\section{Supplemental Material}
\subsection{Per-Photon (Quantum) Chernoff Exponents}
For a binary hypothesis test between two states $\eta_1$ and $\eta_2$ on $\mathcal{H}$, the QCE is defined as \cite{Audenaert2007,Nussbaum2009}
\begin{equation}
	\xi_{\rm Q}=-\log\left[\min_{0\leq s \leq 1} \Tr\big(\eta_1^s\eta_2^{1-s}\big)\right].
	\label{eq:QCE_eta}
\end{equation}
Expanding the states into their Fock-space components, i.e., $\eta_j=(1-\epsilon)\outerproduct{0}{0}+\epsilon\rho_j+O\big(\epsilon^2\big)$, we have
\begin{equation}
	\eta_j^s = (1-\epsilon)^s\outerproduct{0}{0}+\epsilon^s\rho_j^s+O\big(\epsilon^{2s}\big),
	\label{eq:eta_j^s}
\end{equation}
where each term can be individually exponentiated due to the orthogonality of the photon-number subspaces in the Fock decomposition and where $\outerproduct{0}{0}^s=\outerproduct{0}{0}$. Thus,
\begin{equation}
	\begin{aligned}
		\Tr\big(\eta_1^s\eta_2^{1-s}\big)&=\Tr\Big\{\big[(1-\epsilon)^s\outerproduct{0}{0}+\epsilon^s\rho_1^s+O\big(\epsilon^{2s}\big)\big]\\
		&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\times\big[(1-\epsilon)^{1-s}\outerproduct{0}{0}+\epsilon^{1-s}\rho_2^{(1-s)}+O\big(\epsilon^{2(1-s)}\big)\big]\Big\}\\
		&=1-\epsilon+\epsilon\Tr\big(\rho_1^s\rho_2^{1-s}\big)+\Tr\big[O\big(\epsilon^2\big)\big],
	\end{aligned}
	\label{eq:Treta1seta21-s}
\end{equation}
where orthogonality of the photon-number subspaces is used for the second equality (i.e., $\mel{0}{\rho_j}{0}=0$). Including the minimization over $s$ and using the definitions of $\xi_{\rm Q}$ and $\xi_{\rm Q}^{(1)}$ in Eq.~\eqref{eq:QuantumChernoff} from the main text and Eq.~\eqref{eq:QCE_eta}, we rearrange terms to find
\begin{equation}
	1-e^{-\xi_{\rm Q}}=\epsilon\left(1-e^{-\xi^{(1)}_{\rm Q}}\right)+O\big(\epsilon^2\big).
	\label{eq:xiQ_exponential_relation}
\end{equation}
With deeply sub-Rayleigh objects, the regime of focus for our results, the dominating effect of diffraction causes the collected optical field from each of the two objects to be nearly identical. Thus, the states $\eta_1$ and $\eta_2$ converge to one another such that $\xi_{\rm Q}\approx 0$; similarly, $\rho_1$ and $\rho_2$ are nearly identical such that $\xi_{\rm Q}^{(1)}\approx 0$. In this case, under the weak-source approximation with $\epsilon\ll1$ we find the result $\xi_{\rm Q}\approx\epsilon\xi_{\rm Q}^{(1)}$.

For the same hypothesis test, the definition of the CE for a POVM $\{\Pi_z\}_{\mathcal{Z}}$ on $\mathcal{H}$ is~\cite{VanTrees2013}
\begin{equation}
	\xi_{\rm Meas}=-\log\left[\min_{0\leq s \leq 1} \sum_{z\in\mathcal{Z}}P(z\vert\eta_1)^s P(z\vert\eta_2)^{1-s}\right],
	\label{eq:CE_eta}
\end{equation}
where $P(z\vert\eta_j)=\Tr(\Pi_z\eta_j)$. For a measurement that performs photon counts in each individual temporal mode, the single-temporal-mode measurement outcome space $\mathcal{Z}$ can be decomposed into a countably infinite number of subspaces $\mathcal{Z}^{(n)}$, $n\in[0,\infty]$, each corresponding to the detection of $n$ photons in the given temporal mode. The CE therefore becomes 
\begin{equation}
	\xi_{\rm Meas}=-\log\left[\min_{0\leq s \leq 1} \sum_{n=0}^{\infty}\sum_{z\in\mathcal{Z}^{(n)}}P(z\vert\eta_1)^s P(z\vert\eta_2)^{1-s}\right].
	\label{eq:CE_eta_decomposed}
\end{equation}
Under the weak-source approximation, when multi-photon detection events can be neglected, the CE expression retains only the first two terms in the outermost sum corresponding to the $\mathcal{Z}^{(0)}$ and $\mathcal{Z}^{(1)}$ outcome subspaces. Clearly, the detection of zero photons can be considered as a single outcome, which will occur with probability $1-\epsilon$ under both hypotheses. The detection of one photon will occur within the set of outcomes $\mathcal{Z}^{(1)}$ and will depend only on the single-photon state $\rho_j$ and on the component of the POVM contained within the single-photon subspace of $\mathcal{H}$ subspace. Defining this truncated POVM on $\mathcal{H}^{(1)}$ as $\{\Pi_z^{(1)}\}_{\mathcal{Z}^{(1)}}$, the measurement outcome probabilities conditioned on detection of a single photon become $P(z\vert\rho_j)=\Tr(\Pi^{(1)}_z\rho_j)$. Since the probability of single photon detection within each temporal mode is $\epsilon$, for any $z\in\mathcal{Z}^{(1)}$ we have $P(z\vert\eta_j)=\epsilon P(z\vert{\rho_j})$, and thus the full CE becomes
\begin{equation}
	\begin{aligned}
		\xi_{\rm Meas}\approx&-\log\Bigg[1-\epsilon \\
		&+ \epsilon\min_{0\leq s \leq 1} \sum_{z\in\mathcal{Z}^{(1)}}P(z\vert\rho_1)^s P(z\vert\rho_2)^{1-s}\Bigg].
	\end{aligned}
	\label{eq:CE_eta_0and1}
\end{equation}
Using Eq.~\eqref{eq:Chernoff} from the main text and Eq.~\eqref{eq:CE_eta}, we find
\begin{equation}
	1-e^{-\xi_{\rm Meas}}\approx\epsilon\left(1-e^{-\xi^{(1)}_{\rm Meas}}\right).
	\label{eq:xiMeas_exponential_relation}
\end{equation}
Again, for sub-Rayleigh objects the measurement outcome probabilities will be nearly identical for the two hypotheses so that $\xi_{\rm Meas}\approx0$ and $\xi_{\rm Meas}^{(1)}\approx0$, and therefore $\xi_{\rm Meas}\approx\epsilon\xi_{\rm Meas}^{(1)}$.

\subsection{Point Source vs Arbitrary Object Discrimination}
Let the candidate objects be a point source at position $\vec{x}_{1, \rm obj}=\vec{x}_1/\mu$, described by a pure state $\rho_1=\ket{\psi_{\vec{x}_1}}\bra{\psi_{\vec{x}_1}}$, and a second object $m_2(\vec{x}_{\rm obj})$ with state $\rho_2$~(Eq.~\eqref{eq:rhoj} from the main text). For pure-vs-mixed state hypothesis tests, the projective measurement with POVM elements $\Pi_0=\rho_1$ and $\Pi_1= \mathcal{I}-\rho_1$ is known to be quantum-optimal, and the QCE takes the simplified form  $\xi_{\textrm{Q}}^{(1)}=-\log[\Tr(\rho_1\rho_2)]$~\cite{Kargin2005}. In the incoherent imaging context, this measurement can be performed with a 2D BSPADE aligned to the point source location $\vec{x}_{1,\rm obj}$ \cite{Tsang2016a,Boucher2020,Ang2017}, which implements the POVM with elements $\Pi_0=\outerproduct{\psi_{\vec{x}_1}}{\psi_{\vec{x}_1}}$ and $\Pi_1=\mathcal{I}-\outerproduct{\psi_{\vec{x}_1}}{\psi_{\vec{x}_1}}$. Assuming a shift invariant imaging system and paraxial optics~\cite{Goodman2005}, the distinguishability of the two objects will not be changed by a global coordinate shift such that $\rho_1=\outerproduct{\psi_{\vec{\Omega}}}{\psi_{\vec{\Omega}}}$. The shifted second object is then $m_2(\vec{x}_{\rm obj}-\vec{x}_{1, \rm obj})$. Using the shifted coordinates, the QCE canbe written exactly in the form of Eq.~\eqref{eq:quantum_Chernoff_exact} from the main text for any arbitrary object model $m_2(\vec{x}_{\rm obj})$. 

One special case is discrimination between a single point source, modeled by the state $\rho_1=\outerproduct{\psi_{\vec{\Omega}}}{\psi_{\vec{\Omega}}}$, and two point sources, modeled by $\rho_2=(1/2)(\outerproduct{\psi_{\{-d/2,0\}}}{\psi_{\{-d/2,0\}}}+\outerproduct{\psi_{\{d/2,0\}}}{\psi_{\{d/2,0\}}})$, where $d$ is the spatial separation between the two point sources along the $x$ axis. Using a 2D Gaussian PSF $\psi(\vec{x})=(2\pi \sigma^2)^{-1/2}\exp(-(x^2+y^2)/4\sigma^2)$, our result in Eq.~\eqref{eq:quantum_Chernoff_exact} with $\mu=1$ and $\sigma=1$ reproduces the reported QCE of $\xi_{\rm Q}^{(1)}=d^2/16$~\cite{Lu2018}, which is equaled by the CE of the quantum-optimal 2D BSPADE measurement. For comparison, typical receivers for incoherent imaging use the direct imaging measurement (Fig.~\ref{fig:diagram}a.), which consists of an image-plane detector array (e.g., a camera sensor) that records arrival times and positions of incident photons. Idealized 2D direct imaging can be described as a continuously sampled projective measurement with differential POVM elements $d\Pi_{\vec{x}}=\outerproduct{\vec{x}}{\vec{x}}d^2\vec{x}$, where $\int_{\mathbb{R}^2}d\Pi_{\vec{x}}=\mathcal{I}$.
Ref.~\cite{Lu2018} found that the CE for ideal direct imaging is $\xi_{\rm Direct}^{(1)}\approx d^4/256$ to lowest order in $d\ll1$, indicating a quadratic scaling gap in the achievable error exponent compared with the quantum limit as the point-source separation under $H_2$ is made small.

