# Quantum limits for resolving Gaussian sources

Giacomo Sorelli  Laboratoire Kastler Brossel, Sorbonne Université, ENS-
Université PSL, CNRS, Collège de France, 4 Place Jussieu, F-75252 Paris,
France  Manuel Gessner  ICFO-Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860,
Castelldefels (Barcelona), Spain  Mattia Walschaers  Laboratoire Kastler
Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collège de France, 4
Place Jussieu, F-75252 Paris, France  Nicolas Treps  Laboratoire Kastler
Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collège de France, 4
Place Jussieu, F-75252 Paris, France

###### Abstract

We determine analytically the quantum Cramér-Rao bound for the estimation of
the separation between two point sources in arbitrary Gaussian states. Our
analytical expression is valid for arbitrary sources brightness, and it allows
to determine how different resources, such as mutual coherence (induced by
thermal correlations or displacement) or squeezing affect the scaling of the
ultimate resolution limit with the mean number of emitted photons. In
practical scenarios, we find coherent states of the sources to achieve quantum
optimal resolution.

Introduction.— Resolving two point sources, i.e. establishing their separation
with diffraction-limited imaging systems becomes more difficult the closer the
sources are. This intuitive statement lies at the basis of historical
resolution criteria [1, 2], which established minimal distances beyond which
diffraction renders it impossible to resolve two sources. However, in the last
decades a great number of superresolution techniques allowed to resolve
separations beyond the diffraction limit, by either intervening on the
properties of the sources [3, 4, 5] (active imaging) or of the measurements
[6, 7, 8, 9] (passive imaging).

The possibility to overcome the diffraction limit inspired the search for
ultimate resolution limits. The natural framework for this investigation is
that of quantum metrology [10, 11, 12, 13, 14]. The latter establishes that
the quantum limit on the estimation of the source separation $d$ is given by
the quantum Cramér-Rao bound $(\Delta d)^{2}\geq F_{d}^{-1}$, with $F_{d}$ the
quantum Fisher information (QFI) that quantifies the sensitivity of the
quantum state of the sources to variations of the separation $d$. The quantum
estimation approach showed that for two dim incoherent point sources the QFI
is independent of their separation [15]. Accordingly, in this case, when
optimal measurements are performed, arbitrary separations can be resolved with
the same sensitivity. Moreover, it was proved theoretically [15, 16] and
verified experimentally [17, 18, 19, 20, 21, 22], that this ultimate
resolution limit can be achieved by spatial mode demultiplexing followed by
intensity measurements. These results have been extended to incoherent thermal
sources [23] and other photon-number diagonal states [24]. More recently, the
role of partial coherence has been widely discussed [25, 26, 27, 28, 29, 30,
31, 32, 33, 34].

Despite this vast body of research works, ultimate resolution limits are known
for a very limited class of quantum states of the sources. Furthermore, with
the notable exceptions of the bounds in [23, 24], all known results are valid
only for low photon fluxes in the image plane. In this letter, we overcome
these limitations presenting an analytical expression for the QFI for the
estimation of the separation between two point sources in arbitrary Gaussian
states. Such states include fully coherent and incoherent (thermal) states,
but also those (quantum or classically) correlated states that are most widely
accessible in experiments [35, 36, 37]. Accordingly, the associated resolution
limits are practically relevant for both passive and active imaging.

Studying Gaussian states, we have access to different types of mutual
coherence between the sources, and we can evaluate their impact beyond the low
flux limit. In this high brightness regime, we are able to continuously
interpolate between fully mutually coherent and thermal sources, and to show
that mutual coherence originating from thermal correlations and displacement
are non-equivalent. They lead to different scaling of the QFI with the mean
photon number emitted by the sources, with displacement always achieving the
best performances. Accordingly, coherent states of the sources enable a
resolution that surpasses that of every classically correlated thermal state.
In fact, we show that, in all practically relevant scenarios, coherent states,
with an optimized phase relation, approach the ultimate achievable resolution,
as given in [24].

