# Sub-Planck structures and sensitivity of the superposed photon-added or
photon-subtracted squeezed-vacuum states

Naeem Akhtar  0000-0002-6756-2898<EMAIL_ADDRESS>Department of
Physics, Zhejiang Normal University, Jinhua 321004, China  Jizhou Wu
0000-0003-4732-1437 Corresponding author<EMAIL_ADDRESS>Department of
Physics, Southern University of Science and Technology, Shenzhen 518055, China
Jia-Xin Peng  0000-0002-0119-0139 State Key Laboratory of Precision
Spectroscopy, Quantum Institute for Light and Atoms, Department of Physics,
East China Normal University, Shanghai 200062, China  Wu-Ming Liu
0000-0002-5373-4417 Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan 523808,
Guangdong, China  Gao Xianlong  0000-0001-6914-3163<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University, Jinhua 321004, China

###### Abstract

The Wigner function of the compass state (a superposition of four coherent
states) develops phase-space structures of dimension much less than the Planck
scale $\hbar$, which are crucial in determining the sensitivity of these
states to phase-space displacements. In the present work we introduce
compasslike states that may have connection to the contemporary experiments,
which are obtained by either adding photons to or subtracting photons from the
superposition of two squeezed-vacuum states. We show that when a significant
quantity of photons is added (or subtracted), the Wigner functions of these
states are shown to have phase-space structures of an area that is
substantially smaller than the Planck scale. In addition, these states exhibit
sensitivity to displacements that is much higher than the standard quantum
limit. Finally, we show that both the size of the sub-Planck structures and
the sensitivity of our states are strongly influenced by the average photon
number, with the photon-addition case having a higher average photon number
leading to the smaller sub-Planck structures and, consequently, being more
sensitive to displacement than the photon-subtraction case. Our states offer
unprecedented resolution to the external perturbations, making them suitable
for quantum sensing applications.

##  I Introduction

Quantum mechanical states can be visualized in the phase space via the Wigner
quasiprobability distribution [1, 2, 3, 4, 5]. The term “Gaussian state”
refers to a state having the Gaussian Wigner function [6, 7]. The coherent
state [8] is an example of a Gaussian state. The Wigner function of the
coherent state exhibits the Planck limit [9, 10] in the phase space, which is
also known as the standard quantum limit (SQL) or shot-noise limit. The Wigner
function of certain non-Gaussian states may attain negative values [11, 12,
13, 14], indicating that these states are nonclassical. The quantum
superposition is the source of intriguing nonclassical properties of quantum
states, such as quantum coherence [15, 11], squeezing [16, 17], and
entanglement [18, 19, 20]. Nonclassical quantum states play a significant role
in quantum-information processing [21], tests of fundamental of physics [22,
23, 24], and applications in sensing and metrology [25, 26].

Nonclassical states are not always non-Gaussian: however, nonclassical states
can be Gaussian in some cases. For example, a squeezed-vacuum state (SVS) is a
common nonclassical state, but it possesses a Gaussian Wigner function [12,
13]. The Wigner function of the superposition of SVS is non-Gaussian and may
have negative amplitudes [27, 28]. Squeezed quantum states play an important
role in performing enhanced quantum metrology [29, 16]. Squeezed light has
been utilized experimentally to carry out improved measurements [30, 31].

The superpositions of two coherent states with opposite phases (cat states)
also possess non-Gaussian Wigner functions [32, 33, 34, 35]. Moreover, the
superposition of four coherent states, which is known as the “compass state”
[36], exhibits nonclassical features in the Wigner function with dimensions
far smaller than the SQL. The quantum states with sub-Planck structures are
found to be very sensitive to environmental decoherence [37] and have achieved
prominent theoretical attentions in quantum metrology [37, 38, 39, 40, 41].
The connection between sub-Planck structures and teleportation fidelity has
been established [42]. Sub-Fourier sensitivity is a classical analog of the
sub-Planck structures [43]. Compasslike states have been thoroughly
investigated in several situations [44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59]. Both theoretical [60, 61, 62, 63, 64, 65] and
experimental studies [66, 67, 68, 69] have been undertaken to achieve the
controlled generation of such states.

In recent years, there has been a lot of focus on subtracting photons from or
adding photons to the quantum states [70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
80, 81, 82]. A non-Gaussian state can also be generated by adding or
subtracting photons from a Gaussian state. For example, when photons are added
to or subtracted from the Gaussian SVS, one may obtain two non-Gaussian
squeezed states that have non-positive Wigner functions [75, 76, 77, 78, 80,
82, 83]: photon-added squeezed-vacuum states (PASVS) [72] and photon-
subtracted squeezed-vacuum states (PSSVS) [78]. Both PASVS and PSSVS have
attracted theoretical interest in quantum metrology [84, 85, 86].

The first theoretical investigation into the photon-addition operation is
accomplished by adding photons to the coherent states [87]. Later, the photon-
addition operation was successfully adopted in experiments by using a non-
degenerate parametric amplifier with a weak coupling [88]. The PSSVS is
currently the most successfully experimentally observed non-Gaussian SVS in
quantum optics [89, 90].

Schrödinger cat states with higher amplitude can be used as qubits in quantum
computing or as resources for quantum error-correcting coding [91, 92].
Conventional methods are unable to produce Schrödinger-cat-like states with
the necessary amplitudes [93, 94, 95]. Numerous theoretical [96, 97, 98, 99]
and experimental [100, 101, 102] research involving addition or subtractions
of photons from the states have been carried out to achieve larger-amplitude,
catlike states and even multicomponent cat states [103].

In the present work we introduce a few non-Gaussian SVSs that may also hold
the properties of the compass state. In particular, we show that the Wigner
function of the superpositions of two PASVSs (or PSSVSs) exhibit phase-space
structures of an area, which varies inversely with the number of photons added
(or subtracted). When a large amount of the photons are added to or subtracted
from our states the support area of these structures is substantially smaller
than that found for coherent states. Similar sub-Planck structures are also
found in the phase space of the mixed states related to the PASVSs and PSSVSs.
We demonstrate that the average photon number in the states significantly
influences the size of these sub-Planck structures, with the photon-addition
case having higher average photon number, leading to smaller sub-Planck
structures in the phase space than the photon-subtraction case.

To investigate the potential applications of these non-Gaussian states in
quantum metrology, we analyze the overlap between these states and their
slightly shifted analogs [104]. The degree to which the state is sensitive
against perturbations in the phase space can be determined from this overlap.
The sensitivity associated with coherent states cannot be improved by
increasing the number of photons. Techniques using probes prepared in such
states have the sensitivity at the SQL [105, 106]. Here, we show that the
sensitivity of our states is much higher than the SQL when the quantity of
added (or subtracted) photons is relatively high. Furthermore, our
superpositions exhibit this enhanced sensitivity in all phase-space
directions, whereas the mixtures only do so for specific displacements. The
varying average photon number in the states also contributed to the variation
in the sensitivities between the photon-addition and -subtraction cases; it is
shown that the photon-addition cases have higher sensitivity than the
subtracted ones.

The structure of our paper is as follows. In Sec. II we review the concept of
the sub-Planck structures associated to the compass state. In Sec. III we
review the Wigner functions of PASVS and PSSVS. In Sec. IV we introduce our
states and analyze their phase space by using the Wigner function, where we
also discuss the sensitivity of our states against the phase-space
perturbations. In Sec. V we provide our conclusion.

##  II Theory of Sub-Planck structures

This section provides the background of the sub-Planck structures and is
organized as follows. Section II.1 introduces the basic concepts that will be
used in this article. In Sec. II.2 we review the sub-Planck structures that
build in the phase space of the compass state. Section II.3 explains the
sensitivity to phase-space displacements associated to this compass state.

###  II.1 Basic concepts

The position operator $\hat{x}$ and the momentum operator $\hat{p}$ acts on an
infinite-dimensional Hilbert space, forming the so-called Heisenberg-Weyl (HW)
algebra $\mathfrak{hw}(1)$ [107, 108, 109] for a single degree of freedom. For
convenience, we use the unit convention $\hbar=c=1$ in this paper. The quantum
uncertainty principle [9, 10] arising from commutator relations
$\left[\hat{x},\hat{p}\right]=\text{i}$ limits the size of a phase-space
structure [10], for example, represented by the Wigner function [1] for
$\mathfrak{hw}(1)$ algebra and, more generally, by Moyal symbols [110] for
other symmetries [108]. For convenience, we use the vector

$\bm{\zeta}:=(x,p)^{\top}$ (1)

to represent the position-momentum pair in the following.

