# Over-smoothing Effect of Graph Convolutional Networks

Fang Sun
EECS, Peking University
<EMAIL_ADDRESS>

###### Abstract

Over-smoothing is a severe problem which limits the depth of Graph
Convolutional Networks. This article gives a comprehensive analysis of the
mechanism behind Graph Convolutional Networks and the over-smoothing effect.
The article proposes an upper bound for the occurrence of over-smoothing,
which offers insight into the key factors behind over-smoothing. The results
presented in this article successfully explain the feasibility of several
algorithms that alleviate over-smoothing.

##  1 Introduction

Graph data are ubiquitous: from social networks like Weibo and Twitter, to
citation graphs connecting knowledge production in academia, they provide a
natural and flexible way of presenting the information we generate everyday.
Graph Convolutional Network (GCN) [Kipf & Welling, 2017] is a successful
attempt to generalize the powerful convolutional networks (CNNs) in coping
with Euclidean data to modeling graph structured data. GCN is simple and
elegant, out-performing previous works by a large margin on semi-supervised
classification tasks.

Recently, I performed an experiment on vanilla GCN. I stacked up the
convolutional layers in GCN from 2 to 3, 4, 5, 6, and tested their power on
Cora citation network data-set. The result (Figure 1) was astounding: although
GCN falls in the category of ’deep’ learning, its power quickly diminishes as
its layers stack up to merely 6. Several possible explanations for this
phenomenon were quickly ruled out: Not over-fitting, because training accuracy
and testing accuracy degraded synchronously. Not vanishing gradient, since a
6-layer network is too shallow for such an effect to occur.

Figure 1: Accuracy of GCN on Cora w.r.t. # of convolution layers.

In fact, this effect is unique to GCNs, called over-smoothing. [Li et al.,
2018] proposes the concept, pointing out that the intrinsic smoothing nature
of graph Laplacians limits the deepening of GCNs. Much work has been done in
the last 2 years to address the over-smoothing problem.

These are the 3 main objectives I am trying to establish in this article:

  * •

Giving mathematical formulations of ’smoothness’;

  * •

Analyzing the mechanisms for over-smoothing and the scenarios under which
over-smoothing could happen;

  * •

Explaining different methods for alleviating over-smoothing.

##  2 Preliminaries for Graph Convolution

###  2.1 Basic Architecture of GCN

##### Graph Laplacian

GCN is essentially a neighborhood-augmented MLP. Inspired by signal
processing, GCN uses the Laplacian matrix to aggregate the neighborhood
information. For graph $G$, Laplacian matrix
$\boldsymbol{L}:=\boldsymbol{D}-\boldsymbol{A}$. $\boldsymbol{D}$ is the
degree matrix of $G$, and $\boldsymbol{A}$ is the adjacency matrix of $G$.

The spectral convolution of GCN is presented as:

$\boldsymbol{H}^{(l+1)}=\sigma\left(\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{\tilde{A}}\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{H}^{(l)}\boldsymbol{\Theta}^{(l)}\right).$

$\boldsymbol{H}^{(l)},\boldsymbol{H}^{(l+1)}$ are the outputs of the
previous/present layer, $\boldsymbol{\Theta}^{(l)}$ is the tune-able
convolution kernel, $\sigma$ is the non-linear activation function (ReLU).

##### Derivation of GCN Model

The GCN model is derived via 4 steps of approximation:

$\displaystyle g_{\theta}\star\boldsymbol{x}$
$\displaystyle=g_{\theta}\boldsymbol{U}^{\top}\boldsymbol{x}$
$\displaystyle(\text{spectral convolution})$
$\displaystyle\approx\sum_{k=0}^{K}\theta_{k}^{\prime}T_{k}(\boldsymbol{\tilde{L}})\boldsymbol{x}$
$\displaystyle(k\text{-th order Chebyshev apprm.})$
$\displaystyle\approx\theta_{0}^{\prime}\boldsymbol{x}+\theta_{1}^{\prime}\left(\boldsymbol{L}-\boldsymbol{I}_{N}\right)\boldsymbol{x}$
$\displaystyle(k=1,\text{considering only 1st order neighbor})$
$\displaystyle\approx\theta\left(\boldsymbol{I}_{N}+\boldsymbol{D}^{-\frac{1}{2}}\boldsymbol{A}\boldsymbol{D}^{-\frac{1}{2}}\right)\boldsymbol{x}$
$\displaystyle\approx\theta\left(\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{\tilde{A}}\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\right)\boldsymbol{x}$
(the re-normalization trick)

The re-normalization trick is
$\boldsymbol{I}_{N}+\boldsymbol{D}^{-\frac{1}{2}}\boldsymbol{A}\boldsymbol{D}^{-\frac{1}{2}}\rightarrow\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{\tilde{A}}\boldsymbol{\tilde{D}}^{-\frac{1}{2}}$.
$\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{\tilde{A}}\boldsymbol{\tilde{D}}^{-\frac{1}{2}}$
can be viewed as the normalized Laplacian matrix of graph $\widetilde{G}$,
i.e. $G$ with self-loop. The implication behind taking $1$st order Chebyshev
approximation is that, in each layer of convolution, the model only considers
the $1$st order neighbor of each node. Nevertheless, higher orders of neighbor
information can be aggregated via stacking more convolution layers.

###  2.2 Laplacian Smoothing is the Key Power of GCN

[Li et al., 2018] proposes that Laplacian smoothing is central to GCN’s power
in classification tasks. The layer-wise propagation rule of the simplest
fully-connected networks (FCNs) is

$\boldsymbol{H}^{(l+1)}=\sigma(\boldsymbol{H}^{(l)}\boldsymbol{\Theta}^{(l)}).$

We observe that the sole difference between GCN and FCN is the normalized
Laplacian matrix
$\boldsymbol{S}=\boldsymbol{\tilde{D}}^{-\frac{1}{2}}\boldsymbol{\tilde{A}}\boldsymbol{\tilde{D}}^{-\frac{1}{2}}$.
By comparison, even a 1-layer GCN can out-perform a 1-layer FCN by a large
margin. This is because Laplacian smoothing makes the output features of nodes
in the same cluster more similar and eases the classification task.

The aggregating abilities of Laplacian smoothing is further demonstrated by
Simple Graph Convolution (SGC) [Wu et al., 2019]:

$\boldsymbol{\hat{{Y}}}_{\mathrm{SGC}}=\operatorname{softmax}\left({\boldsymbol{S}}^{K}{\boldsymbol{X}}{\boldsymbol{\Theta}}\right),$

where $K$ is the number of Laplacian matrices stacked. SGC shows that even if
we remove the redundant ReLU (non-linearity) and MLP layers between
aggregators, the multi-layer Laplacian smoothing yields the same degree of
accuracy with GCN.

Yet, by applying Laplacian smoothing many times, the feature of nodes in the
same connected component will converge to the same value and thus become
indistinguishable. As is shown in Figure 2, while the two types of points are
well-separable under the 2-layer scenario, they all become squashed up in the
5-layer GCN.

Figure 2: Vertex embeddings of Zachary’s karate club network with GCNs of
1,2,3,4,5 layers.

Thus, we give the natural definition of over-smoothing.

##### Definition 1

Over-smooothing is the effect that node features become indistinguishable
after multiple rounds of Laplacian smoothing.

##  3 Deeper Insight into Over-smoothing via Mathematical Formulation

###  3.1 Spectral Analysis of GCN

Recent works addressing the over-smoothing issue tend to regard GCN as low-
pass filtering [NT & Maehara, 2019], inspired by signal processing. The
spectral analysis on GCN has yielded some qualitative insight into the issue.

##### Theorem 1

Given a connected graph $G$, for the normalized Laplacian $\boldsymbol{S}$,

$\lim_{k\to\infty}{\boldsymbol{S}}^{k}=\boldsymbol{\Pi},$

where
$\boldsymbol{\Pi}=\Phi\left(\boldsymbol{\widetilde{{D}}}^{\frac{1}{2}}\boldsymbol{e}^{\top}\right)\left(\Phi\left(\boldsymbol{\widetilde{{D}}}^{\frac{1}{2}}\boldsymbol{e}^{\top}\right)\right)^{\top}$
, $\Phi(\mathbf{x})=\frac{\mathbf{x}}{||\mathbf{x}||}$.

Proof: Because $\boldsymbol{S}$ is symmetric, we orthogonally diagonalize
$\boldsymbol{S}=\boldsymbol{Q}\boldsymbol{\Lambda}\boldsymbol{Q}^{\top}$.
Thus,

$\boldsymbol{S}^{k}=\boldsymbol{Q}\boldsymbol{\Lambda}\boldsymbol{Q}^{\top}\cdots\boldsymbol{Q}\boldsymbol{\Lambda}\boldsymbol{Q}^{\top}=\boldsymbol{Q}\boldsymbol{\Lambda}^{k}\boldsymbol{Q}^{\top}=\sum_{i=1}^{k}\lambda_{i}^{n}\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\top},$

where $\boldsymbol{v}_{i}$ is the normalized eigenvector of $\lambda_{i}$.
Laplacian $\boldsymbol{S}$ always has an eigenvalue $1$ with unique associated
eigenvector $\boldsymbol{\widetilde{{D}}}^{\frac{1}{2}}\boldsymbol{e}^{\top}$,
and all other eigenvalues $\lambda$ satisfy $|\lambda|<1$. Thus, as
$k\to\infty$,
$\boldsymbol{S}^{k}\to\Phi\left(\boldsymbol{\widetilde{{D}}}^{\frac{1}{2}}\boldsymbol{e}^{\top}\right)\left(\Phi\left(\boldsymbol{\widetilde{{D}}}^{\frac{1}{2}}\boldsymbol{e}^{\top}\right)\right)^{\top}=\boldsymbol{\Pi}$.

Theorem 1 demonstrates that over-smoothing is inevitable in very deep models,
where $\boldsymbol{S}^{k}$ converges to $\boldsymbol{\Pi}$. In this scenario,
only the degree information of graph $G$ is retained.