Imaging Gaussian sources.—  We consider two point sources, that are
transversally separated by a distance $d$ in the object plane, and are
observed through a diffraction-limited imaging system with a real point spread
function (PSF) $u_{0}({\bf r})$. Without loss of generality, we assume that
the sources are aligned along the $x-$axis, and populate the two localized
source modes at $\pm{\bf r}_{0}=(\pm d/2,0)$ with associated quadratures
$\hat{\bf
x}^{(s)}=(\hat{q}_{1}^{(s)},\hat{p}_{1}^{(s)},\hat{q}_{2}^{(s)},\hat{p}_{2}^{(s)})$
satisfying the commutation relations
$[\hat{q}^{(s)}_{j},\hat{p}^{(s)}_{k}]=2i\delta_{jk}$, and
$[\hat{q}^{(s)}_{j},\hat{q}^{(s)}_{k}]=[\hat{p}^{(s)}_{j},\hat{p}^{(s)}_{k}]=0$.
We restrict ourselves to Gaussian states $\rho$ of the sources, which are
fully determined by the first two moments of the quadratures [35, 36, 37]: the
mean field $\bar{\bf x}^{(s)}=\tr[\hat{x}^{(s)}\rho]$, and the covariance
matrix
$V^{(s)}_{jk}=\tr\left[\left\\{\hat{x}^{(s)}_{j}-\bar{x}^{(s)}_{j},\hat{x}^{(s)}_{k}-\bar{x}^{(s)}_{k}\right\\}\rho\right]/2$,
with $\\{\cdot,\cdot\\}$ denoting the anti-commutator.

Figure 1: Graphical illustration of the propagation model: quadrature
operators, $(\hat{q}_{1,2}^{(s)},\hat{p}_{1,2}^{(s)})$, of the localized
source modes are transformed into their symmetric and antisymmetric
superpositions $(\hat{q}_{\pm}^{(s)},\hat{p}_{\pm}^{(s)})$. The latter evolve
through independent loss channels, with transmissivities $\kappa_{\pm}$ into
the image-plane quadrature operators $(\hat{q}_{\pm},\hat{p}_{\pm})$.

In the far field, the imaging system is characterized by a transmission
coefficient $\kappa\ll 1$, and maps the localized sources modes into the
overlapping modes $u_{0}({\bf r}+{\bf r}_{0})$, (with associated quadratures
$\hat{q}_{1}$ and $\hat{p}_{1}$) and $u_{0}({\bf r}-{\bf r}_{0})$, (with
associated quadratures $\hat{q}_{2}$ and $\hat{p}_{2}$). The diffraction-
induced non-orthogonality of the modes $u_{0}({\bf r}\pm{\bf r}_{0})$ induces
the non-commutativity of the associated quadratures:
$[\hat{q}_{1},\hat{p}_{2}]=[\hat{q}_{2},\hat{p}_{1}]=2i\delta$, with
$\delta=\int u_{0}({\bf r}-{\bf r}_{0})u_{0}({\bf r}+{\bf r}_{0})d{\bf r}$. We
therefore introduce the orthonormal image modes $u_{\pm}({\bf r})=[u_{0}({\bf
r}-{\bf r}_{0})\pm u_{0}({\bf r}+{\bf r}_{0})]/\sqrt{2(1\pm\delta)}$ with
associated quadratures

$\hat{q}_{\pm}=\frac{\hat{q}_{1}\pm\hat{q}_{2}}{\sqrt{2(1\pm\delta)}},\quad\hat{p}_{\pm}=\frac{\hat{p}_{1}\pm\hat{p}_{2}}{\sqrt{2(1\pm\delta)}}.$
(1)

