# Quantum precision limits of displacement noise free interferometers

Tuvia Gefen  Institute for Quantum Information and Matter, California
Institute of Technology, Pasadena, CA 91125, USA  Rajashik Tarafder
Theoretical Astrophysics, Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA  LIGO Laboratory,
California Institute of Technology, Pasadena, CA 91125, USA  Rana X. Adhikari
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
Yanbei Chen  Theoretical Astrophysics, Walter Burke Institute for Theoretical
Physics, California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

Current laser-interferometric gravitational wave detectors suffer from a
fundamental limit to their precision due to the displacement noise of optical
elements contributed by various sources. Several schemes for Displacement-
Noise Free Interferometers (DFI) have been proposed to mitigate their effects.
The idea behind these schemes is similar to decoherence-free subspaces in
quantum sensing i.e. certain modes contain information about the gravitational
waves but are insensitive to the displacement noise. In this paper we derive
quantum precision limits for general DFI schemes, including optimal
measurement basis and optimal squeezing schemes. We introduce a triangular
cavity DFI scheme and apply our general bounds to it. Precision analysis of
this scheme with different noise models shows that the DFI property leads to
interesting sensitivity profiles and improved precision due to noise
mitigation and larger gain from squeezing. Further extensions of this scheme
are presented.

Introduction— Quantum metrology studies fundamental precision limits imposed
by quantum physics and seeks to use non-classical effects to enhance
precision. Recent progresses in this field have led to formulation of
precision limits for a variety of sensing devices: gravitational wave (GW)
detectors [1, 2, 3, 4, 5, 6, 7, 8], magnetometers [9, 10], atomic clocks [11,
12, 13, 14], nano-NMR [15, 16, 17, 18, 19] etc.

We focus here on optomechanical sensors and laser interferometers. These
platforms have emerged as the primary instruments for detection of GW , with
successful observations conducted by several of these detectors [20, 21, 22,
23, 24, 25, 26, 27]. They are however severely limited by noise sources that
displace the mirror positions in the interferometer: thermal noise, radiation
pressure noise, seismic noise and Newtonian noise [28, 29, 30]. These noises
are in particular dominant in the low frequency regime ($<10$ Hz), thus
limiting the sensitivity at this range and prevent detection of various
signals such as: intermediate mass black holes, stochastic GW background,
young Neutron Stars, Super Massive Binary Black Hole in-spirals etc.
Overcoming displacement noise is thus an outstanding challenge for GW
detection and optomechnical sensors in general.

Remarkably it has been shown that for certain interferometrs and with an
appropriate signal processing, displacement noise can be removed [31]. This
approach towards cancelling displacement noise while not losing the GW signal
is referred to as Displacement-noise Free Interferometry (DFI). DFI for laser
interferometry was originally proposed in [31] using a toy system and later
expanded to more physical systems such as speed meters and 3D-interferometers
[32, 33, 34, 35, 36, 37]. A similar approach for laser phase noise
cancellation has also been proposed for LISA using Time Delay Interferometry
[38]. However, DFI systems with requisite sensitivities remain elusive.
Furthermore a rigorous study of the quantum precision limits of these
interferometers have not been conducted.

Here, we use techniques of quantum metrology to derive general precision
limits, optimal measurements and optimal squeezing quadratures for DFI
schemes. We develop a simple DFI scheme, a triangular cavity, and apply these
results to analyze it. In small frequencies, where displacemen noise is large,
the displacement free subspace (DFS) becomes dominant, i.e. most of the
information comes from it, leading to an improved sensitivity. We observe
further interesting effects that motivate the use of DFI and multichannel
interferometers. We identify pseudo-DFS, i.e. subspaces that are displacement
free for a limited range of frequencies. These subspaces lead to unexpected
sensitivity profiles and further improvement. Lastly, we show that DFI
increases the sensitivity gain from squeezing in high displacement noise
regime, and present extensions of the triangular scheme.

Formalism and Model—The following formulation is general and holds for any
system with $n$ mirrors and $k>n$ input and output fields. We demonstrate it
and solve for a triangular configuration: three mirrors and six lasers, as
depicted in Fig. 1. The input lasers are aligned such that there are six
fields circulating inside the cavity - split between clockwise and anti-
clockwise direction. All illustrations in this letter correspond to this
symmetric triangular scheme, where the relevant parameters can be found in
111Power transmissivity of all the mirrors: $T=0.1,$. Arm length: $L=4$ km.
Carrier angular frequency: $5\times 10^{14}$ s-1. Input Power: $100$ kW.
Mirrors mass: $5$ kg..

Figure 1: Top: Sketch of the DFI scheme. A triangular cavity is formed by
three mirrors. Three clockwise circulating input and output fields exist;
alongside three anti-clockwise input and output fields. Bottom: Sensitivity as
a function of frequency for different levels of displacement noise for the
simplified model. The solid gray line (bottom-most curve) corresponds to the
shot noise limit, while the black solid line (top-most curve) corresponds to
$\delta\rightarrow\infty.$ The dashed curves correspond to different levels of
$\delta$: the lower, middle and upper dashed curves correspond to
$\delta=10^{-20}\;\text{m},10^{-19}\;\text{m},10^{-18}\;\text{m}$
respectively.

We describe the system using the input-output formalism [40, 41]. We denote
the quadratures vector of the input and output fields as
$Q_{\text{in}}=\left(\begin{array}[]{c}\vec{a_{1}}\\\
\vec{a_{2}}\end{array}\right),Q_{\text{out}}=\left(\begin{array}[]{c}\vec{b_{1}}\\\
\vec{b_{2}}\end{array}\right)$ respectively. $\vec{a_{1}},\vec{a_{2}}$
correspond to $k$-dimensional vectors of amplitude and phase quadratures
respectively, and they satisfy the commutation relations
$\left[\left(\begin{array}[]{c}\vec{a}_{1}\\\
\vec{a}_{2}\end{array}\right),\left(\vec{a}_{1}^{\dagger},\vec{a}_{2}^{\dagger}\right)\right]=i\left(\begin{array}[]{cc}0&\mathbbm{1}_{k}\\\
-\mathbbm{1}_{k}&0\end{array}\right)$. The displacement of the mirrors is
denoted as $\left\\{\Delta x_{i}\right\\}_{i=1}^{n},$ and the amplitude of the
GW polarization vector is $\vec{h}=\left(h_{+},h_{\times}\right)^{T}.$ The
input-output relations in the frequency domain are then given by:

$\vec{b}\left(\Omega\right)=M\left(\Omega\right)\vec{a}\left(\Omega\right)+\mathcal{V}\left(\Omega\right)\vec{h}\left(\Omega\right)+A\left(\Omega\right)\vec{\Delta
x}\left(\Omega\right)$ (1)

where $M\left(\Omega\right),$ $A\left(\Omega\right),$
$\mathcal{V}\left(\Omega\right)$ are the transfer matrices of the input modes,
displacement noise, and the GW vector respectively. $\Omega=2\pi f$ is the
angular frequency, and hereafter we will suppress the $\Omega$ notation. The
displacement noise of the mirrors is thus coupled to a $n$ dimensional
subspace: the linear span of the column vectors of $A$. We refer to this
subspace as the coupled subspace and denote it as $M_{C}.$ Its projection
operator is $\Pi_{\text{C}}=AA^{-1},$ with $A^{-1}$ being the pseudo-inverse
of $A.$ The orthogonal complement of $M_{C}$ is thus completely decoupled from
displacement noise, we denoted it as $M_{\text{DFS}}$ and refer to it as the
displacement free subspace (DFS). Therefore the response to the GW that is
unaffected by the displacement noise is given by
$\left(\mathbb{1}-AA^{-1}\right)\mathcal{V}.$ Throughout this letter we assume
resonance, therefore $A=\left(\begin{array}[]{cc}0\\\
A_{\text{ph}}\end{array}\right)$ and
$\mathcal{V}=\left(\begin{array}[]{cc}0\\\
\mathcal{V}_{\text{ph}}\end{array}\right)$ act only on the phase quadratures.
Information about the GW is thus encoded only in the phase quadratures, and we
can thus express the QFIM using $A_{\text{ph}},\mathcal{V}_{\text{ph}}.$ A
non-trivial DFS, i.e. a DFS that contains information about the GW, can exist
only if $M_{\text{DFS}}$ contains some distributed phase quadratures. Since
the dimension of the phase quadratures is $k$ and $A$ is of rank $n$, a
necessary condition for a non trivial DFS is $k>n$.

Quantum precision limits— The precision in estimating $\vec{h}$ is quantified
by Cov$\left(\vec{h}\right)$, the covariance matrix of the estimators.
According to the Cramér-Rao bound, given a specific readout scheme with
outcomes distribution $\left\\{p\left(x\right)\right\\}_{x},$ any unbiased
estimators satisfy $\text{Cov}\left(\vec{h}\right)\geq F^{-1},$ with
$F_{i,j}=\langle\partial_{h_{i}}\ln\left(p\right)\partial_{h_{j}}\ln\left(p\right)\rangle$
being the Fisher Information Matrix (FIM) and the inequality is tight [42].

In quantum context, further optimization over the detection schemes yields the
Quantum Fisher Information Matrix (QFIM):

$\displaystyle
I_{i,j}=\frac{1}{2}\text{tr}\left(\left\\{L_{i},L_{j}\right\\}\rho\right)\;\;\text{with}\;\;\partial_{h_{i}}\rho=\frac{1}{2}\left(L_{i}\rho+\rho
L_{i}\right),$

where $L_{i}$ is the symmetric logarithmic derivative with respect to the i-th
parameter [43]. Hence for any readout scheme
$\text{Cov}\left(\vec{h}\right)\geq I^{-1},$ this bound will be calculated and
shown to be tight for our case.

In this letter we assume input Gaussian states. Since the evolution in the
interferometer corresponds to a Gaussian channel, the state of the output
modes is also Gaussian. It can be shown that the QFIM in such a scenario reads
[44, 8]:

$\displaystyle
I=2\left(\partial_{\vec{h}}\vec{d}_{q}\right)^{\dagger}\Sigma_{q}^{-1}\left(\partial_{\vec{h}}\vec{d}_{q}\right).$
(2)

where $d_{\vec{q}}$ and $\Sigma_{q}$ are the mean vector and covariance matrix
of $\vec{Q}_{\text{out}}$ respectively:

$\displaystyle\vec{d_{q}}=\langle\vec{Q}_{\text{out}}\rangle,\;\,\Sigma_{q}=\langle\vec{Q}_{\text{out}}\vec{Q}_{\text{out}}^{\dagger}\rangle-\langle\vec{Q}_{\text{out}}\rangle\langle\vec{Q}_{\text{out}}^{\dagger}\rangle.$
(3)

We remark that this is in fact a complex compact form of the real valued QFIM.
The real valued QFIM about the $4$ real parameters
$\left(\text{Re}\left(\vec{h}\right),\text{Im}\left(\vec{h}\right)\right)$
satisfies a circular symmetry, and thus can be represented by the $2\times 2$
complex matrix of eq. 2 [45]. The full derivation of the QFIM is presented in
the supplemental. It relies on the use of the Hermitian quadratures
$\vec{S}=\sqrt{2}\left(\begin{array}[]{cc}\text{Re}\left(\vec{Q}\right),&\text{Im}\left(\vec{Q}\right)\end{array}\right)^{T}$,
on the fact that the covariance matrix of the Hermitian quadratures has a
circular symmetry and on the fact that all the information about $\vec{h}$ is
encoded in $\vec{d_{q}}.$

From eq.1 we observe that $\partial_{\vec{h}}d_{q}=\mathcal{V},$ and that
$\Sigma_{q}=M\Sigma_{i}M^{\dagger}+A\Sigma_{\vec{\Delta x}}A^{\dagger},$ where
$\Sigma_{i},$ $\Sigma_{\vec{\Delta x}}$ are the covariance matrices of the
input quadratures and the displacement noise $\vec{\Delta x}$ respectively.
Unless mentioned otherwise, the input state is vacuum and the distribution of
the displacement noise is Gaussian i.i.d: $\vec{\Delta x}\sim
N\left(0,\frac{1}{2}\delta^{2}\mathbb{1}\right),$ leading to
$\Sigma_{q}=\frac{1}{2}\left(MM^{\dagger}+\delta^{2}AA^{\dagger}\right).$ The
QFIM therefore reads:

$\displaystyle
I=4\mathcal{V}^{\dagger}\left(MM^{\dagger}+\delta^{2}AA^{\dagger}\right)^{-1}\mathcal{V}.$
(4)

In eq. 4, the radiation pressure noise is encoded in the transfer matrix $M$
and the rest of the displacement noise is encoded by the additional
$AA^{\dagger}$ term. The shot noise limit can be obtained by nullifying the
radiation pressure noise and the displacement noise i.e. $M$ is unitary and
$\delta=0$, which yields:
$I=4\mathcal{V}_{\text{ph}}^{\dagger}\mathcal{V}_{\text{ph}}.$ This limit
serves as an upper bound to any noisy QFIM scenario.

Our figure of merit is the minimal standard deviation (std) in estimating any
polarization: $\sigma=1/\sqrt{||I||}$,where $||\bullet||$ is the operator
norm, i.e. the maximal eigenvalue 222From numerical analysis, we observe that
the maximal eigenvalue typically corresponds to $h_{+}$.. This is the minimal
detectable amplitude in any given polarization, we refer to it as the
sensitivity or the standard deviation (std).

We further prove in the supplemental, using Cauchy-Schwarz inequality, that
the QFIM, eqs. 2,4, can be saturated with a homodyne measurement of the
quadratures $\Sigma_{q}^{-1}\mathcal{V}$. 333A similar proof was shown
recently also in [50].