\subsection{Non-Dimensionalized Quantities: $\tilde{m}_j(\vec{x}_{\rm obj})$, $\tilde{\psi}(\vec{x})$, $\ket{\tilde{\psi}_{\vec{x}}}$,  $\tilde{\Gamma}(\vec{x})$, $\tilde{\phi}_m(\vec{x})$, $\ket{\tilde{\phi}_m}$, and $\tilde{c}_{m,n}(\vec{x})$}
We first justify the normalized definitions for the non-dimensionalized quantities $\tilde{m}_j(\vec{x}_{\rm obj})$, $\tilde{\psi}(\vec{x})$ and $\tilde{\Gamma}(\vec{x})$. We seek normalization factors $a$ and $b$ such that $\tilde{m}_j(\vec{x}_{\rm obj})=a m_j(\theta \vec{x}_{\rm obj})$ and $\tilde{\psi}(\vec{x})=b \psi(\sigma \vec{x})$ satisfy the normalization conditions $1=\iint_{-\infty}^{\infty}\tilde{m}_j(\vec{x}_{\rm obj})d^2\vec{x}_{\rm obj}$ and $1=\iint_{-\infty}^{\infty}\tilde{\psi}^\ast(\vec{x})\tilde{\psi}(\vec{x})d^2\vec{x}$. Making the coordinate transformations $\vec{x}_{\rm obj}'=\theta\vec{x}_{\rm obj}$ and $\vec{x}'=\sigma\vec{x}$, we have
\begin{equation}
	\begin{aligned}
		1=&\frac{a}{\theta^2}\iint_{-\infty}^{\infty}m_j(\vec{x}_{\rm obj}')d^2\vec{x}_{\rm obj}'\\
		1=&\frac{\abs{b}^2}{\sigma^2}\iint_{-\infty}^{\infty}\psi^\ast(\vec{x}')\psi(\vec{x}')d^2\vec{x}'.	
	\end{aligned}
	\label{eq:normalizations}
\end{equation}
Since each of the integrals in Eq. \eqref{eq:normalizations} is equal to unity, we confirm the normalizations $a=\theta^2$ and $b=\sigma$. Using the definition $\tilde{\Gamma}(\vec{x})=\Gamma(\sigma\vec{x})$, we then have
\begin{equation}
	\begin{aligned}
		\tilde{\Gamma}(\vec{x})=&\iint_{-\infty}^{\infty}\psi^\ast(\vec{a})\psi(\vec{a}-\sigma\vec{x})d^2\vec{a}\\
		=&\iint_{-\infty}^{\infty}\sigma^2\psi^\ast(\sigma\vec{a})\psi(\sigma\vec{a}-\sigma\vec{x})d^2\vec{a}\\
		=&\iint_{-\infty}^{\infty}\tilde{\psi}^\ast(\vec{a})\tilde{\psi}(\vec{a}-\vec{x})d^2\vec{a}\\
		=&\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\tilde{\psi}_{\vec{x}}},
	\end{aligned}
	\label{eq:GammaTilde}
\end{equation}
where the coordinate transformation $\vec{a}\to\sigma\vec{a}$ is made in the second equality, the definition of $\tilde{\psi}(\vec{x})$ is used for the third equality, and we define $\ket{\tilde{\psi}_{\vec{x}}}=\iint_{-\infty}^{\infty}\tilde{\psi}(\vec{x})\ket{\vec{x}}d^2\vec{x}$ for the final equality.

Likewise, given an arbitrary orthonormalized basis with eigenvectors $|\phi_m\rangle$, the normalization condition $\iint_{-\infty}^{\infty}\phi_{m}(\vec{x})^{\ast}\phi_n(\vec{x})d^2\vec{x}=\delta_{m,n}$ results in the non-dimensionalized basis functions $\tilde{\phi}_m(\vec{x})=\sigma \phi_m(\sigma\vec{x})$. Defining $\tilde{c}_{m,n}(\vec{x})=c_{m,n}(\sigma\vec{x})$, analogous steps to those in Eq.~\eqref{eq:GammaTilde} can be used to show that
\begin{equation}
	\begin{aligned}
		\tilde{c}_{m,n}(\vec{x})=&\iint_{-\infty}^{\infty}\tilde{\phi}_m^\ast(\vec{a})\tilde{\psi}(\vec{a}-\vec{x})d^2\vec{a}\\
		&\times\iint_{-\infty}^{\infty}\tilde{\psi}^\ast(\vec{a}-\vec{x})\tilde{\phi}_n(\vec{a})d^2\vec{a}\\
		=&\innerproduct{\tilde{\phi}_m}{\tilde{\psi}_{\vec{x}}}\innerproduct{\tilde{\psi}_{\vec{x}}}{\tilde{\phi}_n},
	\end{aligned}
	\label{eq:cmntilde}
\end{equation}
where $\ket{\tilde{\phi}_m}=\iint_{-\infty}^{\infty}\tilde{\phi}_{m}(\vec{x})\ket{\vec{x}}d^2\vec{x}$.

\subsection{The QCE for Binary Sub-Rayleigh Object Discrimination}
\begin{table}[b]
	\centering
	\begin{tabular}{|c|cccccc|}
		\hline
		$m$ & 0 & 1 & 2 & 3 & 4 & 5\\
		\hline
		$k_m$ & 0 & 1 & 0 & 2 & 1 & 0\\
		$l_m$ & 0 & 0 & 1 & 0 & 1 & 2\\
		\hline
	\end{tabular}
	\caption{Indexing convention used for the first six PAD basis vectors (indexed by $m$) and their relationships with the $x$ and $y$ derivatives of the PSF (indexed by $k_m$ and $l_m$).}
	\label{tab:indices}	
\end{table}

The PAD basis representation on an even PSF ensures that all basis functions are either even or odd in both $x$ and $y$, i.e., $\phi_m(\vec{x})=(-1)^{p_m}\phi_m(-\vec{x})$, where $p_m=k_m+l_m$ is determined by $m$ according to Table~\ref{tab:indices}. Since this implies $c_{m,n}(\vec{x})=(-1)^{p_{m}+p_n}\big[c_{m,n}(\vec{a}-\vec{x})\big]_{\vec{a}=\vec{\Omega}}$, the density matrix elements of a quantum state of form Eq. \eqref{eq:rhojPAD} from the main text can be written as
\begin{equation}
	d_{j,m,n}=(-1)^{p_m+p_n}\bigg[\frac{1}{\mu^2}m_j\bigg(\frac{\vec{x}}{\mu}\bigg)\conv2D  c_{m,n}(\vec{x})\bigg]_{\vec{x}=\vec{\Omega}},
	\label{eq:rhoconv}
\end{equation} 
where the operator $\conv2D$ represents a 2D convolution. By the convolution theorem along two dimensions, 
\begin{equation}
	\frac{1}{\mu^2}m_j\bigg(\frac{\vec{x}}{\mu}\bigg)\conv2D  c_{m,n}(\vec{x})=\mathcal{F}_{\vec{x}}^{-1}\big\{M_j(\vec{X}_{\rm obj})C_{m,n}(\vec{X})\big\},
	\label{eq:convThm}
\end{equation}
where $\mathcal{F}_{\vec{x}}^{-1}$ represents an inverse 2D Fourier transform over the image-plane Fourier domain coordinates $\vec{X}=\{X,Y\}$, where $\vec{X}_{\rm obj}=\mu\vec{X}$, and where $M_j(\vec{X}_{\rm obj})$ and $C_{m,n}(\vec{X})$ denote the respective 2D characteristic functions of $m_j(\vec{x}_{\rm obj})$ and $c_{m,n}(\vec{x})$. 