As illustrated in Fig. 1, the image-plane quadratures $\hat{q}_{\pm}$ are
related to those of the symmetric and antisymmetric superpositions of the
localized sources modes $\hat{q}^{(s)}_{\pm}$ according to
$\hat{q}_{\pm}=\sqrt{\kappa_{\pm}}\hat{q}^{(s)}_{\pm}+\sqrt{1-\kappa_{\pm}}\hat{q}^{(v)}_{\pm}$
(and analogously for $\hat{p}_{\pm}$), with
$(\hat{q}^{(v)}_{\pm},\hat{p}^{(v)}_{\pm})$ the quadratures associated with
two auxiliary vacuum modes, and $\kappa_{\pm}=(1\pm\delta)\kappa$ are
parameter-dependent transmission coefficients of two independent loss channels
[24]. Therefore, Gaussian states of the sources evolve through the imaging
system into Gaussian states of the orthonormal image modes $u_{\pm}({\bf r})$.
The covariance matrix $V$ and the mean field $\bar{\bf x}$ of such image-plane
states are connected with those in the localized source modes according to
$\bar{\bf x}=T\bar{\bf x}^{(s)}$ and $V=TV^{(s)}T^{\top}+N$, with

$\displaystyle
T=\begin{pmatrix}\sqrt{\frac{\kappa_{+}}{2}}\mathds{1}_{2}&\sqrt{\frac{\kappa_{+}}{2}}\mathds{1}_{2}\\\
\sqrt{\frac{\kappa_{-}}{2}}\mathds{1}_{2}&-\sqrt{\frac{\kappa_{-}}{2}}\mathds{1}_{2}\end{pmatrix},\;N=\begin{pmatrix}(1-\kappa_{+})\mathds{1}_{2}&0\\\
0&(1-\kappa_{-})\mathds{1}_{2}\\\ \end{pmatrix}.$ (2)

The state of the sources in the image plane depends on the separation $d$
through the shape of the populated modes $u_{\pm}({\bf r})$, and, because of
the parameter-dependent transmission coefficients $\kappa_{\pm}$, through the
mean field $\bar{\bf x}$ and covariance matrix $V$. Aided by the Williamson
decomposition of the image-plane covariance matrix,
$V=S(\nu_{+}\mathds{1}_{2}\oplus\nu_{-}\mathds{1}_{2})S^{\top}$, with
symplectic eigenvalues $\nu_{\pm}$, and $S$ a symplectic matrix [36, 37], we
account for these different dependencies and compute the QFI $F_{d}$ for the
estimation of the separation $d$ [38]:

$\displaystyle F_{d}$ $\displaystyle=F_{V}+F_{\bar{\bf x}},\quad{\rm with}$
(3a) $\displaystyle F_{V}$
$\displaystyle=\frac{1}{2}\sum_{l=0}^{3}\sum_{jk=\pm}\left[\frac{\left(a_{jk}^{(l)}\right)^{2}}{\nu_{j}\nu_{k}-(-1)^{l}}+2\frac{\left(\tilde{a}_{k,j}^{(l)}\right)^{2}}{\nu_{j}-(-1)^{l}}\right],$
(3b) $\displaystyle F_{\bar{\bf x}}$ $\displaystyle=(\partial_{d}\bar{\bf
x})^{\top}V^{-1}(\partial_{d}\bar{\bf x})+\bar{\bf
x}^{\top}D_{\partial}^{2}\bar{\bf x},$ (3c)

where $a^{(l)}_{ij}=\tr\left[A^{(l)}_{ij}S^{-1}(\partial_{d}V)S\right]$,
$\tilde{a}^{(l)}_{ij}=\tr\left[A^{(l)}_{ij}S^{-1}(V-\mathds{1}_{4})D_{\partial}\right]$,
and $D_{\partial}=\eta_{+}\mathds{1}_{2}\oplus\eta_{-}\mathds{1}_{2}$ with
$\eta^{2}_{\pm}=\int|\partial_{d}u_{\pm}({\bf r})|^{2}d{\bf r}$, and
$A^{(l)}_{jk}$ are a basis of the space of $4\times 4$ matrices 111The
matrices $A^{(l)}_{jk}$ are zero everywhere except in the $jk$ block where
they are given by $i\sigma_{y}/\sqrt{2}$ (for $l=0$), $\sigma_{z}/\sqrt{2}$
(for $l=1$), $\mathds{1}_{2}/\sqrt{2}$ (for $l=2$) and $\sigma_{x}/\sqrt{2}$
(for $l=3$), with $\sigma_{j=x,y,z}$ the standard Pauli matrices.

