# Quantum phase estimations with spin coherent states superposition

Yusef Maleki 1
1Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843-4242, USA E-mail<EMAIL_ADDRESS>
###### Abstract

The quantum metrological performance of spin coherent states superposition is
considered, and conditions for measurements with the Heisenberg-limit (HL)
precision are identified. It is demonstrated that the choice of the parameter-
generating operator can lead to physically different estimation outcomes. In
particular, closed-form analytical descriptions for the performance of spin
cat states are derived. These findings show the routes to careful control of
parameters necessary for achieving HL precision and provide insightful
information on the geometry of the specific coherent state superposition and
its relevance to the performance of the states for parameter estimations.

##  I. Introduction

High-precision measurements are of utmost importance in fundamental physics,
and various areas of technology [1, 2, 3, 4, 5, 6, 7, 8] . In the classical
realm, the precision of measurements can be enhanced by repeated measurements.
More specifically, with $N$ independent measurements performed to estimate a
parameter $\xi$, the error of $\xi$ estimation is bounded by the shot-noise
limit, $\Delta\xi\propto\frac{1}{\sqrt{N}}$ [1, 2]. In contrast, quantum
physics provides tools for circumventing this limit [8, 9], enabling
measurements with precisions within the limit of
$\Delta\varphi\propto\frac{1}{N}$, referred to as the Heisenberg Limit (HL)
[9, 10].

As an important example of such improvements, cat states of the form
$\frac{1}{\sqrt{2}}(|N\rangle_{a}\otimes|0\rangle_{b}+|0\rangle_{a}\otimes|N\rangle_{b})$,
known as N00N states [11, 12], have been shown [1, 12] to provide the HL
sensitivity. However, the problem is that such states are difficult to
generate and are extremely prone to decoherence [10, 12, 13]. To resolve this,
specific schemes have been proposed [14, 15]. Alternatively, other quantum
states have been considered, which can provide better than shot-noise limit
precision. An important example of such a cat state is the superpositions of
two SU(2) coherent states, which attracted much attention [16, 17, 18, 19].

Spin cat states are promising candidates for quantum metrology due to their
strong capability for beating the shot-noise limit and their robustness
against decoherence [16, 17, 18]. As an important feature, these cat states
can be prepared in a wide variety of settings such as dynamical evolution of
$N$ two-level systems [20], Bose-Einstein condensates of two-level atoms [21,
22] or two-mode photon system in cavity QED [19]. Therefore, parameter
estimation with spin cat state is compatible with both Ramsey [23] and Mach-
Zehnder interferometry [24]. In all of these scenarios, the coherent state
basis could be considered as a two-mode bosonic-harmonic-oscillator basis [16,
17, 18, 19].

Here, we consider the quantum metrological performance of spin coherent states
superposition and identify conditions for measurements with a Heisenberg-limit
(HL) precision. We demonstrate that the choice of the parameter-generating
operator plays a critical role in the performance of such a superposition
state and consider physically relevant parameters induced by different
generators. These findings suggest that careful control of parameters is
necessary for achieving HL precision and provide insightful information on the
geometry of the specific coherent state superposition on the Bloch sphere and
its relevance to the performance of the states for parameter estimations.

##  II. Spin coherent states for phase estimation

This section briefly reviews the spin (SU(2)) coherent states and their
essential properties. Spin coherent states are defined through the action of
rotation operator [25]

$R(\theta,\varphi)=\exp{\\{\frac{-\theta}{2}(J_{+}e^{-i\varphi}-J_{-}e^{i\varphi})\\}},$
(1)

to the reference state $|j,-j\rangle$. The rotation operator in Eq. (1) can be
identified in terms of the generators of the su(2) algebra. The generators of
su(2) algebra are denoted by $J_{-}$, $J_{+}$ and $J_{z}$. $J_{-}$ and $J_{+}$
are the spin-state lowering and raising operators, respectively, and $J_{z}$
is the $z$-component of the spin operators. As generators of the su(2) Lie
algebra, the operators $J_{\pm}$ and $J_{z}$ satisfy the commutation relations
$[J_{+},J_{-}]=2J_{z},[J_{z},J_{\pm}]=\pm J_{\pm}.$

The operator in Eq. (1), acting on the ground state of the Dicke basis,
results in the spin coherent state [25, 26]

$\displaystyle|\theta,\varphi,j\rangle=\exp{[\frac{\theta}{2}(J_{+}e^{-i\varphi}-J_{-}e^{i\varphi})]}|j,-j\rangle=\frac{1}{(1+|\gamma|^{2})^{j}}\sum_{m=-j}^{j}\left(\begin{matrix}2j\\\
j+m\end{matrix}\right)^{\frac{1}{2}}\gamma^{j+m}|j,m\rangle,$ (2)

where $\gamma=e^{-i\varphi}\tan(\frac{\theta}{2})$.

