# Supersensitive phase estimation by thermal light in nonlinear
interferometers

Nilakantha Meher<EMAIL_ADDRESS>Department of Chemical and
Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
Eilon Poem<EMAIL_ADDRESS>Department of Physics of Complex
Systems, Weizmann Institute of Science, Rehovot 7610001, Israel  Tomáš Opatrný
Department of Optics, Faculty of Science, Palacký University, 17. listopadu
50, 77146 Olomouc, Czech Republic  Ofer Firstenberg  Department of Physics of
Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
Gershon Kurizki  Department of Chemical and Biological Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel

###### Abstract

By consensus, estimation of phase delay between interferometer arms may
exhibit an error below the standard quantum (shot-noise) limit if the input is
an entangled two-mode state, e.g., a N00N state. We show, by contrast, that
such super-sensitive phase estimation is achievable by incoherent, e.g.,
thermal, input in an interferometer with Kerr-nonlinear two-mode coupler. Not
less remarkably, the Heisenberg precision bound is attainable and even
surpassed in such nonlinear interferometers even for small nonlinear phase-
shifts per photon pair or for significant photon loss. Feasible mode couplers
with giant Kerr nonlinearity that stems either from dipole-dipole interactions
of Rydberg polaritons in a cold atomic gas, or from cavity-enhanced dispersive
atom-field interactions, may exploit such effects to substantially advance
interferometric phase microscopy using incoherent, faint light sources.

Quantum sensing generally denotes sensing protocols that exploit nonclassical
resources or operate in the domain of quantum signals and noise [1, 2, 3, 4,
5, 6, 7, 8, 9]. One of the prominent directions in this emerging field is that
of super-sensitive phase measurements, namely, measurements with uncertainty
less than the standard quantum limit (SQL) [10, 11, 12, 13, 14]. In
conventional two-mode interferometers, the SQL is set by the vacuum noise in
the empty input mode, corresponding to the shot-noise error of phase-delay
estimation $\Delta\phi$ that scales as $\Delta\phi\sim\bar{n}^{-1/2}$,
$\bar{n}$ being the mean input photon number in the populated mode. A lower
error bound is achievable by nonclassical two-mode input states: squeezed
vacuum states [12, 13], even- and odd-coherent states [15, 16], N00N states
[14] and entangled coherent (“cat”) states (ECS) that consist of coherently
superposed N00N states [15]. Both N00N states and ECS yield the phase-error
bound $\Delta\phi\sim 1/\bar{n}$, alias the Heisenberg limit (HL) [10, 11, 13,
15, 12, 14, 17, 18, 19, 20, 16]. To benefit from this nonclassicality,
supersensitive phase estimation (SSPE) must be inferred from high-order
correlation measurements of the two-mode output [21]. Progress over the years
[22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39,
40, 41] has revealed the hurdles involved in the generation of such states:
especially challenging are the giant nonlinear phase shifts of $\pi$ per
photon pair required for their deterministic preparation, as well as their
fragility under decoherence.

Here we break away from the underlying paradigm that SSPE must employ to
interferometers with maximally entangled states. We show that nonclassical or
coherent sources are not required for deterministic SSPE. Instead, it is
achievable through photodetection of the output channels or photo-number
parity measurement [22, 41] of one output mode derived from incoherent, e.g.,
thermal input, shined upon a Mach-Zehnder interferometer whose two-mode
coupler is Kerr-nonlinear. Not less remarkably, the nonlinear phase shift per
photon pair required to beat the SQL limit can be much less than the
$\pi$-shift which transforms a Fock state into a N00N state or a coherent
state into an ECS [20]. Such Kerr-nonlinear SSPE is also impervious to
decoherence and can withstand substantial photon loss inside the
interferometer. The discussed effects may be experimentally realized by
resorting to the strong optical Kerr effects predicted [42, 43] and
demonstrated in setups based on dipole-dipole interacting Rydberg polaritons
in a cold gas [38, 39, 40, 44, 26], or atom-field interactions in high-$Q$
cavities [45, 46].

Figure 1:  (a) Schematic drawing of a nonlinear Mach-Zehnder interferometer
(MZI) with either self-Kerr (SK) or cross-Kerr (CK) nonlinearity. The goal is
to estimate the unknown phase shift $\phi$ through photodetection of the
output channels or parity measurement of one of them. (b) The phase error
(sensitivity) from parity measurements as a function of $\phi$ for various
input states in the SK and CK MZIs (with $\chi=\pi/2,\bar{n}=5$). (c) The
minimal phase error (maximum sensitivity), given by the inverse square root of
the quantum Fisher information (QFI), as a function of $\bar{n}$ for thermal
state input to the MZI interferometer with weak SK or CK nonlinearity
($\chi=\pi/10$).