A Schrödinger coherent state is a nonspreading wave packet of the quantum
harmonic oscillator [8] and is an eigenstate of the annihilation operator:
$\hat{a}\ket{\alpha}=\alpha\ket{\alpha}$ with $\alpha\in\mathbb{C}$. The
coherent states are obtained by displacing the vacuum state $\ket{0}$, i.e.,

$\displaystyle\ket{\alpha}=\hat{D}(\alpha)\ket{0},$ (2)

where

$\displaystyle\hat{D}(\alpha):=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})$
(3)

is the displacement operator [109].

The overlap between two coherent states $\ket{\alpha}$ and $\ket{\beta}$ is
[111]

$\displaystyle\left|\Braket{\alpha}{\beta}\right|^{2}=\text{e}^{-|\alpha|^{2}-|\beta|^{2}+2\beta^{*}\alpha}=\text{e}^{-|\alpha-\beta|^{2}},$
(4)

which implies that two different coherent states are not orthogonal.

(a)  (b)  (c)

Figure 1: Wigner distribution of the compass state with (a) $x_{0}=4$, (b)
$x_{0}=8$, and (c) $x_{0}=12$. Insets represent the central interference
pattern of each case.

The Wigner function for a generic quantum state $\hat{\rho}$ is written as an
expectation value of the parity kernel [6, 4]

$\displaystyle
W_{\hat{\rho}}\left(\bm{\zeta}\right):=\text{tr}\left[\hat{\rho}\hat{\Delta}(\alpha)\right],$
(5)

with

$\displaystyle\hat{\Delta}(\alpha):=2\hat{D}(\alpha)\hat{\Pi}\hat{D}^{\dagger}(\alpha),\,\hat{\Pi}:=\left(-1\right)^{\hat{a}^{\dagger}\hat{a}}$
(6)

being the displaced parity operator.

The Wigner function for a coherent state is a strictly positive function and
appeared as a Gaussian of the form [5] (we omit the normalization of states
and their Wigner functions throughout the paper)

$\displaystyle G(\bm{\zeta};\pm x_{0},\pm p_{0})=\text{e}^{-(x\mp
x_{0})^{2}-(p\mp p_{0})^{2}},$ (7)

where $(x_{0},p_{0})$ is the location of the coherent state in phase space.
The product of uncertainties of position and momentum for a coherent state has
a lower limit $\Delta x\Delta p=\nicefrac{{1}}{{2}}$ [5, 9, 10, 111], which is
also known as the Planck action in the phase space.

It is a common belief that phase-space structures with areas smaller than the
Planck scale either do not exist or have no observational consequences for
physical quantum states. In fact, this is true for all Gaussian states
(coherent, squeezed, thermal, etc.) [6, 7] and even for other non-Gaussian
states like cat states [32, 33] that exhibit rapid oscillations in one
direction of phase space but an infinite Gaussian profile in the orthogonal
direction [57]. However, this notion was refuted by Zurek [36], who
demonstrated that the Wigner function of compass states develops phase-space
structures with dimensions far smaller than the Planck scale, arguing that
these structures play a vital role in determining the sensitivity of these
states against perturbations.

###  II.2 Zurek compass state

The Zurek compass state [36] is obtained from the superposition of the
following four coherent states

$\displaystyle\ket{\psi}:=\ket{\nicefrac{{x_{0}}}{{\sqrt{2}}}}+\ket{\nicefrac{{-x_{0}}}{{\sqrt{2}}}}+\ket{\nicefrac{{\text{i}x_{0}}}{{\sqrt{2}}}}+\ket{\nicefrac{{-\text{i}x_{0}}}{{\sqrt{2}}}},$
(8)

with $x_{0}\in\mathbb{R}$. Figure 1 depicts the Wigner function for this
compass state for the cases of $x_{0}=4,8$, and 12. Note that we normalize the
Wigner functions throughout by using their maximum amplitudes,
$|W_{\hat{\rho}}(0)|$. The Wigner function of the compass state (8) can be
represented as follows:

$\displaystyle
W_{\ket{\psi}}(\bm{\zeta})=W_{\circ}(\bm{\zeta})+W_{\Xi}(\bm{\zeta})+W_{\boxplus}(\bm{\zeta}),$
(9)

where the first term,

$\displaystyle W_{\circ}(\bm{\zeta}):=$ $\displaystyle
G(\bm{\zeta};x_{0},0)+G(\bm{\zeta};-x_{0},0)+G(\bm{\zeta};0,x_{0})$
$\displaystyle+G(\bm{\zeta};0,-x_{0}),$ (10)

represents the Wigner function of four coherent states that appear in the
phase space as Gaussian lobes. The second term in Eq. (9) is

$W_{\Xi}(\bm{\zeta}):=\frac{1}{2}\sum_{i_{1},i_{2}=\pm 1}I(i_{1}x,i_{2}p)$
(11)

with

$\displaystyle
I(\bm{\zeta}):=G(\bm{\zeta};\nicefrac{{x_{0}}}{{2}},\nicefrac{{x_{0}}}{{2}})\cos\Big{[}x_{0}\left(x+p-\frac{x_{0}}{2}\right)\Big{]},$
(12)

reflecting the Gaussian-modulated oscillations that appear far away from the
phase-space origin.

The central pattern resembles a chessboard as shown in the insets of Fig. 1
and is generated by

$\displaystyle
W_{\boxplus}(\bm{\zeta}):=\frac{1}{2}G(\bm{\zeta};0,0)\Big{[}\cos(2x_{0}x)+\cos(2x_{0}p)\Big{]}.$
(13)

This pattern consists of tiles with alternate signs (a central chessboardlike
pattern). The extension of each tile can be roughly estimated by calculating
zeros of Eq. (13), and it is found that it is proportional to $x^{-1}_{0}$ in
all directions of phase space. Numerically, we use the half width at half
maximum (HWHM) of the central phase-space tile to represent the extension. In
Fig. 2 we show the log-log plot of the extension of the central tile along the
$x$ and $p$ directions. It is demonstrated that for $x_{0}\gg 1$, the
extension of the central tile is constrained in all phase-space directions and
can be simultaneously much smaller than the corresponding extensions of a
coherent state. Note that the mixture of two cat states also contains the same
sub-Planck structures that are found in the compass state [57].

Figure 2: Extension of the central phase-space structure versus $x_{0}$ of the
compass state along the $x$ and $p$ directions.

The sub-Planck structures also emerge in the Wigner functions of non-Gaussian
states with the SU(1,1) [56] and SU(2) symmetries [57]. In particular, it has
been found that the Wigner function of the superposition of four SU(1,1) (or
SU(2)) coherent states also have sub-Planck structures similar to the compass
state when represented on the Poincaré disk [56] (or the sphere [57]). The
two-mode bosonic realization of the SU(1,1) implies that the sub-Planck
structures in the phase space of the SU(1,1) compass state can be associated
to the number of photons added to one of the modes of the two-mode squeezed
number states [56], and they arise at greater numbers of these added photons.
The existence of the sub-Planck structures in the Wigner function of the SU(2)
compass state can similarly be linked to the angular momentum; the higher
value of the angular momentum causes sub-Planck structures in the phase space
[57]. In the subsequent sections, we will demonstrate how adding or
subtracting photons from superpositions related to the one-mode Gaussian SVS
can also cause the emergence of sub-Planck structures in the phase space of
those states.

###  II.3 Sensitivity of compass state

The sensitivity of the compass state to displacements is determined by
calculating the overlap between it and its slightly displaced version [104].
The overlap between a state $\hat{\rho}$ and its displaced version
$\hat{D}(\delta\alpha)\hat{\rho}\hat{D}^{\dagger}(\delta\alpha)$ is

$O_{\hat{\rho}}(\delta\alpha):=\text{tr}\\{\hat{\rho}\hat{D}(\delta\alpha)\hat{\rho}\hat{D}^{\dagger}(\delta\alpha)\\}=\left|\braket{\psi}{\hat{D}(\delta\alpha)}{\psi}\right|^{2},$
(14)

where $\delta\alpha\in\mathbb{C}$ is an arbitrary displacement. Note that the
last equality of above expression holds when the state is pure,
$\hat{\rho}=\ket{\psi}\\!\bra{\psi}$. The smaller the displacement
$\delta\alpha$ needs to be in order to bring the overlap to zero, the more
sensitive the state is claimed to be against displacements [40]. This overlap
results in

$O_{\ket{\alpha}}(\delta\alpha)=\text{e}^{-|\delta\alpha|^{2}},$ (15)

for a coherent state $\ket{\alpha}$, indicating that the smallest noticeable
displacement that vanishes this overlap is above the Planck scale,
$|\delta\alpha|>1$. It is interesting to note that the sensitivity to
displacements in coherent states is independent of the quantity of quanta
contained in the state,
$\bar{n}=\braket{\hat{a}^{\dagger}\hat{a}}=|\alpha|^{2}$. Therefore increasing
$\bar{n}$ will not improve the sensitivity and is solely limited by the shot
noise introduced by vacuum fluctuations [105, 106].