The overlap of two spin coherent states is given by [26]

$\displaystyle\langle\theta_{1},\varphi_{1},j|\theta_{2},\varphi_{2},j\rangle=\frac{(1+\bar{\delta}\gamma)^{2j}}{(1+|\delta|^{2})^{j}(1+|\gamma|^{2})^{j}},$
(3)

where $\gamma=e^{-i\varphi_{1}}\tan(\frac{\theta_{1}}{2})$ and
$\delta=e^{-i\varphi_{2}}\tan(\frac{\theta_{2}}{2})$, and $\bar{\delta}$
represents the complex conjugate of $\delta$. The connection between the Dicke
basis and two-mode bosonic-harmonic-oscillator basis can be understood by
considering the Schwinger realization of the su(2) algebra represented in the
two-mode Hilbert space states as

$\displaystyle J_{+}=a^{\dagger}b,\qquad J_{-}=b^{\dagger}a,\qquad
J_{z}=\frac{1}{2}(a^{\dagger}a-b^{\dagger}b).$ (4)

where $a$ and $b$ are the bosonic-harmonic-oscillator annihilation operators
of the first and the second modes. While the realization is compatible with
various settings, we refer to these operators as the photon annihilation
operators. If the two modes $a$ and $b$ contain $N$-photons in total, taking
$j=\frac{N}{2}$ and $m=\frac{N_{a}-N_{b}}{2}$, we can express Dicke basis as
$|N_{a}\rangle|N_{b}\rangle\equiv|j,m\rangle$ ($N=n_{a}+n_{b}$). Applying the
$J_{\pm}$ and $J_{z}$ operators to the Dicke basis yield
$J_{\pm}|j,m\rangle=\sqrt{(j\pm m)(j\pm m+1)}|j,m\pm 1\rangle$ and
$J_{z}|j,m\rangle=m|j,m\rangle.$ Thus, when all of the photons are in one of
the two modes, we attain the two special cases
$|0\rangle\otimes|N\rangle\equiv|j,-j\rangle$ and
$|N\rangle\otimes|0\rangle\equiv|j,j\rangle.$ To demonstrate the connection of
the spin coherent state superposition and metrology we note that the
superposition of these two special cases provides the N00N state
($|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}(|N,0\rangle+|0,N\rangle)$) in the
Dicke basis as [16]

$|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}(|j,j\rangle+|j,-j\rangle).$ (5)

Therefore, the N00N state can be considered the superposition of the north and
south poles of the Bloch sphere. This particular example demonstrates the
utility of the superposition of spin coherent states for quantum metrological
purposes. To illustrate this capability, the superposition of coherent states
of the form

$|\varphi,j\rangle=\mathcal{N}(|\theta,\varphi,j\rangle+|\pi-\theta,\varphi,j\rangle),$
(6)

with $\mathcal{N}$ being the normalization factor, is shown to suppress the
shot-noise limit, thus, being of great importance for quantum metrology [17,
18]. However, a more generalized superposition of such spin coherent states
can be expressed as

$|\psi,j\rangle=\mathcal{N}(|\theta_{1},\varphi_{1},j\rangle+|\theta_{2},\varphi_{2},j\rangle).$
(7)

A natural question is whether these are states beyond the scope of Eq. (6)
that are metrologically useful? If so, what are these states? We are going to
consider these questions, in detail, in this work.

Note that the phase shift in N00N state could be achieved by
$\exp(i\xi(a^{\dagger}a-b^{\dagger}b)/2)$. This indicates that the path $a$
has gained the phase shift $\xi/2$, while the path $b$ has gained the phase
shift $-\xi/2$, resulting in a total phase shift $\xi$. Following this
discipline, the phase shift in spin states is usually introduced by $e^{i\xi
J_{z}}$, which is expected from Schwinger realization of the spin algebra.

Given a spin cat state, we would like to consider the metrological performance
of the state from a somewhat different perspective. In fact, we ask a natural
question of how the spin cat state would perform by introducing parameters
through some other operators such as $e^{i\xi J_{x}}$ or $e^{i\xi J_{y}}$?
This may seem to be a pure mathematical curiosity at first sight; however, the
question is insightful once we consider it more carefully. In fact,
considering the Schwinger representation of the spin operators, $e^{i\xi
J_{z}}$ could be understood as a phase shift in an interferometer, while
$e^{i\xi J_{x}}$ and $e^{i\xi J_{y}}$ could be considered as beam splitting
operators, where the beam splitter transmittance is determined through $\xi$.
Therefore, parameter estimation with the latter determines the transmittance
of a beam splitter rather than the accuracy of phase shift in an
interferometer.

