# Heralded preparation of polarization entanglement via quantum scissors

Dat Thanh Le<EMAIL_ADDRESS>ARC Centre for Engineered Quantum System,
School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia  Thang Long Institute of Mathematics and Applied Sciences
(TIMAS), Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi 10000,
Vietnam  Warit Asavanant  Department of Applied Physics, School of
Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan  Nguyen Ba An  Thang Long Institute of Mathematics and Applied Sciences
(TIMAS), Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi 10000,
Vietnam  Institute of Physics, Vietnam Academy of Science and Technology
(VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi 10000 Vietnam

###### Abstract

Quantum entanglement is at the heart of quantum information sciences and
quantum technologies. In the optical domain, the most common type of quantum
entanglement is polarization entanglement, which is usually created in a
postselection manner involving destructive photon detection and thus hindering
further applications which require readily available entanglement resources.
In this work, we propose a scheme to prepare multipartite entangled states of
polarized photons in a heralded manner, i.e., without postselection. We
exploit the quantum scissors technique to truncate a given continuous-variable
entanglement into the target entangled states which are of hybrid discrete-
continuous or solely discrete types. We consider two implementations of the
quantum scissors: one modified from the original quantum scissors [Pegg et
al., Phys. Rev. Lett. 81, 1604 (1998)] using single photons and linear optics
and the other designed here using a type-II two-mode squeezer. We clarify the
pros and cons of these two implementations as well as discussing practical
aspects of the entanglement preparation. Our work illustrates an interface
between various types of optical entanglement and the proposed quantum
scissors techniques could serve as alternative methods for heralded generation
of polarization entanglement.

##  I Introduction

Quantum entanglement plays an essential role in foundations of quantum
mechanics Einstein  _et al._ (1935) and proves to be an indispensable resource
in quantum technologies Nielsen and Chuang (2000). Quantum entanglement exists
in various forms, among quantum systems of the same natures or of distinct
natures as well as among different degrees of freedom (DoFs) within a single
quantum system. Of particular interest is the hybrid entanglement between
discrete-variable (DV) and continuous-variable (CV) quantum systems Jeong  _et
al._ (2014); Morin  _et al._ (2014); Kwon and Jeong (2015); Man  _et al._
(2017); Rab  _et al._ (2017); An  _et al._ (2018); Sychev  _et al._ (2018); Li
_et al._ (2018); Huang  _et al._ (2019); Podoshvedov and An (2019); Gouzien
_et al._ (2020); Le  _et al._ (2021); Podoshvedov and Podoshvedov (2021),
which has been successfully applied in numerous quantum information protocols
Furusawa and van Loock (2011); Takeda  _et al._ (2013); Lee and Jeong (2013);
Kwon and Jeong (2013); Takeda  _et al._ (2015); Andersen  _et al._ (2015);
Ulanov  _et al._ (2017); Jeannic  _et al._ (2018); Cavaillès  _et al._ (2018);
Guccione  _et al._ (2020).

In the optical domain, the most frequently used quantum entanglement is
between polarized photons, due to their resilience against decoherence and
loss Kok  _et al._ (2007); Pan  _et al._ (2012) and the availability of high-
quality polarization-control elements Kwiat  _et al._ (1995); Kok and Lovett
(2010). Polarization entanglement was exploited in quantum dense coding Mattle
_et al._ (1996), quantum teleportation Bouwmeester  _et al._ (1997), quantum
cryptography Naik  _et al._ (2000), and tests of Bell inequality Giustina  _et
al._ (2015). Entangled polarization pairs, called polarization Bell pair
(PBP), are routinely produced via spontaneous parametric downconversion
(SPDC), in which a photon in a pumping beam is converted into two photons of
lower frequencies obeying both the energy and momentum conservation
constraints Kwiat and Weinfurter (1998). This process, however, is highly
probabilistic and the generated state is dominantly occupied by the unwanted
vacuum component Hnilo (2005), which necessitates photon detection to verify
entanglement and thus destroys the entangled state itself. Such postselection
procedure hurdles subsequent quantum information tasks which require an on-
demand entanglement resource, such as quantum error correction Shor (1995) and
entanglement purification Bennett  _et al._ (1996).

Therefore, many works have been devoted to the preparation of polarization
entanglement without postselection, e.g., by employing ancilla single photons
and linear optics Zou  _et al._ (2002a, b) or a probabilistic controlled-NOT
(CNOT) gate Pittman  _et al._ (2003). In 2010, two groups independently
reported heralded generation of a PBP Barz  _et al._ (2010); Wagenknecht  _et
al._ (2010), based on detection of four photons in a three-pair SPDC emission
event Śliwa and Banaszek (2003). This method nevertheless suffers from a very
low success probability and false detections from a four-pair emission Kok
(2010). Recently, with the development of deterministic, highly pure single-
photon sources Senellart  _et al._ (2017); Tomm  _et al._ (2021), heralded
production of a PBP by fusion gate Bao  _et al._ (2007); Zhang  _et al._
(2008) has been realized Li  _et al._ (2021). The fusion-based method might
face difficulty when generalizing to $n$-partite entanglement with $n\geq 3$,
which involves nontrivial analyses to find the optimal heralded setup Gubarev
_et al._ (2020).

In this paper, we propose a scheme to prepare polarization entanglement in a
heralded fashion, i.e., without relying on postselection. We show that
truncating unwanted components in a CV polarization entanglement, which can be
supplied by currently accessible resources, gives rise to a hybrid DV-CV or
solely DV entangled polarization state of $n$ parties for an arbitrary $n\geq
2$. We present two implementations for such truncation operation: the first
one is a modified version from the quantum scissors originally proposed in
Ref. Pegg  _et al._ (1998) using single photons and linear optics and the
second one is proposed here exploiting a type-II two-mode squeezer Kok and
Lovett (2010). The technological ingredients needed for these implementations
are well-studied subjects and utilized in a variety of applications Özdemir
_et al._ (2001); Babichev  _et al._ (2003); Goyal and Konrad (2013); Winnel
_et al._ (2020); He  _et al._ (2021); Seshadreesan  _et al._ (2020); Lamas-
Linares  _et al._ (2002); Resch  _et al._ (2002); Zavatta  _et al._ (2004);
Parigi  _et al._ (2007). Our scheme highlights the intriguing interface
between different types of quantum entanglement and is realizable within the
present-day optical technologies.