Precision limits of the simplified model— We begin with a simplified model to
develop an understanding of the DFI method. The simplified model is devoid of
radiation pressure, or equivalently the mass of the mirrors is taken to be
infinity. This implies that $M$ is unitary. The displacement noise is also
taken to be a white noise, i.e. $\delta\left(\Omega\right)$ is constant. The
QFIM is therefore:

$\displaystyle
I=4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbb{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$
(5)

The sensitivity for different levels of $\delta,$ ranging from the shot noise
limit ($\delta=0$) to infinite displacement noise ($\delta\rightarrow\infty$),
is presented in Fig. 1. Clearly as $\delta$ gets larger the standard deviation
becomes larger, however the DFI property is manifested in the fact that as
$\delta\rightarrow\infty$ the standard deviation remains finite, denoted by
the black line in Fig. 1.

To understand the behavior of the sensitivity for different levels of
displacement noise and in particular the limit of $\delta\rightarrow\infty$,
we note that the QFIM can be decomposed as:

$\displaystyle I=F_{\text{C}}+F_{\text{DFS}}$
$\displaystyle=4\mathcal{V}^{\dagger}\Pi_{\text{C}}\left(\mathbb{1}+\delta^{2}AA^{\dagger}\right)^{-1}\Pi_{\text{C}}\mathcal{V}+4\mathcal{V}^{\dagger}\Pi_{\text{DFS}}\mathcal{V}.$

The first term ($F_{\text{C}}$) is the information from the coupled subspace
and the second term ($F_{\text{DFS}}$) is the information from the DFS. We now
introduce the coefficient: $\eta=\frac{F_{\text{DFS}}}{I},$ i.e. the fraction
of the QFI that comes from the DFS. $\eta$ thus quantifies the effectiveness
of the DFI.

In the limit of $\delta\rightarrow\infty$ the information from the coupled
subspace vanishes and we are left only with
$F_{\text{DFS}}=4\mathcal{V}^{\dagger}\Pi_{\text{DFS}}\mathcal{V}.$ As
$f\rightarrow 0$ this standard deviation diverges, indicating that in this
regime $\Pi_{\text{DFS}}\mathcal{V}\rightarrow 0.$ This implies that for any
finite $\delta,$ $F_{\text{DFS}}\rightarrow 0$ as $f\rightarrow 0$ and thus
$I\approx
4\frac{1}{\delta^{2}}\mathcal{V}^{\dagger}\left(AA^{\dagger}\right)^{-1}\mathcal{V}.$
This is the DC plateau of the dashed lines in fig. 1, which therefore grows as
$\delta.$

Figure 2: Noise budget: The shot noise limit (yellow solid line) coincides
with the QFI (dashed, black line) in the presence of radiation pressure alone.
The solid (circles) blue line is the phase quadratures measurement FI given
radiation pressure noise, above it is the phase quadratures FI given thermal
and radiation pressure noise (red line with rectangles). Inset: Comparison
between the phase quadratures FI (red, solid line) and the QFI (dashed, black
line) in the presence of thermal and radiation pressure noise. For this
illustration $m=5$ kg.

Figure 3:  (a) Performance with squeezing in the simplified model. Dashed
lines correspond to std with squeezing and solid lines to unsqueezed
($\delta=0,$ $10^{-18}$ m, $10^{-17}$ m correspond to gray, green and blue
respectively). For white displacement noise Squeezing becomes less effective
for smaller frequencies as can be also observed from the plot of $\eta$ in the
inset. (b) Performance with squeezing given thermal and radiation pressure
noises. The solid (dashed) line corresponds to unsqueezed (optimally squeezed)
std with phase quadratures measurement. Inset: $\eta_{\text{gain}}$ (red dots)
and $\eta$ (green line) as a function of frequency. (c) Comparison between the
QFI with squeezing (blue curve, circles) and the FI with maximum signal
combination (red curve, squares).

Precision limits with realistic noise profiles— Let us now consider the
sensitivity with realistic thermal and radiation pressure noises. We begin by
analyzing the effect of radiation presssure noise alone and then study the
combination of the two noises.

The effect of radiation pressure for a resonant cavity is given by a non-
unitary $M$ of the form

$\displaystyle M=\left(\begin{array}[]{cc}M_{\text{int}}&0\\\
M_{21}&M_{\text{int}}\end{array}\right),$ (8)

where $M_{\text{int}}$ is the unitary interferometer transfer matrix and
$M_{21}$ is the coupling matrix between the amplitude and phase quadratures
due to radiation pressure [48]. $M_{21}$ can be written as a concatenation of
transfer matrices: $M_{21}=A_{\text{ph}}D_{x},$ where $D_{x}$ is the transfer
matrix of the amplitude quadratures to displacement noise $\vec{\Delta x}.$ We
assume for simplicity that the mirrors are free masses, hence
$M_{21}\propto\frac{1}{m\Omega^{2}},$ where $m$ is the mass of the mirrors.
This typically leads to a sensitivity profile that scales as $\Omega^{-2}$ [8,
7].

The QFIM in this case saturates the shot noise limit (black dashed line in
fig. 2), i.e. the radiation pressure noise is completely removed by measuring
an appropriate choice of quadratures. This is just a generalized variational
readout [2]: the quadratures

$\displaystyle\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\\\
\mathbbm{1}\end{array}\right)\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)^{-1/2}$
(11)

are decoupled from radiation pressure noise and homodyne measurement of them
saturates the QFIM and the shot noise limit. The optimal combination of these
quadratures that saturates the QFIM is
$\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\mathcal{V}_{\text{ph}}\\\
\mathcal{V}_{\text{ph}}\end{array}\right)$.

The phase quadratures on the other hand are not decoupled from radiation
pressure and measuring them yields the following FIM [48]:

$\displaystyle
F=4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbb{1}+M_{21}M_{21}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}},$
(12)

This expression is analogous to the QFIM of the simplified model (eq. 5),
where the term $M_{21}M_{21}^{\dagger}$ is the displacement noise caused by
radiation pressure. Since $M_{21}=A_{\text{ph}}D_{x},$ the DFS is decoupled
from the radiation pressure as well. The corresponding sensitivity is
presented in fig. 2, where we observe an interesting behavior: unlike with
single output schemes it does not diverge uniformly as $1/\Omega^{2},$ instead
there is a range of frequencies where the divergence disappears. This plateau
is due to a pseudo-DFS, a subspace that is robust to displacement noise in
this range of frequencies. Let us further elaborate on this.

In our triangular cavity scheme the phase quadratures can be decomposed to
three orthogonal eigenspaces of the covariance matrix: $M_{\text{min}}\oplus
M_{\text{max}}\oplus M_{\text{DFS}},$ where $M_{\text{min}}\oplus
M_{\text{max}}$ is a decomposition of $M_{\text{C}}$ to eigenspaces with
minimal and maximal eigenvalues respectively. Since these are eigenspaces of
the covariance matrix the FIM is a sum of the FIM’s achieved with each one
them separately: $F=F_{\text{min}}+F_{\text{max}}+F_{\text{DFS}}.$ For
different frequencies, different subspaces are dominant, this accounts for the
non uniform divergence. The plateau appears when $F_{\text{min}}$ becomes
dominant. $M_{\text{min}}$ is immune to displacement noise in this range of
frequencies, i.e. it is an eigenspace of $M_{21}M_{21}^{\dagger}$ with an
eigenvalue that is much smaller than shot noise, hence the plateau. This is
discussed further in the supplemental.

Let us now consider thermal noise as well. The thermal noise is modeled as
$\vec{\Delta x}\sim N\left(0,\frac{1}{2}\delta^{2}\mathbb{1}\right),$ where
$\delta^{2}(\Omega)=2.7\cdot 10^{-30}/f^{5}~{}m^{2}/\text{Hz}$ [49]. Hence,
the effect of thermal displacement noise is similar to the simplified model
with a frequency dependent $\delta$.

In the presence of both radiation pressure and thermal displacement noise, the
QFIM coincides with the QFIM of thermal noise alone: measuring again the
quadratures decoupled from radiation pressure, eq. 11, saturates the QFIM,
where radiation pressure is completely cancelled and the QFIM takes the form
of eq. 5 with $\delta$ having the frequency dependence mentioned above. A plot
of the corresponding sensitivity is presented in the inset of fig. 2. Further
details about the derivation of the QFIM and its behavior can be found in the
supplemental.

Measuring the phase quadratures, radiation pressure noise is not cancelled and
the FIM reads:
$4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}$.
The plot of the corresponding sensitivity profile and a comparison with the
QFI is presented in fig 2 (red solid line). Three different regimes can be
observed in the plot, that correspond to three eigenspaces of the covariance
matrix. For small enough frequencies the DFS becomes dominant and the standard
deviation diverges as $f^{-2},$ instead of $f^{-5/2}.$ Before that, there is
an intermediate regime where $M_{\text{min}}$ is dominant and a short plateau.
A comparison between the phase quadratures FIM and QFIM shows that they
coincide for small frequencies where the thermal noise is dominant but the
QFIM clearly outperforms where radiation pressure is dominant.

Effect of squeezing—  We summarize the optimal squeezing schemes and
sensitivities with squeezing. Given a squeezing factor of $e^{-s}$ the optimal
QFIM is:
$4\mathcal{V}_{\text{ph}}^{\dagger}\left(e^{-2s}\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)\mathcal{V}_{\text{ph}},$
it can be achieved with squeezing of the phase quadratures and measuring the
optimal quadratures of eq. 11. For phase quadratures measurement, the optimal
FIM is:
$4\mathcal{V}_{\text{ph}}^{\dagger}\left(e^{-2s}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)\mathcal{V}_{\text{ph}},$
achievable by squeezing the optimal squeezing quadratures
$\left(\begin{array}[]{c}M_{\text{int}}^{\dagger}M_{21}\\\
\mathbbm{1}\end{array}\right)\left(\mathbbm{1}+\left(M_{\text{int}}^{\dagger}M_{21}\right)^{2}\right)^{-1/2}.$
A derivation of these bounds and optimal squeezing quadratures is in the
supplemental.

The performance of the squeezed schemes, and comparison with the unsqueezed
case, is shown in fig. 3. Observe that the gain from squeezing is not uniform
and depends on the efectiveness of the DFI, i.e. on $\eta.$ We can define the
gain from squeezing as
$\eta_{\text{gain}}=\frac{\frac{F_{\text{sq}}}{F}-1}{e^{2s}-1},$ where
$F_{\text{sq}}(F)$ is the FI with(out) squeezing. Clearly
$0\leq\eta_{\text{gain}}\leq 1,$ where $0$ corresponds to no gain and $1$ to
maximal gain. We show in the supplemental that in the limit of large
displacement noise $\eta_{\text{gain}}=\eta,$ hence $\eta_{\text{gain}}$
equals the fraction of information coming from the DFS. This is illustrated in
the insets of fig. 3. DFI is therefore necessary to gain from squeezing in the
presence of large displacement noise. The improvement introduced by DFI is
summarized in fig. 3 (c) where we compare the sensitivity with squeezed input
given different readout combinations: a combination that maximizes the signal
and the optimal combination that saturates QFI. The QFI is considerably better
in small frequencies due to two DFI properties: better scaling with $f$
($f^{-2}$ compared to $f^{-2.5}$), and larger gain from squeezing.

Figure 4: Extensions of the DFI triangular scheme to an arbitrary $n$-gon. The
vertices represent the mirrors and the blue edges represent the light
trajectories.

Extensions and conclusions— The triangular scheme discussed above can be
extended to any $n-$gon with $n$ mirrors, as illustrated in fig. 4. The $n$
mirrors are illustrated as $n$ vertices of an $n$-gon embedded on a circle.
For each polygon the different light trajectories correspond to different
billiard trajectories hitting the vertices, hence there are $n-1$ possible
trajectories that correspond to $n-1$ reflection angles
$\left\\{\frac{2\pi}{n}l\right\\}_{l=1}^{n-1}.$ We perform a numerical
analysis of the schemes described in fig. 4 (a) in the supplemental. In the
presence of thermal displacement noise, we observe that increasing $n$ does
not lead to an improved sensitivity. We leave analysis of the schemes
described by 4 (b) for future work.

In conclusion, we derived quantum precision limits and optimal measurements
and squeezing for DFI schemes. In the protocols analyzed, the DFI mitigates
the divergence of the sensitivity in small frequencies due to displacement
noise, but does not completely remove it. It would be interesting to inquire
whether this divergence can be removed or can be made arbitrarily small. Other
future directions include considering the effect of detuning, schemes with
larger DFS such as in fig. 4 (b), and non-Gaussian states.

Acknowledgments— The authors are thankful to John Preskill, Yehonathan Drori,
Evan Hall, Kevin Kuns and Lee McCuller for helpful discussions. TG
acknowledges funding provided by the Institute for Quantum Information and
Matter and the Quantum Science and Technology Scholarship of the Israel
Council for Higher Education.

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Supplementary Material

##  Appendix 1 Two photon formalism

Assuming an initial phase of $0,$ an EM field can be written as [40]:

$\displaystyle
E\left(t\right)=\left(A\left(t\right)+a_{1}\left(t\right)\right)\cos\left(\omega_{0}t\right)+a_{2}\left(t\right)\sin\left(\omega_{0}t\right)$

where $a_{1,2}\left(t\right)$ are the Hermitian amplitude and phase
quadratures respectively. They describe the amplitude and phase modulation of
the field ($\omega_{0}$ is the carrier frequency). They satisfy $\langle
a_{1,2}\left(t\right)\rangle=0$ and their commutation relations are given by:

$\displaystyle\left[a_{1}\left(t\right),a_{2}\left(t^{\prime}\right)\right]=-i\delta\left(t-t^{\prime}\right)$

We can further define $a_{1,2}\left(\Omega\right)$ as the Fourier transform of
$a_{1,2}\left(t\right)$:

$\displaystyle a_{1,2}\left(\Omega\right)=\frac{1}{\sqrt{2\pi}}\int
a_{1,2}\left(t\right)e^{-i\Omega t}\;\text{dt}.$

Here, we observe that $a_{1,2}\left(\Omega\right)$ are not Hermitian but they
commute with the Hermitian conjugate of themselves. This means that they have
an orthonormal eigenbasis but their eigenvalues are complex. Therefore, the
commutation relations between them is given by:

$\displaystyle\left[\left(\begin{array}[]{c}a_{1}\\\
a_{2}\end{array}\right),\left(a_{1}^{\dagger},a_{2}^{\dagger}\right)\right]=i\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right).$

where the $\Omega$ dependence is suppressed.