##  III. Quantum Metrology with Spin-1/2 Cat States

We first consider the metrology of spin-1/2 cat states and provide a detailed
investigation on the metrological power of such cat states. It is notable that
if $j=\frac{1}{2}$, then $\textrm{su}(2)$ algebra will be generated by Pauli
matrices $\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$, and $\sigma_{z}$. In the
Fock state basis, $j=\frac{1}{2}$ corresponds to having only one photon in the
coherent state. The coherent state, in this case, could be given as

$\displaystyle|\theta,\varphi,1/2\rangle=\exp[\frac{\theta}{2}(\sigma_{+}e^{-i\varphi}-\sigma_{-}e^{i\varphi})]|\frac{1}{2},\frac{-1}{2}\rangle.$
(8)

Therefore, the state can be obtained to be

$\displaystyle|\theta,\varphi,1/2\rangle=\cos\frac{\theta}{2}|\frac{1}{2},\frac{-1}{2}\rangle+e^{-i\varphi}\sin\frac{\theta}{2}|\frac{1}{2},\frac{1}{2}\rangle.$
(9)

In this framework, the general form of the superposition of two spin coherent
states could be represented as

$|\text{Cat},1/2\rangle=\mathcal{N}(|\theta_{1},\varphi_{1},1/2\rangle+|\theta_{2},\varphi_{2},1/2\rangle).$
(10)

Here, $\mathcal{N}$ is the normalization factor of the spin cat states which
can be found to be

$\mathcal{N}=\dfrac{1}{\sqrt{2[1+\cos(\frac{\theta_{1}}{2})\cos(\frac{\theta_{2}}{2})+\cos(\varphi_{1}-\varphi_{2})\sin(\frac{\theta_{1}}{2})\sin(\frac{\theta_{2}}{2})]}}.$
(11)

In the Dicke basis, the cat state above can be expressed as

$|\text{Cat},1/2\rangle=\mathcal{N}[(\cos\frac{\theta_{1}}{2}+\cos\frac{\theta_{2}}{2})|\frac{1}{2},\frac{-1}{2}\rangle+(e^{-i\varphi_{1}}\sin\frac{\theta_{1}}{2}+e^{-i\varphi_{2}}\sin\frac{\theta_{2}}{2})|\frac{1}{2},\frac{1}{2}\rangle].$
(12)

We first analyze the performance of the state $|\text{Cat},1/2\rangle$ for
quantum metrology when the system accumulates a phase through
$e^{i\xi\sigma_{z}}$. Once this phase is introduced, the cat state degenerates
to

$|\text{Cat},1/2\rangle_{\xi}=\mathcal{N}[e^{-i\xi/2}(\cos\frac{\theta_{1}}{2}+\cos\frac{\theta_{2}}{2})|\frac{1}{2},\frac{-1}{2}\rangle+e^{i\xi/2}(e^{-i\varphi_{1}}\sin\frac{\theta_{1}}{2}+e^{-i\varphi_{2}}\sin\frac{\theta_{2}}{2})|\frac{1}{2},\frac{1}{2}\rangle].$
(13)

Now, we are interested in determining the minimum error that could be achieved
by measuring the parameter $\xi$. The ultimate limit of parameter estimation
is determined via unbiased quantum Cramér–Rao bound (CRB) defined as [1, 8]

$\Delta\xi_{CRB}=1/{\sqrt{F_{Q}(\rho(\xi))}},$ (14)

where $F_{Q}(\rho(\xi))=Tr[\rho(\xi)L_{\xi}^{2}]$ is the quantum Fisher
information [1, 8]. The symmetric logarithmic derivative $L_{\xi}$ is defined
by $\partial_{\xi}\rho(\xi)=(1/2)[\rho(\xi)L_{\xi}+L_{\xi}\rho(\xi)]$, and
$\rho(\xi)$ is the density matrix.

For the spin-$1/2$ cat state in Eq. (13), the error of phase measurement
$\Delta\xi$ is limited from below by the CRB, for which we find

$\Delta\xi_{CRB}=\sqrt{\dfrac{2[1+\cos(\frac{\theta_{1}}{2})\cos(\frac{\theta_{2}}{2})+\cos(\varphi_{1}-\varphi_{2})\sin(\frac{\theta_{1}}{2})\sin(\frac{\theta_{2}}{2})]^{2}}{[\cos(\frac{\theta_{1}}{2})+\cos(\frac{\theta_{2}}{2})]^{2}[2-\cos(\theta_{1})-\cos(\theta_{2})+4\cos(\varphi_{1}-\varphi_{2})\sin(\frac{\theta_{1}}{2})\sin(\frac{\theta_{2}}{2})]}}.$
(15)

This expression fully captures the CRB of the generic form of the spin-$1/2$
cat state. Noting that in the Fock basis
$|\frac{1}{2},\frac{-1}{2}\rangle=|0,1\rangle$, and
$|\frac{1}{2},\frac{1}{2}\rangle=|1,0\rangle$, the state in Eq.(12) is in form
of the single-photon superposition. Thus, we expect that its performance for
phase estimation should be limited by the performance of the single photon
N00N state, i.e., $\Delta\xi_{CRB}=1$. As is clear from this result, the CRB
only depends on the phase difference through $\cos(\varphi_{1}-\varphi_{2})$,
and it does not depend on each of the phases individually. To understand the
characteristics of the CRB we consider different situations. First let us take
cat states of the form
$\mathcal{N}(|\theta_{1},\varphi_{1},1/2\rangle+|\pi-\theta_{1},\varphi_{2},1/2\rangle)$.
In this case, the CRB reduces to