(a)  (b)  (c)

Figure 3: Overlap of the compass state with its $\delta\alpha$-displaced part
with $\delta\alpha=(\delta x+\mathrm{i}\delta p)/\sqrt{2}$: (a) $x_{0}=4$, (b)
$x_{0}=8$, and (c) $x_{0}=12$.

We now discuss the sensitivity of the compass state (8) to phase-space
displacements. Assuming $x_{0}\gg 1$ and $|\delta\alpha|\ll 1$ , the overlap
(14) for this compass state results in

$\displaystyle
O_{\ket{\psi}}(\delta\alpha)=\frac{1}{4}\mathrm{e}^{-\frac{1}{2}|\delta\alpha|^{2}}\big{[}\cos\left(x_{0}\delta
x\right)+\cos\left(x_{0}\delta p\right)\big{]}^{2},$ (16)

with

$\delta\alpha=\delta x+\text{i}\delta p,\hskip 8.53581pt\delta x,\delta
p\in\mathbb{R}.$ (17)

It can be concluded that $O_{\ket{\psi}}(\delta\alpha)$ becomes zero when
either of the conditions is satisfied

$\delta x\pm\delta p=\frac{2m+1}{x_{0}}\pi,~{}m\in\mathbb{Z}.$ (18)

As illustrated in Fig. 3 for $O_{\ket{\psi}}(\delta\alpha)$ of cases when
$x_{0}$ is increased from 4 to 12, the overlap vanishes for the displacements
$|\delta\alpha|\sim x^{-1}_{0}$ and the arbitrary directions in the phase
space. As a result, it can be inferred that this sensitivity is proportional
to $x^{-1}_{0}$ and that $\bar{n}=\nicefrac{{x^{2}_{0}}}{{2}}$ ties it to the
number of excitations $\bar{n}$. Therefore, in comparison to coherent states,
a compass state with $\bar{n}$ excitations has shown $\sqrt{\bar{n}}$-enhanced
sensitivity to displacements of any arbitrary directions in the phase space.
Weak force measurements have been performed with Heisenberg-limited
sensitivity using compass states [39, 40]. In contrast, cat states have shown
the sensitivity to displacements along the specific direction in the phase
space [57]. It has been found that cat-state mixtures only exhibit this
enhanced sensitivity for displacements along particular phase-space directions
[57, 56]. Hence, cat-state mixtures with sub-Planck structures in the Wigner
function do not have the potential for metrology of compass states, for which
the additional quantum coherence of the cat-state superposition provided by
the second term in Eq. (9) plays a crucial role.

The SU(1,1) and SU(2) compass states have shown the same sensitivity to
displacements as their HW counterparts [57, 56]. The sensitivity of the
SU(1,1) compass state can be connected with the amount of the number of
photons added to one of the modes of the two-mode squeezed number state, and
this sensitivity improves as the quantity of added photons increases [56].
Similarly, the sensitivity of the SU(2) compass state improves as the angular
momentum goes higher [57]. This addition of the photons increases the average
photon number in the states, and it can be understood in a way similar to that
for compass states of the harmonic oscillator, i.e., injecting more photons in
the states improves its sensitivity.

##  III Non-Gaussian SVS

The non-Gaussian Wigner functions of the PASVS and PSSVS illustrate the non-
Gaussian nature of these states [75, 76, 77, 78, 80, 82]. In this section we
first provide a brief review of two non-Gaussian SVS, the PASVS and the PSSVS,
in Sec. III.1 and III.2, respectively. These two states are heavily used in
our construction of non-Gaussian states that manifest the sub-Planck
structures in Sec. IV. The Wigner functions of both PASVS and PSSVS are
discussed in relation to the amount of photons added or subtracted in the
following sections.

###  III.1 PASVS

First, we review the Wigner function of the PASVS. The creation operator
$\hat{a}^{\dagger}$ is repeatedly applied to SVS $\hat{S}(\pm r)\ket{0}$ to
obtain a single-mode PASVS [72]:

$\displaystyle\ket{\psi^{\pm}_{\text{PA}}}:=\hat{a}^{\dagger n}\hat{S}(\pm
r)\ket{0}\;\text{with}\,n\in\mathbb{N}.$ (19)

The subscript “PA” is the shorthand for “PASVS”, and we introduce“$\pm$” in
the squeezing operator $\hat{S}(\pm r)$ with the definition [112]

$\displaystyle\hat{S}(\pm r):=$
$\displaystyle\exp\bigg{[}\pm\frac{r}{2}\left(\hat{a}^{\dagger
2}-\hat{a}^{2}\right)\bigg{]},$ (20)

which allows us to preserve part of the expressions introduced in this section
for later uses in Sec. IV.

(a)  (b)  (c)

Figure 4: Wigner distribution of the PASVS with (a) $n=10$, (b) $n=15$. and
(c) $n=20$. In all cases $r=0.5$. Insets represent the central interference
pattern of each case.

(a)  (b)  (c)

Figure 5: Wigner distribution of the PSSVS with (a) $n=10$, (b) $n=15$, and
(c) $n=20$. In all cases $r=0.5$. Insets represent the central interference
pattern of each case.

Using Eq. (5), the Wigner function of PASVS is easily found to be [75, 76, 80,
82]

$\displaystyle W_{\ket{\psi^{\pm}_{\text{PA}}}}(\bm{\zeta})=$
$\displaystyle\frac{\exp\left(\chi_{\pm}\right)\left[\pm\sinh(2r)\right]^{n}}{\pi
4^{n}}\sum_{l=0}^{n}\frac{\left(n!\right)^{2}\left[\mp
2\coth(r)\right]^{l}}{l!\left[(n-l)!\right]^{2}}$
$\displaystyle\left|H_{n-l}\left[-\mathrm{i}\sqrt{\pm
2\coth(r)}\bar{\alpha}_{\pm}\right]\right|^{2},$ (21)

where $H_{m}$ represents the Hermite polynomial, and

$\displaystyle\chi_{\pm}:=\pm\sinh(2r)\left(\alpha^{*2}+\alpha^{2}\right)-2|\alpha|^{2}\cosh(2r),$
(22)

with

$\displaystyle\bar{\alpha}_{\pm}:=\alpha\cosh(r)\mp\alpha^{*}\sinh(r).$ (23)

The non-Gaussian shape of the Wigner function
$W_{\ket{\psi^{+}_{\text{PA}}}}(\bm{\zeta})$, which is shown in Fig. 4 for the
cases when $n$ is chosen as 10, 15, and 20, indicates that PASVS is a non-
Gaussian state. We can clearly see the interference pattern that emerges in
the form of an oscillating pattern in the $p$ direction in the phase space. As
the number of photons $n$ rises, this pattern gets more pronounced (the
frequency of the oscillations is increased). Moreover, the existence of these
negative peaks in the Wigner function shows that the PASVS is a nonclassical
state as well. Another indication of the nonclassicality of this state is the
squeezing effect in one of the quadratures, which is visible in the plots.
Note that the PASVS is the Gaussian SVS when $n=0$.

###  III.2 PSSVS

(a)  (b)  (c)

Figure 6: Wigner distribution of the pure SPASVS with (a) $n=10$, (b) $n=15$,
and (c) $n=20$. In all cases $r=0.5$ and $c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$.
Insets represent the central interference pattern of each case.

We now review the Wigner function of the PSSVS. The PSSVS [77, 78] is obtained
by repeatedly applying the annihilation operator $\hat{a}$ to the SVS, as

$\displaystyle\ket{\psi^{{\pm}}_{\text{PS}}}:=\hat{a}^{n}\hat{S}(\pm
r)\ket{0},$ (24)

where the subscript “PS” is the shorthand for “PSSVS”. The Wigner function of
this state is also in a non-Gaussian form and is written as

$\displaystyle W_{\ket{\psi^{\pm}_{\text{PS}}}}(\bm{\zeta})=$
$\displaystyle\frac{\exp\left(\chi_{\pm}\right)[\pm\sinh(2r)]^{n}}{\pi
4^{n}}\sum_{l=0}^{n}\frac{\left(n!\right)^{2}[\mp
2\tanh(r)]^{l}}{l!\left[(n-l)!\right]^{2}}$
$\displaystyle\left|H_{n-l}\left[-\mathrm{i}\sqrt{\pm
2\tanh(r)}\bar{\alpha}_{\pm}\right]\right|^{2}.$ (25)

We plot the Wigner function $W_{\ket{\psi^{+}_{\text{PS}}}}(\bm{\zeta})$ in
Fig. 5, with $r$ being $0.5$ and $n$ being 10, 15, and 20. This Wigner
function exhibits the interference pattern around the origin of the phase
space and oscillates along the $p$ direction in the phase space. As $n$ grows,
the frequency of this oscillating pattern increases. The simplest case $n=0$
of the PSSVS corresponds to the Gaussian SVS. Another indicator of the
nonclassical nature of this state is the squeezing effect in one of the
quadratures, which is indicative of the nonclassicality of state.