Therefore, we can interpret the above as two harmonic oscillators with:

$\displaystyle
X_{R}\left(\Omega\right)=\sqrt{2}\text{Re$\left(a_{1}\right)$,~{}~{}~{}\;$P_{R}\left(\Omega\right)$=$\sqrt{2}\text{Re}\left(a_{2}\right)$}$
$\displaystyle
X_{I}\left(\Omega\right)~{}=\sqrt{2}\text{Im$\left(a_{1}\right)$,~{}~{}~{}\;$P_{I}\left(\Omega\right)$=$\sqrt{2}\text{Im}\left(a_{2}\right)$}$

It follows that the commutation relations of two harmonic oscillators are
given by:

$\displaystyle\left[\left(\begin{array}[]{c}X_{R}\\\ P_{R}\\\ X_{I}\\\
P_{I}\end{array}\right),\left(\begin{array}[]{cccc}X_{R},P_{R},X_{I},P_{I}\end{array}\right)\right]=\left(\begin{array}[]{cc}\sigma_{y}&0\\\
0&\sigma_{y}\end{array}\right),$

where $\sigma_{x/y}$ are the Pauli $X/Y$ matrices and the $\Omega$ dependence
is suppressed.

In a multichannel interferometer, we have a vector of input quadratures
$\vec{Q}=\left(\begin{array}[]{c}\vec{a}_{1},\vec{a}_{2}\end{array}\right)^{T},$
which defines a vector of Hermitian quadratures (position and momentum
operators):
$\vec{S}=\left(\begin{array}[]{c}\vec{X}_{R},\vec{P}_{R},\vec{X}_{I},\vec{P}_{I}\end{array}\right)^{T}.$

##  Appendix 2 Quantum Fisher information matrix

In a multichannel interferometer the output sideband fields are given by

$\displaystyle\vec{Q}_{\text{out}}=\left(\begin{array}[]{c}\vec{b}_{1}\left(\Omega\right)\\\
\vec{b}_{2}\left(\Omega\right)\end{array}\right)$

with commutation relations:

$\displaystyle\left[\left(\begin{array}[]{c}\vec{b}_{1}\\\
\vec{b}_{2}\end{array}\right),\left(\vec{b}_{1}^{\dagger},\vec{b}_{2}^{\dagger}\right)\right]=i\left(\begin{array}[]{cc}0&\mathbbm{1}_{k}\\\
-\mathbbm{1}_{k}&0\end{array}\right)$ (2.5)

where $k$ is the number of output fields and the $\Omega$ dependence is
suppressed.

As above, the Hermitian quadratures are:

$\displaystyle\vec{S}=\left(\begin{array}[]{c}\vec{X}_{R}\\\ \vec{P}_{R}\\\
\vec{X}_{I}\\\
\vec{P}_{I}\end{array}\right)=\sqrt{2}\left(\begin{array}[]{c}\text{Re}\left(\vec{b}_{1}\right)\\\
\text{Re}\left(\vec{b}_{2}\right)\\\ \text{Im}\left(\vec{b}_{1}\right)\\\
\text{Im}\left(\vec{b}_{2}\right)\end{array}\right),$

which is the corresponding quadrature vector of $2k$ harmonic oscillators.

Given the input-output relations of $\vec{Q}$ (eq. (1) in the main text),
these relations for $\vec{S}$ are given by:

$\displaystyle\vec{S}_{\text{out}}=M^{\prime}\vec{S}_{\text{in}}+\mathcal{V}^{\prime}\vec{h}^{\prime}+A^{\prime}\vec{\Delta
x^{\prime}},$ (2.6)

where for all complex vectors $u\;(=\vec{h},\vec{\Delta x})$ we have
$u^{\prime}=\sqrt{2}\left(\begin{array}[]{c}\text{Re}\left(u\right)\\\
\text{Im}\left(u\right)\end{array}\right)$ and for all complex matrices
$\mathcal{M}\;(=M,\mathcal{V},A)$ we have
$\mathcal{M}^{\prime}=\left(\begin{array}[]{cc}\text{Re}\left(\mathcal{M}\right)&-\text{Im}\left(\mathcal{M}\right)\\\
\text{Im}\left(\mathcal{M}\right)&\text{Re}\left(\mathcal{M}\right)\end{array}\right).$
$\vec{h},\vec{\Delta x},M,\mathcal{V},A$ are as defined in the main text.

Since we consider an initial Gaussian state and the evolution is through a
Gaussian channel, the final state of the output modes is also Gaussian and can
be characterized by:

$\displaystyle\vec{d_{s}}\left(\vec{h}\right)=\langle\vec{S}\rangle$
$\displaystyle\Sigma_{i,j}=\frac{1}{2}\langle
S_{i}S_{j}+S_{j}S_{i}\rangle-\langle S_{i}\rangle\langle S_{j}\rangle$

where all the information about $\vec{h}$ is encoded in the first moment
vector $\vec{d_{s}}(\vec{h})$ (mean vector) and $\Sigma$ is the covariance
matrix.

The Quantum Fisher Information Matrix (QFIM) can be expressed using these
first two moments [44, 8]:

$\displaystyle
I=(\partial_{h}\vec{d_{s}})^{T}\Sigma^{-1}(\partial_{h}\vec{d_{s}})=\mathcal{V}^{\prime\dagger}\Sigma^{-1}\mathcal{V}^{\prime},$
(2.7)

where we used the fact the state is Gaussian, that all the information is
encoded in $\vec{d_{s}},$ and that
$\partial_{h}\vec{d_{s}}=\mathcal{V}^{\prime}.$

In the following subsection we will show that the QFIM can be expressed in a
more compact form involving only the mean values of $\vec{Q}$ ($\vec{d_{q}}$),
and the covariance matrix of $\vec{Q}$ ($\Sigma_{q}$).

###  2.1 Complex compact form of the QFIM

Following derivation in ref.[45], we show that given that $\Sigma$ obeys the
following symmetry

$\displaystyle\Sigma=\left(\begin{array}[]{cc}\Sigma^{\prime}&\bar{\Sigma}\\\
-\bar{\Sigma}^{t}&\Sigma^{\prime}\end{array}\right)$ (2.10)

the QFIM can be written in a complex compact form:

$\displaystyle I=2\mathcal{V}^{\dagger}\Sigma_{c}^{-1}\mathcal{V}$

where $\Sigma_{c}$ is the complex compact form of $\Sigma$:

$\displaystyle\Sigma_{c}=\Sigma^{\prime}+i\bar{\Sigma}.$

We refer to the symmetry of eq. 2.10 as a circular symmetry since it is
analogous to complex Gaussian random variables with circular symmetry [45].

To show this we use the fact that this mapping between complex matrices and
real matrices with this symmetry
$M\leftrightarrow\left(\begin{array}[]{cc}\text{Re}\left(M\right)&-\text{Im}\left(M\right)\\\
\text{Im}\left(M\right)&\text{Re}\left(M\right)\end{array}\right)$ is a
homomorphism, i.e. it preserves multiplication:

$\displaystyle
A_{c}=B_{c}C_{c}\Longleftrightarrow\left(\begin{array}[]{cc}A^{\prime}&\bar{A}\\\
-\bar{A}^{t}&A^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}B^{\prime}&\bar{B}\\\
-\bar{B}^{t}&B^{\prime}\end{array}\right)\left(\begin{array}[]{cc}C^{\prime}&\bar{C}\\\
-\bar{C}^{t}&C^{\prime}\end{array}\right)$

As a result, identity is mapped to identity:

$\displaystyle\mathbbm{1}_{k}\longleftrightarrow\left(\begin{array}[]{cc}\mathbbm{1}_{k}&0\\\
0&\mathbbm{1}_{k}\end{array}\right)=\mathbbm{1}_{2k}$ (2.13)

and $\left(\Sigma^{-1}\right)_{c}=\Sigma_{c}^{-1}.$

Therefore given that $\Sigma$ has the symmetry shown in eq. 2.10, the QFIM
also has this symmetry and its complex compact form is given by:

$\displaystyle I=2\mathcal{V}^{\dagger}\Sigma_{c}^{-1}\mathcal{V}$

The covariance matrix of $\vec{Q}$ is defined as:

$\displaystyle\left(\Sigma_{q}\right)_{i,j}=\langle
b_{i}b_{j}^{\dagger}\rangle-\langle b_{i}\rangle\langle
b_{j}^{\dagger}\rangle$

Note that, given our symmetry,

$\displaystyle\left(\Sigma_{q}\right)_{i,j}=\text{Cov$\left(b_{i}b_{j}^{\dagger}\right)$}=2\left[\text{COV$\left(\text{Re}\left(b_{i}\right)\text{Re}\left(b_{j}\right)\right)$}+i\text{COV$\left(\text{Im}\left(b_{i}\right)\text{Re}\left(b_{j}\right)\right)$}\right]=\left(\Sigma_{c}\right)_{i,j}$

Therefore we can write $I$ as:

$\displaystyle I=2\mathcal{V}^{\dagger}\Sigma_{q}^{-1}\mathcal{V}$ (2.14)

.

Note that, the original real $4\times 4$ QFIM becomes in the complex compact
form a complex $2\times 2$ matrix that corresponds to estimation of $2$
complex parameters, i.e. the complex amplitudes of the two polarizations. The
eigenvectors of the complex compact QFIM, denoted as
$\left\\{v_{\text{max}},\;v_{\text{min}}\right\\},$ are the polarizations with
maximal and minimal information respectively. Similarly, the eigenvectors of
the original real QFIM would then be:
$\left\\{\left(\begin{array}[]{c}\text{Re}\left(v_{\text{j}}\right)\\\
\text{Im}\left(v_{j}\right)\end{array}\right),\left(\begin{array}[]{c}\text{Re}\left(iv_{\text{j}}\right)\\\
\text{Im}\left(iv_{j}\right)\end{array}\right)\right\\}_{j=\text{max, min}}$
with the same eigenvalues.

This symmetry of the covariance matrix is satisfied in our problem given that
the displacement noise process is stationary (see section 4 for details). The
initial state is a coherent state, hence $\Sigma_{i}=\frac{1}{2}\mathbbm{1}$,
and the symmetry is of course satisfied. In the interferometer it undergoes a
Gaussian channel which maps the covariance matrix to:

$\displaystyle\Sigma=V\Sigma_{i}V^{\dagger}+\Lambda,$

where
$V=\left(\begin{array}[]{cc}\text{Re}\left(M\right)&-\text{Im}\left(M\right)\\\
\text{Im}\left(M\right)&\text{Re}\left(M\right)\end{array}\right)$ with $M$
being the transfer matrix. $\Lambda$ is due to classical displacement noise
(thermal, seismic etc.). Given that the classical displacement noise is
stationary i.i.d $\Lambda$ takes the form of (see section 4):
$\Lambda=\frac{\delta^{2}}{2}\left(\begin{array}[]{cc}\text{Re}\left(AA^{\dagger}\right)&-\text{Im}\left(AA^{\dagger}\right)\\\
\text{Im}\left(AA^{\dagger}\right)&\text{Re}\left(AA^{\dagger}\right)\end{array}\right),$
with A being the transfer matrix of the displacement noise. Since
$V,\Sigma_{0},\Lambda$ satisfy this symmetry, $\Sigma$ also satisfies this
symmetry and we can thus use:

$\displaystyle\Sigma_{q}=\frac{1}{2}MM^{\dagger}+\frac{\delta^{2}}{2}AA^{\dagger}.$

Inserting this into eq. 2.14 yields:

$\displaystyle
I=4\mathcal{V}^{\dagger}\left(MM^{\dagger}+\delta^{2}AA^{\dagger}\right)^{-1}\mathcal{V}.$

###  2.2 Fisher information with homodyne measurement

For $2k$ output quadratures
$\vec{Q}_{\text{out}}=\left(\begin{array}[]{c}\vec{b_{1}}\\\
\vec{b_{2}}\end{array}\right),$ performing homodyne measurement of $l\leq k$
commuting quadratures yields the following Fisher infirmation matrix (FIM)
[8]:

$\displaystyle
F=2\mathcal{V}^{\dagger}T_{h}\left(T_{h}^{\dagger}\Sigma_{q}T_{h}\right)^{-1}T_{h}^{\dagger}\mathcal{V},$
(2.15)

where $T_{h}$ is a $2k\times l$ basis transformation from the standard basis
of the quadratures to an orthonormal basis of the $l$ measured quadrature:
$T_{h}=\left(\begin{array}[]{cccc}\vec{e}_{1}&\vec{e}_{2}&\cdots&\vec{e}_{l}\end{array}\right),$
with $\left\\{\vec{e}_{i}\right\\}_{i}$ being an orthonormal basis of the
measured quadratues. Denoting the projection operator onto the $l$ measured
quadratures as $\Pi_{h}$ we can observe that $\Pi_{h}=T_{h}T_{h}^{\dagger}.$
$\sigma_{h}=T_{h}^{\dagger}\Sigma_{q}T_{h}$ and
$\sqrt{2}T_{h}^{\dagger}\mathcal{V}$ are the covariance matrix and mean vector
of the $l$-dimensional Gaussian distribution obtained after this measurement.
This distribution and FIM correspond to the Gaussian distribution obtained
from eliminating all the quadratures orthogonal to those being measured.