Scenario and analysis principles: Consider a two-mode interferometer that
incorporates a phase-shifter whose unknown phase $\phi$ we wish to estimate.
The two-mode merger/coupler is endowed with self-Kerr (SK) or cross-Kerr (CK)
nonlinearity [Fig. 1(a)]: A nonlinear phase shift $2\chi$ is caused by SK for
two photons in the same mode, the CK counterpart being the phase shift $\chi$
for two photons in different modes. The overall unitary transformation for
such a Mach-Zehnder interferometer (MZI) with SK or CK nonlinearity is,
respectively,

$\displaystyle U_{SK}$ $\displaystyle=U_{BS}U_{PS}(\phi)U_{BS}e^{i\chi
a^{\dagger 2}a^{2}}U_{PS}(\pi/2)U_{BS},$ (1a) $\displaystyle U_{CK}$
$\displaystyle=U_{BS}U_{PS}(\phi)U_{BS}e^{i\chi
a^{\dagger}ab^{\dagger}b}U_{PS}(\pi/2)U_{BS}.$ (1b)

Here, $U_{PS}(\phi)=e^{i\phi a^{\dagger}a}$ is the operator that shifts the
phase by $\phi$ in mode $a$, and $U_{BS}$ is the operator of a 50:50 beam
splitter [47]. We shall henceforth consider the simplest (both experimentally
and calculationally) case of one empty input mode. Analytical results will be
given for the strongest nonlinear phase-shift per photon pair, $\chi=\pi/2$.

An insightful (but not optimal) phase estimation method is based on measuring
the mean value of the parity operator of one of the output modes:
$\Pi_{out}=e^{i\pi b^{\dagger}_{out}b_{out}}$ [22, 41]. Its direct measurement
by counting the corresponding photon number is challenging. Instead, Kerr
nonlinearity enables a simpler parity measurement of an MZI output by sending
it through a second cross-Kerr interferometer along with a vacuum input and
detecting which of the two output detectors has clicked [see Fig. S1, SI]
[22]. The phase error (sensitivity) obtained from the parity-operator mean
value $\langle\Pi_{out}\rangle$ is compactly expressed as [22, 41]

$\displaystyle\Delta\phi=\frac{\sqrt{1-\langle\Pi_{out}\rangle^{2}}}{\left|\frac{\partial\langle\Pi_{out}\rangle}{\partial\phi}\right|}.$
(2)

In an MZI with SK and $\chi=\pi/2$, for number state input $\ket{n,0}$, we
have $\langle\Pi_{out}\rangle=\sin n\phi$ [see SI], which gives
$\Delta\phi=1/n$ (Heisenberg scaling). One owes this supersensitive phase
resolution to the fact that a number state is transformed after the second
beam splitter/merger in Fig. 1(a) into a state akin to a N00N state.
Accordingly, an input thermal state is then transformed into a mixture of N00N
states corresponding to all $n$, which yields

$\displaystyle\langle\Pi_{out}\rangle=\frac{1}{1+\bar{n}}+\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{(1+\bar{n})^{(n+1)}}\sin
n\phi.$ (3)

The resulting minimum phase error [see Fig. 1(b) and SI]

$\displaystyle\Delta\phi\approx\frac{\sqrt{1-\frac{1}{(\bar{n}+1)^{2}}}}{\bar{n}},$
(4)

falls below the Heisenberg limit (HL) $1/\bar{n}$, reflecting the fact that
the HL is not the ultimate limit for phase estimation for input states with
small $\bar{n}$ and large variance $\Delta n>\bar{n}$ [10, 48]. Importantly,
similar arguments hold for a general input state
$\rho_{in}=\sum_{n,m}\rho_{nm}\ket{n}\bra{m}\otimes\ket{0}\bra{0}$ [see SI].
We note that there is only moderate difference in phase sensitivity between CK
and SK, all parameters being equal [Fig. 1(b)].