$\Delta\xi_{CRB}=\frac{1}{2}\sqrt{\frac{[2+\sin(\theta_{1})(1+\cos(\varphi_{1}-\varphi_{2}))]^{2}}{[1+\sin(\theta_{1})][1+\sin(\theta_{1})\cos(\varphi_{1}-\varphi_{2})]}}.$
(16)

From this expression, for $\phi=\varphi_{1}-\varphi_{2}=0$ we immediately find
$\Delta\xi_{CRB}=1$. To understand the reason behind this fact we note that
the state (12) for $\phi=0$ reduces to

$|\text{Cat},1/2\rangle=\mathcal{N}[(\cos\frac{\theta_{1}}{2}+\sin\frac{\theta_{1}}{2})|\frac{1}{2},\frac{-1}{2}\rangle+(e^{-i\varphi_{1}}\sin\frac{\theta_{1}}{2}+e^{-i\varphi_{2}}\cos\frac{\theta_{1}}{2})|\frac{1}{2},\frac{1}{2}\rangle].$
(17)

Figure 1:  Density of the CRB in terms of $\theta_{1}$ and $\theta_{2}$. (a)
$\phi=\varphi_{1}-\varphi_{2}=0$. (b) $\phi=\pi/2$. (c) $\phi=39\pi/40$. (d)
$\phi=\pi$.

This state reduces to the single photon N00N state when
$\phi=\varphi_{1}-\varphi_{2}=0$. This is manifest from Fig. 1(a). In this
plot, the CRB density is given in terms of $\theta_{1}$ and $\theta_{2}$ when
$\varphi=0$. According to Fig. (1) (a), when $\varphi=0$, the CRB is minimum
for $\theta_{2}=\pi-\theta_{1}$. At the regions far from
$\theta_{2}=\pi-\theta_{1}$, the CRB becomes larger. This becomes clear by
noting that for $\varphi=0$, the CRB is simply given by
$\Delta\xi_{CRB}=1/|\sin({(\theta_{1}+\theta_{2})}/{2})|$. For instance taking
$\theta_{1}=0$ and $\theta_{2}=\pi/3$, we get $\Delta\xi_{CRB}=2$, and for
$\theta_{2}=\pi/2$ one has $\Delta\xi_{CRB}=\sqrt{2}$. In Fig. (1) (b) we plot
the same density of CRB when $\varphi=\pi/2$. For this scenario, the CRB
simplifies as

$\Delta\xi_{CRB}=\sqrt{\dfrac{2[1+\cos(\frac{\theta_{1}}{2})\cos(\frac{\theta_{2}}{2})]^{2}}{[\cos(\frac{\theta_{1}}{2})+\cos(\frac{\theta_{2}}{2})]^{2}[2-\cos(\theta_{1})-\cos(\theta_{2})]}}.$
(18)

In this case, the CRB is minimum when $\theta_{1}=0$ and $\theta_{2}=\pi$, or
$\theta_{1}=\pi$ and $\theta_{2}=0$. Taking $\theta_{1}=\theta_{2}=3\pi/4$, we
get $\Delta\xi_{CRB}=1+\sin^{2}\left(\frac{\pi}{8}\right)\approx 1.14645$.
Also taking $\theta_{1}=\theta_{2}=\pi/2$, we get
$\Delta\xi_{CRB}=3/2\sqrt{2}\approx 1.06$. In Fig. (1) (c) we plot the same
density of CRB when $\varphi=39\pi/40$. The CRB is still minimum when
$\theta_{1}=0$ and $\theta_{2}=\pi$, or $\theta_{1}=\pi$ and $\theta_{2}=0$;
however, it becomes large for the regions far from this condition. Specially,
for the region where $\theta_{1}=\theta_{2}\lessapprox 3\pi/4$. For instance,
taking $\theta_{1}=\theta_{2}=\pi/2$, we find $\Delta\xi_{CRB}\approx 12.755$,
which is very large compared to the previous subplots. Finally, we take
$\varphi=\pi$ in Fig. (1) (d) which demonstrates a singularity when
$\theta_{1}=\theta_{2}$. This could be understood by noting that CRB reduces
to $\Delta\xi_{CRB}=1/|\sin({(\theta_{1}-\theta_{2})}/{2})|$ in this scenario.
As a result, in all of these plots the when $\theta_{1}=0$ and
$\theta_{2}=\pi$, or alternatively $\theta_{1}=\pi$ and $\theta_{2}=0$, the
phase estimation becomes optimal. This is due to the fact that at these cases
the cat state become superposition of the north and south poles of the Bloch
sphere, and $\varphi_{1}$ and $\varphi_{2}$ loose their significance at the
poles, making the phase sensitivity independent of these phases and, thus,
optimal.