Let us decompose $\Pi_{h}$ into orthogonal subspaces
$\Pi_{h}=\underset{i}{\sum}\Pi_{h_{i}},$ and denote the FIM given measurement
of $\Pi_{h_{i}}$ quadratures as $F_{i}:$
$F_{i}=2\mathcal{V}^{\dagger}T_{h_{i}}\left(T_{h_{i}}^{\dagger}\sigma_{h}T_{h_{i}}\right)^{-1}T_{h_{i}}^{\dagger}\mathcal{V}.$
Given that $\sigma_{h}$ is block diagonal in this decomposition, then the
measurements of $\Pi_{h_{i}}$ are statistically independent and thus
$F=\underset{i}{\sum}F_{i}.$ Formally:

$\displaystyle
F\overset{*}{=}2\mathcal{V}^{\dagger}\left(\underset{i}{\sum}\Pi_{h_{i}}\sigma_{h}\Pi_{h_{i}}\right)^{-1}\mathcal{V}\overset{\\#}{=}\underset{i}{\sum}2\mathcal{V}^{\dagger}T_{h_{i}}\left(T_{h_{i}}^{\dagger}\sigma_{h}T_{h_{i}}\right)^{-1}T_{h_{i}}^{\dagger}\mathcal{V}=\underset{i}{\sum}F_{i}$
(2.16)

where $\left(*\right)$ is due to the fact that it is block diagonal and
$\left(\\#\right)$ is basically:

$\left(\begin{array}[]{cccc}\mathcal{V}^{\dagger}_{1}&\mathcal{V}^{\dagger}_{2}&\cdots&\mathcal{V}^{\dagger}_{j}\end{array}\right)\left(\begin{array}[]{@{}c@{}c@{}c@{\mkern-5mu}c@{\,}cc*{2}{@{\;}c}@{}}\vskip
6.45831pt\cr\framebox{\,$\begin{matrix}\sigma_{1}\end{matrix}$\,}\\\\[-0.4pt]
&\hskip-0.4pt\framebox{\,$\begin{matrix}\sigma_{2}\end{matrix}$\,}\\\\[-2.15277pt]
&&\hfil\mkern-5.0&\;\ddots\\\\[-2.15277pt]
&&\hfil\mkern-5.0&&\framebox{\,$\begin{matrix}\sigma_{j}\end{matrix}$\,}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{c}\mathcal{V}_{1}\\\
\mathcal{V}_{2}\\\ \vdots\\\
\mathcal{V}_{j}\end{array}\right)=\underset{i}{\sum}\mathcal{V}_{i}^{\dagger}\sigma_{i}^{-1}\mathcal{V}_{i}.$

We can use this fact to analyze DFI schemes. For example, when measuring the
phase quadratures, the displacement free subspace (DFS) is an eigenspace of
the covariance matrix, and thus the covariance matrix is block diagonal in the
decomposition to the coupled subspace and the DFS. We thus have that
$F=F_{\text{C}}+F_{\text{DFS}},$ where $F_{\text{C}}$ is the information from
the coupled subspace and $F_{\text{DFS}}$ is the information from the DFS. The
quantity:

$\displaystyle\eta=\frac{F_{\text{DFS}}}{F_{\text{DFS}}+F_{\text{C}}}$ (2.17)

is the fraction of the information that comes from the DFS and thus quantifies
the effectiveness of the DFI.

##  Appendix 3 Optimal measurement basis

We derive an optimal measurement basis for the general setting of
Interforemetric GW detection with Gaussian states. We assume that the
parameters are encoded in the mean vector, as it is in interferometric GW
detection [41, 8].

For a single parameter estimation, we can always saturate the QFI with a
homodyne measurement of a single distributed quadrature. This is a quantum
generalization of the fact that for a classical Gaussian distribution it
suffices to use only a single combination of the outcomes to saturate the
Fisher Information [51].

For general $l$ parameters, the QFIM can be saturated with a homodyne
measurement of $l$ distributed quadratures given that the relevant quadratures
commute.

Let us begin with the single parameter case. We use here the complex compact
form. the mean vector is $\vec{d}_{q}=\vec{\mathcal{V}}h$ and the covariance
matrix is $\Sigma_{q}.$ Measuring this Gaussian state in a distributed
quadrature $\vec{u}$ yields the following FI about $h$ (special case of eq.
2.15):

$\displaystyle
F=2\frac{|\vec{u}\cdot\vec{\mathcal{V}}|^{2}}{\vec{u}^{\dagger}\Sigma_{q}\vec{u}}.$
(3.18)

It can be observed from the Cauchy-Schwarz inequality,

$\displaystyle|(\sqrt{\Sigma_{q}}\vec{u})\cdot(\sqrt{\Sigma_{q}}^{-1}\vec{\mathcal{V}})|^{2}\leq\left(\vec{u}^{\dagger}\Sigma_{q}\vec{u}\right)\left(\vec{\mathcal{V}}^{\dagger}\Sigma_{q}^{-1}\vec{\mathcal{V}}\right)~{}\Rightarrow~{}2\frac{|\vec{u}\cdot\vec{\mathcal{V}}|^{2}}{\left(\vec{u}^{\dagger}\Sigma_{q}\vec{u}\right)}\leq
2\left(\vec{\mathcal{V}}^{\dagger}\Sigma_{q}^{-1}\vec{\mathcal{V}}\right).$
(3.19)

where the right hand side of the inequality is the expression for QFI and
equality is obtained if and only if,

$\displaystyle\sqrt{\Sigma_{q}}\vec{u}\propto\sqrt{\Sigma_{q}}^{-1}\vec{\mathcal{V}}\Rightarrow\vec{u}\propto\Sigma_{q}^{-1}\vec{\mathcal{V}}$
(3.20)

Hence, the QFI is saturated given that the distributed quadrature
$\Sigma_{q}^{-1}\vec{\mathcal{V}}$ is measured. $\square$

We can observe from eq. 2.15 that for a homodyne measurement of a set of
quadratures described by projection operator $\Pi_{h},$ if
$\Sigma_{q}^{-1}\vec{\mathcal{V}}$ is contained in this subspace then the QFI
is saturated.

For $l$ complex parameters (in GW detection $l=2$), let us denote the mean
vector and covariance matrxi as $\vec{d}=\mathcal{V}\vec{h}$ $\Sigma_{q}$
respectively. The QFIM is saturated in this case by measuring the $l$
quadratures $\Sigma_{q}^{-1}\mathcal{V}.$ It can be proved using a
generalization of Cauchy-schwarz for positive semi-definite matrices (as shown
in ref. [50]), we prove it here with an equivalent intuitive derivation:
Observe that for any projection operator $\Pi$:

$\displaystyle\mathcal{V}^{\dagger}\sqrt{\Sigma_{q}}^{-1}\left(\mathbbm{1}-\Pi\right)\sqrt{\Sigma_{q}}^{-1}\mathcal{V}\geq
0$ (3.21)
$\displaystyle\Rightarrow\mathcal{V}^{\dagger}\sqrt{\Sigma_{q}}^{-1}\Pi\sqrt{\Sigma_{q}}^{-1}\mathcal{V}\leq\mathcal{V}^{\dagger}\Sigma_{q}^{-1}\mathcal{V}$

with equality iff
$\Pi\sqrt{\Sigma_{q}}^{-1}\mathcal{V}=\sqrt{\Sigma_{q}}^{-1}\mathcal{V}$
(hence $\Pi$ needs to be at least $l$ dimensional). Note that
$\Pi=\sqrt{\Sigma_{q}}T_{h}\left(T_{h}^{\dagger}\Sigma_{q}T_{h}\right)^{-1}T_{h}^{\dagger}\sqrt{\Sigma_{q}}$
is a projection operator onto the column space of $\sqrt{\Sigma_{q}}T_{h},$
denoted as $\mathcal{C}\left(\sqrt{\Sigma_{q}}T_{h}\right).$ Inserting this
$\Pi$ into eq. 3.21 we obtain that:

$\displaystyle\mathcal{V}^{\dagger}T_{h}\left(T_{h}^{\dagger}\Sigma_{q}T_{h}\right)^{-1}T_{h}^{\dagger}\mathcal{V}\leq\mathcal{V}^{\dagger}\Sigma_{q}^{-1}\mathcal{V}.$

Equality is therefore obtained iff
$\mathcal{C}\left(\sqrt{\Sigma_{q}}^{-1}\mathcal{V}\right)\subseteq\mathcal{C}\left(\sqrt{\Sigma_{q}}T_{h}\right)$
and thus the minimal space of quadratures that saturate the inequality is:
$T_{h}=\Sigma_{q}^{-1}\mathcal{V}\Lambda,$ where $\Lambda$ is a normalization
and orthogonalization matrix. $\square$

Since the multi-parameter case requires commutativity of the quadratures given
by the column vectors of $\Sigma_{q}^{-1}\mathcal{V},$ we prove a useful
claim: If the quadratures given by the column vectors of $\mathcal{V}$ commute
and $\Sigma_{q}$ is a conjugate symplectic matrix then the quadratures given
by $\Sigma_{q}^{-1}\mathcal{V}$ commute and thus the QFIM is achievable.

Proof: Since conjugate symplectic matrices form a group then $\Sigma_{q}$
conjugate symplectic $\rightarrow\Sigma_{q}^{-1}$ conjugate symplectic , i.e.
denoting $\mathcal{W}=\left(\begin{array}[]{cc}0&\mathbbm{1}_{k}\\\
-\mathbbm{1}_{k}&0\end{array}\right)$:

$\displaystyle\left(\Sigma_{q}^{-1}\right)^{\dagger}\mathcal{W}\Sigma_{q}^{-1}=\mathcal{W}.$

Hence:

$\displaystyle\mathcal{V}\;\text{commute}\leftrightarrow\mathcal{V}^{\dagger}\mathcal{W}\mathcal{V}=0$
$\displaystyle\Rightarrow\left(\Sigma_{q}^{-1}\mathcal{V}\right)^{\dagger}\mathcal{W}\left(\Sigma_{q}^{-1}\mathcal{V}\right)=\mathcal{V}^{\dagger}\mathcal{W}\mathcal{V}=0.$

Therefore $\Sigma_{q}^{-1}\mathcal{V}$ commute.$\square$

In our problem, the column vectors of $\mathcal{V}$ are in the phase
quadratures, hence they commute. Therefore in order to show commutativity of
the optimal quadratures it suffices to show that $\Sigma_{q}$ is symplectic.

##  Appendix 4 QFIM with Thermal displacement noise

Displacement of optical components leads to a noisy displacement of the
quadratures, i.e. in Heisenberg picture:
$\vec{Q}\rightarrow\vec{Q}+A\vec{\Delta x}$, where $\vec{\Delta x}$ is a
multivariate Gaussian random variable.

The new state under the action of this noise is a Gaussian mixture of states

$\displaystyle\rho=\int p\left(\vec{\Delta x}\right)\rho\left(\vec{\Delta
x}\right)\;\text{d}\vec{\Delta x}$ (4.22)

and therefore is also Gaussian.

Note that while $\vec{\Delta x}\left(t\right)$ is a real vector, $\vec{\Delta
x}\left(\Omega\right)$ is complex. The transformation of the Hermitian
quadratures vector $\vec{S}$ is therefore given by,

$\vec{S}\rightarrow\vec{S}+A^{\prime}\vec{\Delta x^{\prime}}$ (4.23)

where

$A^{\prime}=\left(\begin{array}[]{cc}\text{Re}\left(A\right)&\text{-Im}\left(A\right)\\\
\text{Im}\left(A\right)&\text{Re}\left(A\right)\end{array}\right),\;\Delta
x^{\prime}=\sqrt{2}\left(\begin{array}[]{c}\text{Re}\left(\Delta x\right)\\\
\text{Im}\left(\Delta x\right)\end{array}\right)$ (4.24)

Since $\langle\vec{\Delta x}\rangle=0$ , the displacement vector $\vec{d_{s}}$
is unchanged. The covariance matrix however changes to:

$\displaystyle\Sigma=$ $\displaystyle\Sigma_{i}+\langle(A^{\prime}\vec{\Delta
x^{\prime}})(A^{\prime}\vec{\Delta x^{\prime}})^{\dagger}\rangle$ (4.25)
$\displaystyle=$ $\displaystyle\Sigma_{i}+A^{\prime}\Sigma_{\vec{\Delta
x^{\prime}}}A^{\prime\dagger},$

where $\Sigma_{i}$ is the covariance matrix of the states in the absence of
displacement noise and $\Sigma_{\Delta x^{\prime}}=\langle\vec{\Delta
x^{\prime}}(\vec{\Delta x^{\prime}})^{\dagger}\rangle$ is the covariance
matrix of $\vec{\Delta x^{\prime}}.$

We assume that the displacement noise $\left\\{\vec{\Delta
x}\left(t\right)\right\\}_{t}$ is a Gaussian stationary process, where the
different $\Delta x_{i}\left(t\right),\Delta x_{j}\left(t\right)$ are i.i.d.
Therefore $\vec{\Delta x}^{\prime}\left(\Omega\right)$ is also a Gaussian
random variable with a covariance matrix of:

$\displaystyle\langle\text{Re}\left(\Delta
x\right)^{2}\rangle=2\underset{0}{\overset{T}{\int}}\;\underset{0}{\overset{T}{\int}}\langle\Delta
x\left(t_{1}\right)\Delta x\left(t_{2}\right)\rangle\cos\left(\Omega
t_{1}\right)\cos\left(\Omega t_{2}\right)\;dt_{1}dt_{2}\approx
T\underset{0}{\overset{T}{\int}}C\left(\tau\right)\cos\left(\Omega\tau\right)\;d\tau$
$\displaystyle\langle\text{Im}\left(\Delta
x\right)^{2}\rangle=2\underset{0}{\overset{T}{\int}}\;\underset{0}{\overset{T}{\int}}\langle\Delta
x\left(t_{1}\right)\Delta x\left(t_{2}\right)\rangle\sin\left(\Omega
t_{1}\right)\sin\left(\Omega t_{2}\right)\;dt_{1}dt_{2}\approx
T\underset{0}{\overset{T}{\int}}C\left(\tau\right)\cos\left(\Omega\tau\right)\;d\tau$
$\displaystyle\langle\text{Re}\left(\Delta x\right)\text{Im}\left(\Delta
x\right)\rangle=2\underset{0}{\overset{T}{\int}}\;\underset{0}{\overset{T}{\int}}\langle\Delta
x\left(t_{1}\right)\Delta x\left(t_{2}\right)\rangle\cos\left(\Omega
t_{1}\right)\sin\left(\Omega t_{2}\right)\;dt_{1}dt_{2}\approx 0,$

and of course all the correlations of $\Delta x_{i}^{\prime},\Delta
x_{j}^{\prime}(i\neq j)$ vanish. Therefore $\Sigma_{\Delta
x^{\prime}}=\frac{\delta^{2}}{2}\mathbbm{1},$ and thus $\vec{\Delta
x^{\prime}}\sim N\left(0,\frac{\delta^{2}}{2}\mathbbm{1}\right).$