In summary, for nonzero values of $n$, the Wigner function of PASVS and PSSVS
maintains non-Gaussianity. It is interesting to note that both PASVS and PSSVS
exhibit similar phase-space features as catlike states. The addition or
subtraction of photons from the Gaussian SVS has been employed both
theoretically [96, 97, 98, 99] and experimentally [100, 101] to produce
catlike states. Both PASVS and PSSVS have been found very useful for the
quantum metrology [84, 85, 86]. When photons are added to or subtracted from
the Gaussian SVS, the average photon number of the resulting state grows [85,
86]. It has been shown that for the same number of photons applied on the
Gaussian SVS, the subsequent PASVS has a higher average photon number than the
PSSVS [85, 86]. This means that the PASVS has a better potential for metrology
than the PSSVS [85].

##  IV Superposition of non-Gaussian SVS

In this section we introduce the quantum states associated to non-Gaussian
SVSs and present their phase-space analysis by using the Wigner function [1,
2, 3, 4, 5].

The superpositions involving the Gaussian SVSs have been analyzed previously
[28, 27]. Theoretically, a nonlinear harmonic oscillator can be used to create
some specified superpositions of two SVSs. The superposition of two Gaussian
SVSs with opposite phases is given by

$\displaystyle\ket{\psi_{\text{SSV}}}:=c_{1}\hat{S}(r)\ket{0}+c_{2}\hat{S}(-r)\ket{0},$
(26)

with “SSV” in the subscript for the “superposition of SVSs” and the
probability amplitudes $c_{1}$ and $c_{2}$ fulfilling
$|c_{1}|^{2}+|c_{2}|^{2}=1$. The Wigner function corresponding to this state
exhibits non-Gaussian and nonclassical properties [28, 27].

(a)  (b)  (c)

Figure 7: Extension of the central phase-space structure vs the photon number
$n$ of the SPASVS state with $n$ chosen from 5 to 50 for (a)
$c_{1}=\nicefrac{{1}}{{10}}$, (b) $c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$, and (c)
$c_{1}=\nicefrac{{3\sqrt{11}}}{{10}}$.

(a)  (b)  (c)

Figure 8: Wigner distribution of the mixed-state SPASVS with (a) $n=10$, (b)
$n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$. Insets represent the central interference
pattern of each case.

Now we introduce the superposition related to non-Gaussian SVSs. The addition
of $n$ photons to the superposed state (26) leads to the superposition of two
photon-added squeezed-vacuum states (SPASVS), that is,

$\displaystyle\ket{\psi_{\text{SPA}}}:=\hat{a}^{\dagger
n}\ket{\psi_{\text{SSV}}}=c_{1}\ket{\psi^{+}_{\text{PA}}}+c_{2}\ket{\psi^{-}_{\text{PA}}},$
(27)

with the subscript “SPA” a shorthand for “SPASVS”. Similarly, the subtraction
of $n$ photons from the superposition (26) results in the superposition of two
photon-subtracted squeezed-vacuum states (SPSVS) of the following form:

$\displaystyle\ket{\psi_{\text{SPS}}}:=\hat{a}^{n}\ket{\psi_{\text{SSV}}}=c_{1}\ket{\psi^{+}_{\text{PS}}}+c_{2}\ket{\psi^{-}_{\text{PS}}},$
(28)

where the subscript “SPS” is the short form of “SPSSVS”. Here, we concentrate
on the sub-Planck structures in the Wigner function of these states and their
sensitivity to displacements.

(a)  (b)  (c)

Figure 9: Wigner distribution of the pure SPSSVS with (a) $n=10$, (b) $n=15$,
and (c) $n=20$. In all cases $r=0.5$ and $c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$.
Insets represent the central interference pattern of each case.

(a)  (b)  (c)

Figure 10: Extension of the central phase-space structure versus the photon
number $n$ of the SPSSVS state with $n$ chosen from 5 to 50 for (a)
$c_{1}=\nicefrac{{1}}{{10}}$, (b) $c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$, and (c)
$c_{1}=\nicefrac{{3\sqrt{11}}}{{10}}$.

(a)  (b)  (c)

Figure 11: Wigner distribution of the mixed-state SPSSVS with (a) $n=10$, (b)
$n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$. Insets represent the central interference
pattern of each case.

This section is structured as follows. In Sec. IV.1 we discuss Wigner
functions corresponding to $\ket{\psi_{\text{SPA}}}$ and
$\ket{\psi_{\text{SPS}}}$. Here, we describe how the addition and subtraction
of photons lead to the sub-Planck structures in the phase space. In Sec. IV.2
we discuss the sensitivity to displacements associated with these two
superpositions.

###  IV.1 Photons addition versus photons subtraction

The Wigner function of the SPASVS (27) can be obtained by using Eq. (5) as
(see Appendix A for detailed derivations)

$\displaystyle
W_{\ket{\text{SPA}}}(\bm{\zeta})=2\operatorname{Re}\left[I_{\Xi}(\bm{\zeta})\right]+W_{\boxplus}(\bm{\zeta}),$
(29)

and is shown in Fig. 6 for the cases when $n=10,15$, and 20. The first term in
(29)

$\displaystyle I_{\Xi}(\bm{\zeta}):=$
$\displaystyle\frac{c_{1}^{*}c_{2}\exp\left(\xi\right)\left[-\text{i}\tanh(2r)\right]^{n}}{\pi
4^{n}\cosh(r)\sqrt{1+\tanh^{2}(r)}}\sum^{n}_{l=0}\frac{(n!)^{2}\left[-2\text{i}\coth(r)\right]^{l}}{[(n-l)!]^{2}}$
$\displaystyle
H_{n-l}\left[\text{i}\Omega\alpha_{-}\right]H_{n-l}\left[-\Omega\alpha^{*}_{+}\right],$
(30)

provides the interference pattern that appears far away from the phase-space
origin, where

$\displaystyle\Omega:=\frac{\sqrt{\tanh(2r)}}{\sinh(r)},$ (31)

and

$\displaystyle\xi:=-\tanh(2r)\left(\alpha^{2}-\alpha^{2*}\right)-2|\alpha|^{2}\operatorname{sech}(2r),$
(32)

with

$\displaystyle\alpha_{\pm}:=\alpha^{*}\sinh(r)\pm\alpha\cosh(r)$ (33)

being the hyperbolic-rotated $\alpha$.

(a)  (b)  (c)  .

Figure 12: Overlap of the pure SPASVS state with its $\delta\alpha$-displaced
part with $\delta\alpha=(\delta x+\mathrm{i}\delta p)/\sqrt{2}$: (a) $n=10$,
(b) $n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$

For our purposes we concentrate on the second term in Eq. (29), which
contributes to the chessboardlike pattern that is visible at the phase-space
origin for $n\gg 1$. This central interference pattern is equal to the sum of
the individual Wigner functions of PASVSs as

$\displaystyle
W_{\boxplus}(\bm{\zeta}):=|c_{1}|^{2}W_{\ket{\psi^{+}_{\text{PA}}}}(\bm{\zeta})+|c_{2}|^{2}W_{\ket{\psi^{-}_{\text{PA}}}}(\bm{\zeta}).$
(34)

The extension of an individual tile in the chessboardlike pattern is
constrained concurrently along the $x$ and $p$ direction by the proper choice
of weights in Eq. (34). This is illustrated in Fig. 7, where we show log-log
plot of the extension of the central phase-space tile along the $x$ and $p$
directions versus $n$ for few $c_{1}$ and $c_{2}$ selections. The interference
pattern (34) is much similar to the interference (21) of PASVS and its
$\nicefrac{{\pi}}{{2}}$ rotated form in the cases depicted in Figs. 10(a) and
10(c), respectively. It is abundantly obvious from these two examples that
changing $n$ results in a decrease in the extension of the central phase-space
structure along the one-specific direction. On the other hand, the scenario
shown in Fig. 10(b) represents the case where increasing $n$ causes the
central phase-space tile to be constrained in all phase-space dimensions,
which indicates that the sub-Planck structures discovered for the compass
state [36] are also present in SPASVS.

(a)  (b)  (c)  .

Figure 13: Overlap of the mixed-state SPASVS with its $\delta\alpha$-displaced
part with $\delta\alpha=(\delta x+\mathrm{i}\delta p)/\sqrt{2}$: (a) $n=10$,
(b) $n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$

(a)  (b)  (c)

Figure 14: Overlap of the pure SPSSVS state with its $\delta\alpha$-displaced
part with $\delta\alpha=(\delta x+\mathrm{i}\delta p)/\sqrt{2}$: (a) $n=10$,
(b) $n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$.