Now, let us consider the situation where $\theta_{1}=\theta_{2}=\pi/2$.
Geometrically, this corresponds to the superposition of two coherent states
living on the great circle of the Bloch sphere in the plane of $x$ and $y$
axes. In this case, the CRB simplifies to

$\Delta\xi_{CRB}=\frac{3+\cos(\varphi_{1}-\varphi_{2})}{4|\cos[(\varphi_{1}-\varphi_{2})/2]|}.$
(19)

Thus, taking $\varphi_{1}-\varphi_{2}=0$, this reduces to $\Delta\xi_{CRB}=1$,
and for $\varphi_{1}-\varphi_{2}=\pi$, $\Delta\xi_{CRB}$ diverges, which is
fully in agreement with the observations in Fig. (1).

Even though the phase shift on the spin system is usually induced through
$e^{i\xi J_{z}}$, it is shown that phase estimation through $e^{i\xi J_{x}}$
or $e^{i\xi J_{y}}$ can also provide helpful estimation information [16].
These phases physically have different meanings. For instance, considering the
Schwinger representation of the spin operators, $e^{i\xi J_{z}}$ could be
understood as a phase shift in an interferometer. While $e^{i\xi J_{x}}$ is
the beam splitting operator, where its transmittance is determined through
$\xi$.

To consider the performance of the state in this platform, we next investigate
the phase sensitivity introduced through $e^{i\xi\sigma_{x}}$. Following a
similar method as the previous case, the CRB can be obtained for as

$\Delta\xi_{CRB}=\left[1-\frac{[\cos(\frac{\theta_{1}}{2})+\cos(\frac{\theta_{2}}{2})]^{2}[\cos(\varphi_{1})\sin(\frac{\theta_{1}}{2})+\cos(\varphi_{2})\sin(\frac{\theta_{2}}{2})]^{2}}{[1+\cos(\frac{\theta_{1}}{2})\cos(\frac{\theta_{2}}{2})+\cos(\varphi_{1}-\varphi_{2})\sin(\frac{\theta_{1}}{2})\sin(\frac{\theta_{2}}{2})]^{2}}\right]^{-1/2}.$
(20)

From this result, we immediately conclude that the necessary and sufficient
condition for HL phase sensitivity, here, is
$\cos(\varphi_{1})\sin(\frac{\theta_{1}}{2})+\cos(\varphi_{2})\sin(\frac{\theta_{2}}{2})=0$.
It also becomes clear that for $\theta_{1}=\theta_{2}=0$, (or alternatively
$\theta_{1}=\theta_{2}=\pi$) the sensitivity of the cat state becomes optimal.
We note that, at these extreme points, the state reduces to the south or north
pole of the Bloch sphere, with the resultant states
$|\frac{1}{2},-\frac{1}{2}\rangle$ or $|\frac{1}{2},\frac{1}{2}\rangle$. It is
insightful to note that these states correspond to $|0\rangle|1\rangle$ and
$|1\rangle|0\rangle$ in the Fock state representation. Even though these
states provide no information about the phase accuracy for
$e^{i\xi\sigma_{z}}$, they provide the best accuracy about $\xi$ for
$e^{i\xi\sigma_{x}}$. This is due to the fact that sending a single photon
Fock state through a beam splitter transforms it into a superposition state.

We note that, unlike the previous scenario, in this case, the CRB is
controlled by each of the phases individually rather than only the phase
difference being important. An interesting choice would be
$\\{\varphi_{1},\varphi_{2}\\}=\\{\pi/2,3\pi/2\\}$, which also results in the
optimal phase estimation. This becomes more insightful if we note that taking
$\varphi=\pi/2(3\pi/2)$ demonstrates coherent states on the great circle in
$y$ and $z$ plane of the Bloch sphere. Therefore, if we choose the
superposition of any two coherent states on this circle, we attain HL
sensitivity for $\xi$.

Figure 2:  Density of the CRB in terms of $\theta_{1}$ and $\theta_{2}$. We
set $\varphi_{1}=0$ in these plots. (a) $\varphi_{2}=0$. (b)
$\varphi_{2}=\pi/2$. (c) $\varphi_{2}=3\pi/4$. (d) $\varphi_{2}=\pi$.