Let us now comment on the physical origin of the different terms of the QFI 3:
$F_{\bar{\bf x}}$ (3c) describes how the sensitivity is affected by changes of
the mean field $\bar{\bf x}$ in the modes $u_{\pm}({\bf r})$. In particular,
the first term in Eq. (3c) contains the derivative $\partial_{d}\bar{\bf x}$
which account for mean field variations due to parameter-dependent losses, as
quantified by the transmission coefficients $\kappa_{\pm}$. The second term in
Eq. (3c) contains the diagonal matrix $D_{\partial}$ whose elements
$\eta_{\pm}$ quantify how much the shape of the modes $u_{\pm}({\bf r})$
changes with the separation $d$.

In a similar fashion, $F_{V}$ (3b) describes the sensitivity due to variations
of the covariance matrix $V$ with $d$. More specifically, Eq. (3b) contains a
sum running over the components $a_{jk}$ and $\tilde{a}_{jk}$ (on the basis
$A_{jk}^{(l)}$) of the matrices $S^{-1}(\partial_{d}V)S$ and
$S^{-1}(V-\mathds{1}_{4})D_{\partial}$, respectively. The former matrix
contains the derivative $\partial_{d}V$, which describes changes of the
covariance matrix due to the parameter-dependent transmissions $\kappa_{\pm}$,
while the latter matrix contains $D_{\partial}$, which, as discussed above,
takes into account the $d-$dependence of the modes $u_{\pm}({\bf r})$. In the
following, we discuss various examples to illustrate the impact of these
different contributions on our ability to resolve two point sources.

Mutual coherence.—

Figure 2: Quantum Fisher information $F_{d}$ for two partially mutually
coherent sources ($\gamma=0.7$), with coherence originating from thermal
correlations (dashed lines) or displacement (solid lines) as a function of
their transverse separation $d$. Top and bottom panels correspond to $\kappa
N_{0}=1$ and $\kappa N_{0}=100$, respectively. Different colors represent
different phases of the degree of mutual coherence ($\phi=0$ in green,
$\phi=\pi/e$ in blue, $\phi=\pi$ in red). Black and purple curves correspond
to the mutual incoherent case ($\gamma=0$), and to coherent sources with an
optimized mutual phase $(\phi=0,\pi)$ depending on $d$), respectively.

In optics, two light sources are said to be (partially) mutually coherent, if
they present a definite phase relationship that allows them to interfere [40,
41]. This capacity to interfere can be quantified, in quantum mechanical
terms, through the complex degree of mutual coherence (DMC)

$\gamma
e^{i\phi}=\frac{\langle\hat{s}^{\dagger}_{1}\hat{s}_{2}\rangle}{\sqrt{\langle\hat{s}^{\dagger}_{1}\hat{s}_{1}\rangle\langle\hat{s}^{\dagger}_{2}\hat{s}_{2}\rangle}}.$
(4)

The amplitude of the DMC ranges from $\gamma=0$ for mutually incoherent, i.e.
uncorrelated thermal, sources, to $\gamma=1$ in the case of perfect mutual
coherence. On the other hand, the phase in Eq. (4) distinguishes different
kinds of interference, with $\phi=0$ ($\phi=\pm\pi$) corresponding to
perfectly constructive (destructive) interference.