The outline of this paper is the following. In Sec. II we introduce a
particular CV entanglement and show its connection to hybrid DV-CV and solely
DV (i.e., DV-DV) entangled states via truncation of irrelevant terms. We then
in Sec. III propose two distinct methods to implement the desired truncation
operation. Next, in Sec. IV we analyze the performance for the generation of
several types of entanglement using the two truncation techniques. This is
followed by discussions on practical perspectives of the entanglement
preparation in Sec. V. Finally, we conclude the paper in Sec. VI. An appendix
provides detailed calculations for the results in the main text.

##  II Truncation of continuous-variable entanglement

Figure 1:  (a) Schematic setup to prepare the CV polarization entanglement
$|\Xi\rangle_{12}$ in Eq. 1. Here
$|\mathrm{Cat}_{H}\rangle=N_{0}(|\delta_{H}\rangle+e^{i\varphi}|\\!-\delta_{H}\rangle)$
with $N_{0}=[2(1+\cos(\varphi)e^{-2\delta^{2}})]^{1/2}$, BBS is balanced beam
splitter, PBS polarizing beam splitter, $\mathrm{BS}(t_{0})$ beam splitter
with transmissivity $t_{0}$, and HWP half-wave plate. (b) Splitting of mode 2
in the state $|\Xi\rangle_{12}$ by $\mathrm{BS}(t_{1})$ prepares the
tripartite CV polarization entanglement $|\Lambda\rangle_{12\dots n}$ with
$n=3$ in Eq. (7).

In what follows, we show that a CV entanglement contains within itself a
hybrid DV-CV or a DV-DV entanglement. We consider the CV polarization
entanglement Le  _et al._ (2021); Bowen  _et al._ (2002)

$|\Xi\rangle_{12}=N_{0}\left(|\alpha_{H}\rangle_{1}|\beta_{H}\rangle_{2}+e^{i\varphi}|\\!-\alpha_{V}\rangle_{1}|\\!-\beta_{V}\rangle_{2}\right),$
(1)

where $N_{0}=[2(1+\cos(\varphi)e^{-(\alpha^{2}+\beta^{2})})]^{-1/2}$ and
$|\gamma_{H/V}\rangle=\sum_{n=0}^{\infty}f_{n}(\gamma)|n_{H/V}\rangle$ is a
horizontally or vertically polarized coherent state of real amplitude $\gamma$
with $f_{n}(\gamma)=e^{-\gamma^{2}/2}\gamma^{n}/\sqrt{n!}$ and
$|n_{H/V}\rangle$ a Fock state containing $n$ horizontally or vertically
polarized photons. Note that in this paper without loss of generality we
consider coherent states of real amplitudes only. For convention the
polarization-mode and spatial-mode subscripts are placed respectively inside
and outside of the ket/bra states and from now on, “mode”, when being used,
implies “spatial mode”. Throughout the paper, for brevity we also suppress the
vacuum state when expressing ket/bra states as follows:
$|\alpha_{H},0_{V}\rangle\equiv|\alpha_{H}\rangle$,
$|0_{H},\alpha_{V}\rangle\equiv|\alpha_{V}\rangle$,
$|n_{H},0_{V}\rangle\equiv|n_{H}\rangle$, and
$|0_{H},n_{V}\rangle\equiv|n_{V}\rangle$. The vacuum state in some equations
will be made visible to perform relevant calculations.

A schematic setup for the preparation of the state $|\Xi\rangle_{12}$ is shown
in Fig. 1a and can be briefly described as follows. The required inputs
include a polarized cat state
$|\mathrm{Cat}_{H}\rangle=N_{0}(|\delta_{H}\rangle+e^{i\varphi}|\\!-\delta_{H}\rangle)$
with $N_{0}=[2(1+\cos(\varphi)e^{-2\delta^{2}})]^{1/2}$ in mode $1$ and a
polarized coherent state $|\delta_{H}\rangle$ in mode $1^{\prime}$, both of
which have the same horizontal polarization. The two states first interact at
a balanced beam splitter (BBS). Since the action of a general beam splitter
(BS) with transmissivity $t$ on a pair of coherent states of the same
polarization is

$\mathrm{BS}_{ab}(t)|\mu\rangle_{a}|\nu\rangle_{b}=\big{|}\mu\sqrt{t}+\nu\sqrt{1-t}\big{\rangle}_{a}\big{|}\mu\sqrt{1-t}-\nu\sqrt{t}\big{\rangle}_{b},$
(2)

the two inputs become a NOON-like state Joo  _et al._ (2011)
$N_{0}(|\delta\sqrt{2}_{H}\rangle_{1}|0\rangle_{1^{\prime}}+e^{i\varphi}|0\rangle_{1}|\\!-\delta\sqrt{2}_{H}\rangle_{1^{\prime}})$.
A half-wave plate (HWP) placed at mode $1^{\prime}$, followed by a polarizing
beam splitter (PBS) to merge two modes 1 and $1^{\prime}$ into one mode 1,
changes the the NOON-like state into
$N_{0}(|\delta\sqrt{2}_{H}\rangle+e^{i\varphi}|\\!-\delta\sqrt{2}_{V}\rangle)_{1}$.
A final BS with transmissivity $t_{0}$ splits such state into two spatial
modes, resulting in the state $|\Xi\rangle_{12}$ in Eq. 1 with

$\alpha=\delta\sqrt{2t_{0}},\hskip 28.45274pt\beta=\delta\sqrt{2(1-t_{0})}.$
(3)

Using Fock-state representation for coherent states, we decompose
$|\Xi\rangle_{12}$ as