Fig. (2) presents CRB in terms of $\theta_{1}$ and $\theta_{2}$ for various
$\varphi_{2}$. In all of the sub-figures we have chosen $\varphi_{1}=0$. In
Fig. 2(a) we have considered $\varphi_{2}=0$, where the CRB becomes optimal
for $\theta_{1}=\theta_{2}=0,\pi$, and it is maximal for
$\theta_{1}+\theta_{2}=\pi$. To understand this we note that setting
$\varphi_{1}=\varphi_{2}=0$ in Eq. (22), the CRB reduces to
$\Delta\xi_{CRB}=1/{|{\cos[(\theta_{1}+\theta_{2})/2]}|}$. The same plot with
$\varphi_{2}=\pi/2$, is given in Fig. 2(b). Accordingly, $\theta_{1}=0$, or
$\theta_{1}=\theta_{2}=\pi$, provides optimal CRB. To have a better insight
into this figure we should note that for this situation the CRB reduces to

$\Delta\xi_{CRB}=\left[1-\frac{[\cos(\frac{\theta_{1}}{2})+\cos(\frac{\theta_{2}}{2})]^{2}[\sin(\frac{\theta_{1}}{2})]^{2}}{[1+\cos(\frac{\theta_{1}}{2})\cos(\frac{\theta_{2}}{2})]^{2}}\right]^{-1/2}.$
(21)

Thus, taking $\theta_{1}=0$, or $\theta_{1}=\theta_{2}=\pi$, in this equation,
gives $\Delta\xi_{CRB}=1$. Assuming $\theta_{1}=\theta_{2}=\theta$, we can
further simply this equation to

$\Delta\xi_{CRB}=\frac{\sqrt{2}[3+\cos({\theta})]}{\sqrt{15+12\cos({\theta})+5\cos({2\theta})}},$

resulting in $\Delta\xi_{CRB}=\frac{3}{\sqrt{5}}$ for $\theta=\pi/2$. In Fig.
2(c), we take $\varphi_{2}=3\pi/4$. Accordingly, the phase estimation can
reach optimal value for specific choices of the parameters, and it diverges
for $\theta_{1}=\pi$. If we choose $\theta_{2}=0$ the CRB shall simplify to
$\Delta\xi_{CRB}=1/{{\cos(\theta_{1}/2)}}$. Thus, as the plot presents the CRB
increases by increasing $\theta_{1}$. Alternatively, taking $\theta_{1}=0$,
the CRB simplifies to $\Delta\xi_{CRB}=2/\sqrt{3+\cos(\theta_{2})}$, ranging
between 1 and $\sqrt{2}$.

Now, we take $\varphi_{2}=\pi$, in Fig. 2(d), which presents optimal phase
estimation for $\theta_{1}=\theta_{2}$. This could be understood by noting
that $\Delta\xi_{CRB}=1/{|{\cos[(\theta_{1}-\theta_{2})/2]}|}$, for this
situation. From Fig. (2) it is clear that for $\theta_{1}=\theta_{2}=0,\pi$
the phase sensitivity becomes independent of $\varphi_{1}$ and $\varphi_{2}$,
and it gives $\Delta\xi_{CRB}=1$. As the last case, we take
$\theta_{1}=\theta_{2}=\pi/2$. The CRB reduces then to

$\Delta\xi_{CRB}=\left[1-\frac{4[\cos(\varphi_{1})+\cos(\varphi_{2})]^{2}}{[3+\cos(\varphi_{1}-\varphi_{2})]^{2}}\right]^{-1/2}.$
(22)

Thus, $\varphi_{1}=\varphi_{2}=0$ gives the maximal, and
$\varphi_{1}=\varphi_{2}=\pi$ gives the minimal value for the CRB, as can be
seen from the plots.

##  IV. Quantum Metrology with Spin-1 Cat States

Now, we generalize the investigations of the previous section to the cat
states with $j=1$. To this end, taking $j=1$, the coherent state reduces to

$|\theta,\varphi,1\rangle=\cos^{2}(\theta/2)|1,-1\rangle+\frac{1}{\sqrt{2}}e^{i\varphi}\sin({\theta})|1,0\rangle+e^{2i\varphi}\sin^{2}(\theta/2)|1,1\rangle,$
(23)

Noting that in the Fock basis $|1,-1\rangle=|0,2\rangle$,
$|1,0\rangle=|1,1\rangle$, and $|1,1\rangle=|2,0\rangle$, the state in Eq.(23)
is in form of the two photon superposition. Similar to the previous section,
we consider a general form of the superposition of two spin-1 coherent states
as

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi_{1},1\rangle+|\theta_{2},\varphi_{2},1\rangle).$
(24)

Here, $\mathcal{N}$ is the normalization factor of the spin cat states.

Figure 3:  Density of the CRB in terms of $\theta_{1}$ and $\theta_{2}$. (a)
$\phi=0$. (b) $\phi=\pi/2$. (c) $\phi=3\pi/4$. (d) $\phi=\pi$.