Another important thing to consider is the convergence rate. From the above
deduction, the convergence rate is associated with the largest eigenvalue of
$\boldsymbol{S}$ other than $1$. If we view from another angle and look at how
the features of each node $\boldsymbol{v}_{i}$ in GCN is aggregated with its
local neighbors:

${\mathbf{h}}_{i}^{(k)}\leftarrow\frac{1}{d_{i}+1}\mathbf{h}_{i}^{(k-1)}+\sum_{i=1}^{N}\frac{a_{ij}}{\sqrt{\left(d_{i}+1\right)\left(d_{j}+1\right)}}\mathbf{h}_{j}^{(k-1)}.$

The above propagation suggests that the higher the node degree $d_{i}$ is, the
quicker feature $\boldsymbol{h}_{i}$ would converge. Thus we have the
following claim:

##### Claim 1

Nodes with higher degree $d_{i}$ are more likely to suffer from over-
smoothing.

###  3.2 Quantifying Smoothness: a Topological View

In addressing the over-smoothing effect, many papers have proposed their own
metric for smoothness, either to quantify and prove their hypothesis, or to
validate the effectiveness of their method. JKNet [Xu et al., 2018] defined
Influence Score to measure the sensitivity of node $x$ to node $y$, and uses
the Influence Distribution to capture the relative influences of all other
nodes. PairNorm [Zhao & Akoglu, 2020] focuses on node-wise smoothing and
feature-wise smoothing, and defined two metrics for smoothness: rol-diff and
col-diff. [Chen et al., 2020] proposes Mean Average Distance (MAD). MAD
reflects the smoothness of graph representation by calculating the mean of the
average distances between nodes. However, these metrics are generally task-
specific and incompatible to further theoretical analysis.

By contrast, the metric proposed by [Oono & Suzuki, 2020] provides a general
framework for measuring smoothness, which solely relies on the topological
information of the underlying graph $G$. Denote the maximum singular value of
convolution kernel $\boldsymbol{\Theta_{l}}$ by $s_{l}$ and set
$s:=\sup_{l\in\mathbb{N}_{+}}s_{l}$. Denote the distance induced as the
Frobenius norm from $\boldsymbol{X}$ to $\mathcal{M}$ by
$d_{\mathcal{M}}(\boldsymbol{X}):=\inf_{\boldsymbol{Y}\in\mathcal{M}}\|\boldsymbol{X}-\boldsymbol{Y}\|_{\mathrm{F}}$,
where
$\mathcal{M}:=\left\\{\boldsymbol{E}\boldsymbol{C}\mid\boldsymbol{C}\in\mathbb{R}^{M\times
C}\right\\}$, and $\boldsymbol{E}$ is the eigenspace associated with
$\lambda_{N-M},\lambda_{N-M+1},\cdots,\lambda_{N}$. We define the
$\epsilon$-smoothing metric:

##### Definition 1

($\epsilon$-smoothing) If there exists a layer $L$, such that for any hidden
layer $l$ beyond $L$, output feature $\boldsymbol{H}^{(l)}$ has a distance
smaller than $\epsilon$ w.r.t. subspace $\mathcal{M}$, we call the GCN suffers
from $\epsilon$-smoothing, i.e.,

$\exists L,\forall l\geq L,d_{\mathcal{M}}(\boldsymbol{H^{(l)}})<\epsilon.$

From [Oono & Suzuki, 2020], we have the following lemma:

##### Lemma 1

Let $\lambda_{1}\leq\cdots\leq\lambda_{N}$ be the eigenvalues of graph
Laplacian $\boldsymbol{S}$, sorted in ascending order. Suppose the
multiplicity of the largest eigenvalue $\lambda_{N}=1$ is $M(\leq N)$, i.e.
$\lambda_{N-M}<\lambda_{N-M+1}=\cdots=\lambda_{N}=1$. The second largest
eigenvalue is defined as

$\lambda:=\max_{n=1}^{N-M}\left|\lambda_{n}\right|<\left|\lambda_{N}\right|.$

Then we have $\lambda<\lambda_{N}=1$, and

$d_{\mathcal{M}}\left(\boldsymbol{H}^{(l)}\right)\leq s_{l}\lambda
d_{\mathcal{M}}\left(\boldsymbol{H}^{(l-1)}\right).$

If all the kernel $\boldsymbol{\Theta_{l}}$ have been initialized such that
$s_{l}<1$, we have $s_{l}\lambda<1$, and the output feature
$\boldsymbol{H}^{(l)}$ exponentially approaches $\mathcal{M}$ w.r.t. layer
depth $l$. We derive Theorem 2 from Lemma 1:

##### Theorem 2

If $s\lambda<1$, then $\epsilon$-smoothing would happen whenever layer depth
$l$ satisfies

$l\geq\hat{l}=\left\lceil\frac{\log\frac{\epsilon}{d_{\mathcal{M}}(\boldsymbol{H^{(0)}})}}{\log(s\lambda)}\right\rceil.$

Proof: From Lemma 1, we have

$\displaystyle d_{\mathcal{M}}\left(\boldsymbol{H}^{(l)}\right)$
$\displaystyle\leq s_{l}\lambda
d_{\mathcal{M}}\left(\boldsymbol{H}^{(l-1)}\right)$
$\displaystyle\leq\left(\prod_{i=1}^{l}s_{i}\right)\lambda^{l}d_{\mathcal{M}}(\boldsymbol{H^{(0)}})$
$\displaystyle\leq s^{l}\lambda^{l}d_{\mathcal{M}}(\boldsymbol{H^{(0)}})$
$\displaystyle<\epsilon.$

The inequality is equivalent to

$l>\frac{\log\frac{\epsilon}{d_{\mathcal{M}}(\boldsymbol{H^{(0)}})}}{\log(s\lambda)}.$

Taking the ceiling on RHS, we have $l\geq\hat{l}$.

##  4 Main Result: Factors Contributing to Over-smoothing

###  4.1 Upper Bound for the Occurrence of Over-smoothing

When would the exponential over-smoothing occur? According to Theorem 2, We
only need to guarantee that $s\lambda<1$. [Oono & Suzuki, 2020] has studied
the issue on Erdos–Renyi graph $G_{N,p}$. Here we study the issue in a more
generalized setting.

##### Theorem 3

For $N$-order graph $G$ with no isolated nodes, denote its largest node degree
as $d_{max}$, and denote its diameter as $D$. The GCN satisfies the condition
of Theorem 2, i.e. $s\lambda<1$, providing that

$s<(1-\frac{4}{NDd_{max}})^{-1}.$

Proof: We carry out our discussion on graph $\widetilde{G}$, which adds a
self-loop to each node in $G$.

First, we consider the smallest eigenvalue other than $0$ of the unnormalized
Laplacian
$\boldsymbol{L}=\boldsymbol{\widetilde{D}}-\boldsymbol{\widetilde{A}}$,
denoted as $\lambda(\widetilde{G})$. $\lambda(\widetilde{G})$ is the famous
algebraic connectivity (Fiedler eigenvalue, [Fiedler, 1973]). According to
[Mohar, 1991, p. 25], $\lambda(\widetilde{G})$ is bounded by

$\lambda(\widetilde{G})\geq\frac{4}{ND}.$

Then we consider the relation between $\lambda(\widetilde{G})$ and $\lambda$.
According to [Cavers, 2010], eigenvalues of the normalized Laplacian
$\boldsymbol{S}=\boldsymbol{\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}}$
is bounded by their corresponding eigenvalues in $\boldsymbol{L}$ and
$d_{max}$:

$\lambda_{k}(\boldsymbol{S})\leq
1-\frac{\lambda_{k}(\boldsymbol{L})}{d_{max}},$

where $\lambda_{k}(\boldsymbol{S})$ is the $k$-th largest eigenvalue of
$\boldsymbol{S}$, and $\lambda_{k}(\boldsymbol{L})$ is the $k$-th smallest
eigenvalue of $\boldsymbol{L}$. Because $\lambda$ and $\lambda(\widetilde{G})$
are corresponding eigenvalues, take them into the above inequality to derive

$\displaystyle\lambda$ $\displaystyle\leq
1-\frac{\lambda(\widetilde{G})}{d_{max}}$ $\displaystyle\leq
1-\frac{4}{NDd_{max}}.$

It suffices to show $s\lambda<1$ if we set

$s<(1-\frac{4}{NDd_{max}})^{-1}.$

Thus, the condition of Theorem 2 is satisfied, and exponential over-smoothing
could happen in this scenario.

###  4.2 What Factors Contribute to Over-smoothing?

##### Large and Dense Graphs

Large and dense graphs suffer from over-smoothing. The conclusion is in line
with [Oono & Suzuki, 2020], which formulates the issue on Erdos–Renyi graphs.
It also confirms the sensibility of graph sparsification methods for combating
over-smoothing, e.g. DropEdge [Rong et al., 2020].

##### Small-World Graphs

Small-world graphs, with $D\propto log\ N$, have already achieved relatively
high performance on GCNs with only $2\sim 3$ layers, since these $2\sim 3$
hops are sufficient to aggregate neighboring information from a large portion
of the whole graph. By contrast, tasks like Point Cloud Classification
requires deeper convolutions to capture long-range information.

##### GCN with Residual Connection

In theory, adding residual connections alone cannot address the over-smoothing
issue. If we regard graph convolution as a Markov process [Oono & Suzuki,
2020], the residual connection only leads to a lazy version of the Markov
process. The graph Laplacian would still converge, as is shown in Theorem 1.
Effective versions of residual connections will be discussed in the next
section.

##  5 Methods for Alleviating Over-smoothing

###  5.1 Leveraging between Different Convolution Depths

##### DAGNN

[Liu et al., 2020] This SGC-based [Wu et al., 2019] work is straight-forward,
simple and elegant. With insight from Claim 1 that node features are smoothed
at different rates w.r.t. node degree, DAGNN stacks up the features output
from different convolution depths. By adaptively adjusting these features,
DAGNN exploits the advantage of deeper Laplacian convolutions without
suffering from performance degradation. The adaptive adjustment process of
DAGNN is shown above, where $\boldsymbol{s}$ is a trainable projection vector.