$\displaystyle|\Xi\rangle_{12}$ $\displaystyle=$ $\displaystyle
N_{0}f_{1}(\alpha)\left(|1_{H}\rangle_{1}|\beta_{H}\rangle_{2}-e^{i\varphi}|1_{V}\rangle_{1}|\\!-\beta_{V}\rangle_{2}\right)+\dots$
(6) $\displaystyle=$ $\displaystyle
N_{0}f_{1}(\beta)\left(|\alpha_{H}\rangle_{1}|1_{H}\rangle_{2}-e^{i\varphi}|\\!-\alpha_{V}\rangle_{1}|1_{V}\rangle_{2}\right)+\dots$
$\displaystyle=$ $\displaystyle
N_{0}f_{1}(\alpha)f_{1}(\beta)\left(|1_{H}\rangle_{1}|1_{H}\rangle_{2}+e^{i\varphi}|1_{V}\rangle_{1}|1_{V}\rangle_{2}\right)$
$\displaystyle+\dots,$

where in the first equation we hide terms in which mode 1 is not a single-
photon state, in the second equation we hide terms in which mode 2 is not a
single-photon state, and in the third equation we hide terms in which both
modes 1 and 2 are not single-photon states. The visible terms in Eq.
(LABEL:eq:Decom1) or (LABEL:eq:Decom2) constitute a hybrid DV-CV entangled
state between single photons and coherent states Le  _et al._ (2021); Kwon and
Jeong (2015), whereas those in Eq. (6) represent a DV-DV entangled state in
the form of a PBP. The decompositions suggest that we can prepare these types
of entangled states by performing on mode $1$ and/or mode $2$ of the state
$|\Xi\rangle_{12}$ an operation that (i) truncates the non-single-photon
components (which include the vacuum state and the more-than-one-photon
components), (ii) preserves coherence of two single-photon states of
orthogonal polarizations, and (iii) is carried out in a heralded fashion. We
call such an operation the ideal (or perfect) polarized-single-photon quantum
scissors and for brevity we abbreviate them as PQS. The first two requirements
for the PQS are obvious, while the last one ensures that the entanglement
preparation of interest is heralded given that the input entangled state
$|\Xi\rangle_{12}$ is supplied on-demand. labeled Intriguingly, the above
observation can be generalized to the case of an entangled state among $n$
parties for an arbitrary $n$. This is due to the fact that the $n$-partite
version of the state $|\Xi\rangle_{12}$, which is of the form

$\displaystyle|\Lambda\rangle_{12\dots n}$ $\displaystyle=$ $\displaystyle
M_{n}\big{(}|\alpha^{(1)}_{H}\rangle_{1}|\alpha^{(2)}_{H}\rangle_{2}\dots|\alpha^{(n)}_{H}\rangle_{n}$
(7)
$\displaystyle+e^{i\varphi}|\\!-\alpha^{(1)}_{V}\rangle_{1}|\\!-\alpha^{(2)}_{V}\rangle_{2}\dots|\\!-\alpha^{(n)}_{V}\rangle_{n}\big{)},\quad$

with
$M_{n}=[2(1+\cos(\varphi)\exp({-\sum_{j=1}^{n}(\alpha^{(j)})^{2}})]^{-1/2}$,
can be straightforwardly prepared employing $(n-2)$ BSs to keep splitting
$|\Xi\rangle_{12}$. For example, as depicted in Fig. 1b we split mode 2 of the
state $|\Xi\rangle_{12}$ by one BS with transmissivity $t_{1}$ to produce a
tripartite CV polarization entangled state

$N_{0}\big{(}|\alpha^{(1)}_{H}\rangle_{1}|\alpha^{(2)}_{H}\rangle_{2}|\alpha^{(3)}_{H}\rangle_{3}+e^{i\varphi}|\\!-\alpha^{(1)}_{V}\rangle_{1}|\\!-\alpha^{(2)}_{V}\rangle_{2}|\\!-\alpha^{(3)}_{V}\rangle_{3}\big{)},$
(8)

where $\alpha^{(1)}=\delta\sqrt{2t_{0}}$,
$\alpha^{(2)}=\delta\sqrt{2(1-t_{0})t_{1}}$, and
$\alpha^{(3)}=\delta\sqrt{2(1-t_{0})(1-t_{1})}$. Generalization to a higher
number of parties is straightforward. Given the state
$|\Lambda\rangle_{12\dots n}$, we perform the PQS on $j$ parties of it, say
from party $1$ to party $j$, to obtain an $n$-partite hybrid DV-CV entangled
state of the form

$\displaystyle|\Omega\rangle_{12\dots n}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{\sqrt{2}}(|1_{H}\rangle_{1}\\!\dots\\!|1_{H}\rangle_{j}|\alpha^{(j+1)}_{H}\rangle_{j+1}\\!\dots\\!|\alpha^{(n)}_{H}\rangle_{n}$
$\displaystyle+e^{i\varphi}|1_{V}\rangle_{1}\\!\dots\\!|1_{V}\rangle_{j}|\\!-\alpha^{(j+1)}_{V}\rangle_{j+1}\\!\dots\\!|\\!-\alpha^{(n)}_{V}\rangle_{n}).$

When $j=n$ the above state becomes the $n$-partite DV GHZ polarization
entanglement.

##  III Implementations of the polarized-single-photon quantum scissors

In this section we present two implementations of the PQS that truncates the
CV polarization entanglements in Eqs. (1) and (7) into hybrid DV-CV and DV-DV
entangled states. We show that the two implementations actually realize a
nonideal PQS in the sense that they perfectly satisfy the requirements (ii)
(preserving coherence of two orthogonally polarized single photons) and (iii)
(succeeding in a heralded way) but partially meet the requirement (i) (see the
requirements for the ideal PQS in Sec. II). That is, they do not completely
truncate all the non-single-photon components but retain the vacuum and a two-
photon state, resulting in an unwanted imperfection. This imperfection,
however, can be made arbitrarily small by suitably adjusting relevant
parameters.

###  III.1 Using single photons and linear optics

The first implementation of the desired PQS, denoted as PQS1, involves the
original quantum scissors proposed in Refs. Pegg  _et al._ (1998); Winnel  _et
al._ (2020) which truncate the more-than-one-photon components and amplify the
one-photon component compared to the vacuum one in a quantum state

$|\psi\rangle=\sum_{n=0}^{\infty}c_{n}|n\rangle,$ (10)

where $|n\rangle$ is a Fock state containing $n$ photons and
$\sum_{n=0}^{\infty}|c_{n}|^{2}=1$. We refer to such scissors as QS to
distinguish from the PQS. The QS deals with photons having only one
polarization, whereas the PQS is supposed to work with inputs having two
orthogonal polarizations. Therefore, the QS is not readily applicable for
extracting the desired entanglements out of the CV entanglement in Eq. 1 or
(7). In the following, we first present some details of the QS and then show
how to construct the PQS1 from it.