We analyze the performance of the state $|\text{Cat},1\rangle$ for quantum
metrology when the system has accumulated a phase through $e^{i\xi J_{z}}$.
For such a phase shift, we are going to provide analytic results for various
scenarios. It turns out that the CRB depends on the
$\cos(\varphi_{1}-\varphi_{2})$, and not each of the phases individually. This
is similar to the spin-1/2 case, and hence we can conclude that the
metrological performance of the cat states through $e^{i\xi J_{z}}$ is
controlled by the phase differences. We present the CRB density as a function
of $\theta_{1}$ and $\theta_{2}$ for various $\phi=\varphi_{1}-\varphi_{2}$ in
Fig. (3). Accordingly, the phase difference has a significant role in the
performance of the states. We take $\phi=0$, in Fig. 3(a). In this case, the
cat state reduces to

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\theta_{2},\varphi,1\rangle).$
(25)

For such a superposition state, the CRB of the system could be found to be

$\Delta\xi_{CRB}=\left[\frac{2(\cos(\theta_{1}-\theta_{2})+3)^{2}}{\mathcal{A}-8\mathcal{B}+2\mathcal{C}-18\mathcal{D}+30}\right]^{1/2},$
(26)

where,

$\displaystyle\mathcal{A}$
$\displaystyle=\cos(3\theta_{1}-\theta_{2})+\cos(3\theta_{2}-\theta_{1}),$
(27) $\displaystyle\mathcal{B}$
$\displaystyle=\cos(2\theta_{1})+\cos(2\theta_{2}),$ (28)
$\displaystyle\mathcal{C}$ $\displaystyle=\cos(2(\theta_{1}-\theta_{2})),$
(29) $\displaystyle\mathcal{D}$ $\displaystyle=\cos(\theta_{1}+\theta_{2}).$
(30)

As was expected, $\Delta\xi_{CRB}$ is symmetric in terms of $\theta_{1}$ and
$\theta_{2}$. Particularly, if one of them is chosen to be 0 and the other is
set to be $\pi$, we attain HL sensitivity $\Delta\xi_{CRB}=1/2$. This
observation is expected due to the fact that one of the coherent states, at
these cases, is at the north and the other at the south pole of the Bloch
sphere. However, superposition of the poles provide N00N states, providing HL
precision. This is also clear from Fig. 3(a). According to this figure,
$\Delta\xi_{CRB}$ is optimal only at these specific choices. If we consider
$\theta_{1}=\theta_{2}$, then $\Delta\xi_{CRB}=1/\sqrt{2}|\sin(\theta_{1})|$.
At this scenario, the best performance is achieved for $\theta_{1}=\pi/2$, and
$\Delta\xi_{CRB}$ diverges for $\theta_{1}=0$ or $\pi$. This is clearly shown
in Fig. 3(a). The other interesting situation in this class of cat states is
considering $\theta_{2}=\pi-\theta_{1}$. From Fig. 3(a), it can be realized
that the $\Delta\xi_{CRB}$ provides a better sensitivity along this line. With
this assumption, the explicit form of the cat state will be

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\pi-\theta_{1},\varphi,1\rangle).$
(31)

This type of state is considered in [17, 18]. The CRB provided by this state
is given in Fig. 3(a). For such a superposition state, we find for the CRB of
the system

$\Delta\xi_{CRB}=\left[\frac{3-\cos(2\theta_{1})}{8}\right]^{1/2}.$ (32)

Accordingly, for $\theta_{1}=0,\pi$ we attain HL sensitivity, while the
minimum sensitivity happens for $\theta_{1}=\pi/2$ which gives
$\Delta\xi_{CRB}=1/\sqrt{2}$, which is equal to the shot-noise-limit. Thus,
$\Delta\xi_{CRB}$ is always better than shot-noise-limit, except for
$\theta_{1}=\pi/2$.

In Fig. 3(b) we consider the performance of the cat states of the form

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\theta_{2},\varphi+\pi/2,1\rangle).$
(33)

Geometrically, the two coherent states are on the two orthogonal circles on
the Bloch sphere. For instance, if one of them is on the circle in the plane
of $x$ and $z$ axes, the other must be on the circle crossing $y$ and $z$
axes. For this cat state, $\Delta\xi_{CRB}$ could be found to be

$\Delta\xi_{CRB}=\left[\frac{\mathcal{A}^{2}}{\mathcal{A}\mathcal{B}-\mathcal{C}}\right]^{1/2},$
(34)

where,

$\displaystyle\mathcal{A}$
$\displaystyle=\cos(\theta_{1})+\cos(\theta_{2})+2,$ (35)
$\displaystyle\mathcal{B}$
$\displaystyle=\cos(2\theta_{1})+\cos(2\theta_{2})-8\cos(\theta_{1})\cos(\theta_{2})+6,$
(36) $\displaystyle\mathcal{C}$
$\displaystyle=4(\cos(\theta_{1})\cos(\theta_{2})-1)^{2}.$ (37)

It is easy to verify that for $\theta_{1}=\theta_{2}=0,\pi$ $\Delta\xi_{CRB}$
diverges since $\mathcal{A}\mathcal{B}-\mathcal{C}=0$. While, if one of the
angles is 0 and the other is $\pi$, we immediately attain the HL sensitivity.
These could also be seen from Fig. 3(b). Now, let us consider the specific
case of $\theta_{1}=\theta_{2}$. This geometrically indicates the situation
where both coherent states are on the same circle that is orthogonal to $z$
axe. For this state, we find $\Delta\xi_{CRB}=1/|\sin(\theta_{1})|$. Clearly,
this is above the shot-noise limit, and hence, the state does not provide any
quantum enhancement for metrological applications.