$\displaystyle\boldsymbol{Z}=\text{MLP}(\boldsymbol{\Theta});$
$\displaystyle\boldsymbol{H}_{l}=\boldsymbol{S}^{l}\boldsymbol{Z},l=1,2,\cdots,k;$
$\displaystyle\boldsymbol{H}=\text{stack}(\boldsymbol{Z},\boldsymbol{H}_{1},\cdots,\boldsymbol{H}_{k});$
$\displaystyle\boldsymbol{S}=\sigma(\boldsymbol{Hs}).$

###  5.2 Graph Sparsification

##### DropEdge

[Rong et al., 2020] As has been discussed in Section 4.2, graph sparsification
can slow down the convergence rate of over-smoothing by reducing information
passage between layers. DropEdge randomly removes a certain number of edges
from the input graph at each training epoch, and can be equipped to many other
backbone models. The removal of edges in DropEdge is dynamic and layer-wise:

$\boldsymbol{H}^{(l+1)}=\sigma\left(\mathfrak{N}\left(\boldsymbol{A}\odot\boldsymbol{Z}^{(l)}\right)\boldsymbol{H}^{(l)}\boldsymbol{\Theta}^{(l)}\right),$

where $\boldsymbol{Z}^{(l)}$ is the binary random mask, and
$\mathfrak{N}(\cdot)$ is the normalization operator, i.e.
$\mathfrak{N}(\boldsymbol{A})=\boldsymbol{I}_{N}+\boldsymbol{D}^{-1/2}\boldsymbol{A}\boldsymbol{D}^{-1/2}.$

Other ’smarter’ ways of dropping edges include Graph DropConnect (GDC)
[Hasanzadeh et al., 2020], which drops edges both layer-wise and channel-wise.
Also, NeuralSparse [Zheng et al., 2020] uses neural networks to drop out
edges.

###  5.3 Adding Residual Connections

##### GCNII

[Chen et al., 2020] This is the first work that successfully trains deep GCNs
on knowledge graphs, with up to $64$ layers of Laplacian. The propagation rule
of GCNII is

$\boldsymbol{H}^{(\ell+1)}=\sigma\left(\left(\left(1-\alpha_{\ell}\right){\boldsymbol{S}}\boldsymbol{H}^{(\ell)}+\alpha_{\ell}\boldsymbol{H}^{(0)}\right)\left(\left(1-\beta_{\ell}\right)\boldsymbol{I}_{n}+\beta_{\ell}\boldsymbol{\Theta}^{(\ell)}\right)\right)$

The identity mapping
($\left(1-\beta_{\ell}\right)\boldsymbol{I}_{n}+\beta_{\ell}\boldsymbol{\Theta}^{(\ell)}$)
resembles that of ResNet, yet the initial residual connection
($\left(1-\alpha_{\ell}\right){\boldsymbol{S}}\boldsymbol{H}^{(\ell)}+\alpha_{\ell}\boldsymbol{H}^{(0)}$)
is the highlight. By integrating the most ’unsmooth’ layer
$\boldsymbol{H}^{(0)}$ during each round of propagation, GCNII circumvents the
pitfall described in Theorem 1. In fact, the output feature can still carry
information from both the input feature and the graph structure, even as
$K\to\infty$, which is guaranteed by Theorem 4.

##### Theorem 4

A $K$-layer GCNII can express a $K$ order polynomial filter
$\left(\sum_{\ell=0}^{K}\theta_{\ell}\boldsymbol{\tilde{L}}^{\ell}\right)\boldsymbol{x}$
with arbitrary coefficients.

According the above theorem, by fine-tuning the hyper-parameters
$\alpha_{\ell}$ and $\beta_{\ell}$, GCNII can well preserve node features even
at high depths. The tuning process would be tedious for a deep network,
though.

##  6 Future Work

Although Theorem 3 has yielded much theoretical insight into over-smoothing, a
tighter bound w.r.t. node features like number of nodes, diameter and sparsity
is direly needed. This objective can be better served with comprehensive
experiments measuring the effects of those factors on over-smoothing. Also,
the interesting properties of residual architectures like GCNII call for
further theoretical analysis.

#### Acknowledgments

This paper is a term project for the Machine Learning course at Peking
University, which is taught by Prof. Liwei Wang. The author wishes to express
his sincere gratitude to Prof. Wang, who gives him this invaluable opportunity
to probe into the exciting world of GCNs.

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# Estimation with ultimate quantum precision of the transverse displacement
between two photons via two-photon interference sampling measurements

Danilo Triggiani<EMAIL_ADDRESS>School of Mathematics and
Physics, University of Portsmouth, Portsmouth PO1 3QL, UK  Vincenzo Tamma
<EMAIL_ADDRESS>School of Mathematics and Physics, University of
Portsmouth, Portsmouth PO1 3QL, UK  Institute of Cosmology and Gravitation,
University of Portsmouth, Portsmouth PO1 3FX, UK

(August 28, 2024)

###### Abstract

We present a quantum sensing scheme achieving the ultimate quantum sensitivity
in the estimation of the transverse displacement between two photons
interfering at a balanced beam splitter, based on transverse-momentum sampling
measurements at the output. This scheme can possibly lead to enhanced high-
precision nanoscopic techniques, such as super-resolved single-molecule
localization microscopy with quantum dots, by circumventing the requirements
in standard direct imaging of cameras resolution at the diffraction limit, and
of highly magnifying objectives. Interestingly, the ultimate spatial precision
in nature is achieved irrespectively of the overlap of the two displaced
photonic wavepackets. This opens a new research paradigm based on the
interface between spatially resolved quantum interference and quantum-enhanced
spatial sensitivity.

###### pacs:

Valid PACS appear here

The Hong-Ou-Mandel (HOM) effect Hong  _et al._ (1987); Shih and Alley (1988)
is an emblematic quantum phenomenon that proved to be instrumental for the
development of novel quantum technologies Bouchard  _et al._ (2021). When two
identical photons impinge on the two faces of a balanced beam splitter, and
photodetectors are employed at both the outputs, no coincidence detection is
recorded, as the two photons always ’bunch’ in the same output channel. This
is caused by the quantum interference occurring between the two possible, but
indistinguishable paths undertaken by the two identical photons through the
beam splitter. If instead some distinguishability between the photons is
introduced, caused for example by a relative time delay, different
polarizations or central frequencies, coincidence events can be observed, and
the coincidence rate changes with the distinguishability between the photons
Hong  _et al._ (1987); Shih and Alley (1988); Bouchard  _et al._ (2021). For
this reason, HOM interferometry has been largely employed for high-precision
measurements, e.g., of optical lengths and time delays Hong  _et al._ (1987);
Lyons  _et al._ (2018); Chen  _et al._ (2019), polarisations Harnchaiwat  _et
al._ (2020); Sgobba  _et al._ (2023), and for quantum-enhanced imaging
techniques, such as quantum optical coherence tomography Abouraddy  _et al._
(2002); Nasr  _et al._ (2009). Furthermore, the analysis of the bounds on the
precision achievable in an estimation protocol given by the Cramér-Rao bound
Cramér (1999); Rohatgi and Saleh (2000), and of the ultimate precision
achievable in nature through the quantum Cramèr-Rao bound Helstrom (1969);
Holevo (2011), has become a useful tool to determine the sensitivity of two-
photon interferometry techniques for metrological applications Lyons  _et al._
(2018); Scott  _et al._ (2020); Fabre and Felicetti (2021); Johnson  _et al._
(2023).

Interestingly, it has been recently shown that inner-variables resolved two-
photon interference, in which two delayed or frequency-shifted photons
impinging on the two faces of the beam splitter are detected by frequency- or
time-resolving detectors respectively, circumvents the requirement of large
overlap in the photonic wavepackets typical of HOM interferometry Legero  _et
al._ (2003, 2004); Tamma and Laibacher (2015); Jin  _et al._ (2015); Yepiz-
Graciano  _et al._ (2020); Triggiani  _et al._ (2023); Wang  _et al._ (2018);
Hiemstra  _et al._ (2020); Prakash  _et al._ (2021). Indeed, this technique
allows us to observe beating, i.e. oscillations, in probabilities of
coincidence and bunching of the two photons, with a period that is inversely
proportional to the difference in the colours or in the incidence times at the
beam splitter, hence preserving its sensitivity also in the case of non-
overlapping wavepackets Triggiani  _et al._ (2023). Another practical
advantages of inner-variables resolved two-photon interference are the
possibility to reconstruct the interference pattern without slow mechanical
scannings Yepiz-Graciano  _et al._ (2020).

Two-photon interference has also been performed in the spatial domain, i.e.
while varying transversal properties of the two photonic wavepackets, for
example slightly rotating the momentum of the photons before they interfere Ou
and Mandel (1989); Kim  _et al._ (2006), manipulating the spatial overlap
between the photons Lee and van Exter (2006), or simultaneously introducing a
temporal delay to observe spatio-temporal coherence properties of the two-
photon state Lee and van Exter (2006); Devaux  _et al._ (2020). Spatial HOM
interferometry so far has mostly been employed to assess the spatial coherence
of highly entangled photons, e.g., produced by spontaneous parametric down-
conversion. However, to the best of our knowledge, the metrological potential
of spatial two-photon interference for high precision imaging and sensing
applications has not been investigated yet. A study in this direction is made
even more compelling, for an experimental and technological point of view, due
to the recent development of high-precision nanoscopic techniques that already
employ single-photon emitters and single-photon cameras Hanne  _et al._
(2015); Bruschini  _et al._ (2019); Urban  _et al._ (2021).

Figure 1: Two quasi-monochromatic photons, centred in positions $x_{01}$ and
$x_{02}$ in their respective transverse planes, impinge onto the two faces of
a balanced beam splitter (BS), so that the displacement between the probe
photon $\mathrm{P}_{2}$ and the symmetric image $\mathrm{P}_{1}^{\prime}$ of
the reference photon $\mathrm{P}_{1}$ is $\Delta x=x_{01}-x_{02}$. The two
photons are then detected by two cameras ($C_{1}$ and $C_{2}$) in the far-
field regime, which resolve their transverse momenta $k$ and $k^{\prime}$ for
joint detections either in the same (bunching events) or opposite (coincidence
events) output channels.