The characteristic function of the normalized radiant exitance profile is defined by the object-plane Fourier transform
\begin{equation} 
	\begin{aligned}
		M_j(\vec{X}_{\rm obj})&=\iint_{-\infty}^{\infty}m_j(\vec{x}_{\rm obj})e^{-i(\vec{x}_{\rm obj}\cdot\vec{X}_{\rm obj})}d^2\vec{x}_{\rm obj}  \\
		&=\iint_{-\infty}^{\infty}\tilde{m}_j(\vec{x}_{\rm obj})e^{-i\theta(\vec{x}_{\rm obj}\cdot\vec{X}_{\rm obj})}
		d^2\vec{x}_{\rm obj},
	\end{aligned}
	\label{eq:characteristic1}
\end{equation}
where the second line is found by making the coordinate transformation $\vec{x}_{\rm obj}\to\theta\vec{x}_{\rm obj}$ and applying the definition of $\tilde{m}_j(\vec{x}_{\rm obj})$. Taking a two-dimensional Taylor series expansion about $\vec{X}_{\rm obj}=\vec{\Omega}$,
\begin{equation}
	\begin{aligned}
		M_j(\vec{X}_{\rm obj})=&\sum_{k,l=0}^{\infty}\frac{X_{\rm obj}^kY_{\rm obj}^l}{k!l!}\bigg[\frac{\partial^{k+l}M_j(\vec{X}_{\rm obj})}{\partial X_{\rm obj}^k \partial Y_{\rm obj}^l}\bigg]_{\vec{X}_{\rm obj}=\vec{\Omega}} \\
		=&\sum_{k,l=0}^{\infty}\frac{X_{\rm obj}^kY_{\rm obj}^l}{k!l!}(-i\theta)^{k+l}\\
		&\times\iint_{-\infty}^{\infty}x_{\rm obj}^ky_{\rm obj}^l\tilde{m}_j(\vec{x}_{\rm obj})d^2\vec{x}_{\rm obj}\\
		=&\sum_{k,l=0}^{\infty}\frac{X^kY^l}{k!l!}(-i\mu\theta)^{k+l}m_{j,x^ky^l}.
	\end{aligned}
	\label{eq:characteristic2}
\end{equation}
By the linearity of the Fourier transform, the inverse 2D transform in Eq.~\eqref{eq:convThm} can be evaluated term by term in the Taylor series from Eq.~\eqref{eq:characteristic2}. Setting aside all factors with no $\vec{X}$ dependence, the remaining inverse 2D Fourier transform in each term evaluates at the origin of the image plane to
\begin{widetext}
	\begin{equation}
		\Big[\mathcal{F}_{\vec{x}}^{-1}\big\{X^kY^lC_{m,n}(\vec{X})\big\}\Big]_{\vec{x}=\vec{\Omega}}=(-i)^{k+l}\bigg[\frac{\partial^{k+l}c_{m,n}(\vec{x})}{\partial x^k \partial y^l}\bigg]_{\vec{x}=\vec{\Omega}}
		=\frac{(-i)^{k+l}}{\sigma^{k+l}}\bigg[\frac{\partial^{k+l}\tilde{c}_{m,n}(\vec{x})}{\partial x^k \partial y^l}\bigg]_{\vec{x}=\vec{\Omega}},
		\label{eq:IFT2}
	\end{equation}
\end{widetext}
where the second equality comes from the coordinate transformation $\vec{x}\to\sigma\vec{x}$ and applying the definition $\tilde{c}_{m,n}(\vec{x})=c_{m,n}(\sigma\vec{x})$. These steps taken together result in \begin{equation}
	\begin{aligned}
		d_{j,m,n}=&\sum_{k,l=0}^{\infty}(-1)^{p_m+p_n+k+l}\frac{\gamma^{k+l}}{k!l!}\\
		&\times m_{j,x^ky^l} \bigg[\frac{\partial^{k+l}\tilde{c}_{m,n}(\vec{x})}{\partial x^k \partial y^l}\bigg]_{\vec{x}=\vec{\Omega}},
	\end{aligned}
	\label{eq:rhofull}
\end{equation}
where $p_m=k_m+l_m$. 

Since we seek a lowest order result in the regime where $\gamma\ll 1$, we will focus on the terms in $d_{j,m,n}$ up to $O\big(\gamma^2\big)$, recalling the assumption that the first moments in $x$ and $y$ of the normalized radiant exitance distributions are identical (i.e., $m_{j,x}=0$ and $m_{j,y}=0$, and therefore $m_{j,xy}=0$, without loss of generality). In the PAD basis, where $\vert \phi_0\rangle=\vert\psi_{\vec{x}}\rangle$, we have  $c_{0,0}(\vec{x})=\abs*{\Gamma(\vec{x})}^2$ and $[c_{m,n}(\vec{x})]_{\vec{x}=\vec{\Omega}}=\delta_{m,0}\delta_{n,0}$. We therefore find that a state $\rho_j$ can be expanded in powers of $\gamma$ as
\begin{equation}
	\begin{aligned}
		\rho_j=&\outerproduct{\phi_0}{\phi_0}+\frac{\gamma^2}{2}\sum\limits_{m,n=0}^{\infty}\outerproduct{\phi_{m}}{\phi_{n}}(-1)^{p_m+p_n}\\
		&\times\Bigg(m_{j,x^2}\bigg[\frac{\partial^2 \tilde{c}_{m,n}(\vec{x})}{\partial x^2}\bigg]_{\vec{x}=\vec{\Omega}}\\
		&+m_{j,y^2}\bigg[\frac{\partial^2 \tilde{c}_{m,n}(\vec{x})}{\partial y^2}\bigg]_{\vec{x}=\vec{\Omega}}\Bigg)+O(\gamma^3).
	\end{aligned}
	\label{eq:rhoapprox}
\end{equation}
To analyze the states $\rho_1$ and $\rho_2$, we must evaluate the second-order derivatives in Eq.~\eqref{eq:rhoapprox}. Using Eq.~\eqref{eq:cmntilde}, we find
\begin{equation}
	\begin{aligned}
		\bigg[\frac{\partial^2\tilde{c}_{m,n}(\vec{x})}{\partial x^2}\bigg]_{\vec{x}=\vec{\Omega}}=&\bigg[\frac{\partial^2}{\partial x^2}\Big(\innerproduct{\tilde{\phi}_{m}}{\tilde{\psi}_{\vec{x}}}\innerproduct{\tilde{\psi}_{\vec{x}}}{\tilde{\phi}_{n}}\Big)\bigg]_{\vec{x}=\vec{\Omega}}\\
		=&\innerproduct{\tilde{\phi}_{m}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\tilde{\phi}_{n}}\\
		&+2\innerproduct{\tilde{\phi}_{m}}{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}\innerproduct{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}{\tilde{\phi}_{n}}\\
		&+\innerproduct{\tilde{\phi}_{m}}{\tilde{\psi}_{\vec{\Omega}}}\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\tilde{\phi}_{n}}
	\end{aligned}
	\label{eq:D2cmn_expanded}
\end{equation}
and the analogous quantity involving derivatives with respect to $y$, where we have defined PSF derivative vectors 
\begin{equation}
	\vert\partial_{x^{k_m}y^{l_m}}\tilde{\psi}_{\vec{\Omega}}\rangle=\bigg[\iint_{-\infty}^{\infty}\frac{\partial^{k_m+l_m}}{\partial x^{k_m}\partial y^{l_m}}\tilde{\psi}(\vec{a}-\vec{x}) \ket{\vec{a}}d^2\vec{a}\bigg]_{\vec{x}=\vec{\Omega}}.
	\label{eq:derivative_kets}
\end{equation}

The inner products in Eq.~\eqref{eq:D2cmn_expanded} can be evaluated by performing the Gram-Schmidt orthogonalization procedure to relate the PAD basis functions to the PSF. The Gram-Schmidt process is equivalent to a QR decomposition $\Psi=\Phi W$ in the single-photon Hilbert space $\mathcal{H}^{(1)}$, where the PSF derivative vectors $	\vert\partial_{x^{k_m}y^{l_m}}\tilde{\psi}_{\vec{\Omega}}\rangle$ define the columns of $\Psi$, $\Phi$ is a unitary matrix with the PAD basis vectors $\vert\tilde{\phi}_m\rangle$ as its columns, and $W$ is a non-singular upper diagonal matrix. In the PAD basis representation, $\Phi$ is an infinite-dimensional identity matrix, so the columns of $\Psi=W$ give the PSF derivative vectors in terms of the PAD basis vectors. This decomposition leads to a natural indexing of the PAD basis elements (Table \ref{tab:indices}) according to ascending total number of derivatives $p_m$ taken in the vectors $\vert\partial_{x^{k_m}y^{l_m}}\tilde{\psi}_{\vec{\Omega}}\rangle$. 

Performing the Gram-Schmidt procedure defines the PAD basis vectors
\begin{equation}
	\vert\tilde{\phi}_{m}\rangle=\sum\limits_{n=0}^{m}w_{n,m} \vert\partial_{x^{k_n}y^{l_n}}\tilde{\psi}_{\vec{\Omega}}\rangle,
	\label{eq:GS-vectors}
\end{equation}
where the coefficients
\begin{equation}
	\begin{aligned}
		w_{0,0}=&1\\
		w_{1,1}=&\frac{1}{\sqrt{\innerproduct{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}}}\\
		w_{2,2}=&\frac{1}{\sqrt{\innerproduct{\partial_{y}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y}\tilde{\psi}_{\vec{\Omega}}}}}\\
		w_{0,3}=&\frac{-\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}}{\sqrt{\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}}\\
		w_{3,3}=&\frac{1}{\sqrt{\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}}\\
		w_{4,4}=&\frac{1}{\sqrt{\innerproduct{\partial_{xy}\tilde{\psi}_{\vec{\Omega}}}{\partial_{xy}\tilde{\psi}_{\vec{\Omega}}}}}\\
		w_{0,5}=&\frac{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}{\sqrt{\innerproduct{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2-\abs{\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}}\\
		w_{3,5}=&\frac{-\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}{\sqrt{\innerproduct{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2-\abs{\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}}\\
		w_{5,5}=&\frac{-1}{\sqrt{\innerproduct{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2-\abs{\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}}
	\end{aligned}
	\label{eq:winvs}
\end{equation}
give the matrix elements of $W^{-1}$ in the PAD basis. All other coefficients $w_{m,n}$ for $0\leq m\leq 5$ and $n\geq m$ are zero due to mismatched even-odd parity between the integer pairs $(k_m,k_n)$ and/or $(l_m,l_n)$ in Eq.~\eqref{eq:GS-vectors}. In matrix form,
\begin{equation}
	W^{-1}=
	\begin{pmatrix}
		w_{0,0} & 0 & 0 & w_{0,3} &0 & w_{0,5} &\cdots\\
		0 & w_{1,1} & 0 & 0 & 0 & 0 &\cdots\\
		0 & 0 & w_{2,2} & 0 & 0 & 0 &\cdots\\
		0 & 0 & 0 & w_{3,3} & 0 & w_{3,5} &\cdots\\
		0 & 0 & 0 & 0 &  w_{4,4} & 0 &\cdots \\
		0 & 0 & 0 & 0 & 0 & w_{5,5} &\cdots \\
		\vdots &\vdots &\vdots  &\vdots & \vdots &\vdots& \ddots
	\end{pmatrix},
	\label{eq:Winv}
\end{equation}
where we display just the subspace $\mathcal{H}_6^{(1)}$ corresponding to the first six PAD basis vectors according to Table~\ref{tab:indices}. Since the Schur complement of any block of an upper triangular matrix is simply given by that matrix block, the inverse of any diagonal block of an upper triangular matrix is equal to the same diagonal block of the inverse of the matrix. Inverting $W^{-1}$, we find that the matrix  