In this work, we focus on two different states of two sources, each emitting
on average $N_{0}$ photons, and featuring (partial) mutual coherence. First,
we consider correlated thermal states $\rho_{c}$, which are zero mean Gaussian
states, with covariance matrix

$V^{(s)}=\begin{pmatrix}(2N_{0}+1)\mathds{1}_{2}&2N_{0}\gamma R(\phi)\\\
-2N_{0}\gamma R(\phi)&(2N_{0}+1)\mathds{1}_{2}\end{pmatrix}$ (5)

with $R(\phi)$ a rotation matrix. Second, we consider the two sources to be in
identical uncorrelated thermal states equally displaced along two directions
separated by an angle $\phi$ in phase space. These displaced thermal states
$\rho_{d}$ have a diagonal covariance matrix
$V^{(s)}=[2N_{0}(1-\gamma)+1]\mathds{1}_{4}$, and a mean field $\bar{\bf
x}^{(s)}=2\sqrt{\gamma N_{0}}(1,0,\cos\phi,\sin\phi)$ 222Here, without loss of
generality, we assumed the first source to be displaced along the $q-$axis of
phase space.. Both $\rho_{c}$ and $\rho_{d}$ have a DMC $\gamma e^{i\phi}$.
However, for $\rho_{c}$, $\gamma$ corresponds to the strength of the thermal
correlations, while $\phi$ is their relative phase. On the other hand, for
$\rho_{d}$, $\gamma$ is the displacement amplitude (in unit of the mean photon
number), and $\phi$ is the relative phase between the two displacements. The
QFI (3), for correlated and displaced thermal states, is presented in Fig. 2,
for an imaging system with Gaussian PSF $u_{0}({\bf r})=\sqrt{2/(\pi
w^{2})}\exp(-|{\bf r}|^{2}/w^{2})$.

On the one hand, $\rho_{c}$ and $\rho_{d}$ produce similar dependencies of the
QFI on the source separation: $F_{d}\to 2N_{0}\kappa/w^{2}$ for distances
larger than the PSF width ($d\gg w$), and $F_{d\to 0}=2\kappa
N_{0}(1-\gamma\cos\phi)/w^{2}$. Accordingly, for small separations, when
compared to the fully incoherent case ($\gamma=0$, black curves in Fig. 2)
partial coherence enhances the QFI in the case of destructive interference
($|\phi|>\pi/2$), and reduces it for constructive interference
($|\phi|<\pi/2$).

On the other hand, the QFI for displaced thermal sources (solid lines in Fig.
2) is always larger than that for correlated thermal sources (dashed lines in
Fig. 2). In particular, for coherent states of the sources ($\rho_{d}$ with
$\gamma=1$), we have

$F_{d}=2\kappa N_{0}\left((\Delta k)^{2}-\beta\cos\phi\right),$ (6)

where we have introduced

$\displaystyle(\Delta k)^{2}$ $\displaystyle=\int[\partial_{x}u_{0}({\bf
r})]^{2}d{\bf r},$ (7a) $\displaystyle\beta$
$\displaystyle=\int\partial_{x}u_{0}({\bf r}-{\bf
r}_{0})\partial_{x}u_{0}({\bf r}+{\bf r}_{0})d{\bf r}.$ (7b)

From Eq. (6), we see that the QFI per emitted photon $F_{d}/N_{0}$ does not
depend on the mean photon number $N_{0}$. Conversely, for perfectly correlated
thermal sources ($\gamma=1$), for intermediate separations of the order of the
PSF width $w$, the QFI decreases when increasing $N_{0}$. Accordingly, for two
sources in coherent states, the separation-estimation sensitivity presents a
shot-noise scaling $\Delta d\sim N_{0}^{-1/2}$, while for perfectly correlated
thermal sources ($\rho_{c}$ with $\gamma=1$) the scaling is less favorable.