This work introduces a quantum interference technique based on spatially
resolved sampling measurements, and investigates its properties from the
metrological point of view. We show that resolving and sampling over the
transverse momenta of two interfering photons, generated for example by a
reference and a target single-photon emitter respectively, is an optimal
metrological scheme for the estimation of the transverse separation of their
wavepackets, achieving the ultimate precision given by the quantum Cramér-Rao
bound. This can be done by employing two cameras that spatially sample over
all the possible two-photon interference events in the far field, and
simultaneously record whether the two photons ended up in the same or
different output channels of the beam splitter (see Figure 1). This can
therefore be seen as an interesting metrological application for spatial
estimation of multiboson correlation sampling based on inner-variable sampling
measurements Laibacher and Tamma (2015); Tamma and Laibacher (2016, 2023). In
particular, we show that the precision achieved is independent of the
separation between the spatial wavepackets of the two photons that we aim to
estimate, which makes this technique relevant for localization and tracking
applications. Furthermore, the precision of this scheme can be, in principle,
increased arbitrarily, for any fixed displacement between the photons, by
employing photons with broader and broader transverse-momentum distributions.
Since our technique does not involve a direct detection of the single photons
in the position domain, it removes the need of high-resolution cameras capable
to directly resolve sources at the diffraction limit. As a result, this
technique does not require in principle highly magnifying objectives,
typically employed in super-resolution localization microscopy techniques,
therefore decreasing the sources of aberration Ober  _et al._ (2004); Lelek
_et al._ (2021). Moreover, for the estimation of small separations of the two
wavepackets, such that these are mostly overlapping, the resolving cameras can
be replaced with bucket detectors that only count coincidences and bunching
events, without affecting the sensitivity of the scheme. Other than single
molecule localization nanoscopy, this technique finds possible applications in
astrophysical bodies localization, or for the measurement of any transverse
displacement of a probe single-photon beam with respect to a reference beam,
e.g. caused by refraction or by optical devices such as tuneable beam
displacers Salazar-Serrano  _et al._ (2015).

#### Experimental setup.

We consider pairs of quasi-monochromatic photons, with wavenumber $k_{0}$,
produced by two independent sources impinging on the two faces of a balanced
beam-splitter, and then detected by two cameras positioned at the beam-
splitter outputs in the far-field regime at a distance $d$ from their sources.
For simplicity, we will only describe one transverse dimension of the setup,
but the same analysis can be easily generalised to the two-dimensional case.
We suppose that the transverse position of the $S$-th photon is described by a
wavepacket $\psi_{S}(x)=\psi(x-x_{0,S})$ centred around the position
$x_{0,S}$, with $S=1,2$. A practical example of such setup is shown in the
schematic representation of Figure 1. We can thus write the two-photon state
as

$\ket{\Psi}=\int\mathrm{d}x_{1}\
\psi_{1}(x_{1})\hat{a}_{1}^{\dagger}(x_{1})\ket{0}_{1}\otimes\int\mathrm{d}x_{2}\
\psi_{2}(x_{2})\hat{a}_{2}^{\dagger}(x_{2})\ket{0}_{2},$ (1)

where $\hat{a}^{\dagger}_{S}(x_{S})$, with $S=1,2$, is the bosonic creation
operator associated with the $S$-th input mode of the beam-splitter at
transverse position $x_{S}$, satisfying the commutation relation
$\commutator{\hat{a}_{S}(x)}{\hat{a}^{\dagger}_{S^{\prime}}(x^{\prime})}=\delta_{SS^{\prime}}\delta(x-x^{\prime})$,
where $\delta_{SS^{\prime}}$ and $\delta(x-x^{\prime})$ denote the Kronecker
and Dirac delta respectively. For example, the displacement $\Delta
x=x_{01}-x_{02}$ that we aim to measure can be caused by refraction or any
other beam-displacing optical device Salazar-Serrano  _et al._ (2015), or it
can represent the position of a single-photon source that we want to localise,
e.g. a probe quantum dot attached to a molecule Hanne  _et al._ (2015);
Bruschini  _et al._ (2019); Urban  _et al._ (2021), with respect to an
identical reference photon emitted at a known position.

Figure 2: Examples of joint probability $P(k,k^{\prime},X)$ in Eq. (2) plotted
along the axes $k-k^{\prime}$ and $k+k^{\prime}$ for two different transverse-
momentum distributions $\absolutevalue{\varphi(k)}^{2}$, Gaussian on the left,
a) and c), and rectangular on the right, b) and d). The plots a) and b)
represent same-camera ($X=B$) detection probabilities, while c) and d) two-
cameras ($X=A$) detection probabilities. The variance of the distribution
$\absolutevalue{\varphi(k)}^{2}$ is set to $\sigma_{k}^{2}=1$, and it fixes a
natural scale for the transverse momenta $k$ and $k^{\prime}$, and for the
separation $\Delta x$, set to $\Delta x=4/\sigma_{k}$. Plots e) and f) show
such probabilities, integrated in $k+k^{\prime}$.

After impinging on the beam-splitter, each pair of photons is randomly
detected at the pixels in positions $y$ and $y^{\prime}$ either of a single or
distinct cameras. Since the detection occurs in the far-field regime, it
corresponds to resolving the transverse momenta $k=yk_{0}/d$ and
$k^{\prime}=y^{\prime}k_{0}/d$ of the two photons, i.e. the conjugate
variables to the photon transverse positions. This results in the observation
of quantum beats (see Figure 2) with periodicity inversely proportional to
$\Delta x=x_{0,1}-x_{0,2}$ in the joint probabilities

$P(k,k^{\prime},X)=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\left(1+\alpha(X)\cos((k-k^{\prime})\Delta
x)\right),$ (2)

of the two photons detected by different cameras ($X=A$), or the same camera
($X=B$), with momenta $k$ and $k^{\prime}$, where $\alpha(A)=-$ and
$\alpha(B)=+$, and $\varphi$ denotes the Fourier transform of $\psi$, i.e. the
transverse-momentum probability-amplitude distribution of the photons
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.
The estimation is thus carried out by sampling the outcomes $(k,k^{\prime},X)$
from the probability in Eq. (2). In Eq. (2), we assumed that the size $\delta
y$ of each pixel is small enough to sufficiently resolve the transverse-
momentum distributions of the two photons and the beating oscillations, with
period inversely proportional to $\Delta x$, i.e.

$\delta y\ll\frac{d}{k_{0}}\sigma_{k},\quad\delta
y\ll\frac{d}{k_{0}}\frac{1}{\Delta x},$ (3)

where $\sigma_{k}^{2}$ is the variance of the transverse-momentum distribution
of the photons. Noticeably, we will show that employing transverse-momentum
distributions $\absolutevalue{\varphi(k)}^{2}$ with large variances is
advisable, as they yield a better sensitivity. Since this technique requires
cameras only able to resolve broad transverse-momentum distributions, and
oscillations of period $\propto 1/\Delta x$, it circumvents the need to employ
objectives with high magnifying factors to resolve directly the position of
the single-photon emitters at the diffraction limit, typically required in
imaging techniques such as single-molecule localization microscopy Ober  _et
al._ (2004); Lelek  _et al._ (2021).

#### Ultimate quantum sensitivity.

Any unbiased estimator $\widetilde{\Delta x}$ associated with the estimation
of a physical property, e.g. the separation $\Delta x$ between the photon pair
in Eq. (1), has a variance $\mathrm{Var}[\widetilde{\Delta x}]$ that is
bounded from below by quantum Cramér-Rao bound, which sets the ultimate
quantum limit of the achievable precision in the estimation of $\Delta x$
Helstrom (1969); Paris (2009); Holevo (2011); Liu  _et al._ (2019). At the
same time, the precision achievable with a specific measurement scheme is also
limited by a second lower bound, namely the (classical) Cramér-Rao bound
Cramér (1999); Rohatgi and Saleh (2000). These bounds generate the chain of
inequalities

$\mathrm{Var}[\widetilde{\Delta x}]\geqslant\frac{1}{NF(\Delta
x)}\geqslant\frac{1}{NH(\Delta x)},$ (4)

where $H(\Delta x)$ is the quantum Fisher information, $F(\Delta x)$ the
(classical) Fisher information, and $N$ is the number of independent
repetitions of the measurement Cramér (1999); Helstrom (1969). Importantly,
for a given measurement scheme, the first inequality in Eq. (4) can always be
asymptotically saturated for large $N$ employing the maximum-likelihod
estimation Cramér (1999); Rohatgi and Saleh (2000). Therefore, a measurement
scheme is (asymptotically) optimal when the Fisher information associated with
it equates the quantum Fisher information.

Indeed, we find out that our measurement scheme is optimal, and in particular

$F(\Delta x)=H(\Delta x)=2\sigma^{2}_{k},$ (5)

with the Fisher information directly proportional to the variance
$\sigma_{k}^{2}$ of the transverse-momentum distribution
$\absolutevalue{\varphi(k)}^{2}$
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.
Furthermore, a number of sampling measurements only of the order of $\sim
1000$ is enough to reach the desired sensitivity through maximum likelihood
estimation, without the need of fully retrieving the output probability
distribution with a large number of measurements Triggiani  _et al._ (2023).

Remarkably, the ultimate precision expressed by the quantum Fisher information
in Eq. (5) achieved with our technique is independent of the transverse
separation $\Delta x$ between the two sources, regardless of the overlap of
the two photons wavepackets in the space domain, and can be, in principle,
arbitrarily increased by increasing the transverse-momentum variance
$\sigma_{k}^{2}$ of the photons. Moreover, in addition to yielding a greater
sensitivity, larger variances $\sigma_{k}^{2}$ in the transverse-momentum also
reduce the resolution requirements in Eq. (3) so that magnifying objectives
are not required.