\begin{equation}
	W=
	\begin{pmatrix}
		W_{0,0} & 0 & 0 & W_{0,3} &0 & W_{0,5} &\cdots\\
		0 & W_{1,1} & 0 & 0 & 0 & 0 &\cdots\\
		0 & 0 & W_{2,2} & 0 & 0 & 0 &\cdots\\
		0 & 0 & 0 & W_{3,3} & 0 & W_{3,5} &\cdots\\
		0 & 0 & 0 & 0 &  W_{4,4} & 0 &\cdots \\
		0 & 0 & 0 & 0 & 0 & W_{5,5} &\cdots \\
		\vdots &\vdots &\vdots  &\vdots & \vdots &\vdots& \ddots
	\end{pmatrix}
	\label{eq:W}
\end{equation}
with
\begin{equation}
	\begin{aligned}
		W_{0,0}=&\frac{1}{w_{0,0}}=1\\
		W_{1,1}=&\frac{1}{w_{1,1}}=\sqrt{\innerproduct{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}{\partial_{x}\tilde{\psi}_{\vec{\Omega}}}}\\
		W_{2,2}=&\frac{1}{w_{2,2}}=\sqrt{\innerproduct{\partial_{y}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y}\tilde{\psi}_{\vec{\Omega}}}}\\
		W_{0,3}=&-\frac{w_{0,3}}{w_{0,0}w_{3,3}}=\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}\\
		W_{3,3}=&\frac{1}{w_{3,3}}=\sqrt{\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}\\
		W_{4,4}=&\frac{1}{w_{4,4}}=\sqrt{\innerproduct{\partial_{xy}\tilde{\psi}_{\vec{\Omega}}}{\partial_{xy}\tilde{\psi}_{\vec{\Omega}}}}\\
		W_{0,5}=&\frac{w_{0,3}w_{3,5}-w_{0,5}w_{3,3}}{w_{0,0}w_{3,3}w_{5,5}}=\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}\\
		W_{3,5}=&-\frac{w_{3,5}}{w_{3,3}w_{5,5}}\\
		=&\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\innerproduct{\partial_{x^{2}}\tilde{\psi}_{\vec{\Omega}}}{\tilde{\psi}_{\vec{\Omega}}}\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}\\
		W_{5,5}=&\frac{1}{w_{5,5}}\\
		=&\sqrt{\innerproduct{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}-\abs{\innerproduct{\tilde{\psi}_{\vec{\Omega}}}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2-\abs{\innerproduct{\tilde{\phi}_3}{\partial_{y^{2}}\tilde{\psi}_{\vec{\Omega}}}}^2}
	\end{aligned}
	\label{eq:ws}
\end{equation}
represents on its columns the non-orthonormal PSF derivative vectors in the orthonormal PAD basis, as in
\begin{equation}
	\vert\partial_{x^{k_m}y^{l_m}}\tilde{\psi}_{\vec{\Omega}}\rangle=\sum\limits_{n=0}^{m}W_{n,m}\vert\tilde{\phi}_{n}\rangle.
	\label{eq:derivative-vectors}
\end{equation}
These steps show that the $m^{\rm th}$ PSF derivative vector depends on only the basis vectors $\vert\tilde{\phi}_n\rangle$ for which $n\leq m$. 

Using Eq.~\eqref{eq:derivative-vectors} and $\langle \tilde{\phi}_m\vert\tilde{\phi}_n\rangle=\delta_{m,n}$, we rewrite the density matrix elements [Eq.~\eqref{eq:rhofull}] in the PAD basis as
\begin{equation}
	\begin{aligned}
		d_{j,0,0}=&1+\textrm{Re}\big(m_{j,x^2}W_{0,3}+m_{j,y^2}W_{0,5}\big)\gamma^2+O(\gamma^3)\\
		=&1-(m_{j,x^2}\Gamma_{x^2}+m_{j,y^2}\Gamma_{y^2})\gamma^2+O(\gamma^3)\\
		d_{j,1,1}=&m_{j,x^2}W_{1,1}^2\gamma^2+O(\gamma^3)\\
		=&m_{j,x^2}\Gamma_{x^2}\gamma^2+O(\gamma^3)\\
		d_{j,2,2}=&m_{j,y^2}W_{2,2}^2\gamma^2+O(\gamma^3)\\
		=&m_{j,y^2}\Gamma_{y^2}\gamma^2+O(\gamma^3)\\
		d_{j,3,0}=&\frac{1}{2}\Big(m_{j,x^2}W_{3,3}+m_{j,y^2}W_{3,5}\Big)\gamma^2+O(\gamma^3)\\
		d_{j,0,3}=&\frac{1}{2}\Big(m_{j,x^2}W^\ast_{3,3}+m_{j,y^2}W^\ast_{3,5}\Big)\gamma^2+O(\gamma^3)\\
		d_{j,5,0}=&\frac{1}{2}m_{j,y^2}W_{5,5}\gamma^2+O(\gamma^3)\\
		d_{j,0,5}=&\frac{1}{2}m_{j,y^2}W^\ast_{5,5}\gamma^2+O(\gamma^3),
	\end{aligned}
	\label{eq:rho_j_sparse}
\end{equation}
where all other matrix elements are $O\big(\gamma^3\big)$. The relationships $\textrm{Re}(W_{0,3})=-\Gamma_{x^2}$ and $\textrm{Re}(W_{0,5})=-\Gamma_{y^2}$ are given as an identity in Ref. \cite{Kerviche2017a}, while $W_{1,1}^2=\Gamma_{x^2}$ (and likewise $W_{2,2}^2=\Gamma_{y^2}$) can be derived by taking 2nd derivatives on either side of the PSF normalization condition 
and exchanging the order of integration and differentiation:
\begin{equation}
	\begin{split}
		0
	
		=&\iint_{-\infty}^{\infty}\frac{\partial^2}{\partial x^2}\abs*{\tilde{\psi}(\vec{x})}^2d^2\vec{x}\\
		=&\iint_{-\infty}^{\infty}2\abs*{\frac{\partial\tilde{\psi}(\vec{x})}{\partial x}}^2+2\textrm{Re}\bigg(\tilde{\psi}^\ast(\vec{x})\frac{\partial^2\tilde{\psi}(\vec{x})}{\partial x^2}\bigg)d^2\vec{x}\\
		=&2(W_{1,1}^2-\Gamma_{x^2}).
	\end{split}
\end{equation} 
Note that if the PSF is separable in $x$ and $y$ i.e., $\psi(\vec{x})=\Xi(x)\Upsilon(y)$, then $W_{3,5}=0$.

With the states $\rho_1$ and $\rho_2$ [Eq.~\eqref{eq:rhojPAD} in the main text] represented in the finite dimensional PAD basis defined on the truncated subspace $\mathcal{H}_6^{(1)}$, we can directly evaluate the quantum Chernoff exponent [Eq.~\eqref{eq:QuantumChernoff} in the main text] to lowest non-vanishing order in $\gamma$ using our accompanying work on perturbation theory for common entropic and distance measures in quantum information theory \cite{Grace2021}. We first decompose both density matrices into $\rho_j=\rho_0+\nu_j$, where $\rho_0=\outerproduct{\phi_0}{\phi_0}$ is a pure state and $\nu_1$ and $\nu_2$ are zero-trace perturbation matrices with spectral norms $\norm{\nu_j}=O(\gamma^2)$. We additionally block decompose the truncated Hilbert space as $\mathcal{H}_6^{(1)}=\mathcal{H}_1^{(1)}\oplus\mathcal{H}_5^{(1)}$, where $\mathcal{H}_1^{(1)}$ is the 1-dimensional subspace of $\mathcal{H}_6^{(1)}$ corresponding to the support of the pure state $\rho_0$ and $\mathcal{H}_5^{(1)}$ is the kernel of $\rho_0$ on $\mathcal{H}_6^{(1)}$. We denote the resulting matrix decompositions using the block matrices
\begin{equation}
	\begin{split}
		\rho_0 =& 
		\begin{pmatrix}
			1 & 0\\
			0 & 0
		\end{pmatrix}\\
		\nu_j=&
		\begin{pmatrix}
			\nu_{j,\rm B} & \nu_{j,\rm C}\\
			\nu_{j,\rm C}^{\dagger} & \nu_{j,\rm D}
		\end{pmatrix}.
	\end{split}
	\label{eq:BlockDecomposition_nu}
\end{equation}
Following the perturbation theory for perturbations that extend the support of the original states \cite{Grace2021}, the QCE
\begin{equation}
	\begin{split}
		\xi_{\rm Q}^{(1)}=& -\min_{s\in[0,1]}s\Tr[\nu_{1,\rm B}]+(1-s)\Tr[\nu_{2, \rm B}]\\
		&+\Tr\big[\nu_{1,\rm D}^s\nu_{2, \rm D}^{1-s}\big]+O\big(\max(\norm{\nu_1},\norm{\nu_2})^2\big)
	\end{split}
	\label{eq:QCEperturbation}
\end{equation}
can be easily evaluated in the PAD basis by reading off the matrix elements of $\nu_{1,\rm B}$, $\nu_{2, \rm B}$, $\nu_{1,\rm D}$ and $\nu_{2, \rm D}$ using Eq.~\eqref{eq:rho_j_sparse} because $\nu_{1,\rm D}$ and $\nu_{2, \rm D}$ are diagonal. The result is Eq.~\eqref{eq:GeneralQCE} in the main text.

