{16}\sum_{\vec{k},\vec{k^{\prime}}}|g_{k}|^{2}|g_{k^{\prime}}|^{2}\frac{1-\cos\Omega_{k}t}{\Omega_{k}^{2}}\coth\left(\beta\Omega_{k}/2\right)\frac{1-\cos\Omega_{k^{\prime}}t}{\Omega_{k^{\prime}}^{2}}\coth\left(\beta\Omega_{k^{\prime}}/2\right)$
(122) $\displaystyle\qquad e^{-\epsilon^{2}(k^{2}+k^{\prime
2})}\prod_{i=1}^{D}\cosh(\epsilon^{2}(3-\ell)k_{i}k_{i}^{\prime})-1],\quad\ell\in\\{1,2\\},$

with $\ell=1$ when the qubit pairs have indexes in $\mathcal{S}_{1}$, and
$\ell=2$ when they belong to $\mathcal{S}_{2}$. Note that if we restrict
ourselves to the isotropic $D=1$ case, as we do here, we can exploit the
linear dispersion relationship $\Omega_{k}=kv$ to write $F_{\ell}(t)$ in terms
of a double frequency integral, Eq. (112). Substituting in Eq. (120), this
leads to the following expression for the decay variance:

$\displaystyle\Delta\bar{\gamma}(t)^{2}$
$\displaystyle=F_{1}(t)\Big{(}\sum_{\mathcal{S}_{1}}\mathcal{Q}(\alpha,\beta)\Big{)}+F_{2}(t)\Big{(}\sum_{\mathcal{S}_{2}}\mathcal{Q}(\alpha,\beta)\Big{)},$

where
$\mathcal{Q}(\alpha,\beta)=(\alpha_{n}-\beta_{n})(\alpha_{m}-\beta_{m})(\alpha_{n^{\prime}}-\beta_{n^{\prime}})(\alpha_{m^{\prime}}-\beta_{m^{\prime}})$
gives the state-dependent numerical factor. One may show that the sums over
the restricted index subsets $\mathcal{S}_{1},\mathcal{S}_{2}$ can be written
in terms of $m,m^{\prime}$ and $\theta_{\alpha\beta}$:

$\displaystyle\sum_{\mathcal{S}_{1}}\mathcal{Q}(\alpha,\beta)=32\left[N\sin\left(\theta_{\alpha\beta}/2\right)^{2}(N\sin\left(\theta_{\alpha\beta}/2)^{2}-1\right)\right],$
$\displaystyle\sum_{\mathcal{S}_{2}}\mathcal{Q}(\alpha,\beta)=64\left[(m-m^{\prime})^{2}(N\sin\left(\theta_{\alpha\beta}/2\right)^{2}-2)-N\sin\left(\theta_{\alpha\beta}/2\right)^{2}(N\sin\left(\theta_{\alpha\beta}/2\right)^{2}-1)\right],$

leading directly to Eq. (C.2).

The phase variance $\Delta\bar{\varphi}_{0}(t)^{2}$ and covariance
$\text{Cov}(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))$ can be evaluated
analogously by setting up the equivalent of Eq. (120). The phase
$\varphi_{0}(\vec{r},t)$ is given by Eq. (23), with the dynamic coefficient
$\xi_{nm}(\vec{r},t)$ of Eq. (63). It is then clear that the spatial
dependence of the ensuing covariance
$\text{Cov}\left(\xi_{nm}(\vec{r},t),\,\xi_{n^{\prime}m^{\prime}}(\vec{r},t))\right)$
and
$\text{Cov}\left(\kappa_{nm}(\vec{r},t),\,\xi_{n^{\prime}m^{\prime}}(\vec{r},t)\right)$
is also governed by
$\text{Cov}\\!\left(\cos(\vec{k}\cdot\left(\vec{r}_{n}-\vec{r}_{m})\right)\cos(\vec{k}\cdot\left(\vec{r}_{n^{\prime}}-\vec{r}_{m^{\prime}})\right)\right)$.
Thus, the average is non-vanishing for the same restricted index subsets
$\mathcal{S}_{1},\mathcal{S}_{2}$ we found for $\Delta\bar{\gamma}(t)^{2}$.
The second-order time-dependent coefficients entering
$\Delta\bar{\varphi}_{0}(t)^{2}$ and
$\text{Cov}(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))$ are then:

$\displaystyle G_{\ell}(t)=$
$\displaystyle\frac{1}{16}\sum_{\vec{k},\vec{k^{\prime}}}|g_{k}|^{2}|g_{k^{\prime}}|^{2}\,\frac{\Omega_{k}t-\sin\Omega_{k}t}{\Omega_{k}^{2}}\,\frac{\Omega_{k^{\prime}}t-\sin\Omega_{k^{\prime}}t}{\Omega_{k^{\prime}}^{2}}\,e^{-\epsilon^{2}(k^{2}+k^{\prime
2})}\\!\prod_{i=1}^{D}\\!\cosh(\epsilon^{2}(3-\ell)k_{i}k_{i}^{\prime})-1],$
$\displaystyle FG_{\ell}(t)=$
$\displaystyle\frac{1}{16}\sum_{\vec{k},\vec{k^{\prime}}}|g_{k}|^{2}|g_{k^{\prime}}|^{2}\frac{1-\cos\Omega_{k}t}{\Omega_{k}^{2}}\coth\left(\beta\Omega_{k}/2\right)$
$\displaystyle\qquad\frac{\Omega_{k^{\prime}}t-\sin\Omega_{k^{\prime}}t}{\Omega_{k^{\prime}}^{2}}e^{-\epsilon^{2}(k^{2}+k^{\prime
2})}\prod_{i=1}^{D}\\!\cosh(\epsilon^{2}(3-\ell)k_{i}k_{i}^{\prime})-1],$

for $\ell=1,2$. When $D=1$ these expressions are isotropic and can be cast
into Eqs. (113)-(114). All that is left to do is to write the resulting state-
dependent structures in the summations of $\Delta\bar{\varphi}_{0}(t)^{2}$ and
Cov$(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))$ in terms of $m,m^{\prime}$ and
$\theta_{\alpha\beta}$. Let us define
$\mathcal{G}(\alpha,\beta)\equiv(\alpha_{n}\alpha_{m}-\beta_{n}\beta_{m})(\alpha_{n^{\prime}}\alpha_{m^{\prime}}-\beta_{n^{\prime}}\beta_{m^{\prime}})$
and
$\mathcal{J}(\alpha,\beta)\equiv(\alpha_{n}-\beta_{n})(\alpha_{m}-\beta_{m})(\alpha_{n^{\prime}}\alpha_{m^{\prime}}-\beta_{n^{\prime}}\beta_{m^{\prime}})$.
Direct calculation shows that

$\displaystyle\sum_{\mathcal{S}_{1}}\mathcal{G}(\alpha,\beta)$
$\displaystyle=4N^{2}\sin\left(\theta_{\alpha\beta}\right)^{2},\qquad\sum_{\mathcal{S}_{1}}\mathcal{J}(\alpha,\beta)=0,$
$\displaystyle\sum_{\mathcal{S}_{2}}\mathcal{G}(\alpha,\beta)$
$\displaystyle=8\left[2N\left(m^{2}-2mm^{\prime}\cos\left(\theta_{\alpha\beta}\right)+m^{\prime
2}\right)-N^{2}\sin\left(\theta_{\alpha\beta}\right)^{2}\right],$
$\displaystyle\sum_{\mathcal{S}_{2}}\mathcal{J}(\alpha,\beta)$
$\displaystyle=32\left[(m^{2}-m^{\prime
2})\left(N\sin\left(\theta_{\alpha\beta}/2\right)-1\right)\right],$

which leads to the expressions for $\Delta\bar{\varphi}_{0}(t)^{2}$ and
Cov$(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))$ given by Eqs. (C.2)-(111).