Independently of the measurement setting, the minimal phase error attainable
by a given input state is set by the quantum Cramér-Rao bound [49], which is
related to the quantum Fisher information (QFI) $F_{Q}$ through [41]

$\displaystyle\Delta\phi_{min}\geq\frac{1}{\sqrt{F_{Q}}}.$ (5)

The $F_{Q}$ for thermal, coherent and number states input satisfy a remarkable
analytical inequality [see SI]

$\displaystyle(F_{Q})_{thermal}=2\bar{n}^{2}+\bar{n}>(F_{Q})_{coherent}=\bar{n}^{2}+2\bar{n}>(F_{Q})_{number}=\bar{n}^{2},~{}~{}~{}\text{for
SK}$ (6a)
$\displaystyle(F_{Q})_{thermal}=\bar{n}^{2}+\bar{n}>(F_{Q})_{number}=\bar{n}^{2}>(F_{Q})_{coherent}=\tfrac{1}{2}\bar{n}^{2}+2\bar{n},~{}~{}~{}\text{for
CK}$ (6b)

with $\chi=\pi/2$ and $\bar{n}>1$ in Eq. (6a) or $\bar{n}>4$ in Eq. (6b). It
shows that the minimal phase error $\Delta\phi_{min}=1/\sqrt{F_{Q}}$ is below
the HL $1/\bar{n}$ for thermal input, and thermal input allows higher phase
sensitivity than coherent and number inputs. As shown below, this holds only
true for highly efficient photodetection.

SK outperforms CK for $\chi=\pi/2$ in terms of QFI, since all number states
input are transformed to N00N states for SK, whereas only odd number states do
so for CK [see SI]. Surprisingly, numerical calculations of the minimal phase
error $(1/\sqrt{F_{Q}})$, for weak $\chi\simeq\pi/10$ and small average photon
number $\bar{n}\simeq 1$, show that CK can beat the HL and outperforms SK, and
that both allow beating the SQL [see Fig. 1(c)].

Figure 2:  Fisher information for SK MZI obtained by photon-number resolving
detectors: (a) number-state ($n=5$) input and (b) thermal state ($\bar{n}=5)$
input for $\chi=\pi/2$ and detector efficiencies $\eta_{\text{det}}=0.999$
(left) and $\eta_{\text{det}}=0.95$ (right). The SQL (dot-dashed line) here
corresponds to $F_{SQL}=\bar{n}=5$ and the HL (dashed line) to
$F_{HL}=\bar{n}^{2}=25$. The QFI (magenta line) for thermal input is 55 and
for number input is 25. Red thick line: Single-detector photodetection, Blue
thick line: intensity-difference detection, Black thick line: photodetection
of both detectors. For $\eta_{\text{det}}\geq 0.95$, thermal input yields
$F\geq 10$ and thus beats the SQL.

Photodetection efficiency: Numerical investigations of the Fisher information
(Fig. 2) for phase estimation, obtained either from the count difference of
the two photon-number resolving detectors of $a_{out}$ and $b_{out}$ in Fig.
1(a) or from their complete data, yield the following results for nonlinearity
$(\chi=\pi/2)$ and imperfect detectors with efficiency $\eta_{\text{det}}$:
(a) The Fisher information $(F)$ based on complete data from both detectors is
always higher than or equal to the Fisher information based on partial data.
(b) A decrease in the detection efficiency leads to a fast drop of $F$ and its
more pronounced dependence on $\phi$. (c) With perfect detectors
$\eta_{\text{det}}=1$, photodetection on both output channels saturates the
QFI limit for thermal input and number input [Fig. 2], but not for coherent
state input [see SI]. With near-perfect detectors $\eta_{\text{det}}\geq
0.97$, mean photon numbers being equal, the highest $F$ is reached by thermal
states, which then beat the HL and surpass coherent and Fock states
consistently with the inequalities in Eq. (6). For $\eta_{\text{det}}\leq
0.97$ the order is reversed: the highest $F$ is reached by Fock states and the
lowest by thermal. Yet, thermal input still beats the SQL for
$\eta_{\text{det}}\geq 0.9$ [Fig. S3 in SI]. We note that the FI, based on
parity measurement, reaches the QFI for CK MZI and not for SK MZI [see SI].

Figure 3: (a) Inset: Lossy nonlinear MZI mimicked by a fictitious beam
splitter in one arm. Plot: Minimal phase error, calculated for thermal state
from Eq. (5), as a function of the percentage of photon loss in one arm of
either CK or SK MZI. Here, $\bar{n}=10$ and $\chi=\pi/2$. (b) Sufficient
entanglement conditions, characterized by the negativity of $E_{HZ}$ and
$E_{SV}$ [see text and SI] for the two-mode states after the second beam
splitter of the SK MZI in Fig. 1(a), are not satisfied for thermal input for
any nonlinear Kerr strength $\chi$. Here $\bar{n}_{a}$=5. Inset: Similarly for
CK MZI. (c) The zero time-delay second-order coherence function
$g^{(2)}_{a}(0)$ of the mode $a$ after the second beam splitter as a function
of $\chi$ for thermal input ($\bar{n}_{a}=5$) does not feature sub-Poissonian
[$g^{(2)}_{a}(0)<1]$ photon statistics.