Figure 2: (a) Black-box representation with the label $\mathrm{QS}(t)$ and
physical setup of the quantum scissors proposed in Refs. Pegg  _et al._
(1998); Winnel  _et al._ (2020). Here $t$ denotes the transmissitivity of the
BS, which characterizes the performance of the scissors. The input
$|\psi\rangle=\sum_{n=0}^{\infty}c_{n}|n\rangle$, going through the
$\mathrm{QS}(t)$, is truncated into the (unnormalized) output state
$\sqrt{1-t}c_{0}|0\rangle\pm\sqrt{t}c_{1}|1\rangle$, conditioned by detection
of a single photon at detector D1 and no photons at detector D2 or no photons
at detector D1 and a single photon at detector D2. Choosing $t$ properly this
output can be made very close to a single-photon state. (b) Linear-optics-
based implementation of a nonideal PQS with the black-box labeled as
$\mathrm{PQS}1(t)$, comprising two modules of the quantum scissors in panel
(a). The $\mathrm{PQS1}(t)$ truncates a general polarized state
$|\psi_{\mathrm{p}}\rangle=\sum_{n,m=0}^{\infty}c_{nm}|n_{H},m_{V}\rangle$
into an output state that is very close to the (unnormalized) state
$|\psi_{\mathrm{p}}^{(1)}\rangle=c_{10}|1_{H},0_{V}\rangle+c_{01}|1_{V},0_{H}\rangle$.
The box with the label $\mathrm{QS}_{H}(t)$ ($\mathrm{QS}_{V}(t)$) represents
the physical setup in panel (a) with the ancilla single photon being
horizontally (vertically) polarized.

The black-box representation of the QS and its physical implementation are
depicted in Fig. 2a. Concretely, following the schematic setup in Fig. 2a the
QS transforms the state $|\psi\rangle$ as

$|\psi\rangle=\sum_{n=0}^{\infty}c_{n}|n\rangle\xrightarrow{\mathrm{QS}(t)}\sqrt{1-t}c_{0}|0\rangle\pm\sqrt{t}c_{1}|1\rangle,$
(11)

where the output state is unnormalized and the relative sign “$+$” (“$-$”)
corresponds to detection of a single photon (vacuum) at detector D1 and vacuum
(a single photon) at detector D2. We choose the BS transmissitivity $t$ such
that $\sqrt{1-t}c_{0}$ is much smaller than $\sqrt{t}c_{1}$, making the output
state close enough to a single photon. The setup in Fig. 2a resembles an
error-corrected quantum teleportation circuit, of which the state
$|\psi\rangle$ serves as the input, the quantum channel is a single-rail
entangled state created by splitting the single-photon input via the
$\mathrm{BS}(t)$, and the Bell measurement is performed using the BBS and two
detectors D1 and D2. The output state is corrected, compared to the input one
$|\psi\rangle$, in the sense that more-than-one-photon components are erased,
recovering $|\psi\rangle$ back to the single-rail basis. The heralding
probability and the fidelity of the QS technique are respectively

$\displaystyle P_{\mathrm{QS}}$ $\displaystyle=$
$\displaystyle(1-t)|c_{0}|^{2}+t|c_{1}|^{2},$ (12) $\displaystyle
F_{\mathrm{QS}}$ $\displaystyle=$
$\displaystyle\frac{t|c_{1}|^{2}}{(1-t)|c_{0}|^{2}+t|c_{1}|^{2}}.$ (13)

The probability $P_{\mathrm{QS}}$ here is nothing but the inverse square of
the normalization factor of the output state in Eq. 11.

The above result also holds when the to-be-truncated mode is entangled with
others. To verity this, let us consider a quantum state comprising spatially
distinguishable modes $a,b,c,\dots$ in the form

$|\Psi\rangle_{abc\dots}=\sum_{n=0}^{\infty}c_{n}|n\rangle_{a}|\phi_{n}\rangle_{bc\dots},$
(14)

where $a$ is the to-be-truncated mode and $|\phi_{n}\rangle_{bc\dots}$ is a
(normalized) joint quantum state of modes $b,c\dots$. We can check that the QS
in Fig. 2a applying to only mode $a$ reduces $|\Psi\rangle_{abc\dots}$ in the
following way

$\displaystyle|\Psi\rangle_{abc\dots}=\sum_{n=0}^{\infty}c_{n}|n\rangle_{a}|\phi_{n}\rangle_{bc\dots}$
$\displaystyle\xrightarrow[\text{on\,mode}\,a]{\mathrm{QS}(t)}\sqrt{1-t}c_{0}|0\rangle_{a}|\phi_{0}\rangle_{bc\dots}+\sqrt{t}c_{1}|1\rangle_{a}|\phi_{1}\rangle_{bc\dots}.\qquad$
(15)

This is just the same as the result in Eq. 11 if we formally make the
replacement $c_{n}\to c_{n}|\phi_{n}\rangle_{bc\dots}$. We also note that
different from Eq. 11 the relative sign in the output state of Eq. (15) has
been chosen to be “$+$” for definiteness.

We now turn to the design of the PQS1. We consider a single-spatial-mode
polarized quantum state of a general form

$|\psi_{\mathrm{p}}\rangle=\sum_{n,m=0}^{\infty}c_{nm}|n_{H},m_{V}\rangle,$
(16)

where $|n_{H},m_{V}\rangle$ describes a quantum state of the single spatial
mode of interest having $n$ horizontally polarized photons and $m$ vertically
polarized photons and $\sum_{n,m=0}^{\infty}|c_{nm}|^{2}=1$. The concerned
PQS1 should operate in such a way that it truncates the state
$|\psi_{\mathrm{p}}\rangle$ into (exactly or very close to) the target output
state

$|\psi_{\mathrm{p}}^{(1)}\rangle=c_{10}|1_{H},0_{V}\rangle+c_{01}|0_{H},1_{V}\rangle\equiv
c_{10}|1_{H}\rangle+c_{01}|1_{V}\rangle.$ (17)