###  C.3 Proof of claim 6.2.1

Claim. Assume that the short-time-large-dispersion limit $\eta\ll 1$ is
obeyed. Then:

(i) First-order cumulants provide a good approximation to the full spatial
average provided that the additional condition
$\eta\left(\omega_{c}t\right)^{2}N\ll 1$ is met.

(ii) _In the regime of validity of (i), all terms with exponents proportional
to powers of $\eta$ can be made arbitrarily close to unity provided that the
condition $\eta^{s+1}(\omega_{c}t)^{2}N^{2}\ll 1$ is met._

Proof. Let us start from the second-order cumulant expansion of a reduced
matrix element,

$\langle\vec{\alpha}|\rho(t)|\vec{\beta}\rangle\approx\langle\vec{\alpha}|\rho_{0}|\vec{\beta}\rangle\,e^{-\bar{\gamma}(t)+\frac{1}{2}\left[\Delta\bar{\gamma}^{2}(t)-\Delta\bar{\varphi}_{0}^{2}(t)\right]}\,e^{i\bar{\varphi}_{0}(t)+i\text{Cov}(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))}.$

To establish part (i), we need to prove that for all
$|\vec{\alpha}\rangle,|\vec{\beta}\rangle$ with non trivial phase and decay,

$\displaystyle|\bar{\varphi}_{0}(t)|\gg|\text{Cov}(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))|,\quad|\bar{\gamma}(t)|\gg\frac{1}{2}|\Delta\bar{\gamma}^{2}(t)-\Delta\bar{\varphi}_{0}^{2}(t)|.$
(123)

To see where the first inequality is satisfied, we write down the expressions
for $\bar{\varphi}_{0}(\vec{r},t)$,
$\text{Cov}(\gamma(\vec{r},t),\varphi_{0}(\vec{r},t))$:

$\displaystyle\left|(m^{2}-m^{\prime 2})\xi(t)\right|$ $\displaystyle\gg
8\left|(m^{2}-m^{\prime
2})\left(N\sin\left(\theta_{\alpha\beta}\right)^{2}-1\right)FG_{2}(t)\right|\Leftrightarrow$
$\displaystyle\mathcal{O}\Big{(}\eta^{s+2}\;(\text{$\omega$}_{c}t)^{3}\Big{)}$
$\displaystyle\gg
8\left|\,N\sin(\theta_{\alpha\beta}/2)^{2}-1\,\right|\,\mathcal{O}\Big{(}\eta^{2s+3}\;(\omega_{c}t)^{5}\Big{)}.$

In order for this to hold, it suffices that $\eta^{s+1}\;(\omega_{c}t)^{2}N\ll
1$. Proving the second inequality in Eq. (123) proceeds along similar lines.
If $\theta_{\alpha\beta}\neq 0$, the $\eta$-independent $\bar{\kappa}_{0}(t)$
term in $\bar{\gamma}(t)$, proportional to $\sin(\theta_{\alpha\beta}/2)$,
dominates as long as $\eta\left(\omega_{c}t\right)^{2}N\ll 1$ is satisfied.
For the $\theta_{\alpha\beta}=0$ case, the second term in $\bar{\gamma}(t)$
dominates whenever
$\eta^{s+1}(\omega_{c}t)^{2}[1+\eta^{2}(\omega_{c}t)^{2}N]\ll 1$, which is
already fulfilled provided the condition for the $\theta_{\alpha\beta}\neq 0$
case is met. Note how this is also true for the first inequality, and hence i
is satisfied when $\eta\left(\omega_{c}t\right)^{2}N\ll 1$.