Photon loss: We mimic the loss of photons by a fictitious beam splitter (BS)
in one of the interferometer arms before the phase-shifter [Fig. 3(a), inset],
in order to check whether the phase sensitivity is reduced by the vacuum noise
incurred [50, 51]. This loss is given by the BS reflectivity,
$\eta_{\text{Loss}}=\sin^{2}\theta$; $\tan^{2}\theta$ being the intensity
splitting ratio. The phase error obtained for thermal input to either SK or CK
MZI grows with the loss of photons, but the minimal error still beats the SQL
up to 25% loss in the interferometer arm, thus featuring high resilience to
loss.

Discussion: We have shown that Kerr-phase elements in interferometry can yield
supersensitive phase estimation (SSPE), down to the Heisenberg limit (HL) and
even below it, using incoherent input, particularly thermal. In the absence of
loss, the nonlinear Mach-Zehnder interferometer (MZI) conserves the total
number of photons, and acts on each Fock layer separately. Thus, fluctuations
of the phases between Fock layers do not have any effect on the output
intensities, only phases within a Fock layer affect the output photodetection.
Hence, thermal input states are well suited for SSPE with nonlinear MZI, as
long as the input phase and amplitude fluctuate slower than the time
difference between the interferometer arms. Consequently, the nonlinear MZI
described here are resilient to low-frequency input noise, allowing the use of
unlocked lasers, or other spectrally filtered thermal light sources.

Several important and unexpected results have emerged from our analysis: (1)
For any degree of nonlinearity and low photon numbers [Eqs. (3),(4), Fig.
1(c)], the HL is attained by any input distribution wherein the photon number
variance grows as the mean photon number increases, be it pure or mixed. (2)
Phase sensitivity of the interferometer below the standard quantum limit (SQL)
does not entail adherence of the output to the sufficient conditions for
inter-mode entanglement, as verified by the positivity of the functions that
express the Hillery-Zubairy ($E_{HZ}$) [52] and Shchukin-Vogels $(E_{SV})$
[53] entanglement criteria for any nonlinear strengths $\chi$ [ Fig. 3(b),
SI]. Nor does any of the output modes exhibit sub-Poissonian
($g^{(2)}_{a}(0)<1)$ photon statistics, which is a nonclassical characteristic
of the field, as seen in Fig. 3(c). Yet the phase sensitivity of thermal input
surpasses that of pure states in terms of quantum Fisher information [Eq. (6)]
for highly efficient photodetection. For thermal input, SSPE is achievable for
detection efficiency down to 0.9 and up to photon loss levels of 25%. (3) Even
for weak nonlinearity, the nonlinear interferometer creates states that can
strongly improve phase sensitivity [Fig. 1(c)]. (4) While SSPE is achievable
for both cross-Kerr (CK) and self-Kerr (SK) nonlinearities, CK provides better
sensitivity when the nonlinear strength per photon pair is much smaller than
$\pi/2$ [Fig. 1(c)].

Suitable nonlinear MZI may be implemented based on the giant dipole-dipole
interaction between counter-propagating few-photon beams in a cloud of cold
alkali atoms excited to a Rydberg state [42, 43]. The scheme invokes
electromagnetically induced transparency (EIT) in the slightly-off resonant
case. The detuning can be chosen such that, while both with and without
interaction the transmission is the same, the group velocity, and hence, the
induced phase, are different depending on the photon numbers in the beam. This
scheme has been implemented to yield a Kerr nonlinear phase of over $\pi$ for
a single photon pair [39, 40]. Similar two-photon phase shifts have also been
demonstrated in the dispersive regime of atom-photon interactions in ultra
high-$Q$ cavities [45, 46] and tapered fibers [54].

A Kerr-nonlinear phase element can be used not only as mode coupler within the
MZI, but also as a parity filter of the measured output mode [SI]. Such
filters may bypass the need for photon-number resolved detectors. We have
previously shown [55] that similar devices can act as nonlinear coherent heat
machines.

The present findings constitute guidelines for nonlinear interferometer
implementation that can open new avenues in phase sensing: the ability to
achieve SSPE with classical, incoherent, extremely faint light may bring about
a paradigm shift in this field.