Since $\Sigma_{i}=\frac{1}{2}M^{\prime}M^{\prime\dagger}$ we get that:

$\displaystyle\Sigma=\frac{1}{2}\left(M^{\prime}M^{\prime\dagger}+\delta^{2}A^{\prime}A^{\prime\dagger}\right).$
(4.26)

The QFIM therefore reads,

$\displaystyle I=$ $\displaystyle
2\partial_{\vec{h}}\vec{d_{s}}^{\dagger}\left(M^{\prime}M^{\prime\dagger}+\delta^{2}A^{\prime}A^{\prime\dagger}\right)^{-1}\partial_{\vec{h}}\vec{d_{s}}$
(4.27) $\displaystyle=$ $\displaystyle
2\mathcal{V}^{\prime\dagger}\left(M^{\prime}M^{\prime\dagger}+\delta^{2}A^{\prime}A^{\prime\dagger}\right)^{-1}\mathcal{V}^{\prime}$

Alternatively, the form of the covariance matrix can be also directly derived
from the Wigner function. Due to the displacement noise, we have:

$\displaystyle W\left[\rho\right]=\int p\left(\vec{\Delta
x}\right)W\left[\rho\left(\vec{\Delta x}\right)\right]\;\text{d}\vec{\Delta
x}.$ (4.28)

For $k$ output quadratures, the Wigner function per realization of
$\vec{\Delta x}$ is:

$\displaystyle W(\vec{\Delta
x})=\frac{1}{\left(2\pi\right)^{2k}\sqrt{\text{Det}\left(\sigma\right)}}\exp\left[-\frac{1}{2}\left(\vec{S}-d_{\vec{h}}-d_{\vec{\Delta
x}}\right)^{\dagger}\Sigma_{i}^{-1}\left(\vec{S}-d_{\vec{h}}-d_{\vec{\Delta
x}}\right)\right],$

Since $p\left(\vec{\Delta x}\right)$ is Gaussian, then the averaging (eq.
4.28) is basically a convolution of two Gaussian distributions,

$\displaystyle N(\vec{d_{s}},\Sigma_{i})\circledast
N\left(0,\text{Cov$\left(d_{\vec{\Delta
x}}\right)$}\right)=N(\vec{d_{s}},\Sigma_{i})\circledast
N\left(0,\delta^{2}A^{\prime}A^{\prime\dagger}\right)=N(\vec{d_{s}},\Sigma_{i}+\delta^{2}A^{\prime}A^{\prime\dagger})$

Therefore,

$\displaystyle
W=\frac{1}{\left(2\pi\right)^{2n}\sqrt{\text{Det}\left(\Sigma\right)}}\exp\left[-\frac{1}{2}\left(\vec{S}-d_{\vec{h}}\right)^{\dagger}\Sigma^{-1}\left(\vec{S}-d_{\vec{h}}\right)\right],$

with $\Sigma=\Sigma_{i}+\delta^{2}A^{\prime}A^{\prime\dagger}.$

We can therefore observe that the full covariance matrix, eq. 4.26, satisfies
the symmetry of eq. 2.10. We have shown in sec. 2.1 that if the covariance
matrix satisfies the circular symmetry then the QFIM can be written in the
complex compact form, this justifies our use of this form:

$\displaystyle
I=4\mathcal{V}^{\dagger}\left(MM^{\dagger}+\delta^{2}AA^{\dagger}\right)\mathcal{V}.$
(4.29)

A brief note regarding the DFS. A DFS is defined as the kernel of the
(general) noise term $A\Sigma_{\Delta x}A^{\dagger}.$ Note that
$\text{ker}\left(A^{\dagger}\right)$ is contained in this subspace.
furthermore if $\Sigma_{\Delta x}$ is a full rank matrix, the DFS is equal to
$\text{ker}\left(A^{\dagger}\right).$

##  Appendix 5 FIM and QFIM with radiation pressure

###  5.1 Derivation of transfer matrix

Let us first write how radiation pressure noise enters into the equations
[41]. Resonance conditions are assumed. Given $\bf{a},\bf{d}$ fields that hit
a mirror, $\bf{a}=\left(\begin{array}[]{c}a_{1}\\\ a_{2}\end{array}\right),$
$\bf{d}=\left(\begin{array}[]{c}d_{1}\\\ d_{2}\end{array}\right)$, the
reflected fields $\bf{b},\bf{c}$ satisfy :

$\displaystyle\left(\begin{array}[]{c}\bm{\bm{b}}\\\
\bm{c}\end{array}\right)=M_{\text{mirror}}\left(\begin{array}[]{c}\bm{a}\\\
\bm{d}\end{array}\right)-2\left(\frac{\omega_{0}}{c}\right)\frac{\sqrt{R}}{\sqrt{\hbar\omega_{0}}}\Delta
x\left(\begin{array}[]{c}\bm{D_{a}^{*}}\\\ \bm{D_{d}^{*}}\end{array}\right),$
(5.36)

with $M_{\text{mirror}}=\left(\begin{array}[]{cccc}\sqrt{R}&\sqrt{T}\\\
\sqrt{R}&\sqrt{T}\\\ &&-\sqrt{T}&\sqrt{R}\\\
&&-\sqrt{T}&\sqrt{R}\end{array}\right)$ being the mirror transformation,
$\Delta x$ is the displacement due to radiation pressure and assuming
resonance ($\omega_{0}L/c=2\pi n$, where $n$ is an integer):
$\bm{D_{j}}=\sqrt{2I_{j}}\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$
$\bm{D_{j}^{*}}=\sqrt{2I_{j}}\left(\begin{array}[]{c}0\\\
-1\end{array}\right)$. Eq. 5.36 is the general way displacement noise $\Delta
x$ is being propagated. In radiation pressure $\Delta x$ is given by :

$\displaystyle\Delta
x=\frac{1}{m\Omega^{2}}\sqrt{\frac{\hbar\omega_{0}}{c^{2}}}\left[\left(\begin{array}[]{cc}\bm{D_{a}^{t}}&\bm{-D_{d}^{t}}\end{array}\right)\left(\begin{array}[]{c}\bm{a}\\\
\bm{d}\end{array}\right)+\left(\begin{array}[]{cc}\bm{D_{b}^{t}}&-\bm{D_{c}^{t}}\end{array}\right)\left(\begin{array}[]{c}\bm{b}\\\
\bm{c}\end{array}\right)\right]$
$\displaystyle=\frac{1}{m\Omega^{2}}\sqrt{\frac{2\hbar\omega_{0}}{c^{2}}}\left[\sqrt{I_{a}}a_{1}-\sqrt{I_{d}}d_{1}+\sqrt{I_{b}}b_{1}-\sqrt{I_{c}}c_{1}\right].$

Inserting this $\Delta x$ into eq. 5.36 we get two coupled sets of equations.
For the amplitude quadratures:

$\left(\begin{array}[]{c}b_{1}\\\
c_{1}\end{array}\right)=M_{\text{mirror}}\left(\begin{array}[]{c}a_{1}\\\
d_{1}\end{array}\right),$

and for the phase quadratures:

$\displaystyle\left(\begin{array}[]{c}b_{2}\\\
c_{2}\end{array}\right)=M_{\text{mirror}}\left(\begin{array}[]{c}a_{2}\\\
d_{2}\end{array}\right)-\frac{2\sqrt{2}\omega_{0}\sqrt{R}}{m\Omega^{2}c^{2}}\left[\sqrt{I_{a}}a_{1}-\sqrt{I_{d}}d_{1}+\sqrt{I_{b}}b_{1}-\sqrt{I_{c}}c_{1}\right]\left(\begin{array}[]{c}\sqrt{2I_{a}}\\\
\sqrt{2I_{d}}\end{array}\right).$ (5.43)

This implies a general structure for the multichannel case: the equations for
the amplitude quadratures are closed and their solution is given by

$\displaystyle\vec{b_{1}}=M_{\text{int}}\vec{a_{1}},$

where $M_{\text{int}}$ is the unitary transfer matrix of the interferometer
which does not depend on the radiation pressure. The equations for the
amplitude quadratures are coupled to the phase quadratures and their solution
is given by:

$\displaystyle\vec{b_{2}}=M_{\text{int}}\vec{a_{2}}+M_{21}\vec{a_{1}}.$

Hence amplitude noise is being propagated into phase noise with a transfer
matrix $M_{21}$. The input-output relations therefore read:

$\displaystyle\left(\begin{array}[]{c}\vec{b_{1}}\\\
\vec{b_{2}}\end{array}\right)=\left(\begin{array}[]{cc}M_{\text{int}}&0\\\
M_{21}&M_{\text{int}}\end{array}\right)\left(\begin{array}[]{c}\vec{a_{1}}\\\
\vec{a_{2}}\end{array}\right),$ (5.50)

Two observations regarding the transfer matrix of eq. 5.50 will be useful
later:

1\. $M_{21}$ can be expressed as a concatenation of two transfer matrices:
$M_{21}=AD_{x},$ where $A$ is the transfer matrix of mirror displacement
vector $\vec{\Delta x},$ and $D_{x}$ is the transfer matrix of the amplitude
noise to the displacement vector: $\vec{\Delta x}=D_{x}\vec{a_{1}}.$ This fact
will be used to characterize the DFS in section 6.

2\. Commutation relations have to be preserved (see eq. 2.5). This implies
that $M$ is a conjugate symplectic matrix and thus
$M_{\text{int}}^{\dagger}M_{21}$ is Hermitian.

3\. From eq. 5.43 we observe that $M_{21}\propto 1/m\Omega^{2}.$

###  5.2 QFIM

Given the general form of the transfer matrix $M$ (eq. 5.50), we can calculate
the general form of the QFIM and FIM. The QFIM is given by:
$4\mathcal{V}^{\dagger}\left(MM^{\dagger}\right)^{-1}\mathcal{V}.$
$MM^{\dagger}$ $(=2\Sigma_{q})$ equals to:

$\displaystyle
MM^{\dagger}=\left(\begin{array}[]{cc}\mathbbm{1}&M_{\text{int}}M_{21}^{\dagger}\\\
M_{21}M_{\text{int}}^{\dagger}&\mathbbm{1}+M_{\text{21}}M_{21}^{\dagger}\end{array}\right).$
(5.53)

Since $M_{\text{int}}$ is unitary we can observe that:

$\displaystyle\left(MM^{\dagger}\right)^{-1}=\left(\begin{array}[]{cc}*&-M_{\text{int}}M_{21}^{\dagger}\\\
&\mathbbm{1}\end{array}\right)$
$\displaystyle\overset{\\#}{\Rightarrow}I=4\mathcal{V}^{\dagger}\mathcal{V}=4\mathcal{V}_{\text{ph}}^{\dagger}\mathcal{V}_{\text{ph}},$

where in $(\\#)$ we used the fact that
$\mathcal{V}=\left(\begin{array}[]{c}0\\\
\mathcal{V}_{\text{ph}}\end{array}\right)$. The QFIM thus obtains the shot-
noise limit. This is a generalized variational readout scheme: by measuring
certain distributed quadratures we can overcome the radiation pressure noise.
Using the results of section 3, we know that the optimal quadratures are (up
to normalization): $\Sigma_{q}^{-1}\mathcal{V},$ i.e.:

$\displaystyle
u\propto\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\mathcal{V}_{\text{ph}}\\\
\mathcal{V}_{\text{ph}}\end{array}\right).$ (5.56)

This optimal quadratures are a linear combination of the quadratures:

$\displaystyle\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\\\
\mathbbm{1}\end{array}\right),$ (5.59)

hence instead of measuring the single or two optimal quadratures (for single
or multivariate estimation respectively), which depends on $M$ and
$\mathcal{V},$ we can measure this set of quadratures, more precisely an
orthonormal basis: $\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\\\
\mathbbm{1}\end{array}\right)\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)^{-1/2},$
which depends only on $M.$ In complete analogy to the standard variational
readout, since the covariance matrix reads:

$\displaystyle\left(\begin{array}[]{cc}\mathbbm{1}&M_{\text{int}}M_{21}^{\dagger}\\\
M_{21}M_{\text{int}}^{\dagger}&\mathbbm{1}+M_{21}M_{21}^{\dagger}\end{array}\right)=\left(\begin{array}[]{cc}0&0\\\
0&\mathbbm{1}\end{array}\right)+\left(\begin{array}[]{c}\mathbbm{1}\\\
M_{21}M_{\text{int}}^{\dagger}\end{array}\right)\left(\begin{array}[]{cc}\mathbbm{1}&M_{\text{int}}M_{21}^{\dagger}\end{array}\right),$

this space of quadratures is decoupled from radiation pressure noise. Hence
any combination of these quadratures is decoupled from this noise and thus
yields an FI that does not diverge in the $f\rightarrow 0$ limit, the optimal
combination of eq. 5.56 obtains the shot nois limit.