Interestingly, this interferometric technique is completely unaffected by the
lack of overlap between the spatial wavepackets of the two photons. Indeed,
performing transverse-momentum resolving detections, i.e., resolving the
variables in the conjugate domain to the position, renders the detectors
“blind” to the position of emission of the two photons, and thus it enables
the observation of two-photon quantum interference. This feature is in stark
contrast with standard non-resolving two-photon interference, where the
distinguishability of photons with non-overlapping wavepackets is not erased
at the detectors, hindering the precision of the estimation
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.

#### Fisher information analysis.

Figure 3: Sum $f(k,k^{\prime},A;\Delta x)+f(k,k^{\prime},B;\Delta x)$ of the
contributions from the outcomes $(k,k^{\prime},A)$ and $(k,k^{\prime},B)$ to
the Fisher information found in Eq. (7) for Gaussian (left) and uniform
(right) transverse-momentum distributions $\absolutevalue{\varphi(k)}^{2}$
with unit variance $\sigma_{k}^{2}=1$, which fixes a natural unit for the
transverse momenta $k$ and $k^{\prime}$. Most of the information on $\Delta x$
can be obtained by sampling measurements in the transverse momenta $k$ and
$k^{\prime}$ only in a range of $\sim\\!3\sigma_{k}$, and it is concentrated
around two maxima at specific values of $k-k^{\prime}$. Noticeably, these
plots are independent of $\Delta x$.

The Fisher information $F(\Delta x)$ in Eq. (5), whose outcomes
$(k,k^{\prime},X)$ are governed by the probability distribution
$P(k,k^{\prime},X)$ in Eq. (2), can be written as the sum

$F(\Delta x)=\sum_{X=A,B}\int\mathrm{d}k\mathrm{d}k^{\prime}\
f(k,k^{\prime},X;\Delta x)$ (6)

of the contributions

$f(k,k^{\prime},X;\Delta
x)=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}\\\
\times(1-\alpha(X)\cos((k-k^{\prime})\Delta x)),$ (7)

each one representing the information yielded by the outcome
$(k,k^{\prime},X)$ of a single pair of photons
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.
The expression of the contributions $f(k,k^{\prime},X;\Delta x)$ in Eq. (7)
allows us to understand, of all the possible outcomes $(k,k^{\prime},X)$,
which ones yield more information on the separation $\Delta x$, and conversely
which ones are less informative and can be discarded with a negligible change
of the overall precision. This type of analysis is useful from an experimental
standpoint since, for example, it can help to choose efficiently the sensing
range and the position of the two cameras.

In general, the term $f(k,k^{\prime},X;\Delta x)$ strongly depends on the
shape of the transverse-momentum distribution
$\absolutevalue{\varphi(k)}^{2}$, but some interesting universal properties of
$f(k,k^{\prime},X;\Delta x)$ can be found. Firstly, the presence of the
envelope
$\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}$
as shown in Eq. (7) guarantees that, if the tails of the distribution
$\absolutevalue{\varphi(k)}^{2}$ have negligible contribution (e.g. for
distribution with finite support, or decreasing exponentially for large $k$),
then it is possible to approximate the integration domain in Eq. (6) to a few
units of $\sigma_{k}$ ($\sim 3\sigma_{k}$ for Gaussian and uniform transverse-
momentum distributions), hence limiting the required cameras sensing range
(see Figure 3). Noticeably, this envelope does not depend on the displacement
$\Delta x$, meaning that the size of the cameras does not need to be adjusted
according to the unknown separation between the two photons.

An interesting insight regarding the role of the two-camera ($X=A$) and same-
camera ($X=B$) detection events in our transverse-momentum resolving technique
is given by separately quantifying their overall contributions to the
information on $\Delta x$. These contributions can be evaluated integrating
the expression in Eq. (7) over all possible transverse momenta $k$ and
$k^{\prime}$, obtaining

$F(X;\Delta x)=\sigma_{k}^{2}-\frac{1}{2}\alpha(X)\Phi(\varphi;\Delta x),$ (8)

with

$\Phi(\varphi;\Delta x)=\int\mathrm{d}k\mathrm{d}k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}\\\
\times\cos((k-k^{\prime})\Delta x)).$ (9)

From Eq. (8), we see that the same-camera and two-camera detection events
contribute differently to the overall sensitivity, with the term
$\Phi(\varphi;\Delta x)\equiv F(A,\Delta x)-F(B,\Delta x)$ quantifying the
difference of the two contributions (see Figure 4).

Figure 4: Plots of the Fisher information $F(\Delta x)$ in Eq. (5) (solid
line), and of the contributions $F(A;\Delta x)$ (dash-dotted line) and
$F(B;\Delta x)$ (dashed lined) in Eq. (8) from joint detection at both cameras
$(X=A)$, and at a single camera $(X=B)$, normalised to $H(\Delta x)$, for
Gaussian (left) and uniform (right) transverse-momentum distributions
$\absolutevalue{\varphi(k)}^{2}$ with unit variance $\sigma_{k}^{2}=1$, which
fixes a natural unit for $\Delta x$.

For small separations $\Delta x\ll 1/\sigma_{k}$, such that the spatial
wavepackets are mostly overlapping, the difference $\Phi(\varphi;\Delta
x\simeq 0)\simeq 2\sigma_{k}^{2}$ reaches its maximum value, therefore most of
the information is obtained with joint detection events at both cameras.
Interestingly, in this regime it is possible to replace, without any loss in
sensitivity, the resolving cameras with simple bucket detectors, as the Fisher
information

$F^{\mathrm{(nr)}}(\Delta x)\simeq 2\sigma_{k}^{2}$ (10)

associated with the detection of bunching and coincidence events without
resolving the transverse momenta is approximately equal to the quantum Fisher
information $H(\Delta x)$
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.

In the opposite regime of large separations $\Delta x\gg 1/\sigma_{k}$, such
that essentially the two spatial wavepacket do not overlap, the beating
oscillations in Eq. (2), as well as the oscillations in Eq. (7), become
increasingly fast. In this regime, if $\absolutevalue{\varphi(k)}^{2}$ is not
highly concentrated around certain values of $k$, it can be considered
essentially constant within every small interval $2\pi/\Delta x$ corresponding
to one beating period. In this case, the integral $\Phi(\varphi,\Delta x)$ in
Eq. (9) vanishes
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl,
and thus

$F(A,\Delta x)=F(B,\Delta x)=\sigma_{k}^{2}.$ (11)

Noticeably, since the two-camera and same-camera detection events yield the
same amount of information for large $\Delta x$, it is possible to resolve
only one of these two types of events, e.g. only the two-camera events,
reducing the Fisher information by only a factor of two
[SeeSupplementalMaterialat(URLwillbeinsertedbypublisher)forthederivationsof\lx@cref{creftypeplural~refnum}{eq:Prob};
\lx@cref{refnum}{eq:QFI}; \lx@cref{refnum}{eq:FIContr};
\lx@cref{refnum}{eq:FINRMain}and\lx@cref{refnum}{eq:FIABLarge}whichincludes\cite[cite]{\@@bibref{AuthorsPhrase1YearPhrase2}{Glauber1963}{\@@citephrase{(}}{\@@citephrase{)}}}]Suppl.

#### Conclusions.

We presented a spatial quantum interference technique allowing the estimation
of the transverse separation of two single-photons beams at the ultimate
precision allowed by the quantum Cramér-Rao bound, based on transverse-
momentum resolved sampling measurements. We showed that the precision achieved
with this technique is independent of the separation that we aim to measure,
and it can be, in principle, arbitrarily increased by employing photons with
more broadly distributed transverse momenta.

These features yield a number of important implications for experimental
applications of our measurement technique. Firstly, despite its
interferometric nature, this technique achieves the same precision
irrespectively of the overlap between the two photonic wavepackets in the
spatial domain. Indeed, resolving the transverse momenta erases the spatial
distinguishability between the photons, allowing us to observe interference
also in regimes where the separation is much larger than the spatial
wavepackets widths. Secondly, the achieved ultimate precision is in principle
unbounded, in the sense that is only limited by the highest value of the
transverse-momentum distribution variance experimentally feasible with current
technologies. Thirdly, by using indirect sampling measurements which resolve
the photonic transverse momenta instead of the positions directly, this
technique removes the requirement of cameras resolution at the diffraction
limit typical of direct imaging techniques, as well as highly magnifying
objectives, decreasing the sources of aberration. Furthermore, in the regime
of transverse separation smaller than the spatial width of the wavepackets,
the two cameras can even be replaced with non-resolving bucket detectors
without any loss of sensitivity.

These results shed new light on the metrological power of two-photon spatial
interference and can pave the way to new high-precision sensing techniques.
This work has the potential to enhance super-resolution imaging techniques
that already employ single-photon sources as probes for the localization and
tracking of biological samples, such as single-molecule localization
microscopy with quantum dots Hanne  _et al._ (2015); Bruschini  _et al._
(2019); Urban  _et al._ (2021). Other possible applications can be found in
the development of quantum sensing techniques for high-precision refractometry
and astrophysical bodies localization. Finally, such a technique could, in
principle, be also extended to the estimation of spatial parameters in a more
general N-photon network by performing sampling measurements in the photonic
transverse momenta.

## Acknowledgements

VT acknowledges support from the Air Force Office of Scientific Research under
award number FA8655-23-1-7046.

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##  Appendix A Joint detection probabilities in Eq. (2)

In this section we will evaluate the probability distribution
$P(k,k^{\prime},X)$ in Eq. (2) in the main text associated with the event of
detecting the two photons with transverse-momenta $k$ and $k^{\prime}$ in the
same ($X=B$) and in different ($X=A$) cameras.