###### Acknowledgements.

Acknowledgements: The authors acknowledge support from the US-Israel
Binational Science Foundation (BSF) and the US National Science Foundation
(NSF). EP and OF acknowledge support from the Estate of Louise Yasgour. GK is
supported by the Deutsche Forschung Geselschaft (DFG) FOR 2724, by the
European commission (FET Open, PATHOS) and by QUANTERA (PACE-IN). TO was
supported by the Czech Science Foundation, grant 20-27994S. NM acknowledges
the support of the Feinberg Graduate School (FGS) Dean Postdoctoral
Fellowship.

##  I Supplementary Information

##  II Parity operator mean values in nonlinear MZIs

###  II.1 Self-Kerr (SK) MZI

For an input two-mode number state $\ket{n,0}$, the state right before the
final beam splitter [Fig. 1(a) of text], is

$\displaystyle\ket{\psi(\phi)}=\frac{i^{n}}{\sqrt{2}}e^{i\pi/4}(i^{n-1}e^{in\phi}\ket{n,0}+\ket{0,n}).$
(S1)

To find the expectation value of the corresponding parity operator at the
output, we apply to the parity operator the beam splitter (BS) transformation

$\displaystyle\Pi_{out}=U_{BS}^{\dagger}\Pi
U_{BS}=\sum_{M}(-i)^{M}\sum_{k=0}^{M}(-1)^{k}\ket{k,M-k}\bra{M-k,k}.$ (S2)

We then have for its expectation value

$\displaystyle\langle\Pi_{out}\rangle$
$\displaystyle=\frac{1}{2}((-i)^{n-1}e^{-in\phi}\bra{n,0}+\bra{0,n})\left[\sum_{M}(-i)^{M}\sum_{k=0}^{M}(-1)^{k}\ket{k,M-k}\bra{M-k,k}\right](i^{n-1}e^{iN\phi}\ket{n,0}+\ket{0,n})$
$\displaystyle=\frac{1}{2}((-i)^{n-1}e^{-in\phi}\bra{n,0}+\bra{0,n})\left[(-i)^{n}i^{n-1}e^{in\phi}\ket{0,n}+(-i)^{n}(-1)^{n}\ket{n,0}\right]$
$\displaystyle=\sin n\phi.$ (S3)

The phase error is then given by

$\displaystyle\Delta\phi=\frac{\sqrt{1-\langle\Pi_{out}\rangle^{2}}}{\left|\frac{\partial\langle\Pi_{out}\rangle}{\partial\phi}\right|}=\frac{1}{n}.$
(S4)

For coherent state $\ket{\alpha,0}$ input, we get

$\displaystyle\langle\Pi_{out}\rangle$
$\displaystyle=e^{-|\alpha|^{2}}\left[1+\sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\sin
n\phi\right]$
$\displaystyle=e^{-|\alpha|^{2}}\left[1+e^{|\alpha|^{2}\cos\phi}\sin(|\alpha|^{2}\sin\phi)\right],$
(S5)

As $\phi\rightarrow 0$, this expression becomes

$\displaystyle\langle\Pi_{out}\rangle\approx
e^{-|\alpha|^{2}}+\sin(|\alpha|^{2}\phi),$ (S6)

so that for large $|\alpha|^{2}$

$\displaystyle\Delta\phi\approx\frac{\sqrt{1-e^{-|\alpha|^{2}}+\sin(|\alpha|^{2}\phi)}}{\left||\alpha|^{2}\cos(|\alpha|^{2}\phi)\right|}\approx\frac{1}{|\alpha|^{2}}=\frac{1}{\bar{n}}.$
(S7)

For thermal state input, by averaging Eq. (II.1) over the thermal number-state
distribution with average $\bar{n}$, we have

$\displaystyle\langle\Pi_{out}\rangle=\frac{1}{1+\bar{n}}+\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{(1+\bar{n})^{(n+1)}}\sin
n\phi,$ (S8)

As $\phi\rightarrow 0$, this expression becomes

$\displaystyle\langle\Pi_{out}\rangle\approx\frac{1}{1+\bar{n}},$ (S9)

and

$\displaystyle\langle\frac{\partial\Pi_{out}}{\partial\phi}\rangle\approx\bar{n},$
(S10)

which yields, for large $\bar{n}$,

$\displaystyle\Delta\phi\approx\frac{\sqrt{1-\frac{1}{(\bar{n}+1)^{2}}}}{\bar{n}}\approx\frac{1}{\bar{n}}.$
(S11)

For a general input state

$\displaystyle\rho_{in}=\sum_{n}\rho_{nm}\ket{n}\bra{m}\otimes\ket{0}\bra{0},$
(S12)

the output expectation value of the parity operator becomes

$\displaystyle\langle\Pi_{out}\rangle=|\rho_{00}|^{2}+\sum_{n=0}^{\infty}|\rho_{nn}|^{2}\sin
n\phi.$ (S13)