The same sub-Planck structures are also contained by the following incoherent
mixture of two PASVSs:

$\displaystyle\hat{\rho}_{\text{PA}}:=|c_{1}|^{2}\ket{\psi^{+}_{\text{PA}}}\\!\bra{\psi^{+}_{\text{PA}}}+|c_{2}|^{2}\ket{\psi^{-}_{\text{PA}}}\\!\bra{\psi^{-}_{\text{PA}}},$
(35)

and its Wigner function, which is the same as the one given in Eq. (34), is
shown in Fig. 8.

Similarly, for the SPSSVS (28), the Wigner function is also written into two
terms (see Appendix A for detailed derivations):

$\displaystyle
W_{\ket{\text{SPS}}}(\bm{\zeta})=2\operatorname{Re}\left[I_{\Xi}(\bm{\zeta})\right]+W_{\boxplus}(\bm{\zeta}),$
(36)

which is plotted in Fig. 9. With

$\displaystyle\omega:=\frac{\sqrt{\tanh(2r)}}{\cosh(r)},$ (37)

the term that contains

$\displaystyle I_{\Xi}(\bm{\zeta}):=$
$\displaystyle\frac{c_{1}^{*}c_{2}\exp\left(\xi\right)\left[\text{i}\tanh(2r)\right]^{n}}{\pi
4^{n}\cosh(r)\sqrt{1+\tanh^{2}(r)}}\sum^{n}_{l=0}\frac{(n!)^{2}\left[2\text{i}\tanh(r)\right]^{l}}{[(n-l)!]^{2}}$
$\displaystyle
H_{n-l}\left[-\omega\alpha_{-}\right]H_{n-l}\left[-\text{i}\omega\alpha^{*}_{+}\right],$
(38)

causing the interference pattern that manifests as oscillating peaks far from
the phase-space origin. Again, we concentrate on the second term of Eq. (36),
which results in the chessboardlike pattern at the phase-space origin. This
term is the sum of the Wigner functions (25) of PSSVSs, that is,

$\displaystyle
W_{\boxplus}(\bm{\zeta}):=|c_{1}|^{2}W_{\ket{\psi^{+}_{\text{PS}}}}(\bm{\zeta})+|c_{2}|^{2}W_{\ket{\psi^{-}_{\text{PS}}}}(\bm{\zeta}).$
(39)

This pattern manifests sub-Planck oscillations around the origin of the phase
space by the proper choices of $c_{1}$ and $c_{2}$. This is illustrated in
Fig. 10, where we show log-log plot of the extension of the central phase-
space tile. The case depicted in Fig. 10(b) is the situation in which the
SPSSVS have sub-Planck structures in phase space, whereas the cases shown in
Figs. 10(a) and 10(c) represent the central phase-space pattern (39)
resembling that of the sole PSSVSs. It is interesting to note that for a given
$r$ and $n$, the central tile in the chessboardlike pattern is larger than
that of the SPASVS.

Additionally, we demonstrate that the following incoherent mixture of two
PSSVSs likewise has the same sub-Planck structures:

$\displaystyle\hat{\rho}_{\text{PS}}:=|c_{1}|^{2}\ket{\psi^{+}_{\text{PS}}}\\!\bra{\psi^{+}_{\text{PS}}}+|c_{2}|^{2}\ket{\psi^{-}_{\text{PS}}}\\!\bra{\psi^{-}_{\text{PS}}}.$
(40)

Similar to the case of the mixture of PASVSs (35), the Wigner function of
$\hat{\rho}_{\text{PS}}$ shown in Fig. 11 is identical to Eq. (39).

In summary, the photon-addition or photon-subtraction operations on the
superpositions of the Gaussian SVS produce the sub-Planck structures in the
phase space. The same sub-Planck structures are also present in mixtures
related to PASVSs or PSSVSs. The main difference between our states and the
compass states [36] is that our states are built by superposition of only two
non-Gaussian SVSs rather than four coherent-state superpositions.
Interestingly, adding or even subtracting photons from the superposition of
the Gaussian SVS actually increases the average photon number of the resultant
states, with the addition case having a greater average photon number than the
subtraction case [85]. The size of their sub-Planck structures is likewise
impacted by this variation in the average photon quantity in the states. For
instance, given the same number of photons used, the SPASVS and its related
mixture have smaller sub-Planck structures than their equivalents in the
photon-subtracted case.

(a)  (b)  (c)

Figure 15: Overlap of the mixed-state SPSSVS with its $\delta\alpha$-displaced
part with $\delta\alpha=(\delta x+\mathrm{i}\delta p)/\sqrt{2}$: (a) $n=10$,
(b) $n=15$, and (c) $n=20$. In all cases $r=0.5$ and
$c_{1}=\nicefrac{{1}}{{\sqrt{2}}}$.

###  IV.2 Sub-shot noise sensitivity of our states

In this section we discuss the susceptibility of our proposed states to phase-
space displacement. Let us first consider SPASVS (27). The overlap (14) for
this state under the approximation $|\delta\alpha|\ll 1$ and $n\gg 1$ leads to
(see Appendix Overlaps of SPASV and SPSSV for the detailed derivations)

$\displaystyle O_{\text{SPA}}(\delta\alpha)$
$\displaystyle=\left[|c_{1}|^{2}\bra{\psi^{+}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{+}_{\text{PA}}}+|c_{2}|^{2}\bra{\psi^{-}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PA}}}\right]^{2}.$
(41)

Each term of this overlap is calculated as

$\displaystyle\bra{\psi^{\pm}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\frac{\left[\mp\sinh(2r)\right]^{n}\mathrm{e}^{-\nicefrac{{|\eta_{\pm}|^{2}}}{{2}}}}{4^{n}}\sum^{n}_{l=0}\frac{(n!)^{2}}{l![(n-l)!]^{2}}$
$\displaystyle[\mp
2\text{\text{coth}}(r)]^{l}H_{n-l}\left[\Theta_{\pm}\right]H_{n-l}\left[\Theta_{\pm}^{*}\right],$
(42)

where

$\displaystyle\Theta_{\pm}=\text{i}\sqrt{\pm\frac{\text{coth}(r)}{2}}\eta_{\pm},$
(43)

and

$\eta_{\pm}=\delta\alpha\text{cosh}(r)\mp\delta\alpha^{*}\text{sinh}(r),$ (44)

with

$\delta\alpha=\delta x+\text{i}\delta p.$ (45)

In Fig. 12 we plot this overlap for $n=10$, 15, and 20. For the large $n$, a
small displacement $|\delta\alpha|\ll 1$ can turn the SPASVS into a state
orthogonal to its original state, and this orthogonality occurs in all phase-
space directions. We have normalized overlaps to their maximum amplitudes,
$O_{{\hat{\rho}}}(0)$.

Let us now consider the mixture of PASVSs (19), for which the overlap (14) is
calculated as

$\displaystyle O_{{\hat{\rho}_{\text{PA}}}}(\delta\alpha)$
$\displaystyle=\left|c_{1}^{2}\bra{\psi^{+}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{+}_{\text{PA}}}\right|^{2}+\left|c_{2}^{2}\bra{\psi^{-}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PA}}}\right|^{2}.$
(46)

We plot this overlap with $n=10$, 15, and 20 in Fig. 13. Again, we see that
the overlap $O_{\hat{\rho}_{\text{PA}}}(\delta\alpha)$ disappears for the
displacement $|\delta\alpha|\ll 1$, but unlike the SPASVS, this orthogonality
now takes place when $\delta x=\pm\delta p$ in the phase space.

Similarly, overlap (14) for SPSSVS is obtained as

$\displaystyle O_{\text{SPS}}(\delta\alpha)$
$\displaystyle=\left[|c_{1}|^{2}\bra{\psi^{+}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{+}_{\text{PS}}}+|c_{2}|^{2}\bra{\psi^{-}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PS}}}\right]^{2},$
(47)

where

$\displaystyle\bra{\psi^{\pm}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{\pm}_{\text{PS}}}=$
$\displaystyle\frac{\left[\mp\sinh(2r)\right]^{n}\mathrm{e}^{-\nicefrac{{|\eta_{\pm}|^{2}}}{{2}}}}{4^{n}}\sum^{n}_{l=0}\frac{(n!)^{2}}{l![(n-l)!]^{2}}$
$\displaystyle[\mp
2\text{\text{tanh}}(r)]^{l}H_{n-l}\left[\theta_{\pm}\right]H_{n-l}\left[\theta_{\pm}^{*}\right],$
(48)

with

$\displaystyle\theta_{\pm}=\text{i}\sqrt{\pm\frac{\text{tanh}(r)}{2}}\eta_{\pm}.$
(49)

In Fig. 14 we plot this overlap with $n$ equal to 10, 15, and 20. The SPSSVS
overlap plot exhibits the same behavior as the SPASVS: the overlap disappears
in any direction in phase space when $|\delta\alpha|\ll 1$. The only
distinction is that, for the same $n$ and $r$, the central pattern of the
overlap of the SPSSVS is larger than that of the SPASVS, showing that the
SPSSVS is less sensitive than the SPASVS.