Since the detection occurs by means of cameras resolving the position of the
photons in the far-field regime, we will first evaluate the second-order
correlation functions Glauber (1963)

$G^{(2)}_{CC^{\prime}}(y,y^{\prime})=\Tr[\outerproduct{\Psi}{\Psi}\hat{E}_{C}(y)\hat{E}_{C^{\prime}}(y^{\prime})\hat{E}^{+}_{C}(y)\hat{E}^{+}_{C^{\prime}}(y^{\prime})],$
(12)

associated with the detection of the two photons at the pixels at the
positions $y$ and $y^{\prime}$ of the cameras $C$ and $C^{\prime}$
respectively, with $C,C^{\prime}=C_{1},C_{2}$ and we denote with

$\hat{E}^{+}_{C}(y)=\sum_{S=1,2}\int\mathrm{d}x_{S}\
g(x_{S},S;y,C)\hat{a}_{S}(x_{S})$ (13)

the positive frequency part of the electric field operator at position $y$ of
the camera $C$, with $\hat{E}^{-}_{C}(y)=(\hat{E}^{+}_{C}(y))^{\dagger}$ the
negative frequency part, and

$g(x,S;y,C)=\frac{\mathrm{e}^{\mathrm{i}\phi(S,C)}}{\sqrt{2}}\mathrm{e}^{\mathrm{i}k_{0}d}\mathrm{e}^{-\mathrm{i}\frac{k_{0}}{d}xy},$
(14)

is the Fraunhofer transfer function, describing the propagation of the photon
$S=1,2$ from transverse position $x$ through the beam-splitter and up to
position $y$ at the detector $C=C_{1},C_{2}$ in the far-field, where $k_{0}$
is the wavenumber, $d$ is the longitudinal distance travelled by the photons
to the detectors, $\phi$ is the phase acquired through the beam-splitter. In
Eq. (14), we have chosen a dimensionless normalisation that guarantees the
unitary propagation of the photons through the beam splitter.

To evaluate Eq. (12), we first introduce the electric field operators

$\displaystyle\hat{E}_{SC}^{+}(y)$ $\displaystyle=\int\mathrm{d}x_{S}\
g(x_{S},S;y,C)\hat{a}_{S}(x_{S}),\ S=1,2$ (15)

associated with the $S$-th photon, so that

$\hat{E}^{+}_{C}(y)=\sum_{S=1,2}\hat{E}^{+}_{SC}(y).$ (16)

We can thus expand the second-order correlation function in Eq. (12) by
employing Eq. (16), obtaining

$G^{(2)}_{CC^{\prime}}(y,y^{\prime})=\sum_{\begin{subarray}{c}S_{1},S_{2}\\\
S_{3},S_{4}\end{subarray}=1,2}\Tr[\outerproduct{\Psi}{\Psi}\hat{E}^{-}_{S_{1}C}(y)\hat{E}^{-}_{S_{2}C^{\prime}}(y^{\prime})\hat{E}^{+}_{S_{3}C}(y)\hat{E}^{+}_{S_{4}C^{\prime}}(y^{\prime})].$
(17)

Of the 16 terms in the summation in Eq. (17), we easily see that only a few
survive. Indeed, since the state $\ket{\Psi}$ has a single photon at each
input channel, the terms with $S_{1}=S_{2}$ or $S_{3}=S_{4}$ vanish. Moreover,
since the positive-frequency part of the electric field annihilates a photon,
and the negative-frequency part creates a photon, evaluating the trace in Eq.
(17) is equivalent to projecting onto the vacuum
$\outerproduct{\mathrm{vac}}{\mathrm{vac}}$, so we have

$\displaystyle G^{(2)}_{CC^{\prime}}$
$\displaystyle(y,y^{\prime})=\sum_{S_{1},S_{3}=1,2}\matrixelement{\Psi}{\hat{E}^{-}_{S_{1}C}(y)\hat{E}^{-}_{\sigma(S_{1})C^{\prime}}(y^{\prime})}{\mathrm{vac}}$
$\displaystyle\quad\times\matrixelement{\mathrm{vac}}{\hat{E}^{+}_{S_{3}C}(y)\hat{E}^{+}_{\sigma(S_{3})C^{\prime}}(y^{\prime})}{\Psi},$
$\displaystyle=\absolutevalue{\sum_{S_{1}=1,2}\matrixelement{\mathrm{vac}}{\hat{E}^{+}_{S_{1}C}(y)\hat{E}^{+}_{\sigma(S_{1})C^{\prime}}(y^{\prime})}{\Psi}}^{2}$
(18)

where $\sigma(1)=2,\sigma(2)=1$. We can now proceed with evaluating the scalar
products in Eq. (18)

$\displaystyle\matrixelement{\mathrm{vac}}{\hat{E}^{+}_{S_{1}C}(y)\hat{E}^{+}_{\sigma(S_{1})C^{\prime}}(y^{\prime})}{\Psi}=$
$\displaystyle=\int\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}x^{\prime}_{1}\mathrm{d}x^{\prime}_{2}\
g(x_{1},S_{1};y,C)g(x_{2},\sigma(S_{1});y^{\prime},C^{\prime})\psi_{1}(x^{\prime}_{1})\psi_{2}(x^{\prime}_{2})\matrixelement{\mathrm{vac}}{\hat{a}_{S_{1}}(x_{1})\hat{a}_{\sigma(S_{1})}(x_{2})\hat{a}^{\dagger}_{1}(x^{\prime}_{1})\hat{a}^{\dagger}_{2}(x^{\prime}_{2})}{\mathrm{vac}}$
$\displaystyle=\int\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}x^{\prime}_{1}\mathrm{d}x^{\prime}_{2}\
g(x_{1},S_{1};y,C)g(x_{2},\sigma(S_{1});y^{\prime},C^{\prime})\psi_{1}(x^{\prime}_{1})\psi_{2}(x^{\prime}_{2})\left(\delta_{1S_{1}}\delta(x_{1}-x_{1}^{\prime})\delta(x_{2}-x_{2}^{\prime})+\delta_{2S_{1}}\delta(x_{1}-x_{2}^{\prime})\delta(x_{2}-x_{1}^{\prime})\right)$
$\displaystyle=\delta_{1S_{1}}\int\mathrm{d}x_{1}\mathrm{d}x_{2}\
g(x_{1},1;y,C)g(x_{2},2;y^{\prime},C^{\prime})\psi_{1}(x_{1})\psi_{2}(x_{2})+\delta_{2S_{1}}\int\mathrm{d}x_{1}\mathrm{d}x_{2}\
g(x_{1},2;y,C)g(x_{2},1;y^{\prime},C^{\prime})\psi_{1}(x_{2})\psi_{2}(x_{1}).$
(19)

The integrals appearing in Eq. (19) can be written in terms of the Fourier
transform of the position wave-functions $\psi_{S}(x)$. Indeed

$\displaystyle\int\mathrm{d}x\
g(x,S;y,C)\psi_{S}(x)=\frac{\mathrm{e}^{\mathrm{i}\phi(S,C)}}{\sqrt{2}}\mathrm{e}^{\mathrm{i}k_{0}d}\int\mathrm{d}x\
\mathrm{e}^{-\mathrm{i}\frac{k_{0}}{d}xy}\psi(x-x_{0S})=\frac{\mathrm{e}^{\mathrm{i}\phi(S,C)}}{\sqrt{2}}\mathrm{e}^{\mathrm{i}k_{0}d}\mathrm{e}^{-\mathrm{i}\frac{k_{0}}{d}yx_{0s}}\varphi\left(\frac{k_{0}}{d}y\right).$
(20)

We can thus write all the second-order correlation functions in Eq. (18) by
plugging in the expressions in Eq. (19)-(20), obtaining

$\displaystyle G^{(2)}_{CC^{\prime}}(y,y^{\prime})$
$\displaystyle=\frac{1}{4}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\big{|}\mathrm{e}^{\mathrm{i}(\Phi(1,C)+\Phi(2,C^{\prime}))}\mathrm{e}^{-\mathrm{i}(kx_{01}+k^{\prime}x_{02})}+\mathrm{e}^{\mathrm{i}(\Phi(2,C)+\Phi(1,C^{\prime}))}\mathrm{e}^{-\mathrm{i}(kx_{02}+k^{\prime}x_{01})}\big{|}^{2}$
$\displaystyle=\frac{1}{4}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(2+2\real(\mathrm{e}^{\mathrm{i}(\Phi(1,C)+\Phi(2,C^{\prime})-\Phi(2,C)-\Phi(1,C^{\prime}))}\mathrm{e}^{-\mathrm{i}(k-k^{\prime})\Delta
x})),$ (21)

where we introduced the notation $k=k_{0}y/d$, $k^{\prime}=k_{0}y^{\prime}/d$
and $\Delta x=x_{01}-x_{02}$. Due to the constraint
$\Phi(1,C_{1})+\Phi(2,C_{2})-\Phi(1,C_{2})-\Phi(2,C_{1})=\pm\pi$ imposed by
the unitarity of the beam-splitter, we notice that the phase term
$\mathrm{e}^{\mathrm{i}(\Phi(1,C)+\Phi(2,C^{\prime})-\Phi(2,C)-\Phi(1,C^{\prime})}$
is always $+1$ for $C=C^{\prime}$ and $-1$ for $C\neq C^{\prime}$. We thus
obtain

$\displaystyle P(k,k^{\prime},A)$
$\displaystyle=G^{(2)}_{12}(y,y^{\prime})=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\left(1-\cos((k-k^{\prime})\Delta
x)\right),$ $\displaystyle P(k,k^{\prime},B)$
$\displaystyle=\frac{G^{(2)}_{11}(y,y^{\prime})+G^{(2)}_{22}(y,y^{\prime})}{2}=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\left(1+\cos((k-k^{\prime})\Delta
x)\right),$ (22)

where the factor $1/2$ for the same-camera probabilities is needed to
correctly take into account the symmetry for the exchange $k\leftrightarrow
k^{\prime}$, and thus to consider $P(k,k^{\prime},B)$ as defined over every
$k,k^{\prime}$ and not, e.g., only for $k\leqslant k^{\prime}$.