For this purpose, we arrange a setup including two modules of the QS,
$\mathrm{QS}_{H}(t)$ and $\mathrm{QS}_{V}(t)$, as sketched in Fig. 2b, where
$\mathrm{QS}_{H/V}(t)$ operates as $\mathrm{QS}(t)$ in Fig. 2a when the input
photon is horizontally or vertically polarized. According to Fig. 2b, we first
separate spatially two polarizations of the single-spatial-mode state
$|\psi_{\mathrm{p}}\rangle$ by a PBS to obtain a two-spatial-mode state

$\sum_{n,m=0}^{\infty}c_{nm}|n_{H}\rangle_{a}|m_{V}\rangle_{a^{\prime}}\equiv\sum_{n=0}^{\infty}|n_{H}\rangle_{a}\sum_{m=0}^{\infty}c_{nm}|m_{V}\rangle_{a^{\prime}}.$
(18)

Following the transformation in Eq. (15), the module $\mathrm{QS}_{H}(t)$
acting on mode $a$ truncates the state in Eq. 18 into

$\displaystyle\\!\sqrt{1-t}|0_{H}\rangle_{a}\sum_{m=0}^{\infty}c_{0m}|m_{V}\rangle_{a^{\prime}}\\!+\\!\sqrt{t}|1_{H}\rangle_{a}\sum_{m=0}^{\infty}c_{1m}|m_{V}\rangle_{a^{\prime}}$
(19) $\displaystyle\equiv$
$\displaystyle\\!\sum_{m=0}^{\infty}|m_{V}\rangle_{a^{\prime}}(\sqrt{1-t}c_{0m}|0_{H}\rangle\\!+\\!\sqrt{t}c_{1m}|1_{H}\rangle)_{a}.$

At the same time, the module $\mathrm{QS}_{V}(t)$ on mode $a^{\prime}$, also
according to Eq. (15), shortens the state in Eq. (19) into

$\displaystyle\sqrt{1-t}|0_{V}\rangle_{a^{\prime}}(\sqrt{1-t}c_{00}|0_{H}\rangle+\sqrt{t}c_{10}|1_{H}\rangle)_{a}$
$\displaystyle+\sqrt{t}|1_{V}\rangle_{a^{\prime}}(\sqrt{1-t}c_{01}|0_{H}\rangle+\sqrt{t}c_{11}|1_{H}\rangle)_{a}.\qquad$
(20)

This truncated two-spatial-mode state going through another PBS and after
rearrangements is recast to a single-spatial-mode state

$\sqrt{(1-t)t}|\psi_{\mathrm{p}}^{(1)}\rangle_{a}+(1-t)c_{00}|0_{H},0_{V}\rangle_{a}+tc_{11}|1_{H},1_{V}\rangle_{a}.$
(21)

We extend the above result to the case when the mode to be truncated is in
entanglement with others. That is, instead of the state
$|\psi_{\mathrm{p}}\rangle$ in Eq. 16 we consider

$|\Psi_{\mathrm{p}}\rangle_{abc\dots}=\sum_{n,m=0}^{\infty}c_{nm}|n_{H},m_{V}\rangle_{a}|\phi_{nm}\rangle_{bc\dots},$
(22)

where $a$ is the to-be-truncated mode and $|\phi_{nm}\rangle_{bc\dots}$ is a
(normalized) joint quantum state of modes $b,c\dots$. We aim to get the ideal
output state after truncation as

$\displaystyle|\Psi_{\mathrm{p}}^{(1)}\rangle_{abc\dots}$ $\displaystyle=$
$\displaystyle
c_{10}|1_{H},0_{V}\rangle_{a}|\phi_{10}\rangle_{bc\dots}\\!+\\!c_{01}|0_{H},1_{V}\rangle_{a}|\phi_{01}\rangle_{bc\dots}$
(23) $\displaystyle\equiv$ $\displaystyle
c_{10}|1_{H}\rangle_{a}|\phi_{10}\rangle_{bc\dots}\\!+\\!c_{01}|1_{V}\rangle_{a}|\phi_{01}\rangle_{bc\dots}.$

Similar to the output in Eq. (21), application of the PQS1 on mode $a$ in the
state $|\Psi_{\mathrm{p}}\rangle_{abc\dots}$ gives

$\displaystyle\sqrt{(1\\!-\\!t)t}|\Psi_{\mathrm{p}}^{(1)}\rangle_{abc\dots}+(1-t)c_{00}|0_{H},\\!0_{V}\rangle_{a}|\phi_{00}\rangle_{bc\dots}$
$\displaystyle+tc_{11}|1_{H},1_{V}\rangle_{a}|\phi_{11}\rangle_{bc\dots}.$
(24)

The success probability and the corresponding fidelity between this output
state and the ideal one $|\Psi_{\mathrm{p}}^{(1)}\rangle_{abc\dots}$ are

$\displaystyle P_{\mathrm{PQS1}}$ $\displaystyle=$
$\displaystyle(1-t)t(|c_{10}|^{2}\\!+\\!|c_{01}|^{2})\\!+\\!(1-t)^{2}|c_{00}|^{2}\\!+\\!t^{2}|c_{11}|^{2},$
$\displaystyle F_{\mathrm{PQS1}}$ $\displaystyle=$
$\displaystyle\frac{(1-t)t(|c_{10}|^{2}+|c_{01}|^{2})}{(1-t)t(|c_{10}|^{2}\\!+\\!|c_{01}|^{2})\\!+\\!(1-t)^{2}|c_{00}|^{2}\\!+\\!t^{2}|c_{11}|^{2}}.$

As will be shown later, for our initial input entanglement of interest, i.e.,
the state $|\Psi_{\mathrm{p}}\rangle_{abc\dots}$ in Eq. 22, $c_{11}=0$, so
that by choosing $t\to 1$ the fidelity $F_{\mathrm{PQS1}}$ is approaching 1
but the success probability $P_{\mathrm{PQS1}}$ is turning out to be very low.