We need of course to show that these quadratures commute, otherwise they
cannot be measured simultaneously. This can be shown from direct calculation:

$\displaystyle\left(\begin{array}[]{cc}-M_{21}M_{\text{int}}^{\dagger}&\mathbbm{1}\end{array}\right)\left(\begin{array}[]{cc}0&\mathbbm{1}\\\
-\mathbbm{1}&0\end{array}\right)\left(\begin{array}[]{c}-M_{\text{\text{int}}}M_{21}^{\dagger}\\\
\mathbbm{1}\end{array}\right)=-M_{\text{\text{int}}}M_{21}^{\dagger}+M_{21}M_{\text{\text{int}}}^{\dagger}=0,$

where in the last equality we used the fact that
$M_{21}M_{\text{int}}^{\dagger}$ is Hermitian.

###  5.3 FIM

Let us now consider the case of measuring the phase quadratures. The
correspondong covariance matrix is obtained by applying eq. 2.15, i.e. keeping
only the phase quadrature terms in the full covariance matrix (eq. 5.53),
which leaves us with
$\sigma_{h}=1/2\left(\mathbbm{1}+M_{\text{21}}M_{21}^{\dagger}\right)$, and
thus the FI is
$F=4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbbm{1}+M_{\text{21}}M_{21}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$
$M_{\text{21}}M_{21}^{\dagger}$ is basically the displacement noise term.

Since $M_{21}=A_{\text{ph}}D_{x},$ we get:

$\displaystyle
M_{21}M_{21}^{\dagger}=A_{\text{ph}}D_{x}D_{x}^{\dagger}A_{\text{ph}}^{\dagger},$

hence it takes the form of displacement noise (eq. 4.25) with
$D_{x}D_{x}^{\dagger}$ being the covariance matrix of the displacement vector
$\vec{\Delta x}.$ We can immediately observe that the space decoupled from
this noise is $\text{ker}\left(D_{x}^{\dagger}A^{\dagger}\right).$ The DFS,
$\text{ker}A^{\dagger}$ , is thus contained in this subspace and equals to it
if $D_{x}D_{x}^{\dagger}$ is full rank. The DFS is therefore decoupled from
this noise.

In fig. 1 we show the sensitivity profile given radiation pressure noise for
different homodyne measurements: optimal (eq. 5.56), phase quadratures and
maximal signal combination (basically $\mathcal{V},$ which maximizes the
signal and thus optimal in case there is only shot noise). For the optimal
combination (eq. 5.56), the std coincides with the shot noise limit as
expected.

Figure 1:  sensitivity profile with radiation pressure noise for different
measurement bases. The QFI coreesponds to optimal measurement (orange, solid
line) and saturates the shot noise limit (black, dashed line). The solid blue
line corresponds to phase quadratures measurement (and thus optimal
combination of phase quadratures) and the dashed red line to the max-signal
combination of phase quadratures, i.e. a combination that is optimal given
only shot noise. For this illustration $m=5$ kg.

It is interesting to compare the behavior of the FI with phase quadratures
measurement and the behavior with the max-signal combination in fig. 1. Both
diverge in the DC limit and clearly since the max-signal combination is not
optimal its sensitivity is worse than the sensitivity of the phase quadratures
measurement. While the max-signal combination diverges uniformly as $1/f^{2},$
the optimal phase quadratures combination has an intermediate range where the
divergence is stopped and there is a plateau. This creates two orders of
magnitude difference between the sensitivity with max-signal combination and
with the optimal phase quadratures combination.

This plateau is due to a pseudo-DFS contained in the coupled subspace,
$M_{\text{C}}.$ For any frequency, $M_{21}M_{21}^{\dagger}$ has $3$
eigenvalues: $0,t_{\text{min}},t_{\text{max}},$ where $t_{\text{min}}\ll
t_{\text{max}}.$ The phase quadratures can be thus decomposed to the
corresponding eigenspaces. The eigenspace of $0$ is the DFS, $M_{\text{DFS}},$
and the eigenspaces of $t_{\text{min}},t_{\text{max}}$ are denoted as
$M_{\text{min}},M_{\text{max}}$ respectively. Note that
$M_{\text{C}}=M_{\text{min}}\oplus M_{\text{max}}.$ Since the covariance
matrix is diagonal w.r.t these subspaces the FI is a sum of the FI’s of these
subspaces:

$\displaystyle
F=F_{\text{DFS}}+F_{\text{C}}=F_{\text{DFS}}+F_{\text{max}}+F_{\text{min}},$

where $F_{\text{max}},F_{\text{min}}$ are the FI’s achieved with
$M_{\text{max}},M_{\text{min}}$ respectively. $\mathcal{V}$ lies mostly in
$M_{\text{max}},$ hence for frequencies higher than the plateau range
$F\approx F_{\text{max}}.$ The std that corresponds to $F_{\text{max}}$ goes
as $1/f^{2}$ and coincides with the max-signal combination (see fig. 2). As
$f$ gets smaller, $F_{\text{max}}$ drops as $f^{4}$ while $F_{\text{min}}$
remains the same (since $t_{min}\ll 1$). In this regime $M_{\text{min}}$
functions as a pseudo-DFS since the effect of displacement noise is much
smaller than the shot noise. Therefore at some point
$F_{\text{min}}>F_{\text{max}},$ and the FI coincides with the
$F_{\text{min}}$ which remains the same. This is the plateau that can be
observed in figs. 1, 2. $t_{\text{min}}$ however also goes as $1/f^{4}$ and
thus for low enough frequencies $t_{\text{min}}\gg 1$ and the std continues to
diverge as $1/f^{2}.$ This is shown in fig 2 where the FI is shown along with
the contribution of $F_{\text{max}},F_{\text{min}},$ $F_{\text{DFS}}.$ We can
see the crossing between $F_{\text{max}}$ and $F_{\text{min}}$ that takes
place at the beginning of the plateau.

Figure 2:  Sensitivity profile that corresponds to phase quadratures
measurement ($F$) along with $F_{\text{max}}$ (blue dots), $F_{\text{min}}$
(green triangles), and $F_{\text{DFS}}$ (brown squares), as defined in the
text. As shown in the text $F=F_{\text{max}}+F_{\text{min}}+F_{\text{DFS}}.$

##  Appendix 6 FIM and QFIM with radiation pressure and thermal displacement
noise

Figure 3:  Sensitivity profile given thermal noise, with and without radiation
pressure. (a) QFI: QFI with both thermal and radiation pressure noise (red,
solid line) coincides with QFI given only thermal noise (green diamonds). Blue
dashed line corresponds to the shot noise limit. (b) FI with phase quadrature
measurement: red solid line corresponds to both thermal and radiation pressure
noise, green diamonds to only thermal noise and blue, dashed line to shot
noise limit.For this illustration $m=5$ kg.

In realistic scenarios we have both radiation pressure and thermal
displacement noise, i.e. a covariance matrix of
$\Sigma_{q}=1/2\left(MM^{\dagger}+\delta^{2}AA^{\dagger}\right).$ The QFIM in
this case is the same as with only thermal displacement noise since the
radiation pressure can be removed using the same variational readout. Let us
show this formally:

$\displaystyle\Sigma_{q}=1/2\left(\begin{array}[]{cc}\mathbbm{1}&M_{\text{int}}M_{21}^{\dagger}\\\
M_{21}M_{\text{int}}^{\dagger}&\mathbbm{1}+M_{\text{21}}M_{21}^{\dagger}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\end{array}\right).$

With a similar calculation as before:

$\displaystyle\Sigma_{q}^{-1}=2\left(\begin{array}[]{cc}*&-M_{\text{int}}M_{21}^{\dagger}\left(\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\\\
&\left(\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\end{array}\right)$
$\displaystyle\Rightarrow
I=4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$

Hence radiation pressure noise is removed with a generalized variational
readout, this is shown in fig. 3 (a). The optimal combination is

$\displaystyle
u\propto\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\tilde{\mathcal{V}}\\\
\tilde{\mathcal{V}}\end{array}\right),$

where
$\tilde{\mathcal{V}}=\left(\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$
As in section 5, we can achieve this QFIM also by measuring the quadratures of
eq. 5.59, that are decoupled from radiation pressure noise. The only
difference is that now the optimal combination of these quadratures is
$\tilde{\mathcal{V}}$ instead of $\mathcal{V}.$

Measuring the phase quadratures the FI reads:

$\displaystyle
F=4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}$
$\displaystyle\overset{*}{=}4\mathcal{V}_{\text{ph}}^{\dagger}\left(\mathbbm{1}+A_{\text{ph}}\left[D_{x}D_{x}^{\dagger}+\delta^{2}\mathbbm{1}\right]A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$

There are two displacement noise terms: $M_{21}M_{21}^{\dagger}$ due to
radiation pressure and $\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}$ due to
thermal displacement noise. In $(*)$ we use the fact that
$M_{21}=A_{\text{ph}}D_{x},$ therefore the two noise terms can be written as a
single displacement noise $\vec{\Delta x},$ with a covariance matrix of
$1/2\left(\delta^{2}\mathbbm{1}+D_{x}D_{x}^{\dagger}\right).$ Since this
covariance matrix is full rank the DFS is $\text{ker}\left(A^{\dagger}\right)$
(in our triangular cavity example it is $3$ dimensional).

The behavior of the FI for phase quadratures measurement is shown in fig 3
(b): there is an intermediate regime where the radiation pressure noise is
dominant and the FI does not coincide with the FI of only thermal noise. For
low enough frequencies however the thermal noise becomes dominant and the FI
coincides with the FI of only thermal noise.

A more detailed analysis of the behavior of the FI is presented in fig. 4. As
observed with radiation pressure in section 5, there is no uniform divergence
that goes as $1/f^{2.5},$ but there are three regimes. Like in section 5, we
can decompose the phase quadratures to: $M=M_{\text{max}}\oplus
M_{\text{min}}\oplus M_{\text{DFS}},$ and define the corresponding FI’s:
$F_{\text{max}},F_{\text{min}},F_{\text{DFS}}.$ Since these subspaces are
eigenspaces of the covariance matrix we have:
$F=F_{\text{max}}+F_{\text{min}}+F_{\text{DFS}}.$ In fig. 4 we examine the
contribution of the these FI’s. We can observe three regimes: for relatively
high frequencies $F_{\text{max}}$ is dominant and coincides with $F.$
$M_{\text{max}}$ is a single dimensional subspace that has maximal noise
(largest eigenvalue of the covariance matrix), but also yields maximal signal.
Therefore in the limit of small noise $F$ coincides with it. At some point,
$F_{\text{min}}$ becomes larger than $F_{\text{max}},$ this crossing can be
observed in fig. 4. After this crossing $F_{\text{min}}$ becomes the dominant
contribution. Like in fig. 2, there is a small range of frequencies with
plateau, where $v_{\text{min}}\approx 1.$ For smaller frequencies the noise
becomes larger and the dominant contribution is from the DFS: $F\approx
F_{\text{DFS}}.$

Figure 4:  Similar to fig. 2 given thermal displacement and radiation pressure
noise with phase quadratures measurement. Three different regimes of the FI:
the solid red line correspond to $F$, the blue dots to $F_{\text{max}},$ green
triangles to $F_{\text{min}}$ and brown squares to $F_{\text{DFS}}.$ The
different FI’s coincide with total FI in three different regimes.

##  Appendix 7 Effect of squeezing

So far the initial state was taken to be a coherent state, i.e.
$\Sigma_{i}=\frac{1}{2}\mathbbm{1}$. By squeezing some of the quadratures the
sensitivity can be improved. The covariance matrix with squeezing is taken to
be
$\Sigma_{i}=\frac{1}{2}e^{2s}\Pi_{1}+\frac{1}{2}e^{-2s}\left(\mathbbm{1}-\Pi_{1}\right),$
i.e. $k$ commuting quadratures are squeezed and as a result the conjugate $k$
quadratures are antisqueezed. A trivial upper bound of the QFIM corresponds to
squeezing all the quantum noise by $e^{-2s}$, i.e.:

$\displaystyle I\leq
2\mathcal{V}^{\dagger}\left(\Sigma_{\text{min}}\right)^{-1}\mathcal{V},$
(7.60)

where $2\Sigma_{\text{min}}=e^{-2s}MM^{\dagger}+\delta^{2}AA^{\dagger},$ i.e.
the quantum part ,$MM^{\dagger}$, is squeezed by $e^{-2s}$. This bound of the
QFI implies a bound on the FI with any homodyne measurement. The bound for the
FI given phase quadratures measurement is:

$\displaystyle F\leq
2\mathcal{V}^{\dagger}T_{\text{ph}}\left(T_{\text{ph}}^{\dagger}\Sigma_{\text{min}}T_{\text{ph}}\right)^{-1}T_{\text{ph}}^{\dagger}\mathcal{V}$
(7.61)
$\displaystyle=4\mathcal{V}_{\text{ph}}^{\dagger}\left(e^{-2s}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)+\delta^{2}A_{\text{ph}}A^{\dagger}_{\text{ph}}\right)^{-1}\mathcal{V}_{\text{ph}},$

i.e. squeezing the quantum part, $\mathbbm{1}+M_{21}M_{21}^{\dagger},$ by
$e^{-2s}$. We show a brief proof of these bounds. For brevity we omit the
$\delta^{2}AA^{\dagger}$ term which does not change the proof:

$\displaystyle
2\Sigma_{q}=M\left[e^{2s}\Pi_{1}+e^{-2s}\left(\mathbbm{1}-\Pi_{1}\right)\right]M^{\dagger}\geq
e^{-2s}MM^{\dagger}=2\Sigma_{\text{min}}$
$\displaystyle\overset{*}{\Rightarrow}\Sigma^{-1}\leq\Sigma_{\text{min}}^{-1}$
$\displaystyle\overset{**}{\Rightarrow}\mathcal{V}^{\dagger}\left(\Sigma\right)^{-1}\mathcal{V}\leq\mathcal{V}^{\dagger}\left(\Sigma_{\text{min}}\right)^{-1}\mathcal{V}.$

$(*)$ is due to the fact that for any positive semi-definite matrices
$B_{1},B_{2}$: $B_{1}\geq B_{2}\Rightarrow B_{1}^{-1}\leq B_{2}^{-1}.$ $(**)$
is immediate from definition. Similarly:

$\displaystyle\mathcal{V}^{\dagger}T_{\text{ph}}\left(T_{\text{ph}}^{\dagger}\Sigma
T_{\text{ph}}\right)^{-1}T_{\text{ph}}^{\dagger}\mathcal{V}\leq\mathcal{V}^{\dagger}T_{\text{ph}}\left(T_{\text{ph}}^{\dagger}\Sigma_{\text{min}}T_{\text{ph}}\right)^{-1}T_{\text{ph}}^{\dagger}\mathcal{V}.$

Let us now show that both of these bounds are achievable. We first show that
by an optimal choice of squeezed quadratures, $\Pi_{1}$, the phase quadratures
bound of eq. 7.61 is achievable.