\subsection{Direct Imaging for Binary Object Discrimination}
For a continuously valued measurement (i.e., where the outcome space $\mathcal{Z}$ is continuous), the classical Chernoff exponent takes the form 
\begin{equation}
	\xi_{\textrm{Direct}}^{(1)}=-\log\Big[\min\limits_{0\leq s \leq 1} Q_s\Big]
	\label{eq:ContinuousChernoff}
\end{equation}
with
\begin{equation}
	Q_s=\int_{\mathcal{Z}}P(z\vert\rho_1)^s P(z\vert\rho_2)^{1-s}dz.
	\label{eq:ContinuousQs}
\end{equation}
Ideal direct imaging implements a POVM $\{\Pi_{\vec{x}}\}_{\mathbb{R}^2}$, with elements $\Pi_{\vec{x}}=\outerproduct{\vec{x}}{\vec{x}}$ corresponding to photon arrival locations $\vec{x}$ in the outcome space $\mathbb{R}^2$. The classical Chernoff exponent for direct imaging can therefore be found using
\begin{equation}
	Q_s=\iint_{-\infty}^{\infty}\big(\sigma^2 P(\sigma\vec{x}\vert\rho_1)\big)^s\big(\sigma^2 P(\sigma\vec{x}\vert\rho_2)\big)^{1-s}d^2\vec{x},
	\label{eq:QsDirect}
\end{equation}
where we made the coordinate transformation $\vec{x}\to\sigma\vec{x}$ and distributed the factor $\sigma^2$.

From Eq.~\eqref{eq:rhoj}, we have
\begin{equation}
	\begin{aligned}
		P(\vec{x}\vert\rho_j)=&\Tr\big[\rho_j \outerproduct{\vec{x}}{\vec{x}}d^2\vec{x}\big] \\
		=&\iint_{-\infty}^\infty \frac{1}{\mu^2}m_j\bigg(\frac{\vec{a}}{\mu}\bigg)\langle \vec{x}\vert\psi_{\vec{a}}\rangle\langle\psi_{\vec{a}}\vert\vec{x}\rangle d^2\vec{a}\\
		=&\frac{1}{\mu^2}m_j\bigg(\frac{\vec{x}}{\mu}\bigg)\ast\ast\abs*{\psi(\vec{x})}^2,
	\end{aligned}
	\label{eq:DirectP}
\end{equation} 
which is consistent with the standard result from classical image science that the image plane intensity distribution in an incoherent imaging context is given by a 2D convolution of the object radiant exitance and the incoherent PSF \cite{Goodman2005}. By the convolution theorem, $(1/\mu^2)~m_j(\vec{x}/\mu)\conv2D \abs*{\psi(\vec{x})}^2=\mathcal{F}_{\vec{x}}^{-1}\big\{M_j(\vec{X}_{\rm obj})\Psi(\vec{X})\big\}$, where $\Psi(\vec{X})$ is the 2D characteristic function of the incoherent PSF $\abs*{\psi(\vec{x})}^2$. Using Eq.~\eqref{eq:characteristic2} and evaluating the inverse 2D Fourier transform via similar methods to those used for Eq.~\eqref{eq:IFT2}, we find that the relevant probability density functions for the integrand of Eq.~\eqref{eq:QsDirect} have the form
\begin{equation}
	\begin{aligned}
		P(\sigma\vec{x}\vert\rho_j)=&\frac{1}{\sigma^2}\sum_{k,l=0}^{\infty}(-1)^{k+l}\frac{\gamma^{k+l}}{k!l!}m_{j,x^ky^l}\psi_{x^ky^l}(\vec{x})
	\end{aligned}
	\label{eq:DirectPSeries}
\end{equation}
where the derivatives of the incoherent PSF
\begin{equation}
	\psi_{x^ky^l}(\vec{x})=\frac{\partial^{k+l}\abs{\tilde{\psi}(\vec{x})}^2}{\partial x^k \partial y^l}
\end{equation} 
were defined in the main text.

In order to calculate the Chernoff exponent in the limit $\gamma\ll 1$, we generalize the methods from Ref. \cite{Lu2018} and expand the integrand of $Q_s$ using the Taylor series
\begin{equation}
	\begin{aligned}
		f(\gamma)^sg(\gamma)^{1-s}=&f_0+(sf_2+(1-s)g_2)\gamma^2\\
		&+(sf_3+(1-s)g_3)\gamma^3\\
		&+\bigg[sf_4+(1-s)g_4\\
		&-\frac{1}{2f_0}s(1-s)(f_2-g_2)^2\bigg]\gamma^4 + O(\gamma^5)
	\end{aligned}
	\label{eq:IntegrandSeries}
\end{equation}
where $f(\gamma)\equiv\sum_{n=0}^\infty f_n\gamma^n=\sigma^2 P(\sigma\vec{x}\vert\rho_0)$ and $g(\gamma)\equiv\sum_{n=0}^\infty g_n\gamma^n=\sigma^2 P(\sigma\vec{x}\vert\rho_1)$, and where we used $g_0=f_0$ and $g_1=f_1=0$ since $m_{j,x}=m_{j,y}=m_{j,xy}=0$. First, we consider only terms in $Q_s$ up to third order in $\gamma$. Using Eq. \eqref{eq:DirectPSeries} to determine the coefficients $f_n$ and $g_n$, Eqs.~\eqref{eq:QsDirect} and \eqref{eq:IntegrandSeries} result in
\begin{equation}
	\begin{aligned}
		Q_s=&\iint_{-\infty}^{\infty}\abs{\tilde{\psi}(\vec{x})}^2\\
		&+\frac{1}{2}\big[\big(sm_{1,x^2}+(1-s)m_{2,x^2}\big)\psi_{x^2}(\vec{x})\\
		&+\big(sm_{1,y^2}+(1-s)m_{2,y^2}\big)\psi_{y^2}(\vec{x})\big]\gamma^2\\
		&+\frac{1}{6}\big[\big(sm_{1,x^3}+(1-s)m_{2,x^3}\big)\psi_{x^3}(\vec{x})\\
		&+\big(sm_{1,y^3}+(1-s)m_{2,y^3}\big)\psi_{y^3}(\vec{x})\big]\gamma^3d^2\vec{x}+O(\gamma^4).
	\end{aligned}
	\label{eq:ChernoffDirect3}
\end{equation}
After distributing the integral to each of the terms, we note that
\begin{equation}
	\iint_{-\infty}^{\infty}\psi_{x^ky^l}(\vec{x})d^2\vec{x}=\delta_{k,0}\delta_{l,0},
	\label{eq:PSFidentity}
\end{equation} 
which can be shown by exchanging the order of integration and differentiation and using the normalization condition on $\tilde{\psi}(\vec{x})$ \cite{Lu2018}. Thus, $Q_s=1+O(\gamma^4)$ for all values of $s$, and $\xi_{\textrm{Direct}}^{(1)}=O(\gamma^4)$ in the sub-Rayleigh limit, while the quantum Chernoff exponent has nonzero $\gamma^2$ terms [Eq. \eqref{eq:GeneralQCE} in the main text]. This implies at least a quadratic scaling difference between the achievable error exponent when using direct imaging compared with the quantum limit for any sub-Rayleigh binary object discrimination task with an arbitrary 2D aperture.