###  III.2 Using type-II two-mode squeezer

Figure 3: Two-mode-squeezer-based implementation of a nonideal PQS with the
black box labeled as $\mathrm{PQS}2(\Gamma)$ using a type-II two-mode squeezer
represented by a SPDC crystal Lamas-Linares  _et al._ (2002). Here $\Gamma$ is
the squeezing parameter characterizing the performance of the scissors. The
input state
$|\psi_{\mathrm{p}}\rangle=\sum_{n,m=0}^{\infty}c_{nm}|n_{H},m_{V}\rangle$ is
injected into the signal ($s$) mode of the squeezer while the idle ($i$) mode
is in the vacuum. Co-detection of single photons at both detectors D1 and D2
heralds the truncated output at the idle mode which is very close to the
desired state
$|\psi_{\mathrm{p}}^{(1)}\rangle=c_{10}|1_{H},0_{V}\rangle+c_{01}|1_{V},0_{H}\rangle$.
A pump, denoted by p, stimulates squeezing in the SPDC crystal.

Here we propose a different PQS implementation, denoted as PQS2, employing a
type-II two-mode squeezer

$\hat{S}_{si}=\exp(\xi\hat{K}_{si}^{\dagger}-\xi^{*}\hat{K}_{si}),$ (27)

where $s$ and $i$ respectively denote the signal and idle modes, $\xi$ is
proportional to the coupling constant $\chi^{(2)}$ and the intensity of the
classical pump Gerry and Knight (2004), and

$\hat{K}_{si}=\hat{a}_{s,H}\hat{a}_{i,V}+\hat{a}_{s,V}\hat{a}_{i,H}.$ (28)

This squeezer is typically realized by a SPDC crystal Kok and Lovett (2010)
and commonly used in laboratories. The setup to realize the PQS2 for
truncating unwanted components in the state $|\psi_{\mathrm{p}}\rangle$ in Eq.
16 is depicted in Fig. 3 comprising two main steps: (step 1) injecting
$|\psi_{\mathrm{p}}\rangle$ to mode $s$ of the squeezer while leaving mode $i$
in the vacuum, which is mathematically equivalent to acting $\hat{S}_{si}$ on
$|\psi_{\mathrm{p}}\rangle_{s}|0\rangle_{i}$, and (step 2) detecting two
photons in mode $s$, one horizontally polarized and the other vertically
polarized, which heralds the truncated output state in mode $i$ that is
expected to be exactly or very close to the desired state
$|\psi_{\mathrm{p}}^{(1)}\rangle$ in Eq. 17.

To get intuition on how this PQS2 implementation works, let us consider a
scenario in which we inject
$|\psi_{\mathrm{p}}^{(1)}\rangle=c_{10}|1_{H}\rangle+c_{01}|1_{V}\rangle$ in
Eq. 17 to mode $s$ of the squeezer $\hat{S}_{si}$ and leave mode $i$ in the
vacuum. We approximate $\hat{S}_{si}$ to the first order of $|\xi|$, which is
typically of order $10^{-2}$ Kok and Lovett (2010), as $\hat{S}_{si}\simeq
1+(\xi\hat{K}^{\dagger}_{si}-\xi^{*}\hat{K}_{si})$. We find that

$\displaystyle\hat{S}_{si}|\psi_{\mathrm{p}}^{(1)}\rangle_{s}|0\rangle_{i}$
$\displaystyle\simeq$
$\displaystyle|\psi_{\mathrm{p}}^{(1)}\rangle_{s}|0\rangle_{i}++\xi|1_{H},1_{V}\rangle_{s}|\psi_{\mathrm{p}}^{(1)}\rangle_{i}$
$\displaystyle+\xi\sqrt{2}(c_{10}|2_{H}\rangle_{s}|1_{V}\rangle_{i}+c_{01}|2_{V}\rangle_{s}|1_{H}\rangle_{i}).$

We then detect in mode $s$ single photons of both horizontal and vertical
polarizations, i.e., we perform measurement with the projector

$\hat{\Pi}_{s}=|1_{H},1_{V}\rangle_{s}\langle 1_{H},1_{V}|.$ (30)

The projected state in mode $i$ will be $|\psi_{\mathrm{p}}^{(1)}\rangle$,
implying that the initial input state of mode $s$ is completely transferred to
mode $i$. We repeat the same procedure for different inputs in mode $s$,
including the vacuum state $|0\rangle$ and $|n_{H},m_{V}\rangle$ with $n,m\geq
1$, and observe that

$\displaystyle\hat{S}_{si}|0\rangle_{s}|0\rangle_{i}$ $\displaystyle\simeq$
$\displaystyle|0\rangle_{s}|0\rangle_{i}+\xi(|1_{H}\rangle_{s}|1_{V}\rangle_{i}\\!+\\!|1_{V}\rangle_{s}|1_{H}\rangle_{i}),$
$\displaystyle\hat{S}_{si}|n_{H},m_{V}\rangle_{s}|0\rangle_{i}$
$\displaystyle\simeq$ $\displaystyle|n_{H},m_{V}\rangle_{s}|0\rangle_{i}$ (32)
$\displaystyle+\xi\sqrt{n\\!+\\!1}|(n\\!+\\!1)_{H},m_{V}\rangle_{s}|1_{V}\rangle_{i}$
$\displaystyle+\xi\sqrt{m\\!+\\!1}|n_{H},(m\\!+\\!1)_{V}\rangle_{s}|1_{H}\rangle_{i},\qquad$

of which the probability of finding $|1_{H},1_{V}\rangle_{s}$ in mode $s$ is
non-zero only when $n=m=1$. Noticing that the state
$|\psi_{\mathrm{p}}\rangle$ in Eq. 16 is a superposition of
$|\psi_{\mathrm{p}}^{(1)}\rangle$, $|0\rangle$, and $|n_{H},m_{V}\rangle$ with
$n,m\geq 1$, the results in Eqs. (LABEL:eq:SqueezerActingOnPsi1),
(LABEL:eq:SqueezerActingOnVacuum), and (32) thus suggest that if the input
state in mode $s$ is $|\psi_{\mathrm{p}}\rangle$, application of
$\hat{S}_{si}$ on the input $|\psi_{\mathrm{p}}\rangle_{s}|0\rangle_{i}$
combining with the subsequent measurement $\hat{\Pi}_{s}$ on mode $s$ will
yield in mode $i$ an output state consisting of
$\big{|}\psi_{\mathrm{p}}^{(1)}\big{\rangle}$ and $|0\rangle$. When the
contribution from the vacuum is negligible, this can realize the map
$|\psi_{\mathrm{p}}\rangle\to|\psi_{\mathrm{p}}^{(1)}\rangle$, i.e., the PQS2,
with a high fidelity. We also note that an unwanted term from second-order
squeezing can appear in the output, as will be shown below by exact
calculations.