###  7.1 Optimal squeezing for phase quadratures measurements

To saturate eq. 7.61 we need to find a set of commuting quadratures such that
squeezing them yields this bound. Let us first find suitable quadratures and
then verify that they commute. We therefore need to find a projection operator
$\Pi_{1}$ such that

$\displaystyle
M\left(e^{2s}\Pi_{1}+e^{-2s}\left(\mathbbm{1}-\Pi_{1}\right)\right)M^{\dagger}=\left(\begin{array}[]{cc}*&*\\\
&e^{-2s}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)\end{array}\right)$

This implies that $\Pi_{1}$ needs to satisfy:

$\displaystyle M\Pi_{1}M^{\dagger}=\left(\begin{array}[]{cc}*&*\\\
&0\end{array}\right),$

i.e. no antisqueezing in the phase quadratures. Following the derivation for
single input single output [2], we use an ansatz:

$\displaystyle\Pi_{1}=\left(\begin{array}[]{cc}\cos\left(\mathcal{A}\right)^{2}&\cos\left(\mathcal{A}\right)\sin\left(\mathcal{A}\right)\\\
\cos\left(\mathcal{A}\right)\sin\left(\mathcal{A}\right)&\sin\left(\mathcal{A}\right)^{2}\end{array}\right),$

where $\mathcal{A}$ is a Hermitian matrix that we seek to find. It can be
checked that this defines a projection operator ($\Pi_{1}^{2}=\Pi_{1}$) onto
the space spanned by $\left(\begin{array}[]{c}\cos\left(\mathcal{A}\right)\\\
\sin\left(\mathcal{A}\right)\end{array}\right).$ We now observe:

$\displaystyle
M\left(\begin{array}[]{cc}\cos\left(\mathcal{A}\right)^{2}&\cos\left(\mathcal{A}\right)\sin\left(\mathcal{A}\right)\\\
\cos\left(\mathcal{A}\right)\sin\left(\mathcal{A}\right)&\sin\left(\mathcal{A}\right)^{2}\end{array}\right)M^{\dagger}=\left(\begin{array}[]{cc}*&*\\\
&\left(M_{21}\cos\left(\mathcal{A}\right)+M_{\text{int}}\sin\left(\mathcal{A}\right)\right)\left(\cos\left(\mathcal{A}\right)M_{21}^{\dagger}+\sin\left(\mathcal{A}\right)M_{\text{int}}^{\dagger}\right)\end{array}\right)$
$\displaystyle\Rightarrow
M_{21}\cos\left(\mathcal{A}\right)+M_{\text{int}}\sin\left(\mathcal{A}\right)=0$
$\displaystyle\mathcal{A}=\arctan\left(-M_{\text{int}}^{\dagger}M_{21}\right).$

Note that $-M_{\text{int}}^{\dagger}M_{21}$ is Hermitian hence $\mathcal{A}$
is well defined. It is therefore optimal to squeeze the quadratures projected
by $\mathbbm{1}-\Pi_{1}$:

$\displaystyle\left(\begin{array}[]{c}-\sin\left(\mathcal{A}\right)\\\
\cos\left(\mathcal{A}\right)\end{array}\right)=\left(\begin{array}[]{c}-\sin\left(-\arctan\left(M_{\text{int}}^{\dagger}M_{21}\right)\right)\\\
\cos\left(-\arctan\left(M_{\text{int}}^{\dagger}M_{21}\right)\right)\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}M_{\text{int}}^{\dagger}M_{21}\\\
\mathbbm{1}\end{array}\right)\left(\mathbbm{1}+\left(M_{\text{int}}^{\dagger}M_{21}\right)^{2}\right)^{-1/2}.$

It is straightforward to see that these quadratures commute: for any
$\mathcal{A}$

$\displaystyle\left(\begin{array}[]{cc}-\sin\left(\mathcal{A}\right)&\cos\left(\mathcal{A}\right)\end{array}\right)\left(\begin{array}[]{cc}0&\mathbbm{1}_{k}\\\
-\mathbbm{1}_{k}&0\end{array}\right)\left(\begin{array}[]{c}-\sin\left(\mathcal{A}\right)\\\
\cos\left(\mathcal{A}\right)\end{array}\right)=0.$

Numerical results of the std without squeezing and with optimal squeezing are
presented in fig. 5.

###  7.2 QFIM with squeezing

It is simple to show that the QFI satisfies the bound of eq. 7.60. The
calculation is identical to the calculations in sections 5 and 4. Given
squeezing of phase quadratures:

$\displaystyle\Sigma=1/2\left(\begin{array}[]{cc}M_{\text{int}}&0\\\
M_{21}&M_{\text{int}}\end{array}\right)\left(\begin{array}[]{cc}e^{2s}\mathbbm{1}&0\\\
0&e^{-2s}\mathbbm{1}\end{array}\right)\left(\begin{array}[]{cc}M_{\text{int}}^{\dagger}&M_{21}^{\dagger}\\\
0&M_{\text{int}}^{\dagger}\end{array}\right)+\frac{\delta^{2}}{2}AA^{\dagger}$
$\displaystyle=1/2\left(\begin{array}[]{cc}e^{2s}\mathbbm{1}&M_{\text{int}}M_{21}^{\dagger}e^{2s}\\\
M_{21}M_{\text{int}}^{\dagger}e^{2s}&e^{-2s}\mathbbm{1}+M_{21}M_{21}^{\dagger}e^{2s}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\end{array}\right)\Rightarrow\Sigma^{-1}=2\left(\begin{array}[]{cc}*&-M_{\text{int}}M_{21}^{\dagger}\left(e^{-2s}\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\\\
&\left(e^{-2s}\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\end{array}\right)$
$\displaystyle\Rightarrow
I=4\mathcal{V}_{\text{ph}}^{\dagger}\left(e^{-2s}\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}.$

The distributed quadrature that saturates this bound is:

$\displaystyle\left(\begin{array}[]{c}-M_{\text{int}}M_{21}^{\dagger}\\\
\mathbbm{1}\end{array}\right)\left(e^{-2s}\mathbbm{1}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{ph}}$

The QFI sensitivity with and without squeezing is presented in fig. 5. It is
interesting to compare this QFI to the phase quadratures FI. Without squeezing
the QFI is significantly better than the FI in the regime of dominant
radiation pressure (in this illustration $50$ Hz-$500$ Hz). Optimal squeezing
removes this difference almost completely since it squeezes the radiation
pressure noise while keeping the thermal displacement noise intact. We can
also observe that in both cases the gain from squeezing is not uniform: in
some frequency regimes this gain is much larger than in other regimes. This is
explained in the next subsection.

###  7.3 $\eta$ and squeezing

As mentioned in the main text, for a given choice of homodyne measurement we
can define the gain factor

$\displaystyle\eta_{\text{gain}}=\frac{\frac{F_{\text{sq}}}{F}-1}{e^{2s}-1},$

where $F_{\text{sq}}\,(F)$ is the FI with(out) squeezing.

For phase quafratures measurement we defined $\eta$ in the main text as:

$\displaystyle\eta=\frac{F_{\text{DFS}}}{F},$

where $F_{\text{DFS}}$ is the FI coming from the DFS. This is well defined for
phase quadratures measurement, since in this case the quadratures space
contains a DFS, $h_{\text{DFS}}\subseteq h,$ and the DFS is an eigenspace of
the relevant covariance matrix $\sigma_{h}.$ Furthermore for phase quadratures
measurement we can decompose: $h=h_{\text{DFS}}\oplus h_{\text{C}},$ where
$h_{\text{C}}$ is the coupled subspace, and thus
$F=F_{\text{DFS}}+F_{\text{C}},$ where:

$\displaystyle
F_{\text{DFS}}=4\mathcal{V}_{\text{DFS}}^{\dagger}\mathcal{V}_{\text{DFS}},$
$\displaystyle
F_{\text{C}}=4\mathcal{V}_{\text{C}}^{\dagger}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{C}}$

where $\mathcal{V}_{\text{DFS}}=\Pi_{\text{DFS}}\mathcal{V}_{\text{ph}},$
$\mathcal{V}_{\text{C}}=\Pi_{\text{C}}\mathcal{V}_{\text{ph}}.$

Optimal squeezing changes the covariance matrix to
$\frac{e^{-2s}}{2}\left(\mathbbm{1+M_{21}M_{21}^{\dagger}}\right)+\frac{\delta^{2}}{2}A_{\text{ph}}A_{\text{ph}}^{\dagger},$
and thus after squeezing:

$\displaystyle
F_{\text{DFS}}=4e^{2s}\mathcal{V}_{\text{DFS}}^{\dagger}\mathcal{V}_{\text{DFS}}$
$\displaystyle
F_{\text{C}}=4\mathcal{V}_{\text{C}}^{\dagger}\left(e^{-2s}\left(\mathbbm{1}+M_{21}M_{21}^{\dagger}\right)+\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{C}}\overset{*}{\approx}4\mathcal{V}_{\text{C}}^{\dagger}\left(\delta^{2}A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)^{-1}\mathcal{V}_{\text{C}}.$

In $(*)$ we use the fact that
$\mathcal{V}_{\text{C}}\perp\text{ker}\left(A_{\text{ph}}A_{\text{ph}}^{\dagger}\right)$
(by definition of the coupled subspace), we assume large displacement noise,
and $\left(\delta^{2}AA^{\dagger}\right)^{-1}$ refers to the pseudo-inverse of
$\delta^{2}AA^{\dagger}.$ Hence $F_{\text{DFS}}$ is increased by the squeezing
factor $e^{2s},$ while in the limit of large displacement noise $F_{\text{C}}$
remains unchanged. Clearly only the DFS gains from squeezing and thus the gain
from squeezing should go as the effectiveness of the DFI. We can show this
formally with $\eta_{\text{gain}}$:

$\displaystyle\frac{F_{\text{sq}}}{F}=\frac{e^{2s}F_{\text{DFS}}+F_{\text{C}}}{F_{\text{DFS}}+F_{\text{C}}}$
$\displaystyle\eta_{\text{gain}}=\frac{\frac{F_{\text{sq}}}{F}-1}{e^{2s}-1}=\frac{\left(e^{2s}-1\right)F_{\text{DFS}}}{\left(e^{2s}-1\right)\left(F_{\text{DFS}}+F_{\text{C}}\right)}=\frac{F_{\text{DFS}}}{F}=\eta.$

Therefore in the limit of large displacement noise $\eta_{\text{gain}}=\eta,$
this is illustrated in fig. (3) in the main text.

Figure 5: std corresponding to QFI (brown) and phase quadratures FI (green).
The solid lines correspond to unsqueezed quadratures and the dashed to optimal
squeezing. In both cases, the QFI curves are below the corresponding FI
curves. In this illustration $s=1.5$.

##  Appendix 8  Extensions

Figure 6:  Comparison between the sensitivities of the different $n$-gon
schemes: dashed lines correspond to shot-noise only and solid lines to thermal
displacement noise as well. The blue, red and gray lines correspond to
$n=3,5,9$ respectively.

We numerically compare the triangular scheme analyzed above with $n$-gon
extensions described in fig. 4 in the main text. In fig. 6 the triangular
scheme ($n=3$) is compared to larger number of mirrors: $n=5,9,$ where thermal
displacement noise is considered. In all schemes we consider only the two
cyclic trajectories with maximal length as shown in fig. 4 (a) in the main
text (angles $\pi\frac{n-1}{n},\pi\frac{n+1}{n}$ for odd $n$). Therefore the
number of input/output fields is $2n$ and the phase quadratures DFS is
$n$-dimensional. In this comparison, all the $n$-gons have the same radius and
the intracavity power is the same (taken to be
$10^{5}\frac{T}{\left(1-\sqrt{R}\right)^{2}}\approx 3.8\;\text{MW}$).

It can be observed in fig. 6 that assuming only shot noise the sensitivity in
the DC limit becomes better as $n$ is increased. More precisely, in DC the QFI
goes as $I\propto n.$ The reason is that the length of the trajectory goes as
$n$ which accounts for a factor of $n$ in the response vector $\mathcal{V},$
on the other hand there is more loss due to the increased number of mirros
which accounts for a factor of $\frac{1}{\sqrt{n}}.$ Altogether $\mathcal{V}$
goes as $\sqrt{n}$ and thus the QFI as $n.$ However once we add thermal
displacement noise the sensitivities of the schemes coincide and only a very
minor improvement is observed for larger $n.$

Further extensions, such as using all the $n-1$ trajectories as showm in fig.
4(b) hold the promise for improved sensitivities, as they have larger DFS. We
leave these extensions for future work.

##  Appendix 9  Newtonian and Seismic Noise

Apart from thermal and radiation pressure noise, interferometers suffer also
from newtonian and seismic noise [52, 30].

Like thermal noise, they can be modelled as random displacement vector
$\vec{\Delta x}_{\text{NN}},\vec{\Delta x}_{\text{S}}$ propagated to the
output quadratures by a transfer matrix $A.$ Denoting the covariance matrices
of $\vec{\Delta x}_{\text{NN}},\vec{\Delta x}_{\text{S}}$ as
$\sigma_{\text{S}},$ $\sigma_{\text{NN}},$ we get that the QFIM with these
noises reads:

$\displaystyle
I=4\mathcal{V}^{\dagger}\left(MM^{\dagger}+A\left(\delta^{2}\mathbbm{1}+\sigma_{\text{NN}}+\sigma_{\text{S}}\right)A^{\dagger}\right)^{-1}\mathcal{V}.$

Unlike thermal noise, the assumption of uncorrelated noise between mirrors is
no longer correct for these noises. For example, two nearby mirrors will have
correlated seismic and newtonian noises.