As $\phi\rightarrow 0$,

$\displaystyle\langle\Pi_{out}\rangle\approx|\rho_{00}|^{2},$ (S14)

and

$\displaystyle\langle\frac{\partial\Pi_{out}}{\partial\phi}\rangle\approx\bar{n},$
(S15)

which yields for large $\bar{n}$

$\displaystyle\Delta\phi\approx\frac{\sqrt{1-|\rho_{00}|^{2}}}{\bar{n}}\approx\frac{1}{\bar{n}}.$
(S16)

###  II.2 Cross-Kerr (CK) MZI

For input number state $\ket{n,0}$, we have the output state right before the
final BS

$\displaystyle\ket{\psi(\phi)}\propto\frac{1}{\sqrt{2}}(e^{in\phi}\ket{n,0}+\ket{0,n})~{}~{}~{}\text{for
odd n}.$ (S17)

leading to

$\displaystyle\langle\Pi\rangle=-i^{n+1}\sin n\phi~{}~{}~{}\text{for odd n}.$
(S18)

For even $n$, the state $\ket{\psi(\phi)}$ is proportional to a beam splitter
transformation, leading to

$\displaystyle\langle\Pi\rangle=\sin^{n}\phi~{}~{}~{}~{}\text{for even n}.$
(S19)

Therefore, number state with odd $n$ yields the Heisenberg scaling (HL),
$\Delta\phi\sim 1/n$, while even $n$ leads to the standard quantum limit
(SQL), $\Delta\phi\sim 1/\sqrt{n}$.

For coherent state input, we get

$\displaystyle\langle\Pi_{out}\rangle=e^{-|\alpha|^{2}}\left[1-\sum_{\text{odd}~{}n}\frac{|\alpha|^{2n}}{n!}i^{n+1}\sin
n\phi+\sum_{\text{even}~{}n}\frac{|\alpha|^{2n}}{n!}\sin^{n}\phi\right].$
(S20)

For $\phi\approx\pi/2$ with large $\bar{n}$, we get

$\displaystyle\Delta\phi\approx\frac{1}{|\alpha|^{2}}.$ (S21)

Similarly, for thermal input, we get

$\displaystyle\langle\Pi_{out}\rangle=\left[\frac{1}{1+\bar{n}}-\sum_{\text{odd}~{}n}\frac{\bar{n}^{n}}{(1+\bar{n})^{(n+1)}}i^{n+1}\sin
n\phi+\sum_{\text{even}~{}n}\frac{\bar{n}^{n}}{(1+\bar{n})^{(n+1)}}\sin^{n}\phi\right],$
(S22)

and the sensitivity for $\phi\approx\pi/2$, for large $\bar{n}$, is then

$\displaystyle\Delta\phi\approx\frac{1}{\bar{n}}.$ (S23)

Figure S1:  Nonlinear interferometer with cross-Kerr nonlinearity as a parity
filter device.

##  III Cross-Kerr nonlinear MZI as parity measurement device

The parity of the output mode $b_{out}$ [Fig. 1(a) in text] can be measured by
sending this mode through a nonlinear interferometer containing cross-Kerr
nonlinearity [22]. The nonlinear CK MZI shown in Fig. S1, governed by the
unitary operator $\tilde{U}_{CK}=U_{BS}e^{i\chi\hat{n}_{b}\hat{n}_{c}}U_{BS}$,
transforms the two-mode number state $\ket{n_{b},0_{c}}$, for $\chi=\pi$, as

$\displaystyle\tilde{U}_{CK}\ket{n_{b},0_{c}}=i^{n}\ket{n_{1},0_{2}},~{}~{}\text{for
even n}$
$\displaystyle\tilde{U}_{CK}\ket{n_{b},0_{c}}=i^{n}\ket{0_{1},n_{2}}.~{}~{}\text{for
odd n}$ (S24)

The subscripts $b$ and $c$ are the input modes, and $1$ and $2$ are the output
modes of this device. Hence, if the input field contains an even number of
photons, then only detector $D_{1}$ clicks and, only $D_{2}$ clicks if the
input field contains an odd number of photons. The same holds true for an
input that is a mixture of $\ket{n_{b},0_{c}}$. Thus, the nonlinear
interferometer can be used as a parity measurement device merely by checking
which detector clicks.