To provide another situation, we considering $\theta_{2}=\pi-\theta_{1}$. From
Fig. 3(b), it can be realized that $\Delta\xi_{CRB}$ provides a better
sensitivity along this line. Here, the explicit form of the cat state will be

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\pi-\theta_{1},\varphi+\pi/2,1\rangle).$
(38)

For this cat state, $\Delta\xi_{CRB}$ could be found as

$\Delta\xi_{CRB}=\left[\frac{8}{12\cos(2\theta_{1})-\cos(4\theta_{1})+21}\right]^{1/2}.$
(39)

This reduces to HL metrology for $\theta_{1}=0,\pi$, while it gives
$\Delta\xi_{CRB}=1$ for $\theta_{1}=\pi/2$.

In Fig. 3(c), we consider the performance of a cat state given by

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\theta_{2},\varphi+3\pi/4,1\rangle).$
(40)

Similar to the previous cases, the sensitivity is larger as the difference
between $\theta_{1}$ and $\theta_{2}$ is bigger. This means that when one of
the coherent states is close to the north and the other to the south pole of
the Bloch sphere, parameter estimation accuracy is higher. For instance, let
us consider $\theta_{1}=\pi/4$ and $\theta_{2}=3\pi/4$. Then we get
$\Delta\xi_{CRB}\approx 0.525$, which is far below the shot-noise limit. Of
course, further moving the coherent states towards the poles will improve the
performance.

Finally, in Fig. 3(d), we plot $\Delta\xi_{CRB}$ for the cat state

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\theta_{2},\varphi+\pi,1\rangle).$
(41)

To understand the performance we obtain $\Delta\xi_{CRB}$ for this cat state
to be

$\Delta\xi_{CRB}=\left[\frac{2(\cos(\theta_{1}+\theta_{2})+3)^{2}}{\mathcal{A}-8\mathcal{B}+2\mathcal{C}-18\mathcal{D}+30}\right]^{1/2},$
(42)

where,

$\displaystyle\mathcal{A}$
$\displaystyle=\cos(3\theta_{1}+\theta_{2})+\cos(\theta_{1}-3\theta_{2}),$
(43) $\displaystyle\mathcal{B}$
$\displaystyle=\cos(2\theta_{1})+\cos(2\theta_{2}),$ (44)
$\displaystyle\mathcal{C}$ $\displaystyle=\cos(2(\theta_{1}+\theta_{2})),$
(45) $\displaystyle\mathcal{D}$ $\displaystyle=\cos(\theta_{1}-\theta_{2}).$
(46)

To have a better insight into the behavior of $\Delta\xi_{CRB}$ along the line
$\theta_{1}=\theta_{2}$, in Fig. 3(d), we note that with this constrain
$\Delta\xi_{CRB}$ reduces to

$\Delta\xi_{CRB}=\left[\frac{3+\cos(2\theta_{1})}{4|\sin(\theta_{1})|}\right].$
(47)

Therefore, $\Delta\xi_{CRB}$ diverges for $\theta_{1}=0,\pi$, and it reduces
to HL for $\theta_{1}=\pi/2$. The other interesting scenario would be to
consider $\Delta\xi_{CRB}$ along the line $\theta_{1}=\pi-\theta_{2}$. For
this case, the cat state reduces to

$|\text{Cat},1\rangle=\mathcal{N}(|\theta_{1},\varphi,1\rangle+|\pi-\theta_{1},\varphi+\pi,1\rangle).$
(48)

We note that this is a superposition of two antipodal points on the Bloch
sphere, which we refer to it as the antipodal cat state. In this case, the two
coherent states become orthogonal. Interestingly enough, $\Delta\xi_{CRB}$
provides HL sensitivity for any choices of the parameters in this case.
Therefore, antipodal cat state provides HL metrology for $j=1$.

As a result, in all of these plots, when $\theta_{1}=0$ and $\theta_{2}=\pi$,
or alternatively $\theta_{1}=\pi$ and $\theta_{2}=0$, the phase estimation
becomes optimal. This is due to the fact that in these cases, the cat state
becomes a superposition of the north and south poles of the Bloch sphere, and
$\varphi_{1}$ and $\varphi_{2}$ lose their significance at the poles, making
the phase sensitivity independent of these phases and, thus, optimal.

##  V. Conclusion

To summarize, we have considered the quantum metrological performance of spin
coherent states superposition and identified conditions for measurements with
the Heisenberg-limit precision. We demonstrated that the operator’s choice,
which introduces the parameter for the estimation, plays a significant role in
the performance of the quantum probe. We derive closed-form analytical
descriptions for the performance of spin cat states. These findings suggest
that careful control of parameters is necessary for achieving Heisenberg-limit
precision and provide insightful information on the geometry of the specific
coherent state superposition on the Bloch sphere and its relevance to the
performance of the states for parameter estimations.

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