To establish part (ii), note that when the exponents containing $\eta$ factors
are small enough, it is valid to take a first-order Taylor expansion. We can
then write $e^{i\bar{\varphi}_{0}(t)}\approx 1-4(m^{2}-m^{\prime
2})\eta^{s+2}\bar{\xi}_{0}^{\,3}(\omega_{c}t)^{3}$, and similarly
$e^{-\eta^{s+1}\bar{\kappa}_{0}^{2}(\omega_{c}t)^{2}(m-m^{\prime})^{2}}\approx
1-\eta^{s+1}\bar{\kappa}_{0}^{2}(\omega_{c}t)^{2}(m-m^{\prime})^{2}$. Upper-
bounding $(m^{2}-m^{\prime 2})$ by $N^{2}$ and $(m-m^{\prime})$ by $N$ then
yields the desired result. $\blacksquare$

##  Appendix D Non-collective Gaussian dephasing models with
$\bm{\varphi_{1}(t)\neq 0$}$

As we established in Sec. 4, the spin-boson model we focused on (which fully
accounted for the angular dependence in the inner product
$\vec{k}\cdot\vec{r}_{nm}$) has vanishing coefficients $\vartheta_{nm}(t)=0$.
As a consequence, the phase $\varphi_{1}(t)$ reflecting non-commutativity of
the set $\\{U_{\vec{\alpha}}(t)\\}$ is identically zero. However, within the
broader class of ZMGSD noise models, there may be relevant instances in which
$\varphi_{1}(t)$ plays a non-trivial role. It is then reasonable to ask how
the presence of this second phase would affect the conclusions we have drawn
for the even-odd and RC settings.

Even-odd noise. When qubits $n$ and $m$ belong to the same (s) cluster, they
interact with the bath through a single noise operator, leading to a vanishing
dynamic coefficient $\vartheta_{s}(t)$; hence, $\Psi_{s}(t)$ remains
unaltered. However, the phase coefficient corresponding to qubits $n$ and $m$
on different (d) clusters picks up an extra contribution: If the first qubit
of the pair is even, and the second one is odd, then we have
$\Psi_{\text{eo}}(t)=4(\xi_{d}(t)+\vartheta_{d}(t))$, with $\vartheta_{d}(t)$
given by Eq. (27) with $B_{n}(s)\mapsto B_{\text{e}}(s)$ and
$B_{m}(s^{\prime})\mapsto B_{\text{o}}(s^{\prime})$. If the opposite is true,
we have $\Psi_{\text{oe}}(t)=4(\xi_{d}(t)-\vartheta_{d}(t))$ instead. Invoking
Eq. (33), we see that due to the presence of $\vartheta_{d}(t)$, the phase
coefficient short-time behavior is now (to leading order) quadratic rather
than cubic in time: $\Psi_{\text{oe}}(t)\approx-\Psi_{\text{oe}}(t)\approx
4\vartheta_{d}^{2}(x)(\omega_{c}t)^{4}$. Expressions for deriving the scaling
of both CSS and OATS states can be handled similarly as in the text. In both
cases, the resulting $N$-scaling of optimal measurement time and estimation
precision are not affected. However, quantum corrections
$Q^{\,\text{eo}}_{\text{CSS}}(x,t)$ and $Q^{\,\text{eo}}_{\text{OATS}}(x,t)$
do differ. While for an OATS their contribution is still vanishing at
$\tau_{\text{opt, eo}}^{\text{OATS}}$ in the $N\gg 1$ limit, for the CSS we
have
$Q^{\,\text{eo}}_{\text{CSS}}(x,t)=\frac{1}{128}N^{2}\vartheta^{4}_{d}(x)(\omega_{c}t)^{4}$,
which at $\tau_{\text{opt,eo}}^{\text{CSS}}$ is constant with respect to $N$.
However, as long as this constant correction is sufficiently small,
$\frac{1}{128}N^{2}\vartheta^{4}_{d}(x)(\omega_{c}\tau_{\text{opt,eo}}^{\text{CSS}})^{4}\ll
1$, the $N^{-1/4}$ scaling of the uncertainty [Eq. (55) in the main text]
remains accurate.