##  IV Quantum Fisher Information

According to the quantum Cramer-Rao bound [49], the optimal sensitivity is
bounded by

$\displaystyle\Delta\phi_{min}\geq\frac{1}{\sqrt{kF_{Q}}},$ (S25)

where $k$ is the number of measurements that can be taken to be 1 and $F_{Q}$
is the quantum Fisher information that depends neither on the measurement nor
on the estimator and is solely a function of the probing state.

The input state $\rho_{in}$ is transformed after the phase-shift operation
[Fig. 1(a) in the text] to

$\displaystyle\tilde{\rho}=U\rho_{in}U^{\dagger},$ (S26)

where $U=U_{PS}(\phi)U_{BS}e^{i\chi a^{\dagger 2}a^{2}}U_{PS}(\pi/2)U_{BS}$
for SK and $U=U_{PS}(\phi)U_{BS}e^{i\chi
a^{\dagger}ab^{\dagger}b}U_{PS}(\pi/2)U_{BS}$ for CK.

The corresponding QFI is given by the formula [41]

$\displaystyle F_{Q}(\tilde{\rho})=2\sum_{\begin{subarray}{c}k,l\\\
\lambda_{k}+\lambda_{l}>0\end{subarray}}\frac{(\lambda_{k}-\lambda_{l})^{2}}{(\lambda_{k}+\lambda_{l})}|\langle
k|a^{\dagger}a|l\rangle|^{2},$ (S27)

where $\\{\lambda_{k},\ket{k}\\}$ are the eigenvalues and their corresponding
eigenvectors of $\tilde{\rho}$.

For a pure state input [12], the above expression reduces to

$\displaystyle
F_{Q}=4(\langle\dot{\psi}_{\phi}|\dot{\psi}_{\phi}\rangle-|\langle\dot{\psi}_{\phi}|\psi_{\phi}\rangle|^{2}),$
(S28)

where
$|\dot{\psi}_{\phi}\rangle=\frac{\partial}{\partial\phi}\ket{\psi_{\phi}}=ia^{\dagger}a\ket{\psi_{\phi}}$.
Thus,

$\displaystyle
F_{Q}=4(\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle-|\langle\psi_{\phi}|a^{\dagger}a|\psi_{\phi}\rangle|^{2})=4\langle\Delta
a^{\dagger}a\rangle^{2},$ (S29)

where as for mixed state,

$\displaystyle F_{Q}\leq 4\langle\Delta a^{\dagger}a\rangle^{2}.$ (S30)

###  IV.1 Self-Kerr

For number state input $\ket{n,0}$, we get $\ket{\psi(\phi)}$ as given in Eqn.
S1, and the expectation values are

$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle=\frac{1}{2}n^{2},\langle\psi_{\phi}|(a^{\dagger}a)|\psi_{\phi}\rangle=\frac{1}{2}n.$
(S31)

Using these expectation values, we get

$\displaystyle
F^{number}_{Q}=n^{2},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{n}.$
(S32)

Similarly, for coherent state input

$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle=\frac{1}{2}|\alpha|^{4}+\frac{1}{2}|\alpha|^{2},\langle\psi_{\phi}|(a^{\dagger}a)|\psi_{\phi}\rangle=\frac{1}{2}|\alpha|^{2},$
(S33)

and

$\displaystyle
F^{coherent}_{Q}=|\alpha|^{4}+2|\alpha|^{2}=\bar{n}^{2}+2\bar{n},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{\sqrt{\bar{n}^{2}+2\bar{n}}}.$
(S34)

For thermal state input, using Eq. (S27), we get

$\displaystyle
F^{thermal}_{Q}=2\bar{n}^{2}+\bar{n},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{\sqrt{2\bar{n}^{2}+\bar{n}}}.$
(S35)

###  IV.2 cross-Kerr

For number state input $\ket{n,0}$,

$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle=\frac{1}{4}(n^{2}+n)~{}~{}\text{for
even N},$ (S36)
$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle=\frac{1}{2}n^{2}~{}~{}\text{for
odd N},$ (S37)
$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)|\psi_{\phi}\rangle=\frac{1}{2}n,$
(S38)

and

$\displaystyle
F^{number}_{Q}=n^{2},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{n},~{}~{}\text{for
odd N}$ (S39) $\displaystyle
F^{number}_{Q}=n,~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{\sqrt{n}},~{}~{}~{}\text{for
even N}.$ (S40)