To get a closed form expression for the direct detection classical Chernoff exponent to lowest non-zero order in $\gamma\ll 1$ for a restricted class of PSFs (see below), we expand $Q_s$ to fourth order in $\gamma$ using all of the terms given in Eq.~\eqref{eq:IntegrandSeries}. After integrating the $\gamma^2$ and $\gamma^3$ terms of $Q_s$ to zero as before, we have
\begin{equation}
	\begin{aligned}
		Q_s=&\iint_{-\infty}^{\infty}\abs{\tilde{\psi}(\vec{x})}^2\\
		&+\frac{1}{24}\bigg[\big(sm_{1,x^4}+(1-s)m_{2,x^4}\big)\psi_{x^4}(\vec{x})\\
		&+6\big(sm_{1,x^2y^2}+(1-s)m_{2,x^2y^2}\big)\psi_{x^2y^2}(\vec{x})\\
		&+\big(sm_{1,y^4}+(1-s)m_{2,y^4}\big)\psi_{y^4}(\vec{x})\\
		&-\frac{3s(1-s)}{\abs{\tilde{\psi}(\vec{x})}^2}\Big((m_{1,x^2}-m_{2,x^2})\psi_{x^2}(\vec{x})\\
		&+(m_{1,y^2}-m_{2,y^2})\psi_{y^2}(\vec{x})\Big)^2\bigg]\gamma^4d^2\vec{x}+O(\gamma^5).
	\end{aligned}
	\label{eq:ChernoffDirect4}
\end{equation}
Applying Eq.~\eqref{eq:PSFidentity}, we find $Q_s=1-s(1-s)\mathcal{K}\gamma^4/8+O(\gamma^5)$, where $\mathcal{K}$ is given by 
\begin{equation}
	\begin{split}
		\mathcal{K}=&\iint_{-\infty}^{\infty}\abs{\tilde{\psi}(\vec{x})}^{-2}\Big((m_{1,x^2}-m_{2,x^2})\psi_{x^2}(\vec{x})\\
		&+(m_{1,y^2}-m_{2,y^2})\psi_{y^2}(\vec{x})\Big)^2d^2\vec{x}.
	\end{split}
	\label{eq:K}
\end{equation}
Minimizing $Q_s$ is then equivalent to maximizing $s(1-s)$ over $0\leq s\leq 1$, so $s=1/2$ and the Chernoff exponent is given by
\begin{equation}
	\xi_{\textrm{Direct}}^{(1)}=\frac{\mathcal{K}}{32}\gamma^4+O(\gamma^5).
	\label{eq:CEDirectGeneral}
\end{equation}
If the PSF is separable in $x$ and $y$, i.e., $\psi(\vec{x})=\Xi(x)\Upsilon(y)$, we have
\begin{equation}
	\begin{split}
		\psi_{x^2}(\vec{x})\psi_{y^2}(\vec{x})d^2\vec{x}=&\abs{\tilde{\Upsilon}(y)}^2\frac{d\abs{\tilde{\Xi}(x)}^2}{dx^2}\abs{\tilde{\Xi}(x)}^2\frac{d\abs{\tilde{\Upsilon}(y)}^2}{dx^2}\\
		=&\abs{\tilde{\psi}(\vec{x})}^2\psi_{x^2y^2}(\vec{x}),
	\end{split}
	\label{eq:SeparablePSFIdentity}
\end{equation}
and $\iint_{-\infty}^{\infty}\abs{\tilde{\psi}(\vec{x})}^{-2}\psi_{x^2}(\vec{x})\psi_{y^2}(\vec{x})d^2\vec{x}=0$ follows from Eq.~\eqref{eq:PSFidentity}. Therefore, Eq.~\eqref{eq:CEDirectGeneralSeparable} in the main text can be found by expanding terms inside the integral of Eq.~\eqref{eq:K}. 

A very important caveat is that this closed form expression for the direct imaging Chernoff exponent [Eq.~\eqref{eq:CEDirectGeneral} or Eq.~\eqref{eq:CEDirectGeneralSeparable} in the main text] does not hold for all PSFs. Namely, the presence of the factor $\abs{\tilde{\psi}(\vec{x})}^{-2}$ in the integrand of $\mathcal{K}$ (or $\mathcal{K}_x$ and $\mathcal{K}_y$) results in a diverging integral if the coherent PSF $\psi(\vec{x})$ is exactly zero at any point $\vec{x}\in\mathbb{R}^2$. Most significantly, this includes the Airy disk PSF associated with a hard circular aperture \cite{Goodman2005}, a ubiquitous aperture geometry for realistic imaging systems. A similar situation was encountered in Ref.~\cite{Paur2019}, which pointed out that the sub-Rayleigh scaling of the Fisher information for estimating an object parameter cannot be evaluated by expanding the integrand as a Taylor series and integrating individual terms when there are zeros in the PSF. Likewise, our integration of the individual terms of Eq.~\eqref{eq:ChernoffDirect4} is invalid for such PSFs. However, our result differs in kind from that of Ref.~\cite{Paur2019}, which reported a factor of $\gamma$ (in our notation) improvement in the scaling of the direct imaging Fisher information using a hard aperture compared with that using a Gaussian attenuated aperture model. In the case of binary hypothesis testing, we find that the integrals of the individual terms of $Q_s$ up to third order in $\gamma$ always converge [Eq. \eqref{eq:ChernoffDirect3}], so the result $\xi_{\textrm{Direct}}^{(1)}=O(\gamma^4)$ holds for any PSF. Therefore, while we cannot write down a closed form expression for $\xi_{\textrm{Direct}}^{(1)}$ when using an aperture with zeros in its PSF, the scaling gap between the classical and quantum Chernoff exponents is always at least quadratic in the sub-Rayleigh limit, and PSFs with zeros give no fundamental advantage for sub-Rayleigh object discrimination. 

\subsection{Mode-Sorting Measurements for Binary Object Discrimination}
We first consider a binary projective measurement with POVM elements $\Pi_0=\outerproduct{\psi_{\vec{\Omega}}}{\psi_{\vec{\Omega}}}$ and $\Pi_1= \mathcal{I}-\outerproduct{\psi_{\vec{\Omega}}}{\psi_{\vec{\Omega}}}$, which can be implemented by a 2D BSPADE device \cite{Ang2017}. From Eqs.~\eqref{eq:rhojPAD} in the main text and Eq.~\eqref{eq:rho_j_sparse},
\begin{equation}
	\begin{aligned}
		P(0\vert\rho_j)=&d_{j,0,0}\\
		=& 1-(m_{j,x^2}\Gamma_{x^2}+m_{j,y^2}\Gamma_{y^2})\gamma^2+O(\gamma^3)\\
		P(1\vert\rho_j)=&1-d_{j,0,0}\\
		=& (m_{j,x^2}\Gamma_{x^2}+m_{j,y^2}\Gamma_{y^2})\gamma^2+O(\gamma^3).
	\end{aligned}
	\label{eq:BSPADEP}
\end{equation}
We apply Eq.~\eqref{eq:Chernoff} from the main text and utilize the first order expansion $(1+u)^v=1+vu+O(u^2)$ to simplify the terms $P(0\vert\rho_1)^s$ and $P(0\vert\rho_2)^{1-s}$. Expanding the logarithm in that equation using $\ln(1+u)=u+O(u^2)$, the classical Chernoff exponent for the 2D BSPADE measurement is given by
\begin{equation}
	\begin{aligned}
		\xi_{\rm BSPADE}^{(1)}=&\max\limits_{0\leq s \leq 1}\Big[s(m_{1,x^2}\Gamma_{x^2}+m_{1,y^2}\Gamma_{y^2})\\
		&+(1-s)(m_{2,x^2}\Gamma_{x^2}+m_{2,y^2}\Gamma_{y^2})\\
		&-(m_{1,x_{\rm obj}^2}\Gamma_{x^2}+m_{1,y_{\rm obj}^2}\Gamma_{y^2})^s\\
		&\times(m_{2,x^2}\Gamma_{x^2}+m_{2,y^2}\Gamma_{y^2})^{1-s}\Big]\gamma^2\\
		&+O(\gamma^3).
	\end{aligned}
	\label{eq:BSPADECE}
\end{equation}
Unlike hypothesis tests between a point source and an arbitrary object, the 2D BSPADE measurement does not in general achieve the quantum limit [Eq. \eqref{eq:GeneralQCE} in the main text] in the sub-Rayleigh limit for a binary hypothesis test between any two arbitrary objects. A few steps of algebra show that $\xi_{\rm BSPADE}^{(1)}=\xi_{\rm Q}^{(1)}$ when the two objects have the same ellipticity, i.e., when $m_{1,x^2}/m_{1,y^2}=m_{2,x^2}/m_{2,y^2}$, but when this condition is not met the 2D BSPADE measurement is not quantum-optimal. 

Alternatively, the projectors $\Pi_0=\outerproduct{\phi_{0}}{\phi_{0}}$, $\Pi_1=\outerproduct{\phi_{1}}{\phi_{1}}$, $\Pi_2=\outerproduct{\phi_{2}}{\phi_{2}}$, and $\Pi_3=\mathcal{I}-\Pi_0-\Pi_1-\Pi_2$ form a POVM and are constructed from the first three eigenvectors of the PAD basis and the orthogonal complement on $\mathcal{H}^{(1)}$. From Eq.~\eqref{eq:rho_j_sparse}, the measurement outcome probabilities are
\begin{equation}
	\begin{aligned}
		P(0\vert\rho_j)=&d_{j,0,0}\\
		=& 1-(m_{j,x^2}\Gamma_{x^2}+m_{j,y^2}\Gamma_{y^2})\gamma^2+O(\gamma^3)\\
		P(1\vert\rho_j)=&d_{j,1,1}= m_{j,x^2}\Gamma_{x^2}\gamma^2+O(\gamma^3)\\ 
		P(2\vert\rho_j)=&d_{j,2,2}= m_{j,y^2}\Gamma_{y^2}\gamma^2+O(\gamma^3)\\
		P(3\vert\rho_j)=&1-d_{j,0,0}-d_{j,1,1}-d_{j,2,2}=O(\gamma^3).
	\end{aligned}
	\label{eq:3-SPADEP}
\end{equation}
Since $\Pi_3$ has negligible outcome probability in the limit $\gamma\ll1$, the fourth POVM element can be ignored in a practical implementation designed for sub-Rayleigh imaging. Using Eq.~\eqref{eq:Chernoff} from the main text and the same expansions as before, the CE of the resulting TriSPADE measurement formed by the projectors $\Pi_0$, $\Pi_1$, and $\Pi_2$,
\begin{equation}
	\begin{aligned}
		\xi_{\rm TriSPADE}^{(1)}=&\max\limits_{0\leq s \leq 1}\Big[s(m_{1,x^2}\Gamma_{x^2}+m_{1,y^2}\Gamma_{y^2})\\
		&+(1-s)(m_{2,x^2}\Gamma_{x^2}+m_{2,y^2}\Gamma_{y^2})\\
		&-m_{1,x^2}^sm_{2,x^2}^{1-s}\Gamma_{x^2}\\
		&-m_{1,y^2}^sm_{2,y^2}^{1-s}\Gamma_{y^2}\Big]\gamma^2+O(\gamma^3),
	\end{aligned}
	\label{eq:3-SPADECE}
\end{equation}
is exactly equal to the QCE [Eq.~\eqref{eq:GeneralQCE} in the main text]. 