###  A.1 Probabilities for joint detection not resolving the transverse
momenta

If we replace the resolving cameras at the outputs of the beam splitter with
simple bucket detectors, we average out all the information about the
separation $\Delta x$ obtained by resolving the transverse momenta $k$ and
$k^{\prime}$ of the photons. In this case, we would only discriminate two
possible outcome events, i.e., photons detected by the different ($X=A$) or
same ($X=B$) detector, occurring with probabilities

$\displaystyle P(A)$ $\displaystyle=\int\differential k\differential
k^{\prime}\ P(k,k^{\prime},A)=\frac{1}{2}-\frac{1}{2}\int\differential
k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\cos((k-k^{\prime})\Delta
x)$ $\displaystyle P(B)$ $\displaystyle=\int\differential k\differential
k^{\prime}\ P(k,k^{\prime},B)=\frac{1}{2}+\frac{1}{2}\int\differential
k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\cos((k-k^{\prime})\Delta
x),$ (23)

obtained integrating the probabilities in Eq. (22). Notice that the integral
in Eq. (23) defines the shape of the HOM dip (or peak, in case of the same-
camera events probability $P(B)$), which clearly depends on the transverse-
momentum distribution $\absolutevalue{\varphi(k)}^{2}$ of the two photons.

##  Appendix B Quantum Cramér-Rao bound

To see that the quantum Fisher information for the estimation of the
separation $\Delta x=x_{01}-x_{02}$ of the two photons in the state in Eq. (1)
is the one shown in Eq. (5) in the main text, it is sufficient to notice that
the initial state is generated by a unitary translation of a state
$\ket{\Psi_{0}}$, i.e.

$\ket{\Psi}=\hat{U}(x_{01},x_{02})\ket{\Psi_{0}}=\hat{U}(x_{01},x_{02})\int\mathrm{d}x_{1}\
\psi(x_{1})\hat{a}_{1}^{\dagger}(x_{1})\ket{0}_{1}\otimes\int\mathrm{d}x_{2}\
\psi(x_{2})\hat{a}_{2}^{\dagger}(x_{2})\ket{0}_{2},$ (24)

with

$\hat{U}(x_{01},x_{02})=\exp(-\mathrm{i}x_{01}\hat{k}_{1})\otimes\exp(-\mathrm{i}x_{02}\hat{k}_{2})$
(25)

unitary evolution generated by the conjugate variables to the photon
positions, i.e., the transverse momenta

$\displaystyle\hat{k}_{i}\equiv\int\differential k\
k\,\hat{a}_{i}^{\dagger}(k)\hat{a}_{i}(k)$ (26)

and $\hat{a}_{i}(k)$ being defined as $\int\mathrm{d}x\
\mathrm{e}^{-\mathrm{i}kx}\hat{a}_{i}(x)$, for $i=1,2$. Since we can write
$x_{01}=x_{\mathrm{sum}}/2+\Delta x/2$ and $x_{02}=x_{\mathrm{sum}}/2-\Delta
x/2$, with $x_{\mathrm{sum}}=(x_{01}+x_{02})$, substituting these expressions
in Eq. (25), we see that the operators

$\hat{G}^{\pm}=\frac{\hat{k}_{1}}{2}\pm\frac{\hat{k}_{2}}{2}$ (27)

are the generators of the parameters $\Delta x$ and $x_{\mathrm{sum}}$. We
will assume that no information on the parameter $x_{\mathrm{sum}}$ is
available prior to the estimation of $\Delta x$, so that the quantum Cramér-
Rao bound on $\Delta x$ must be evaluated through the two-parameter quantum
Fisher information matrix $H$ Helstrom (1969); Paris (2009); Holevo (2011);
Liu  _et al._ (2019). In this type of scenario, i.e., for pure states and
unitary evolutions, the $2\times 2$ quantum Fisher information matrix of the
parameters $\Delta x$ and $x_{\mathrm{sum}}$ is evaluated as Liu  _et al._
(2019)

$H_{ij}=4\mathrm{Cov}_{\ket{\Psi_{0}}}(\hat{G}^{i},\hat{G}^{j}),\quad
i,j=\pm,$ (28)

where $\mathrm{Cov}_{\ket{\Psi_{0}}}$ represents the covariance over the state
$\ket{\Psi_{0}}$. Since $\hat{k}_{1}$ and $\hat{k}_{2}$ commute, clearly also
$\hat{G}^{+}$ and $\hat{G}^{-}$ commute. We thus evaluate the covariance
$\matrixelement{\Psi_{0}}{\hat{G}^{+}\hat{G}^{-}}{\Psi_{0}}-\matrixelement{\Psi_{0}}{\hat{G}^{+}}{\Psi_{0}}\matrixelement{\Psi_{0}}{\hat{G}^{-}}{\Psi_{0}}=0$,
and the variances $\matrixelement{\Psi_{0}}{\hat{G}^{\pm
2}}{\Psi_{0}}-\matrixelement{\Psi_{0}}{\hat{G}^{\pm}}{\Psi_{0}}^{2}=\sigma_{k}^{2}/2$,
with $\sigma_{k}^{2}$ the transverse-momentum variance of each photon. The
quantum Fisher information matrix in Eq. (28) associated with the estimation
of $x_{\mathrm{sum}}$ and $\Delta x$ is thus diagonal, meaning that these two
parameters can be estimated simultaneously and independently, and in
particular the quantum Fisher information associated with $\Delta x$ is

$H(\Delta
x)=4(\matrixelement{\Psi_{0}}{(\hat{G}^{-})^{2}}{\Psi_{0}}-\matrixelement{\Psi_{0}}{\hat{G}^{-}}{\Psi_{0}}^{2})=2\sigma_{k}^{2},$
(29)

as shown in Eq. (4) in the main text.

##  Appendix C Fisher information in Eq. (5) and its contributions in Eq. (7)

The Fisher information associated with the estimation of the separation
$\Delta x$ through our transverse-momentum resolving technique for generic
photon distinguishability is defined as Cramér (1999); Rohatgi and Saleh
(2000)

$F(\Delta x)=\mathbb{E}\left[\left(\frac{\mathrm{d}}{\mathrm{d}\Delta
x}\log(P(k,k^{\prime},X))\right)^{2}\right]\equiv\sum_{X=A,B}\int\mathrm{d}k\mathrm{d}k^{\prime}\
P(k,k^{\prime},X)\left(\frac{\mathrm{d}}{\mathrm{d}\Delta
x}\log(P(k,k^{\prime},X))\right)^{2},$ (30)

with $P(k,k^{\prime},X)$ found in Eq. (2) in the main text. From Eq. (30), we
see that the contributions to the Fisher information $f(k,k^{\prime},X;\Delta
x)$ as defined in Eqs. (6) in the main text can be written as

$f(k,k^{\prime},X;\Delta x)=\frac{\left(\frac{d}{d\Delta
x}P(k,k^{\prime},X)\right)^{2}}{P(k,k^{\prime},X)}.$ (31)

We now evaluate the numerator of Eq. (31), obtaining

$\left(\frac{d}{d\Delta
x}P(k,k^{\prime},X)\right)^{2}=\frac{1}{4}\absolutevalue{\varphi(k)}^{4}\absolutevalue{\varphi(k^{\prime})}^{4}(k-k^{\prime})^{2}\sin^{2}((k-k^{\prime})\Delta
x),$ (32)

from which, recalling the expression of the probability $P(k,k^{\prime},X)$ in
Eq. (2), can be used to obtain from Eq. (31) the expression

$\displaystyle f(k,k^{\prime},X;\Delta x)$
$\displaystyle=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}\frac{\sin^{2}((k-k^{\prime})\Delta
x)}{1+\alpha(X)\cos((k-k^{\prime})\Delta x)}$
$\displaystyle=\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}(1-\alpha(X)\cos((k-k^{\prime})\Delta
x))$ (33)

which respectively coincide with Eq. (7) in the main text.

##  Appendix D Fisher information for non-resolving detectors

As a comparison, we evaluate here the Fisher information associated with the
estimation of $\Delta x$ through measurements that do not resolve the
transverse momenta of the photons, i.e., arising from the probabilities
evaluated in Eq. (23). In this case, the Fisher information is a sum of only
two contributions

$F^{\mathrm{(nr)}}(\Delta
x)=\sum_{X=A,B}P(X)\left(\frac{\differential}{\differential\Delta x}\log
P(X)\right)^{2}=\sum_{X=A,B}\frac{1}{2}\frac{\left(\int\differential
k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})\sin((k-k^{\prime})\Delta
x)\right)^{2}}{1+\alpha(X)\int\differential k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\cos((k-k^{\prime})\Delta
x)}\\\ =\frac{\left(\int\differential k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})\sin((k-k^{\prime})\Delta
x)\right)^{2}}{1-\left(\int\differential k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}\cos((k-k^{\prime})\Delta
x)\right)^{2}},$ (34)

and it depends on the transverse momentum distribution
$\absolutevalue{\varphi(k)}^{2}$.

###  D.1 Small separations regime

In the regime of small separations such that the two spatial wavepackets are
mostly overlapping, we can neglect higher terms of $\Delta x$ in Eq. (34),
obtaining

$F^{\mathrm{(nr)}}(\Delta x)\simeq\frac{\Delta x^{2}\left(\int\differential
k\differential k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}\right)^{2}}{\Delta
x^{2}\int\mathrm{d}k\mathrm{d}k^{\prime}\
\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}}=2\sigma_{k}^{2}\equiv
H(\Delta x),$ (35)

which is equal to the quantum Fisher information in Eq. (5) in the main text.
Therefore, for small separations, i.e. such that $\Delta x\ll 1/\sigma_{k}$,
it is even possible to replace the resolving cameras with bucket detectors
without any loss of sensitivity.