For coherent state input,

$\displaystyle\langle\psi_{\phi}|(a^{\dagger}a)^{2}|\psi_{\phi}\rangle=\frac{3}{8}|\alpha|^{4}+\frac{1}{2}|\alpha|^{2},\langle\psi_{\phi}|(a^{\dagger}a)|\psi_{\phi}\rangle=\frac{1}{2}|\alpha|^{2},$
(S41)

and

$\displaystyle
F^{coherent}_{Q}=\frac{1}{2}|\alpha|^{4}+2|\alpha|^{2}=\frac{1}{2}\bar{n}^{2}+2\bar{n},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{\sqrt{\frac{1}{2}\bar{n}^{2}+2\bar{n}}}.$
(S42)

For thermal state input (from Eq. (S27))

$\displaystyle
F^{thermal}_{Q}=\bar{n}^{2}+\bar{n},~{}~{}~{}~{}\Delta\phi_{min}=\frac{1}{\sqrt{F_{Q}}}=\frac{1}{\sqrt{\bar{n}^{2}+\bar{n}}}.$
(S43)

##  V Inter-mode entanglement

To check whether the two-mode output states (after the second beam splitter of
the setup given in Fig. 1(a) of the text) are entangled or not, we consider
the following quantities

$\displaystyle E_{HZ}$
$\displaystyle=\langle\hat{n}_{a}\hat{n}_{b}\rangle_{out}-|\langle
a_{out}^{\dagger}b_{out}\rangle|^{2},$ (S44) $\displaystyle E_{SV}$
$\displaystyle=\left|\begin{array}[]{ccc}1&\langle
a_{out}^{\dagger}\rangle&\langle b_{out}^{\dagger}\rangle\\\ \langle
a_{out}\rangle&\langle a_{out}^{\dagger}a_{out}\rangle&\langle
a_{out}^{\dagger}b_{out}^{\dagger}\rangle\\\ \langle b_{out}\rangle&\langle
a_{out}b_{out}\rangle&\langle
b_{out}^{\dagger}b_{out}\rangle\end{array}\right|.$ (S48)

These quantities are constructed based on the Hillery-Zubairy [52], and
Shchukin-Vogels entanglement criteria [53], respectively. The negativity of
these quantities for a two-mode state is a sufficient condition for
entanglement. As shown in Fig. 3(b) in the text, this condition is not
satisfied by output states in Kerr-nonlinear interferometers when the input is
a thermal state.

Figure S2:  Fisher information for SK MZI obtained by photon-number resolving
detectors: (a) number-state ($n=5$), (b) coherent state ($\bar{n}=5)$ (c)
thermal state ($\bar{n}=5)$ inputs for $\chi=\pi/2$ and detector efficiencies
$\eta_{\text{det}}=0.999$ (top) and $\eta_{\text{det}}=0.95$ (down). The SQL
(dot-dashed line) here corresponds to $F_{SQL}=\bar{n}=5$ and the HL (dashed
line) to $F_{HL}=\bar{n}^{2}=25$. The QFI (magenta line) for thermal input is
55, for coherent input is 35 and for number input is 25. Red thick line:
Single-detector photodetection, Blue thick line: intensity-difference
detection, Black thick line: photodetection by both detectors.  Figure S3:
Fisher information for SK MZI with thermal state ($\bar{n}=5)$ input for
$\chi=\pi/2$ and detector efficiency $\eta_{\text{det}}=0.9$. The SQL (dashed
line) here corresponds to $F_{SQL}=\bar{n}=5$. Red thick line: Single-detector
photodetection, Blue thick line: intensity-difference detection, Black thick
line: photodetection on both detectors.  Figure S4:  (a) Fisher information,
based on parity measurement, for SK MZI with thermal state ($\bar{n}=5)$ input
for $\chi=\pi/2$ and detector efficiency $\eta_{\text{det}}=0.999$. (b)
Similarly, for CK MZI. The SQL (dot-dashed line) here corresponds to
$F_{SQL}=\bar{n}=5$ and the HL (dashed line) to $F_{HL}=\bar{n}^{2}=25$. The
QFI (magenta line) for thermal input in SK MZI is 55 and in CK MZI is 30.

##  VI Photodetection efficiency

With perfect detectors $\eta_{\text{det}}=1$, photodetection on both output
channels saturates the QFI limit for thermal and number inputs while not for
coherent input [Fig. S2)]. With near-perfect detectors, mean photon numbers
being equal, the highest $F$ is reached by thermal states, which then beat the
HL and surpass coherent and Fock states. For $\eta_{\text{det}}\leq 0.97$ the
order is reversed: the highest $F$ is reached by Fock states and the lowest by
thermal states. Yet, thermal input still beats the SQL for
$\eta_{\text{det}}\geq 0.9$ [Fig. S3]. We note that the FI, based on parity
measurement, achieves the QFI for CK MZI and not for SK MZI [Fig. S4].

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