As an aside, since the Cartesian derivatives of a function that is even in $x$ and $y$ will all be either odd or even, the PAD-basis vectors inherit even or odd parity. The projectors $\Pi_0$, $\Pi_1$, $\Pi_2$ and $\Pi_3$ can therefore be thought of as sorting the 2D even/odd parity of the captured optical field to lowest order in $\gamma$. As a result, the same quantitative performance will be achieved with a 2D inversion-interferometric measurement (SLIVER)~\cite{Nair2016b,Ang2017}. This device may be easier to implement for some applications.

\subsection{Special Cases for Binary Object Discrimination}
Our results must be modified in the special case of two candidate objects that have exactly the same second moment in $x_{\rm obj}$ and $y_{\rm obj}$. In this case, the quantities given in Eqs. \eqref{eq:GeneralQCE}, \eqref{eq:CEDirectGeneral}, \eqref{eq:BSPADECE}, and \eqref{eq:3-SPADECE} are all zero to the orders in $\gamma$ specified, and higher order terms are needed to represent these Chernoff exponents. Let $\kappa$ be the lowest-order moment in $x_{\rm obj}$ and/or $y_{\rm obj}$ that differs between $m_1(\vec{x}_{\rm obj})$ and $m_2(\vec{x}_{\rm obj})$, i.e., $m_{1,x^ky^l}=m_{2,x^ky^l}$ for all $k+l<\kappa$. For the quantum Chernoff exponent, we reverse the order of summation in Eq.~\eqref{eq:rhojPAD} from the main text and in Eq.~\eqref{eq:rhofull} to write the state of an arbitrary 2D incoherent object as
\begin{equation}
	\begin{aligned}
		\rho_j=&\sum_{k,l=0}^{\infty}\frac{\gamma^{k+l}}{k!l!}\sum_{m,n=0}^{\infty}\outerproduct{\phi_{m}}{\phi_{n}}(-1)^{p_m+p_n+k+l}\\
		&\times m_{j,x^ky^l} \bigg[\frac{\partial^{k+l}\tilde{c}_{m,n}(\vec{x})}{\partial x^k \partial y^l}\bigg]_{\vec{x}=\vec{\Omega}}.
	\end{aligned}
	\label{eq:rhoapprox_kappa}
\end{equation}



	
	
	


We then perform the decomposition of Eq.~\eqref{eq:BlockDecomposition_nu}: $\rho_0$ contains all terms with $k+l<\kappa$ in the outer sum of Eq.~\eqref{eq:rhoapprox_kappa}, while $\nu_j=O(\gamma^\kappa)$ contains the terms with $k+l\geq\kappa$. It is clear from the perturbation theory result of Eq.~\eqref{eq:QCEperturbation} that the QCE will be $\xi_{\rm Q}^{(1)}=O\big(\gamma^\kappa\big)$. On the other hand, the CE for direct imaging can be analyzed using a generalization of the series expansion of Eq. \eqref{eq:IntegrandSeries}. If $f(\gamma)=\sum_{n=0}^{\infty}f_n \gamma^n$ and $g(\gamma)=\sum_{n=0}^{\kappa-1}f_n \gamma^n+\sum_{n=\kappa}^{\infty}g_n \gamma^n$, then 
\begin{equation}
	\begin{aligned}
		f(\gamma)^{s}g(\gamma)^{1-s}=&\sum_{n=0}^{\kappa-1}f_n\gamma^n+\sum_{n=\kappa}^{2\kappa-1}(s f_n+(1-s)g_n)\gamma^n\\
		&-\frac{1}{2f_0}s(1-s)(f_{\kappa}-g_{\kappa})^2\gamma^{2\kappa}+O(\gamma^{2\kappa+1}).
	\end{aligned}
	\label{eq:IntegrandSeriesGeneral}
\end{equation}
Applying this series to the integral given by Eqs.~\eqref{eq:QsDirect} and~\eqref{eq:DirectPSeries}, we perform term-by-term integration of $Q_s$ up to order $2\kappa-1$ in $\gamma$ using the identity in Eq.~\eqref{eq:PSFidentity}, resulting in $Q_s=1+O\big(\gamma^{2\kappa}\big)$ and $\xi_{\rm Direct}^{(1)}=O\big(\gamma^{2\kappa}\big)$. We thus find the scaling gap between the QCE and the CE for direct imaging to be of the order $O(\gamma^\kappa)$. The TriSPADE measurement will not be quantum-optimal in this case, and finding an optimal measurement remains an open question for future work.

In the opposite scenario, when the two objects have different first moments in $x_{\rm obj}$ or $y_{\rm obj}$ (i.e., different centroids), we encounter different behavior: the scaling gap between the quantum limit and direct imaging vanishes. For the quantum Chernoff exponent, the states $\rho_j$ can be approximated using Eq.~\eqref{eq:rhoapprox_kappa}, with $\kappa=1$. Since lateral shifts of the reference frame should not affect the quantum Chernoff exponent, it is valid to move the origin of the object plane coordinate system so that the Cartesian first moments of one of the two objects are zero, i.e., $m_{1,x}=m_{1,y}=0$.
As a result, $\rho_1=\outerproduct{\phi_1}{\phi_1}+O(\gamma^2)$ is a pure state up to first order in $\gamma$, and the QCE is given by $\xi_{\textrm{Q}}^{(1)}=-\log\big[ F(\outerproduct{\phi_1}{\phi_1},\rho_2)\big]+O(\gamma^2)$. Using $\log(1+u)= u+O(u^2)$, the QCE becomes
\begin{equation}
	\begin{split}
		\xi_{\rm Q}^{(1)}=& \Bigg(m_{2,x}\bigg[\frac{\partial\tilde{\Gamma}(\vec{x})}{\partial x}\bigg]_{\vec{x}=\vec{\Omega}}+m_{2,y}\bigg[\frac{\partial\tilde{\Gamma}(\vec{x})}{\partial y}\bigg]_{\vec{x}=\vec{\Omega}}\Bigg)\gamma\\
		&+O(\gamma^2),
	\end{split}
	\label{eq:QCE_differentCentroids}
\end{equation}
which is zero because $\tilde{\Gamma}(\vec{x})$ is even in $x$ and $y$ for a circularly symmetric PSF. Therefore, the lowest nonzero order term of the QCE is still of order $O(\gamma^2)$. On the other hand, for the direct imaging Chernoff exponent, it is trivial to use Eq.~\eqref{eq:IntegrandSeriesGeneral} with $\kappa=1$ to find that $\xi_{\textrm{Direct}}^{(1)}=O(\gamma^2)$. We therefore find that the scaling gap with respect to $\gamma$ between the QCE and the direct imaging CE disappears when the two objects have different centroids. 

\subsection{$M$-ary Object Discrimination}
The $M$-ary QCE and the measurement-specific $M$-CE extend naturally from $M=2$ to $M>2$ candidates via the minimizations $\xi_{\textrm{Q},M}^{(1)}=\min_{i\neq j}\xi_{\textrm{Q},i,j}^{(1)}$ and $\xi_{\textrm{Meas},M}^{(1)}=\min_{i\neq j}\xi_{\textrm{Meas},i,j}^{(1)}$ over all pairwise QCEs and CEs, respectively \cite{Li2016}. We analyze the $M$-ary QCE by approximating the pairwise QCEs using the quantum Bhattacharyya bound \cite{Pirandola2008}:
\begin{equation}
	\xi_{\textrm{Q},i,j}^{(1)}\geq\xi_{\textrm{B},i,j}^{(1)}=-\log\Big[\Tr\big(\rho_i^{1/2}\rho_j^{1/2}\big)\Big].
	\label{eq:QBB}
\end{equation}
The quantum Bhattacharyya bound is proven to be tight, matching the quantum Chernoff bound, for any two quantum states that share the same Hilbert space support and differ by a vanishing operator perturbation \cite{Grace2021}. While this condition is not in general satisfied by the states $\rho_i$ and $\rho_j$ here, the quantum Bhattacharyya bound is always a valid upper bound on the asymptotic error and is often used to approximate the quantum Chernoff bound. Using the quantum Bhattacharyya bound, Eq.~\eqref{eq:GeneralQCE} in the main text can be approximated as
\begin{equation}
	\begin{split}
		\xi_{\textrm{Q},i,j}^{(1)}\approx&\frac{1}{2}\Big[\big(\sqrt{m_{i,x^2}}-\sqrt{m_{j,x^2}}\big)^2\Gamma_{x^2}\\
		&+\big(\sqrt{m_{i,y^2}}-\sqrt{m_{j,y^2}}\big)^2\Gamma_{y^2}\Big]\gamma^2+O(\gamma^3)
	\end{split}
	\label{eq:QBB_Mary}
\end{equation}
This form reveals the dependence of the $M$-ary QCE on differences of square roots of object second moments, constituting a distance measure for quantum-optimal object discrimination.

\begin{figure}[b]
	\centering
	\includegraphics[width=\columnwidth]{QuadraticPackingExplained.png}
	\caption{Depiction of an object database with quadratic packing of objects on a rectangular grid. The two representations are equivalent via a coordinate transformation.}
	\label{fig:quadraticPacking}
\end{figure}