##  Appendix E Large separations regime

###  E.1 Fisher information for spatially resolved measurements

In this section we will evaluate the expression of $F(X,\Delta x)$ in Eq. (11)
in the main text in the regime of large separations $\Delta x$, i.e. for
mostly non-overlapping spacial wavepackets. Then, we will also show that, in
this same regime, the Fisher information for non-resolving technique
$f^{\mathrm{nr}}(\Delta x)$ in Eq. (34) vanishes. The contribution $f(X,\Delta
x)$ in Eq. (7) in the main text can be rewritten, after a change of
integration variables $k-k^{\prime}=\delta$, $(k+k^{\prime})/2=K$, as

$\displaystyle F(X,\Delta x)$
$\displaystyle=\int\mathrm{d}k\mathrm{d}k^{\prime}\
\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}\gamma((k-k^{\prime});\Delta
x,X)=\int\mathrm{d}\delta\ \tau(\delta)\delta^{2}\gamma(\delta;\Delta x,X)$
(36)

where

$\tau(\delta)=\frac{1}{2}\int\mathrm{d}K\absolutevalue{\varphi\left({K}+\frac{\delta}{2}\right)}^{2}\absolutevalue{\varphi\left(K-\frac{\delta}{2}\right)}^{2}$
(37)

is the integral over the average momentum $K$, and

$\gamma(\delta;\Delta x,X)=1-\alpha(X)\cos(\delta\Delta x)$ (38)

is a periodic function in $z$ of period inversely proportional to $\Delta x$,
i.e., $2\pi/\Delta x$. We can thus split the interval of integration in Eq.
(36) in sub-intervals of width $2\pi/\Delta x$, so that the overall integral
can be written as sum

$F(X;\Delta x)=\sum_{n\in\mathbb{Z}}\int_{\frac{2n\pi}{\Delta
x}}^{\frac{2(n+1)\pi}{\Delta x}}\mathrm{d}\delta\
\tau(\delta)\delta^{2}\gamma(\delta;\Delta x,X),$ (39)

with $n$ integer. If the transverse-momentum distribution
$\absolutevalue{\varphi(k)}^{2}$ does not have ‘atoms’, i.e. it has no
discrete part, and thus the same is true for $\tau(\delta)$,
$\tau(\delta)\delta^{2}$ can be eventually be considered constant within each
integration domain $[2n\pi/\Delta x,2(n+1)\pi/\Delta x]$ for large $\Delta x$.
We can thus approximate $\delta^{2}\tau(\delta)\simeq(2\pi n/\Delta
x)^{2}\tau(2\pi n/\Delta x)$ for each integral, so that

$F(X;\Delta x)\simeq\sum_{n\in\mathbb{Z}}\left(\frac{2\pi n}{\Delta
x}\right)^{2}\tau\left(\frac{2\pi n}{\Delta x}\right)\int_{\frac{2n\pi}{\Delta
x}}^{\frac{2(n+1)\pi}{\Delta x}}\mathrm{d}\delta\ \gamma(\delta;\Delta
x,X)=\sum_{n\in\mathbb{Z}}\left(\frac{2\pi n}{\Delta
x}\right)^{2}\tau\left(\frac{2\pi n}{\Delta x}\right)\frac{2\pi}{\Delta
x}\bar{\gamma},$ (40)

where $\bar{\gamma}$ is the average of $\gamma(\delta;\Delta x,X)$ over its
period, which can be evaluated from Eq. (38) to be

$\bar{\gamma}=1,$ (41)

and noticeably it does not depend on $\Delta x$ nor on $X$. We can now rewrite
the summation in Eq. (40) as an integral with the substitutions $2\pi n/\Delta
x\rightarrow\zeta$, $\sum_{n\in\mathbb{Z}}2\pi/\Delta
x\rightarrow\int\differential\zeta$ yielding

$F(X;\Delta x)\simeq\bar{\gamma}\int\mathrm{d}\zeta\
\tau(\zeta)\zeta^{2}\equiv\int\mathrm{d}k\mathrm{d}k^{\prime}\
\frac{1}{2}\absolutevalue{\varphi(k)}^{2}\absolutevalue{\varphi(k^{\prime})}^{2}(k-k^{\prime})^{2}=\sigma_{k}^{2},$
(42)

as shown in Eq. (11) in the main text.

###  E.2 Fisher information for non-resolved measurements

For $F^{\mathrm{(nr)}}(\Delta x)$ in Eq. (34) similar considerations can be
done in order to obtain the asymptotic behaviour of the Fisher information for
the estimation technique that does not resolve the transverse momenta of the
photons. Indeed, as we can see from Eq. (34), we need to obtain the
asymptotics of both integrals, one in the numerator and the other in the
denominator, appearing in $F^{\mathrm{(nr)}}(\Delta x)$. In these integrals,
the oscillating factors are, respectively, $\sin((k-k^{\prime})\Delta x)$ and
$\cos((k-k^{\prime})\Delta x)$, both of which average to zero in one
oscillation period. Repeating the same steps performed for $F(X,\Delta x)$ and
$f(X,\Delta x)$ for large $\Delta x$, which allows us to replace the
oscillating factor inside each integral with its (vanishing) average, we
easily see that $F^{\mathrm{nr}}(\Delta x)\simeq 0$ in the regime of large
$\Delta x$, meaning that a non-resolving two-photon interference technique
does not yield any information on large separations $\Delta x$.

###  E.3 Contributions to the Fisher information from the same or different
cameras

In this section we will show how the Fisher information is affected if, in a
given experiment, some possible events are undetected or neglected. We will
use the results of this section to show that, in the large separation regime,
neglecting all the same-camera (or all the two-camera) events only reduces the
associated Fisher information of a factor $2$.

Let us imagine an experiment whose outcomes follow a discrete probability, and
that we arbitrarily partition the outcomes in two groups. The case of a
continuous probability distribution can be obtained with a simple change of
notation. Each outcome has a given probability to happen that depend on an
unknown parameter $\lambda$ that we want to estimate. Thus, the probability of
a given outcome $i$ of the first group is given by $P_{1,i}(\lambda)$, while
the probability of an outcome $i^{\prime}$ of the second group is
$P_{2,i^{\prime}}(\lambda)$, with $i=1,\dots,n$,
$i^{\prime}=1,\dots,n^{\prime}$, and
$\sum_{i=1}^{n}P_{1,i}(\lambda)+\sum_{i^{\prime}=1}^{n^{\prime}}P_{2,i^{\prime}}(\lambda)=1$.
For simplicity, from now on, we will omit to explicit the dependency from
$\lambda$ of the probabilities $P_{1,i}(\lambda)\equiv P_{1,i}$,
$P_{2,i^{\prime}}(\lambda)\equiv P_{2,i^{\prime}}$.

If none of the possible outcomes are neglected, and the information on
$\lambda$ is retrieved from the outcomes of both groups, the Fisher
information $F(\lambda)$ associated with the estimation of $\lambda$ after $N$
repetitions of the experiment is, by definition,

$F(\lambda)=N\left(\sum_{i=1}^{n}\frac{(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{1,i})^{2}}{P_{1,i}}+\sum_{i^{\prime}=1}^{n^{\prime}}\frac{(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{2,i^{\prime}})^{2}}{P_{2,i^{\prime}}}\right).$
(43)

We want now to compare this Fisher information $F(\lambda)$ with the Fisher
information $F_{1}(\lambda)$ associated with the observation of only the
outcomes of the first group. If we are neglecting the outcomes of the second
group, only $N\sum_{i=1}^{n}P_{1,i}$ outcomes will be observed in average,
while the (conditioned) probability for the $i$-th event to happen is given by
$P_{1,i}/\sum_{j=1}^{n}P_{1,j}$. We can then evaluate $F_{1}(\lambda)$ as

$\displaystyle F_{1}(\lambda)$
$\displaystyle=\left(N\sum_{k=1}^{n}P_{k,i}\right)\left(\sum_{i=1}^{n}\frac{P_{1,i}}{\sum_{j=1}^{n}P_{1,j}}\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}\log(\frac{P_{1,i}}{\sum_{j=1}^{n}P_{1,j}})\right)^{2}\right)$
$\displaystyle=N\left(\sum_{i=1}^{n}P_{1,i}\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}\log(P_{1,i})-\frac{\mathrm{d}}{\mathrm{d}\lambda}\log(\sum_{j=1}^{n}P_{1,j})\right)^{2}\right)$
$\displaystyle=N\left(\sum_{i=1}^{n}\frac{(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{1,i})^{2}}{P_{1,i}}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}\sum_{i=1}^{n}P_{1,i}\right)^{2}}{\sum_{i=1}^{n}P_{1,i}}\right)$
$\displaystyle\equiv
N\left(\sum_{i=1}^{n}\frac{(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{1,i})^{2}}{P_{1,i}}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{1}\right)^{2}}{P_{1}}\right),$
(44)

where we introduced $P_{1}=\sum_{i=1}^{n}P_{1,i}$ the overall probability that
any outcome of the first group is observed. We recognise in Eq. (44) the first
term of the total Fisher information in Eq. (43), given by the sum of the
contributions of each outcome of the first group. However, this is in general
reduced by a term
$N\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}P_{1}\right)^{2}/P_{1}$ that can
be interpreted as the amount of information that is lost due to the fact that
we are neglecting all the events of the second group, and thus we do not gain
any information on the value of $P_{1}$. However, if $P_{1}$ does not depend
on $\lambda$, or in general has a negligible derivative, this term can be
neglected and the Fisher information $F_{1}(\lambda)$ is equal to the sum of
the contributions of the outcomes of the first group.

This result can be applied to our scheme. In the regime of large separations
(i.e. small spacial wavepackets overlap), the overall probability $P(X)$ in
Eq. (2) in the main text for the two photons to be in the same output port
$X=B$, or in opposite output ports $X=A$, is

$P(X)=\int\mathrm{d}k\mathrm{d}k^{\prime}\
P(k,k^{\prime},X)\simeq\frac{1}{2}.$ (45)

This correspond to the regime far from the dip in standard two-photon
interference, since the two photons are completely distinguishable and we are
averaging over all possible observable transverse momenta $k,k^{\prime}$. This
means, that, if we group the outcomes of our scheme as same-camera and two-
camera events, and neglect all the outcomes of one type, e.g. all the same-
camera events, the Fisher information will be given by Eq. (44), where we
identify $P_{1,i}\equiv P(k,k^{\prime},A)$ and
$P_{1}=\int\mathrm{d}k\mathrm{d}k^{\prime}\ P(k,k^{\prime},A)=1/2$. Since
$P_{1}$ does not depend on $\Delta x$, the Fisher information for large
separations when neglecting same-camera events is given by the sum of the two-
camera events contributions, which we found to be $F(A,\Delta x)$ in Eq. (11)
in the main text, i.e. half of the overall Fisher information $F(\Delta x)$
when no outcome is neglected. This proves that, neglecting one type of
detection events for large $\Delta x$, the Fisher information only reduces of
a factor $1/2$.