Finally, we consider mixtures of PSSVSs (24). The overlap (14) for this state
leads to

$\displaystyle O_{{\hat{\rho}_{\text{PS}}}}(\delta\alpha)$
$\displaystyle=\left|c_{1}^{2}\bra{\psi^{+}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{+}_{\text{PS}}}\right|^{2}+\left|c_{2}^{2}\bra{\psi^{-}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PS}}}\right|^{2}.$
(50)

In Fig. 15, we plot the overlap $O_{\hat{\rho}_{\text{PS}}}(\delta\alpha)$
using the same parameter values as that of the SPSSVS. We observe that the
overlap of the mixture of the PSSVSs looks similar to the mixture of the
PASVSs, with the distinction being that the mixture of PSSVSs appears to be
less sensitive for a given number of $n$ and $r$, which is manifested as the
larger chessboardlike pattern in the center of the phase space than that of
the mixture of PASVSs.

In summary, we have demonstrated that the sensitivity associated to our
proposed states depends on the quantity of photons added (or subtracted), and
it is considerably lower than the sensitivity of the coherent state when there
are excessive amounts of photons added (or subtracted). For both SPASVS and
SPSSVS, the enhanced sensitivity is unaffected by the directions of phase-
space displacements. However, for the mixtures related to PASVSs or PSSVSs,
this enhancement only takes place in particular phase-space directions. This
implies that compared to mixed states, our superposition states have more
potential for quantum sensing applications. Moreover, it has been found that
the quantum states associated with photon-addition cases are more sensitive
than their counterparts of the photon-subtraction cases.

In comparison to the original compass state [36], our proposed states perform
similarly in terms of sensitivity to phase-space displacements, but the
sensitivities of our states are impacted either by adding or subtracting
photons from these states. For example, in the compass state, the sensitivity
to displacements is proportional to the parameter $x_{0}$ (separation between
coherent states), which is in fact connected to the average photon number in
the states; that is, increasing $x_{0}$ leads to a higher average photon
number in the states. In our case, enhanced sensitivity is now connected with
the number of added or subtracted photons $n$ in the states, as adding or
removing photons leads in a larger average photon number in the resultant
states.

Our states present an alternative to compass states that have the same
properties but may be prepared more feasibly. It was demonstrated that the
coherent-state superpositions could be constructed deterministically using
third-order Kerr nonlinearity [33]. However, this technique necessitates Kerr
non-linearities of such magnitude that it is not practicable with currently
available Kerr media. Furthermore, states of this type are particularly
susceptible to loss, and because absorption cannot be ignored in currently
available Kerr-media, the capacity to extract coherent-state superpositions
before they decohere is severely constrained [113]. By adding or subtracting
photons from Gaussian SVSs, a comparably practical strategy for constructing
coherent-state superpositions is adopted [101, 100, 99, 98, 97, 103, 102].

##  V Summary and outlook

We have shown that the Wigner function of the SPASVS (or SPSSVS) contains the
chessboardlike pattern around the origin of the phase space. A similar
chessboardlike pattern also emerges by the mixtures related to PASVSs and
PSSVSs. The support area of the phase-space structures contained by this
chessboardlike pattern varies inversely with the photon number added (or
subtracted). When a sizable number of photons are added (or subtracted), the
support area of these structures is noticeably smaller than that of the
coherent state.

The average photon numbers of our states, which are increased either by
photon-addition or photon-subtraction actions on the Gaussian SVS, have an
impact on the size of the sub-Planck structures in the phase space. The sub-
Planck structures associated with the SPASVS are smaller than those of the
SPSSVS for the same number of photons added or subtracted. This is because the
photon-addition operation always leads to the higher average photon number in
the resultant states. The association of the average photon number with the
sub-Planck structures in our states is much similar to that of the compass
states, i.e., a higher average photon number in the compass state corresponds
to the smaller sub-Planck structures in the phase space.

We have demonstrated that the sensitivity of our proposed states is noticeably
higher than that of the coherent state when a significant number of photons
are added (or subtracted). Both the SPASVS and SPSSVS exhibit the enhanced
sensitivity, which is independent of the phase-space directions, indicating
that they hold more promise for quantum metrology. In addition, the difference
in the sensitivities between the photon-addition and -subtraction cases arose
from the different average photon numbers in the states; photon-addition cases
are demonstrated to have greater sensitivity than the subtracted cases.

It is incredibly exciting that sub-Planck structures can possibly build from
photons being added to or subtracted from states. As a result, it will be able
to apply a variety of ways to engineer compasslike states in association with
contemporary experiments. Numerous theoretical and experimental research
utilizing photon-addition or -subtraction operations from states have been
developed to produce catlike states [101, 100, 99, 98, 97, 103, 102].
Theoretical research to construct the catlike states advocated in [97] is
subsequently applied in actual experiments [100, 101]. Another subsequent
theoretical approach introduces the idea of photon subtraction from the
Gaussian SVS to produce the compasslike states [64]. These illustrations
unequivocally demonstrate that some of these techniques can be modified to
generate SPASVS and SPSSVS, which are entirely new research avenues that can
be adapted in the future.

###### Acknowledgements.

N.A. acknowledges the support of the postdoctoral fund of the Zhejiang Normal
University under Grant No. ZC304021937. G.X. acknowledges support by the
National Natural Science Foundation of China under Grants No. 11835011 and No.
12174346. W.M.L. acknowledges support from the National Key R&D Program of
China under Grants No. 2021YFA1400900, No. 2021YFA0718300, and No.
2021YFA1402100, the NSFC under Grants No. 61835013, No. 12174461, and No.
12234012, and the Space Application System of China Manned Space Program.

##  Appendix A Wigner functions of SPASVS and SPSSVS

This section provides the main steps to drive the Wigner functions of SPASVS
and SPSSVS.

###  A.1 Derivations of the Wigner function of SPASVS

Let us first consider the Wigner function of SPASVS given by Eq. (29). The
first term of this Wigner function is given by

$\displaystyle
I_{\Xi}(\bm{\zeta})=\Braket{\psi^{+}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{-}_{\text{PA}}},$
(51)

where the alternative form of the displaced parity operator
$\hat{\Delta}(\alpha)$ is [80]

$\displaystyle\hat{\Delta}(\alpha):=$
$\displaystyle\frac{1}{\pi^{2}}\mathrm{e}^{2\left|\alpha\right|^{2}}\int^{\infty}_{-\infty}\,\mathrm{d}^{2}\beta\mathrm{e}^{-2\alpha^{*}\beta+2\alpha\beta^{*}}\Ket{\beta}\\!\Bra{-\beta}.$
(52)

Equation (51) can be rewritten as

$\displaystyle
I_{\Xi}(\bm{\zeta})=\frac{(-1)^{n}\mathrm{e}^{2|\alpha|^{2}}}{\pi^{2}\cosh(r)}\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta|\beta|^{2n}\exp\bigg{[}-|\beta|^{2}-\frac{\tanh(r)}{2}\big{(}\beta^{2}+\beta^{*2}\big{)}-2\beta\alpha^{*}+2\beta^{*}\alpha\bigg{]}.$
(53)

We incorporate the factor $|\beta|^{2n}$ into a differential equation as

$\displaystyle
I_{\Xi}(\bm{\zeta})=\frac{(-1)^{n}\mathrm{e}^{2|\alpha|^{2}}}{\pi^{2}\cosh(r)}\bigg{.}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta\exp\bigg{[}-|\beta|^{2}-\frac{\tanh(r)}{2}\big{(}\beta^{2}-\beta^{*2}\big{)}-2\beta\alpha^{*}+2\beta^{*}\alpha+s\beta+t\beta^{*}\bigg{]}\bigg{|}_{s=t=0}.$
(54)

Consider the integral formula [114]

$\displaystyle\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta\exp{\bigg{[}a|\beta|^{2}+b\beta+c\beta^{*}+d\beta^{2}+k\beta^{*2}}\bigg{]}=\frac{\pi}{\sqrt{a^{2}-4dk}}\exp{\bigg{[}\frac{-abc+b^{2}k+c^{2}d}{a^{2}-4dk}}\bigg{]},$
(55)

whose convergent conditions are $\operatorname{Re}\left[a\pm d\pm k\right]<0$
and $\operatorname{Re}\left[\nicefrac{{(a^{2}-4dk)}}{{a\pm d\pm k}}\right]<0$.
By using this integral Eq. (54) leads to

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\frac{(-1)^{n}\mathrm{e}^{\xi}}{\pi\cosh(r)\sqrt{1+\tanh^{2}(r)}}\Bigg{.}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\exp\Bigg{[}\frac{\tanh(2r)}{4}s^{2}-\frac{\tanh(2r)}{4}t^{2}+\big{(}1+\tanh^{2}(r)\big{)}^{-1}st-2\cosh(r)\text{sech}(2r)\alpha_{-}s$
$\displaystyle-2\cosh(r)\text{sech}(2r)\alpha^{*}_{+}t\Bigg{]}\Bigg{|}_{s=t=0},$
(56)

with

$\displaystyle\xi:=-(\alpha^{2}-\alpha^{*2})\tanh(2r)-2|\alpha|^{2}\text{sech}(2r).$
(57)

It is challenging to solve Eq. (A.1) because it has $\mathrm{e}^{\gamma st}$
terms. We employ the following sum series [76] to get rid of it.