First, consider $M$ objects whose 2D second moments are quadratically spaced on a rectangular grid, which is equivalent to requiring that the square roots of the second moments are equidistantly packed in two transverse directions $x$ and $y$ (Fig.~\ref{fig:quadraticPacking}). Formally, this condition can be described by the pair of equations
\begin{equation}
	\begin{split}
		\sqrt{m_{i,x^2}}-\sqrt{m_{j,x^2}}=&(l_{i,x}-l_{j,x})\mu_{\sqrt{x}}\\
		\sqrt{m_{i,y^2}}-\sqrt{m_{j,y^2}}=&(l_{i,y}-l_{j,y})\mu_{\sqrt{y}},
	\end{split} 
	\label{eq:quadraticCondition}
\end{equation}
where $\mu_{\sqrt{x}}=(\sqrt{m_{x^2,\rm max}}-\sqrt{m_{x^2,\rm min}})/(M_x-1)$ and $\mu_{\sqrt{y}}=(\sqrt{m_{y^2,\rm max}}-\sqrt{m_{y^2,\rm min}})/(M_y-1)$ represent the quadratic grid spacing in each direction and where $l_{i,x}\in[1,M_x]$ and $l_{i,y}\in[1,M_y]$ are integer-valued indices for the column and row, respectively, of the $i^{\rm th}$ object in the 2D grid of second moments. It is apparent that the minimal pairwise QCE will occur either when $l_{i,x}-l_{j,x}=1$ and $l_{i,y}=l_{j,y}$ or when $l_{i,x}=l_{j,x}$ and $l_{i,y}-l_{j,y}=1$; the $M$-ary QCE therefore becomes
\begin{equation}
	\xi_{\textrm{Q},M}^{(1)}\approx\frac{1}{2}\min\Big[\mu_{\sqrt{x}}^2\Gamma_{x^2},\mu_{\sqrt{y}}^2\Gamma_{y^2}\Big]\gamma^2+O(\gamma^3).
	\label{eq:QBB_Mary_quadratic2}
\end{equation}
Ensuring that the $x$ direction satisfies the minimum, e.g., by rotating the coordinate system, results in Eq.~\eqref{eq:QCE_M_quadratic} in the main text.

Alternatively, a linearly packed 2D grid of object second moments (Fig.~\ref{fig:regionsHatches}b.) is defined by the condition
\begin{equation}
	\begin{split}
		m_{i,x^2}-m_{j,x^2}=&(l_{i,x}-l_{j,x})\mu_{x}\\
		m_{i,y^2}-m_{j,y^2}=&(l_{i,y}-l_{j,y})\mu_{y}
	\end{split} 
	\label{eq:linearCondition}
\end{equation}
with linear grid spacing $\mu_{x}=(m_{x^2,\rm max}-m_{x^2,\rm min})/(M_x-1)$ and $\mu_{y}=(m_{y^2,\rm max}-m_{y^2,\rm min})/(M_y-1)$.
Under this condition, the pairwise QCEs [Eq.~\eqref{eq:QBB_Mary}] become
\begin{equation}
	\begin{split}
		\xi_{\textrm{Q},i,j}^{(1)}\approx&\frac{1}{2}\Bigg[\bigg(\frac{(l_{i,x}-l_{j,x})\mu_{x}}{\sqrt{m_{i,x^2}}+\sqrt{m_{j,x^2}}}\bigg)^2\Gamma_{x^2}\\
		&+\bigg(\frac{(l_{i,y}-l_{j,y})\mu_{y}}{\sqrt{m_{i,y^2}}+\sqrt{m_{j,y^2}}}\bigg)^2\Gamma_{y^2}\Bigg]\gamma^2+O(\gamma^3).
	\end{split}
	\label{eq:QBB_Mary_Linear}
\end{equation}
By inspection, the minimized pairwise QCE will occur when $l_{i,x}=M_x$, $l_{j,x}=M_x-1$ and $l_{i,y}=l_{j,y}=M_y$ or when $l_{i,x}=l_{j,x}=M_x$, $l_{i,y}=M_y$ and $l_{j,y}=M_y-1$. The resulting approximation to the $M$-ary QCE is
\begin{equation}
	\begin{split}
		\xi_{\textrm{Q},M}^{(1)}\approx&\frac{1}{2}\min\Bigg[\bigg(\frac{\mu_{x}}{\sqrt{m_{x^2,\rm max}}+\sqrt{m_{x^2,\rm max}-\mu_x}}\bigg)^2\Gamma_{x^2}\\
		&+\bigg(\frac{\mu_{y}}{\sqrt{m_{y^2,\rm max}}+\sqrt{m_{y^2,\rm max}-\mu_y}}\bigg)^2\Gamma_{y^2}\Bigg]\gamma^2+O(\gamma^3).
	\end{split}
	\label{eq:QBB_Mary_Linear2}
\end{equation}
Specifying a large object database, i.e., $M_x\gg 1$ and $M_y\gg 1$, implies $\mu_{x}\ll m_{x^2,\rm max}$ and $\mu_{y}\ll m_{y^2,\rm max}$, which can be used to simplify the denominators in Eq.~\eqref{eq:QBB_Mary_Linear2}. An appropriate choice of coordinate axes then yields Eq.~\eqref{eq:QCE_M_linear} in the main text.


For comparison with direct imaging, the exact pairwise CEs $\xi_{\textrm{Direct},i,j}^{(1)}$ from Eq.~\eqref{eq:CEDirectGeneral} can be minimized over all pairs of objects to find $\xi_{\textrm{Direct},M}^{(1)}$. The pairwise direct imaging CEs depend on differences of object second moments in $x$ and $y$, forming a distance measure that is different from that for the quantum limit, as described in the main text. Under a linearly packed 2D object database [Eq.~\eqref{eq:linearCondition}], the pairwise CEs will be minimized either when $l_{i,x}-l_{j,x}=1$ and $l_{i,y}=l_{j,y}$ or when $l_{i,x}=l_{j,x}$ and $l_{i,y}-l_{j,y}=1$, resulting in 
\begin{equation}
	\xi_{\textrm{Direct}}^{(1)}=\frac{1}{32}\min\big[\mu_x^2\Psi_{x^2},\mu_y^2\Psi_{y^2}\big]\gamma^4+O(\gamma^5),
	\label{eq:CE_Mary_Linear}
\end{equation}
where $\Psi_{x^2}=\iint_{-\infty}^{\infty}\psi_{x^2}(\vec{x})^2/\abs{\tilde{\psi}(\vec{x})}^2 d^2\vec{x}$ and $\Psi_{y^2}=\iint_{-\infty}^{\infty}\psi_{y^2}(\vec{x})^2/\abs{\tilde{\psi}(\vec{x})}^2 d^2\vec{x}$. \newpage An appropriate choice of coordinate axes gives Eq.~\eqref{eq:CE_M_linear} in the main text.

Finally, under a quadratically packed 2D object database [Eq.~\eqref{eq:quadraticCondition}], the pairwise CEs for direct imaging become
\begin{widetext}
	\begin{equation}
		\xi_{\textrm{Direct},i,j}^{(1)}=\frac{1}{32}\Big[\big((l_{i,x}-l_{j,x})(\sqrt{m_{i,x^2}}+\sqrt{m_{j,x^2}})\mu_{\sqrt{x}}\big)^2\Psi_{x^2}
		+\big((l_{i,y}-l_{j,y})(\sqrt{m_{i,y^2}}+\sqrt{m_{j,y^2}})\mu_{\sqrt{y}}\big)^2\Psi_{y^2}\Big]\gamma^4
		+O(\gamma^5).
		\label{eq:CE_Mary_quadratic}
	\end{equation}
\end{widetext}
The pairwise CEs will be minimized either when $l_{i,x^2}=1$, $l_{j,x^2}=2$ and $l_{i,y^2}=l_{j,y^2}$ or when $l_{i,x^2}=l_{j,x^2}$, $l_{i,y^2}=1$ and $l_{j,y^2}=2$, resulting in
\begin{equation}
	\begin{split}
		\xi_{\textrm{Direct}}^{(1)}=&\frac{1}{32}\min\Big[(2\sqrt{m_{x^2,\rm min}}+\mu_{\sqrt{x}})^2\mu_{\sqrt{x}}^2\Psi_{x^2}\\
		&+(2\sqrt{m_{y^2,\rm min}}+\mu_{\sqrt{y}})^2\mu_{\sqrt{y}}^2\Psi_{y^2}\Big]\gamma^4+O(\gamma^5).
	\end{split}
	\label{eq:CE_Mary_quadratic2}
\end{equation}
Specifying a large object database implies $\mu_{\sqrt{x}}\ll 2\sqrt{m_{x^2,\rm min}}$ and $\mu_{\sqrt{y}}\ll 2\sqrt{m_{y^2,\rm min}}$, simplifying Eq.~\eqref{eq:CE_Mary_quadratic2}. Choosing the coordinate axes gives Eq.~\eqref{eq:CE_M_quadratic} in the main text.

For completeness, we note that the rectangular grid configurations shown in Fig.~\ref{fig:regionsHatches}a. and Fig.~\ref{fig:regionsHatches}b. are not the optimal way to pack $M$ points in a rectangular region. For example, assuming an object database with $x$-$y$ symmetry, the maximized minimum distance between any two points within a square region is known to have the asymptotic behavior $D_M\sim\sqrt{2/(\sqrt{3} M)}$ as $M\to\infty$ \cite{Croft1991}, whereas the square grid configuration yields $D_M\sim1/\sqrt{M}$. Assuming a circularly symmetric aperture, employing the optimal packing configuration would therefore increase the results in Eqs.~\eqref{eq:QCE_M_quadratic}-\eqref{eq:CE_M_linear} in the main text by a factor of $2/\sqrt{3}\approx1.155$ in the large-database regime. We conclude that considering the optimal configuration of objects does not significantly change our results.
\end{document}