##  Appendix 10 Complementary calculations

###  10.1 GW response of the basic scheme: sensitivity analysis

We calculate the GW response, i.e. the displacement vector
$\vec{d}=\mathcal{V}\vec{h}$, from which we can get $\mathcal{V}$ and the shot
noise limit 4$\mathcal{V}_{\text{ph}}^{\dagger}\mathcal{V}_{\text{ph}}.$ Since
the responce to the GW is in the phase quadratures, it suffices to calculate
the $6$-dimensional $\mathcal{V}_{\text{ph}}$ and the corresponding
$\vec{d}_{\text{ph}}=\mathcal{V}_{\text{ph}}\vec{h}$. We calculate it for the
symmtric triangular scheme depicted in fig. (1) in the main text. Let us
introduce the required notation: $L$-length of the arms,
$A_{\text{in}}$-amplitude of the incoming fields (identical for all fields),
$\omega_{0}$-frequency of the carrier fields, $c$-speed of light. All the
mirrors have the same power reflectivity $R$ (and power transmissivity
$T=1-R$).

We assume that the GW signal travels along an arbitrary direction given below,

$\hat{k}=\left(\sin\left(\theta\right)\cos\left(\phi\right),\sin\left(\theta\right)\sin\left(\phi\right),\cos\left(\theta\right)\right),$

and the orthogonal unit vectors to this direction of travel are

$\displaystyle\begin{split}&\hat{u}=\left(-\cos\left(\theta\right)\cos\left(\phi\right),-\cos\left(\theta\right)\sin\left(\phi\right),\sin\left(\theta\right)\right),\\\
&\hat{v}=\left(-\cos\left(\phi\right),\sin\left(\phi\right),0\right).\end{split}$

To calculate the GW response, we begin by considering a cyclic trajectory. We
denote the arms as $\left\\{h_{j}\right\\}_{j=1}^{3}$ while the effect of GW
with a frequency $f$ on the $j$-th arm, $h_{j}\left(f\right)$, is given by:

$h_{j}\left(f\right)=\zeta_{j}\left(f\right)\left(~{}h_{+}\langle
n_{j}|e_{+}|n_{j}\rangle~{}+~{}h_{\times}\langle
n_{j}|e_{\times}|n_{j}\rangle~{}\right),$

where $e_{+}=u\otimes u-v\otimes v,~{}e_{\times}=u\otimes v+v\otimes u$ are
the polarization tensors and $|n_{j}\rangle$ are the unit vector of the $j$-th
arm. The factor $\zeta_{j}\left(f\right)$ contains the dependence on the
frequency and GW direction and is given by

$\displaystyle\begin{split}\zeta_{j}\left(f\right)=A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)L~{}\text{sinc}\left(\epsilon(1-\hat{n}_{j}\cdot\hat{k})\right)\exp\left(i\frac{1}{L}\left(\vec{x}_{j,1}+\vec{x}_{j,2}\right)\cdot\hat{k}\epsilon-i\epsilon\right),\end{split}$

with $\epsilon=2\pi fL/2$ and $\vec{x}_{j,1},\vec{x}_{j,2}$ are the positions
of the mirrors. We can observe that as $f\rightarrow 0$,
$\zeta_{j}\left(f\right)\rightarrow L$ for all arms, and the response is
simplified to

$h_{j}\left(f\right)=A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)L\left(h_{+}\langle
n_{j}|e_{+}|n_{j}\rangle+h_{\times}\langle
n_{j}|e_{\times}|n_{j}\rangle\right)$

Each $h_{j}$ propagates to the $k$-th output mode in the following way:

$\displaystyle\begin{split}\left(d_{\text{ph}}\right)_{k}=\frac{T}{(1-\sqrt{R})(1-R^{\prime
1.5})}\cdot(R^{\prime}h_{k}+\sqrt{R^{\prime}}h_{k+1}+h_{k+2}),\end{split}$

with $R^{\prime}=Re^{i2\Omega L}.$ Hence the full vector of output modes
(clockwise and counter-clockwise) reads:

$\vec{d}_{\text{ph}}=\frac{(1+\sqrt{R})}{(1-R^{\prime
3/2})}\left(\begin{array}[]{c}\left(\begin{array}[]{c}R^{\prime}h_{1}+\sqrt{R^{\prime}}h_{2}+h_{3}\\\
R^{\prime}h_{2}+\sqrt{R^{\prime}}h_{3}+h_{1}\\\
R^{\prime}h_{3}+\sqrt{R^{\prime}}h_{1}+h_{2}\end{array}\right)_{\text{L}}\\\
\left(\begin{array}[]{c}h_{1}+\sqrt{R^{\prime}}h_{2}+R^{\prime}h_{3}\\\
h_{2}+\sqrt{R^{\prime}}h_{3}+R^{\prime}h_{1}\\\
h_{3}+\sqrt{R^{\prime}}h_{1}+R^{\prime}h_{2}\end{array}\right)_{\text{R}}\end{array}\right)$
(10.62)

The QFIM about $h_{+},h_{\times}$ in the shot-noise limit is given by:

$\vec{d}_{\text{ph}}^{\dagger}\vec{d}_{\text{ph}}=\vec{h}^{\dagger}I\vec{h},$
(10.63)

with $\vec{h}=\left(h_{+},h_{\times}\right)^{T}.$

We thus calculate $\vec{d}_{\text{ph}}^{\dagger}\vec{d}_{\text{ph}}$ using eq.
10.62:

$\displaystyle\begin{split}\vec{d}_{\text{ph}}^{\dagger}\vec{d}_{\text{ph}}=A_{\text{in}}^{2}\left(\frac{\omega_{0}}{c}\right)^{2}\frac{T}{\left(1-\sqrt{R}\right)^{2}}\frac{T}{\left|1-R^{\prime
1.5}\right|^{2}}\left[l_{1}\left(R\right)\underset{j=1}{\overset{3}{\sum}}h_{j}^{2}+l_{2}\left(R^{\prime}\right)\underset{j,m>j}{\sum}h_{j}h_{m}\right],\end{split}$
(10.64)

where $l_{1}\left(R\right)=\frac{1-R^{3}}{1-R}$ and
$l_{2}\left(R^{\prime}\right)=2\text{Re}\left(\sqrt{R^{\prime}}\frac{1-R^{\prime
1.5}}{1-\sqrt{R^{\prime}}}\right).$

Let us now focus on the DC limit ($f\ll c/L$ and thus $R=R^{\prime}$ ). In
this limit eq. 10.64 becomes:

$\displaystyle\begin{split}\vec{d}_{\text{ph}}^{\dagger}\vec{d}_{\text{ph}}=2A_{\text{in}}^{2}\left(\frac{\omega_{0}}{c}\right)^{2}\frac{\left(1+\sqrt{R}\right)^{2}}{\left(1-R^{1.5}\right)^{2}}\cdot\left[\frac{1-R^{3}}{1-R}\underset{j}{\sum}h_{j}^{2}+2\sqrt{R}\frac{1-R^{1.5}}{1-\sqrt{R^{\prime}}}\underset{m>j}{\sum}h_{j}h_{m}\right].\end{split}$
(10.65)

Note that in the relevant limit of $T\ll 1:$

$\displaystyle\frac{1-R^{3}}{1-R},\sqrt{R}\frac{1-R^{1.5}}{1-\sqrt{R}}\approx
3R+\mathcal{O}\left(T^{2}\right)$

hence this expression can be further simplified to:

$\vec{d}_{\text{ph}}^{\dagger}\vec{d}_{\text{ph}}\approx
6A_{\text{in}}^{2}\left(\frac{\omega_{0}}{c}\right)^{2}\frac{\left(1+\sqrt{R}\right)^{2}}{\left(1-R^{1.5}\right)^{2}}R\left(\underset{j}{\sum}h_{j}\right)^{2}.$
(10.66)

The coefficients of the bilinear terms of $h_{+},h_{\times}$ in eqs.
10.65,10.66 correspond to the FIM elements (eq. 10.64). It can be shown that
the coefficient of $h_{+}h_{\times}$ vanishes , hence the FIM is diagonal in
the $h_{+},h_{\times}$ basis. Note that

$\displaystyle\begin{split}&\underset{j}{\sum}h_{j}=Lh_{+}\underset{j}{\sum}\langle
n_{j}|e_{+}|n_{j}\rangle+Lh_{\times}\underset{j}{\sum}\langle
n_{j}|e_{\times}|n_{j}\rangle=\\\
&\frac{3}{2}Lh_{+}\text{Tr}_{x-y}\left(e_{+}\right)+\frac{3}{2}Lh_{\times}\text{Tr}_{x-y}\left(e_{\times}\right)=\\\
&\frac{3}{2}Lh_{+}\text{Tr}_{x-y}\left(e_{+}\right).\end{split}$ (10.67)

In the second equality we use the fact that
$\underset{j}{\sum}|n_{j}\rangle\langle n_{j}|$ is proportional to the
projection operator on the $x-y$ plane:
$\underset{j}{\sum}|n_{j}\rangle\langle n_{j}|=\frac{3}{2}\Pi_{x-y},$ , hence
$\underset{j}{\sum}\langle
n_{j}|\bullet|n_{j}\rangle=\frac{3}{2}\text{Tr}_{x-y}\left(\bullet\right).$ In
the third equality we use the fact that
$\text{Tr}_{x-y}\left(e_{\times}\right)=0,$ because:

$\text{Tr}_{x-y}\left(e_{\times}\right)=2\langle u|\Pi_{x-y}|v\rangle=0.$

Hence combining eqs. 10.66, 10.67, we get that the minimal variance is about
$h_{+}$ (maximal eigenvalue of the FIM) and it reads:

$\displaystyle\sigma=\Delta
h_{+}=\frac{1}{2\sqrt{||V^{\dagger}\mathcal{V}||}}$
$\displaystyle\Rightarrow\Delta
h_{+}\approx\frac{1}{2A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)L|\text{Tr}_{x-y}\left(e_{+}\right)|\sqrt{13.5\frac{\left(1+\sqrt{R}\right)^{2}}{\left(1-R^{1.5}\right)^{2}}R}}$
$\displaystyle=\frac{1}{2A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)L\sin\left(\theta\right)^{2}\sqrt{13.5\frac{\left(1+\sqrt{R}\right)^{2}}{\left(1-R^{1.5}\right)^{2}}R}}.$

###  10.2 DFS

Each mirror displacement $\Delta x_{j}$ is coupled to a single output mode:
$\underset{i}{\sum}\left(A_{\text{ph}}\right)_{i,j}\Delta x_{j}$, where
$A_{\text{ph}}$ is the transfer matrix. Let us define the vector $d_{\Delta
x}=A_{\text{ph}}\vec{\Delta x}.$ We find the propagation into the clockwise
cyclic trajectory and from symmetry we get the propagation into the counter
clockwise trajectory: It can be observed that for every $j,k$ the coefficient
of $\Delta x_{j}$ in $(d_{\Delta x})_{j+k,\text{L}}$ is identical to its
coefficient in $(d_{\Delta x})_{j-k,\text{R}}$, where by by $j\pm k$ we mean
mod $3$. Observe that for every $j\neq 0$ the coefficient of $\Delta x_{j}$ in
$(d_{\Delta x})_{j+k,\text{L}}$ is equal to

$-A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)\cos\left(\pi/6\right)\frac{\left(1+\sqrt{R}\right)\left(R^{\prime}\right)^{k/2}}{\left(1-R^{\prime
3/2}\right)},$

the coefficient of $\Delta x_{j}$ in $b_{j}\left(k=0\right)$ is equal to

$\displaystyle\begin{split}&-A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)\cos\left(\pi/6\right)\left(\sqrt{R}+\frac{\left(1+\sqrt{R}\right)\left(R^{\prime}\right)^{3/2}}{\left(1-R^{\prime
3/2}\right)}\right)\\\
&=-A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)\cos\left(\pi/6\right)\left(\frac{\sqrt{R}+R^{\prime
3/2}}{1-R^{\prime 3/2}}\right).\end{split}$

Hence in the DC limit the transfer matrix of the displacement
vector,$A_{\text{ph}}$, reads:

$\displaystyle\begin{split}&A_{\text{ph}}=-A_{\text{in}}\left(\frac{\omega_{0}}{c}\right)\cos\left(\pi/6\right)\frac{\sqrt{R}}{\left(1-R^{3/2}\right)}\cdot\\\
&\left(\begin{array}[]{c}\left(\begin{array}[]{c}1+R\\\ 1+\sqrt{R}\\\
\sqrt{R}+R\end{array}\right)_{\text{L}}\\\ \left(\begin{array}[]{c}1+R\\\
\sqrt{R}+R\\\
1+\sqrt{R}\end{array}\right)_{\text{R}}\end{array}\begin{array}[]{c}\left(\begin{array}[]{c}\sqrt{R}+R\\\
1+R\\\ 1+\sqrt{R}\end{array}\right)_{\text{L}}\\\
\left(\begin{array}[]{c}1+\sqrt{R}\\\ 1+R\\\
\sqrt{R}+R\end{array}\right)_{\text{R}}\end{array}\begin{array}[]{c}\left(\begin{array}[]{c}1+\sqrt{R}\\\
\sqrt{R}+R\\\ 1+R\end{array}\right)_{\text{L}}\\\
\left(\begin{array}[]{c}\sqrt{R}+R\\\ 1+\sqrt{R}\\\
1+R\end{array}\right)_{\text{R}}\end{array}\right)\end{split}$ (10.68)

Comparing with the DC limit of eq. 10.62, we can observe that the GW response
is a linear combination of the vector of $A_{\text{ph}}$. This fact explains
the DC divergence of the std.