We facilitate exact calculations related to the squeezer by using the
following formula Simon  _et al._ (2000)

$\displaystyle\hat{S}_{si}|n_{H},m_{V}\rangle_{s}|0\rangle_{i}$
$\displaystyle=$ $\displaystyle
K_{n+m}\sum_{k,l=0}^{\infty}(-i\Gamma)^{k+l}(C_{n+k}^{n}C_{m+l}^{m})^{\tfrac{1}{2}}\quad$
(33)
$\displaystyle\times|(n\\!+\\!k)_{H},(m\\!+\\!l)_{V}\rangle_{s}|l_{H},k_{V}\rangle_{i},$

where $C^{k}_{n}=n!/[(n-k)!k!]$, $\Gamma=\tanh(i\xi)$ is the characteristic
squeezing parameter, and $K_{n}=(1-|\Gamma|^{2})^{(n+2)/2}$. Acting
$\hat{S}_{si}$ on $|\psi_{\mathrm{p}}\rangle_{s}|0\rangle_{i}$ (step 1) then
produces

$\displaystyle\hat{S}_{si}|\psi_{\mathrm{p}}\rangle_{s}|0\rangle_{i}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{n,m=0}^{\infty}\\!\\!c_{nm}K_{n+m}\sum_{k,l=0}^{\infty}\\!(-i\Gamma)^{k+l}(C_{n+k}^{n}C_{m+l}^{m})^{\tfrac{1}{2}}$
(34)
$\displaystyle\\!\times|(n+k)_{H},(m+l)_{V}\rangle_{s}|l_{H},k_{V}\rangle_{i}.$

Performance of the measurement $\hat{\Pi}_{s}$ in Eq. 30 on mode $s$ of this
state (step 2) projects mode $i$ onto

$K_{1}(-i\Gamma)|\psi_{\mathrm{p}}^{(1)}\rangle+c_{11}K_{2}|0\rangle+c_{00}K_{0}(-i\Gamma)^{2}|1_{H},1_{V}\rangle,$
(35)

where the last term, which is an unwanted two-photon state, results from
second-order squeezing.

We replace the single-mode input $|\psi_{p}\rangle$ by the state
$|\Psi_{\mathrm{p}}\rangle_{abc\dots}$ in Eq. 22 and apply the PQS2 above to
only mode $a$. Analogous to the result in Eq. (35) we obtain the following
output

$\displaystyle
K_{1}(-i\Gamma)|\Psi_{\mathrm{p}}^{(1)}\rangle_{abc\dots}+c_{11}K_{2}|0\rangle_{a}|\phi_{11}\rangle_{bc\dots}$
$\displaystyle+c_{00}K_{0}(-i\Gamma)^{2}|1_{H},1_{V}\rangle_{a}|\phi_{00}\rangle_{bc\dots},$
(36)

where $|\Psi_{\mathrm{p}}^{(1)}\rangle_{abc\dots}$, given in Eq. (23), is the
ideal truncated state. The success probability and the fidelity of the state
preparation for this case are

$\displaystyle P_{\mathrm{PQS2}}$ $\displaystyle=$
$\displaystyle(|c_{10}|^{2}\\!+\\!|c_{01}|^{2})K_{1}^{2}|\Gamma|^{2}\\!+\\!|c_{11}|K^{2}_{2}\\!+\\!|c_{00}|^{2}K_{0}^{2}|\Gamma|^{4},$
$\displaystyle F_{\mathrm{PQS2}}$ $\displaystyle=$
$\displaystyle\frac{(|c_{10}|^{2}\\!+\\!|c_{01}|^{2})K_{1}^{2}|\Gamma|^{2}}{(|c_{10}|^{2}\\!+\\!|c_{01}|^{2})K_{1}^{2}|\Gamma|^{2}\\!+\\!|c_{11}|K^{2}_{2}\\!+\\!|c_{00}|^{2}K_{0}^{2}|\Gamma|^{4}}.$

As typically $|\xi|\sim 10^{-2}\ll 1$ Kok and Lovett (2010), one finds
$|\Gamma|\sim 10^{-2}$ and $K_{0}\approx K_{1}\approx K_{2}\approx 1$. This
combining with the fact that $c_{11}=0$ for our particular input entanglement,
i.e., the state $|\Psi_{\mathrm{p}}\rangle_{abc\dots}$ in Eq. 22, hints that
the fidelity $F_{\mathrm{PQS2}}$ should be very high.

##  IV Performance analysis of the entanglement preparation

Figure 4: (a) Applying the PQS on $j$ modes of the CV entanglement
$|\Lambda\rangle_{12\dots n}$ in Eq. (7) produces the entanglement
$|\Omega\rangle_{12\dots n}$ in Eq. (LABEL:eq:NPartiteHy). For $n=2$ and
$j=1$, panel (a) reduces to panel (b) which shows the preparation of the
hybrid entanglement $|\Phi^{\mathrm{DV\text{-}CV}}\rangle$ in Eq. 39 from the
CV entanglement $|\Xi\rangle_{12}$ in Eq. 1. For $n=2$ and $j=2$, panel (a)
reduces to panel (c) which shows the preparation of the DV PBP
$|\Phi^{\mathrm{DV\text{-}DV}}\rangle$ in Eq. 40 from the CV entanglement
$|\Xi\rangle_{12}$. Here the PQS can be realized by either the PQS1 or the
PQS2.