$\displaystyle\exp(Cs+Dt+Est)=\sum_{l=0}^{\infty}\frac{E^{l}}{l!}\frac{\partial^{2l}}{\partial
C^{l}\partial D^{l}}\left[\exp\left(Cs+Dt\right)\right].$ (58)

Using this formula Eq. (A.1) modifies as

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\frac{(-1)^{n}\mathrm{e}^{\xi}}{\pi\cosh(r)\sqrt{1+\tanh^{2}(r)}}\sum_{l=0}^{\infty}\frac{1}{l!~{}2^{2l}}\frac{\big{[}1+\tanh^{2}(r)\big{]}^{-l}}{\cosh^{2l}(r)\text{sech}^{2l}(2r)}\frac{\partial^{2l}}{\partial\alpha_{+}^{*l}\alpha_{-}^{l}}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\exp\bigg{[}\frac{\tanh(2r)}{4}s^{2}-\frac{\tanh(2r)}{4}t^{2}$
$\displaystyle-2\cosh(r)\text{sech}(2r)(\alpha_{-}s+\alpha^{*}_{+}t)\bigg{]}\bigg{|}_{s=t=0}.$
(59)

Notice the generating function of the Hermite polynomial

$H_{n}(x)=\big{.}\frac{\partial^{n}}{\partial
s^{n}}\exp\left(2xs-s^{2}\right)\big{|}_{s=0}$ (60)

and its recursive relation

$\frac{\mathrm{d}^{l}}{\mathrm{d}x^{l}}H_{n}(x)=\frac{2^{l}n!}{(n-l)!}H_{n-l}(x).$
(61)

The preceding equation can then be simplified in the form of Eq. (IV.1) by
applying the relationships (60) and (61).

Let us now calculate second term of the Wigner function (29). This term can be
written as

$\displaystyle
W_{\boxplus}(\bm{\zeta})=\Braket{\psi^{+}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{+}_{\text{PA}}}+\Braket{\psi^{-}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{-}_{\text{PA}}},$
(62)

where

$\displaystyle\Braket{\psi^{\pm}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\frac{(-1)^{n}}{\pi^{2}}\frac{\mathrm{e}^{2|\alpha|^{2}}}{\cosh(r)}\big{.}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta\exp\big{[}-|\beta|^{2}\pm\frac{1}{2}\tanh(r)(\beta^{2}+\beta^{*2})-2\beta\alpha^{*}+2\beta^{*}\alpha+s\beta+t\beta^{*}\big{]}\big{|}_{s=t=0}.$

Using the integral (55), we get

$\displaystyle\Braket{\psi^{\pm}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\frac{(-1)^{n}}{\pi}\exp\big{[}\pm\sinh(2r)(\alpha^{*2}+\alpha^{2})-4\cosh^{2}(r)|\alpha|^{2}\big{]}\bigg{.}\frac{\partial^{2n}}{\partial
s^{n}\partial t^{n}}\exp\bigg{[}\pm\frac{1}{4}\sinh(2r)(s^{2}+t^{2})$
$\displaystyle+2\cosh(r)(\bar{\alpha}_{\pm}s-\bar{\alpha}_{\pm}^{*}t)+\cosh^{2}(r)st\bigg{]}\bigg{|}_{s=t=0}.$
(63)

Again, use of the sum series (58) eliminates the factors $\mathrm{e}^{\gamma
st}$, that is,

$\displaystyle\Braket{\psi^{\pm}_{\text{PA}}}{\hat{\Delta}(\alpha)}{\psi^{\pm}_{\text{PA}}}=\sum_{l=0}^{\infty}\frac{(-1)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\bar{\alpha}_{\pm}^{l}\partial\bar{\alpha}_{\pm}^{*l}}\left.\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\exp\left[\pm\frac{\sinh(2r)}{4}\left(s^{2}+t^{2}\right)+2\cosh
r\left(\bar{\alpha}_{\pm}s-\bar{\alpha}_{\pm}^{*}t\right)\right]\right|_{s=t=0}.$
(64)

Then, by using the relations (60) and (61), the expression (39) is obtained.

###  A.2 Derivations of the Wigner function of SPSSVS

This section presents the detailed derivation of the Eq. (36), for which the
first term has the following form as below:

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\Braket{\psi^{+}_{\text{PS}}}{\hat{\Delta}(\alpha)}{\psi^{-}_{\text{PS}}}.$
(65)

This term is calculated as

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\frac{1}{\pi^{2}}\frac{\text{e}^{2|\alpha|^{2}}}{\cosh(r)}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\exp\bigg{[}-\frac{\tanh(r)}{2}\big{(}t^{2}-s^{2}\big{)}\bigg{]}\int^{\infty}_{-\infty}\mathrm{d}^{2}\beta\exp\bigg{[}-|\beta|^{2}-\big{(}\tanh(r)t+2\alpha^{*}\big{)}\beta-\big{(}\tanh(r)s$
$\displaystyle-2\alpha\big{)}\beta^{*}-\frac{\tanh(r)}{2}(\beta^{2}-\beta^{*2})\bigg{]}\bigg{|}_{s=t=0}.$
(66)

Using the integral (55), we obtain

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\frac{\mathrm{e}^{\xi}}{\pi\cosh(r)\sqrt{1+\tanh^{2}(r)}}\frac{\partial^{2n}}{\partial
s^{n}\partial
t^{n}}\exp\bigg{[}\frac{\tanh(2r)}{4}s^{2}-\frac{\tanh(2r)}{4}t^{2}+2\text{sech}(2r)\sinh(r)(\alpha^{*}_{+}s-\alpha_{-}t)$
$\displaystyle+\text{sech}(2r)\sinh^{2}(r)st\bigg{]}\bigg{|}_{s=t=0}.$ (67)

Now, we eliminate $\mathrm{e}^{\gamma st}$ terms by using Eq. (58):

$\displaystyle I_{\Xi}(\bm{\zeta})=$
$\displaystyle\frac{\mathrm{e}^{\xi}}{\pi\cosh(r)\sqrt{1+\tanh^{2}(r)}}\sum^{\infty}_{l=0}\frac{1}{l!2^{2l}\text{sech}^{l}(2r)}\frac{\partial^{2l}}{\partial\alpha_{+}^{*l}\partial\alpha_{-}^{l}}\frac{\partial^{2}}{\partial
s^{n}\partial
t^{n}}\exp\bigg{[}\frac{\tanh(2r)}{4}s^{2}-\frac{\tanh(2r)}{4}t^{2}$
$\displaystyle+2\text{sech}(2r)\sinh(r)(\alpha^{*}_{+}s+\alpha_{-}t)\bigg{]}\bigg{|}_{s=t=0}.$
(68)

Then, by using the relations (60) and (61) we obtain expression (IV.1).