In the low photon flux regime ($\kappa N_{0}\ll 1$), the role of partial
coherence has been recently debated [25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
The debate originated from two different models for the state of the sources
in the image plane. The first, proposed by Larson and Saleh in [25], describes
the sources in the image plane as a single photon state, while the other,
introduced by Tsang and Nair in [26], considers a mixture of a single photon
and a vacuum contribution. This vacuum contribution allows to describe losses
induced by the imaging system, which, in presence of mutual coherence, lead to
a separation-dependent mean photon number in the image plane. Such an
additional dependence can increase the QFI [26, 30, 34]. However, this
information is not available if the mean photon number $N_{0}$ emitted by the
sources is unknown, and in this case one recovers the results of the single-
photon model [32, 43]. The two classes of Gaussian states considered here both
contain a vacuum contribution. In particular, correlated thermal states are
photon-number diagonal and can be considered a generalization of the Tsang and
Nair model [26], to which they reduce in the low flux regime. On the contrary,
displaced thermal states are not diagonal in the photon number basis and, even
for $N_{0}\ll 1$, do not correspond to any partial coherence model that has
been studied in the low flux regime.

Squeezing and entanglement.—

Figure 3: Graphical representation of the propagation of two sources squeezed
along the same direction (a) and along opposite directions (b) to the image
plane. (c) Quantum Fisher information $F_{d}$ as a function of their
transverse separation $d$, for two sources squeezed along the same direction
($\theta=0$, solid lines) or along opposite directions ($\theta=\pi/2$, dashed
lines) in phase space. Different colors correspond to different transmission
of the imaging system, $\kappa=0.01$ (blue) and $\kappa=0.1$ (red), and
different shades to different mean photon numbers, $N_{0}=1$ (darker) and
$N_{0}=100$ (lighter).

We have just seen that coherent states of the source modes provide the optimal
resolution among mutual coherent, classically correlated, states. We now
investigate if quantum resources such as squeezing or entanglement can lead to
enhanced performances. To this goal, we consider two zero-mean sources
squeezed along two directions separated by an angle $\theta$ in phase space.
Such sources have covariance matrix $V_{0}^{\theta}=S^{2}(\xi)\oplus
R(\theta)S^{2}(\xi)R^{\top}(\theta)$, with $S(\xi)={\rm
diag}(e^{-\xi},e^{\xi})$ a single-mode squeezing matrix, and $R(\theta)$ a
rotation matrix. The mean photon number $N_{0}$ emitted by each source is
related to the squeezing parameter $\xi$ according to $N_{0}=\sinh^{2}\xi$
[36, 44]. In the following, we focus on two simple, but exemplary cases: two
sources squeezed along the same direction ($\theta=0$) and along opposite
directions ($\theta=\pi/2$).

The evolution from the object to the image plane (see Fig. 1) is given by a
transformation to the symmetric and antisymmetric source modes, followed by
two independent loss channels. As illustrated in Fig. 3 (a), for $\theta=0$,
the mapping to the symmetric and antisymmetric source modes leaves the state
unchanged, while propagation through the loss channels adds noise, such that
the quantum state in the image modes becomes a product of two thermal squeezed
states. Similar results are obtained considering a two-mode squeezed vacuum of
the localized source modes, corresponding to two vacuum states squeezed in
opposite directions in the symmetric and antisymmetric source modes. The
latter case was studied in the supplementary material of [24]. On the
contrary, for $\theta=\pi/2$, see Fig. 3 (b), we obtain a two-mode squeezed
vacuum in the symmetric and antisymmetric source modes, which is entangled,
and it remains entangled upon propagation to the image plane for every non-
zero value of the source separation $d>0$.