RC noise. As the RC model (as we formulated it) hinges specifically upon
having a spin-boson dephasing interaction, in order to have non-zero
$\varphi_{1}(t)$ we must alter the hypotheses of Sec. 4. Specifically, let us
approximate $\vec{k}\cdot(\vec{r}_{n}-\vec{r}_{m})\approx\Omega_{k}t_{nm}$
with $\Omega_{k}=vk$ and $t_{nm}=|\vec{r}_{n}-\vec{r}_{m}|/v$, disregarding
angular dependence in the inner product. This leads to the following
expressions for the dynamic coefficients:

$\displaystyle\chi_{nm}(t)$
$\displaystyle=\sum_{\vec{k}}|g_{k}|^{2}\frac{1-\cos\Omega_{k}t}{\Omega_{k}^{2}}\coth\Big{(}\frac{\beta\Omega_{k}}{2}\Big{)}\cos\left(\Omega_{k}t_{nm}\right),$
(124) $\displaystyle\xi_{nm}(t)$
$\displaystyle=\sum_{\vec{k}}|g_{k}|^{2}\frac{\Omega_{k}t-\sin\Omega_{k}t}{\Omega_{k}^{2}}\cos\left(\Omega_{k}t_{nm}\right),$
(125) $\displaystyle\vartheta_{nm}(t)$
$\displaystyle=\frac{1}{4}\sum_{\vec{k}}|g_{k}|^{2}\frac{1-\cos\Omega_{k}t}{\Omega_{k}^{2}}\sin\left(\Omega_{k}t_{nm}\right).$
(126)

Note that, as the cosine is an even function, Eqs. (124)-(125) match their
Sec. 4 counterparts, Eqs. (62)-(63). However, in this case the argument of the
sine in Eq. (126) is always positive, which does not allow us to exploit the
sines’ odd parity to cancel the contributions corresponding to $\vec{k}$ and
$-\vec{k}$. Thus, we obtain a non-vanishing $\vartheta_{nm}(t)$ and
$\varphi_{1}(\vec{r},t)\neq 0$. As a result of the non trivial second phase,
the reduced matrix elements spatial cumulant expansion, Eq. (65), is modified.
Let us focus on the $D=1$ case. It is straightforward to check that
$\mathbb{E}\\{\vartheta_{nm}(t)\\}$ vanishes, and so
$\mathbb{E}\\{\varphi_{1}(\vec{r},t)\\}=0$. In the same way,
$\text{Cov}(\varphi_{1}(t),\gamma(t))$ and
$\text{Cov}(\varphi_{0}(t),\varphi_{1}(t))$ are vanishing identically due to
the spatial average being zero. There is, however, an additional contribution
to the second-order cumulant coming from the variance of
$\varphi_{1}(\vec{r},t)$. With that, the expression becomes

$\displaystyle\mathbb{E}\left\\{e^{-\gamma(\vec{r},t)}e^{i\varphi_{0}(\vec{r},t)}e^{i\varphi_{1}(\vec{r},t)}\right\\}\approx
e^{-\bar{\gamma}(t)+i\bar{\varphi}_{0}(t)+\frac{1}{2}(\Delta\bar{\gamma}(t)^{2}-\Delta\bar{\varphi}_{0}(t)^{2}-\Delta\bar{\varphi}_{1}(t)^{2}+2i\text{Cov}(\varphi_{0}(t),\gamma(t))},$

where we may write

$\Delta\bar{\varphi}_{1}(t)^{2}=\frac{N^{2}}{4}\sin(\theta_{\alpha\beta})^{2}H_{1}(t)+\bigg{[}{N}(m^{2}-2mm^{\prime}\cos(\theta_{\alpha\beta})+m^{\prime
2})-\frac{N^{2}}{2}\sin(\theta_{\alpha\beta})^{2}\bigg{]}H_{2}(t).$

Here, the time-dependent quantities $H_{1}(t),H_{2}(t)$ can be evaluated in
the short-time limit, leading to $H_{1}(t)\approx
K_{1}\,\eta\,(\omega_{c}t)^{4}$ and $H_{2}(t)\approx
K_{2}\,\eta^{2(s+1)}(\omega_{c}t)^{4}$, with $K_{1},K_{2}$ dimensionless
constants. It is then straightforward to show that Claim 6.2.1 still holds in
the presence of $\Delta\bar{\varphi}_{1}(t)^{2}$, leaving the conclusions
derived in the small-$\eta$ regime – in particular, the $N$-scaling of OATS
and GHZ states – unchanged.

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