Finally, we derive the second term of the Eq. (36). This term can be written
as

$\displaystyle
W_{\boxplus}(\bm{\zeta})=\Braket{\psi^{+}_{\text{PS}}}{\hat{\Delta}(\alpha)}{\psi^{+}_{\text{PS}}}+\Braket{\psi^{-}_{\text{PS}}}{\hat{\Delta}(\alpha)}{\psi^{-}_{\text{PS}}},$
(69)

where

$\displaystyle\Braket{\psi^{\pm}_{\text{PS}}}{\hat{\Delta}(\alpha)}{\psi^{\pm}_{\text{PS}}}=$
$\displaystyle\frac{1}{\pi}\exp\big{[}\pm\sinh(2r)(\alpha^{2}+\alpha^{*2})-2\cosh(2r)|\alpha|^{2}\big{]}\bigg{.}\frac{\partial^{2n}}{\partial
s^{n}t^{n}}\exp\bigg{[}\pm\frac{1}{4}\sinh(2r)(s^{2}+t^{2})$ $\displaystyle\pm
2\sinh(r)(\bar{\alpha}_{\pm}t+\bar{\alpha}_{\pm}^{*}s)-\sinh^{2}(r)st\bigg{]}\bigg{|}_{s,t=0}.$
(70)

Again, we use Eq. (58) to get rid of all $\mathrm{e}^{\gamma st}$ factors,
obtaining

$\displaystyle\Braket{\psi^{\pm}_{\text{PS}}}{\hat{\Delta}(\alpha)}{\psi^{\pm}_{\text{PS}}}=$
$\displaystyle\frac{1}{\pi}\exp\big{[}\pm\sinh(2r)(\alpha^{2}+\alpha^{*2})-2\cosh(2r)|\alpha|^{2}\big{]}\sum_{l=0}^{\infty}\frac{(-1)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\bar{\alpha}_{\pm}^{l}\bar{\alpha}_{\pm}^{*l}}\bigg{.}\frac{\partial^{2n}}{\partial
s^{n}t^{n}}\exp\bigg{[}\pm\frac{\sinh(2r)}{4}\big{(}s^{2}+t^{2}\big{)}$
$\displaystyle\pm
2\sinh(r)\big{(}\bar{\alpha}_{\pm}t+\bar{\alpha}_{\pm}^{*}s\big{)}\bigg{]}\bigg{|}_{s=t=0}.$
(71)

Finally, this equation can be simplified to expression (39) by utilizing the
relations (60) and (61).

## Overlaps of SPASV and SPSSV

In this section we calculate the overlap (14) of SPASVS and SPSSVS. Note that
for $n\gg 1$ and $|\delta\alpha|\ll 1$, the contribution of the cross terms
between the states to the overlap is negligible, that is,

$\displaystyle\bra{\psi^{+}_{\text{PA}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PA}}}=0\,\text{and}\,\bra{\psi^{+}_{\text{PS}}}\hat{D}(\delta\alpha)\ket{\psi^{-}_{\text{PS}}}=0.$
(72)

First, we drive each term of Eq. (IV.2). PASV (19) can be rewritten as [76]

$\displaystyle\ket{\psi^{\pm}_{\text{PA}}}=\hat{S}(\pm
r)\left[\hat{a}^{\dagger}\cosh(r)\pm\hat{a}\sinh(r)\right]^{n}\ket{0}.$ (73)

Then, considering the relation given by [76],

$\displaystyle(f\hat{a}+g\hat{a}^{\dagger}):=\bigg{(}-\text{i}\sqrt{\frac{fg}{2}}\bigg{)}^{n}H_{n}\bigg{(}\text{i}\sqrt{\frac{f}{2g}}\hat{a}+\text{i}\sqrt{\frac{g}{2f}}\hat{a}^{\dagger}\bigg{)},$
(74)

which leads to

$\displaystyle[\hat{a}^{\dagger}\cosh(r)+\hat{a}\sinh(r)]^{n}=\bigg{[}-\text{i}\sqrt{\frac{\sinh(2r)}{4}}\bigg{]}^{n}H_{n}\bigg{[}\text{i}\sqrt{\frac{\tanh(r)}{2}}\hat{a}+\text{i}\sqrt{\frac{\coth(r)}{2}}\hat{a}^{\dagger}\bigg{]},$
(75)
$\displaystyle[\hat{a}\cosh(r)+\hat{a}^{\dagger}\sinh(r)]^{n}=\bigg{[}-\text{i}\sqrt{\frac{\sinh(2r)}{4}}\bigg{]}^{n}H_{n}\bigg{[}\text{i}\sqrt{\frac{\tanh(r)}{2}}\hat{a}^{\dagger}+\text{i}\sqrt{\frac{\coth(r)}{2}}\hat{a}\bigg{]}.$
(76)

By using these relations, we obtain

$\displaystyle\braket{\psi^{\pm}_{\text{PA}}}{\hat{D}(\delta\alpha)}{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\Braket{0}{H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\coth(r)}{2}}\hat{a}\bigg{)}\hat{D}(\eta_{\pm})H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\coth(r)}{2}}\hat{a}^{\dagger}\bigg{)}}{0},$
$\displaystyle=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\int^{\infty}_{-\infty}\frac{\text{d}^{2}\alpha}{\pi}\exp\bigg{(}-\frac{|\alpha|^{2}}{2}-\frac{\alpha}{2}\eta_{\pm}^{*}+\frac{\alpha^{*}}{2}\eta_{\pm}-\frac{|\alpha-\eta_{\pm}|^{2}}{2}\bigg{)}H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\coth(r)}{2}}\alpha\bigg{)}$
$\displaystyle
H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\coth(r)}{2}}(\alpha^{*}-\eta_{\pm}^{*})\bigg{)},$
(77)

where

$\displaystyle\hat{D}(\eta_{\pm})=\hat{S}^{\dagger}(\pm
r)\hat{D}(\delta\alpha)\hat{S}(\pm
r)\;\text{with}\;\eta_{\pm}=\delta\alpha\cosh(r)\mp\delta\alpha^{*}\sinh(r).$
(78)

By using (60), we get

$\displaystyle\braket{\psi^{\pm}_{\text{PA}}}{\hat{D}(\delta\alpha)}{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\frac{\partial^{2n}}{\partial\tau^{n}\partial
t^{n}}\exp\big{(}-\text{i}\sqrt{\pm
2\coth(r)}~{}\eta_{\pm}^{*}\tau\big{)}\exp\big{(}-\tau^{2}-t^{2}\big{)}$
$\displaystyle\int^{\infty}_{-\infty}\frac{\text{d}^{2}\alpha}{\pi}\exp\bigg{(}-\frac{|\alpha|^{2}}{2}-\frac{\alpha}{2}\eta_{\pm}^{*}+\frac{\alpha^{*}}{2}\eta_{\pm}-\frac{|\alpha-\eta_{\pm}|^{2}}{2}+\text{i}\sqrt{\pm
2\coth(r)}~{}\alpha t+\text{i}\sqrt{\pm
2\coth(r)}~{}\alpha^{*}\tau\bigg{)}\bigg{|}_{\tau=t=0}.$ (79)

Using the integral (55), the previous equation yields

$\displaystyle\braket{\psi^{\pm}_{\text{PA}}}{\hat{D}(\delta\alpha)}{\psi^{\pm}_{\text{PA}}}=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\exp\bigg{(}-\frac{|\eta_{\pm}|^{2}}{2}\bigg{)}\frac{\partial^{2n}}{\partial\tau^{n}\partial
t^{n}}\exp\bigg{(}-t^{2}+\text{i}\sqrt{\pm
2\coth(r)}~{}\eta_{\pm}t-\tau^{2}-\text{i}\sqrt{\pm
2\coth(r)}~{}\eta_{\pm}^{*}\tau$ (80) $\displaystyle\mp
2\coth(r)~{}t\tau\bigg{)}\bigg{|}_{t=\tau=0}.$ (81)

First, we rid out the factors $\mathrm{e}^{\gamma\tau t}$ from above equation
by using (58). Then, by using (60) and (61), the preceding equation is
simplified to (IV.2).

Similarly, PSSVS can be rewritten as [76]

$\displaystyle\ket{\psi^{\pm}_{\text{PS}}}=\hat{S}(\pm
r)\left[\hat{a}\cosh(r)\pm\hat{a}^{\dagger}\sinh(r)\right]^{n}\ket{0}.$ (82)

The overlap

$\displaystyle\braket{\psi^{\pm}_{\text{PS}}}{\hat{D}(\delta\alpha)}{\psi^{\pm}_{\text{PS}}}=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\Braket{0}{H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\tanh(r)}{2}}\hat{a}\bigg{)}\hat{D}(\eta_{\pm})H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\tanh(r)}{2}}\hat{a}^{\dagger}\bigg{)}}{0},$
$\displaystyle=$
$\displaystyle\bigg{[}\mp\frac{\sinh(2r)}{4}\bigg{]}^{n}\int^{\infty}_{-\infty}\frac{\text{d}^{2}\alpha}{\pi}\exp\bigg{(}-\frac{|\alpha|^{2}}{2}-\frac{\alpha}{2}\eta_{\pm}^{*}+\frac{\alpha^{*}}{2}\eta_{\pm}-\frac{|\alpha-\eta_{\pm}|^{2}}{2}\bigg{)}H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\tanh(r)}{2}}\alpha\bigg{)}$
$\displaystyle
H_{n}\bigg{(}\text{i}\sqrt{\pm\frac{\tanh(r)}{2}}(\alpha^{*}-\eta_{\pm}^{*})\bigg{)},$
(83)

can be easily simplified to (IV.2).

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