The behaviour of the QFI for these two classes of squeezed states is presented
in Fig. 3 (c). For sources squeezed along the same direction ($\theta=0$), the
high propagation losses ($\kappa\ll 1$ in the far field) render the state
substantially semiclassical. Accordingly, their QFI (solid lines in Fig. 3
(c)) is very similar to that for uncorrelated thermal sources (black lines in
Fig. 2): it equals $F_{d}=2N_{0}\kappa/w$ except for a region, around
separations of the order of the PSF width $w$, where it presents a dip which
gets deeper when increasing the mean number of photons $\kappa N_{0}$ in the
image plane. On the other hand, when the two sources are squeezed along
opposite directions ($\theta=\pi/2$), despite the high losses, their state
preserve its quantum character upon propagation through the imaging system,
i.e. it remains entangled in the orthogonal image modes $u_{\pm}({\bf r})$. As
a consequence, in this case, the QFI per emitted photon $F_{d}/N_{0}$ is
independent of the mean photon number $N_{0}$ (see dashed lines in Fig. 3
(c)). Thus, entanglement enables a shot-noise scaling $\Delta d\sim
N_{0}^{-1/2}$ of the separation estimation sensitivity. Contrarily, non-
entangled squeezed states provide a less favourable scaling for intermediate
separations. Finally, comparing Fig. 3 (c) with Fig. 2, we observe that
several partially mutually coherent states allow for larger values of the QFI.
Thus, squeezing in the sources modes does not improve the separation
estimation sensitivity with respect to semiclassical states.

Approaching the ultimate quantum limit.— Coherent sources ($\rho_{d}$ with
$\gamma=1$) present the same shot-noise scaling $\Delta d\sim N_{0}^{-1/2}$ of
the separation estimation sensitivity as entangled squeezed states, but with a
more favourable prefactor. In fact, we can show that in all practically
relevant scenarios, coherent states are quantum optimal: Recent works
demonstrated that, for the separation estimation sensitivity, it is impossible
to achieve a sub-shot-noise scaling [24, 45], and that for arbitrary quantum
states of the sources the QFI cannot exceed the global upper bound [24]

$\displaystyle F_{d}^{\rm max}$ $\displaystyle=2\kappa
N_{0}\max\left\\{f_{+},f_{-}\right\\},\quad{\rm with}$ (8) $\displaystyle
f_{\pm}$ $\displaystyle=(\Delta
k)^{2}\mp\beta+\frac{\kappa(\partial_{d}\delta)^{2}}{1-\kappa(1\pm\delta)}.$
(9)

For imaging in the far field regime ($\kappa\ll 1$), the last term in Eq. (9)
is negligible ($f_{\pm}\approx(\Delta k)^{2}\mp\beta$), and therefore the
global upper bound (8) is achieved by the QFI (6) for coherent sources either
in or out of phase (purple curves in Fig. 2).

Conclusion.— We determined the ultimate quantum resolution limit for two point
sources in arbitrary Gaussian states. Such a limit is provided in the form of
an analytical expression for the QFI quantifying the sensitivity of the
quantum state of the sources to variations of their separation. With this
general expression for the QFI, we explored the the role of different (non-
equivalent) types of partial coherence, and classical and quantum correlations
in the two source resolution problem. Our results show that semi-classical
states of the sources featuring (partial) coherence can often outperform
sources featuring quantum properties such as squeezing. In fact, coherent
states of the sources can even approach the ultimate quantum resolution, in
the practically relevant far field regime.

Acknowledgments.— This work was partially funded by French ANR under COSMIC
project (ANR-19-ASTR0020-01). This work received funding from the European
Union’s Horizon 2020 research and innovation programme under Grant Agreement
No. 899587. This work was supported by the European Union’s Horizon 2020
research and innovation programme under the QuantERA programme through the
project ApresSF. This work received funding from Ministerio de Ciencia e
Innovación (MCIN) / Agencia Estatal de Investigación (AEI) for Project No.
PID2020-115761RJ-I00 and support of a fellowship from “la Caixa” Foundation
(ID 100010434) and from the European Union’s Horizon 2020 research and
innovation program under Marie Skłodowska-Curie Grant Agreement No. 847648,
fellowship code LCF/BQ/PI21/11830025. This work was partially funded by
CEX2019-000910-S [MCIN/AEI], Fundació Cellex, Fundació Mir-Puig, and
Generalitat de Catalunya through CERCA.

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