At this point, we are ready to exploit the two PQS implementations, the PQS1
and the PQS2, developed in the previous section to prepare the target
entangled states. In Fig. 4a, we show the generation of the $n$-partite
polarization entangled state $|\Omega\rangle_{12\dots n}$ in Eq.
(LABEL:eq:NPartiteHy) using PQSs to truncate $j$ modes of the input CV
entanglement $|\Lambda\rangle_{12\dots n}$ in Eq. (7). In what follows, we
consider two specific cases with $(n,j)=(2,1)$ and $(n,j)=(2,2)$, which
correspond respectively to the preparations of the hybrid DV-CV entangled
state

$|\Phi^{\mathrm{DV\text{-}CV}}\rangle=\frac{1}{\sqrt{2}}(|1_{H}\rangle|\alpha_{H}\rangle+e^{i\varphi}|1_{V}\rangle|\\!-\alpha_{V}\rangle),$
(39)

and the DV-DV PBP

$|\Phi^{\mathrm{DV\text{-}DV}}\rangle=\frac{1}{\sqrt{2}}(|1_{H}\rangle|1_{H}\rangle+e^{i\varphi}|1_{V}\rangle|1_{V}\rangle),$
(40)

via truncating the input entanglement $|\Xi\rangle_{12}$ in Eq. 1. The
schematic diagrams for the entanglement preparations of interest are shown in
Figs. 4b and 4c. We compute the success probability and the fidelity when
preparing these entangled states by means of both the PQS1 and the PQS2.

###  IV.1 Hybrid DV-CV entangled state

We reexpress the state $|\Xi\rangle_{12}$ in Eq. 1 in the form of the state
$|\Psi_{\mathrm{p}}\rangle_{abc\dots}$ in Eq. 22

$|\Xi\rangle_{12}=\sum_{n,m=0}^{\infty}c_{nm}|n_{H},m_{V}\rangle_{2}|\phi_{nm}\rangle_{1},$
(41)

where mode $2$ is to be truncated and

$\left\\{\begin{array}[]{rlrl}c_{10}=&N_{0}f_{1}(\beta),&|\phi_{10}\rangle=&|\alpha_{H}\rangle,\\\
c_{01}=&N_{0}e^{i\varphi}f_{1}(-\beta),&|\phi_{01}\rangle=&|\\!-\alpha_{V}\rangle,\\\
c_{00}=&N_{0}L_{\alpha}^{-1}f_{0}(\beta),&|\phi_{00}\rangle=&L_{\alpha}(|\alpha_{H}\rangle\\!+\\!e^{i\varphi}|\\!-\alpha_{V}\rangle),\\\
c_{11}=&0,&|\phi_{11}\rangle=&\varnothing,\\\
\vdots&&\vdots&\end{array}\right.$ (42)

with $L_{\alpha}=[2(1+\cos(\varphi)e^{-\alpha^{2}})]^{-1/2}$. Using the
results in Subsec. III.1 [i.e., Eqs. (24) to (LABEL:eq:FidePQS1PsiP)]
application of the PQS1 on mode 2 of the state $|\Xi\rangle_{12}$ written in
the form of Eq. 41 gives the (unnormalized) output

$\displaystyle\sqrt{(1-t)t}N_{0}f_{1}(\beta)(|1_{H}\rangle_{2}|\alpha_{H}\rangle_{1}-e^{i\varphi}|1_{V}\rangle_{2}|\\!-\alpha_{V}\rangle_{1})$
$\displaystyle+(1-t)N_{0}L_{\alpha}^{-1}f_{0}(\beta)|0\rangle_{2}L_{\alpha}(|\alpha_{H}\rangle+e^{i\varphi}|\\!-\alpha_{V}\rangle)_{1}.\qquad$
(43)

The heralding probability and the fidelity of the prepared state to the hybrid
DV-CV entanglement
$(|1_{H}\rangle_{2}|\alpha_{H}\rangle_{1}-e^{i\varphi}|1_{V}\rangle_{2}|\\!-\alpha_{V}\rangle_{1})/\sqrt{2}$,
which is up to a local unitary transformation equivalent to the state
$|\Phi^{\mathrm{DV\text{-}CV}}\rangle$ in Eq. 39, are

$\displaystyle P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ $\displaystyle=$
$\displaystyle
2(1-t)tN_{0}^{2}f_{1}^{2}(\beta)+(1-t)^{2}N_{0}^{2}L_{\alpha}^{-2}f_{0}^{2}(\beta),$
$\displaystyle F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ $\displaystyle=$
$\displaystyle\frac{2tf_{1}^{2}(\beta)}{2tf_{1}^{2}(\beta)+(1-t)L_{\alpha}^{-2}f_{0}^{2}(\beta)}.$
(45)

The unwanted term containing the vacuum in mode 2 of the output state in Eq.
(43) results in an additional contribution to the heralding probability
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ [i.e., the second term in the RHS
of Eq. (LABEL:eq:ProbHy1)] but induces a less-than-one fidelity
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ [i.e., the presence of a second
term in the denominator of the RHS of Eq. (45)]. Notably, as $\alpha$ and
$\beta$ are defined via $\delta$ and $t_{0}$ as in Eq. 3 and $N_{0}$ is
defined via $\delta$ and $\varphi$,
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ and
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ in effect are functions of
$\delta$, $t_{0}$, $\varphi$, and $t$.

As for the PQS2 proposed in Subsec. III.2, applying it on mode 2 of the state
$|\Xi\rangle_{12}$ yields the (unnormalized) output (see Eqs. (36) to
(LABEL:eq:FidePQS2PsiP))

$\displaystyle
K_{1}(-i\Gamma)N_{0}f_{1}(\beta)(|1_{H}\rangle_{2}|\alpha_{H}\rangle_{1}\\!-\\!e^{i\varphi}|1_{V}\rangle_{2}|\\!-\alpha_{V}\rangle_{1})$
$\displaystyle+\\!K_{0}(-i\Gamma)^{2}N_{0}L_{\alpha}^{-1}f_{0}(\beta)|1_{H},\\!1_{V}\rangle_{2}L_{\alpha}(|\alpha_{H}\rangle\\!+\\!e^{i\varphi}|\\!-\alpha_{V}\rangle)_{1}.$
(46)

The success probability and the fidelity of the entanglement preparation for
this case are

$\displaystyle P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ $\displaystyle=$
$\displaystyle
2K_{1}^{2}|\Gamma|^{2}N_{0}^{2}f_{1}(\beta)^{2}+K_{0}^{2}|\Gamma|^{4}N_{0}^{2}L_{\alpha}^{-2}f_{0}^{2}(\beta),$
$\displaystyle F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ $\displaystyle=$
$\displaystyle\frac{2K_{1}^{2}f_{1}(\beta)^{2}}{2K_{1}^{2}f_{1}(\beta)^{2}+K_{0}^{2}|\Gamma|^{2}L_{\alpha}^{-2}f_{0}^{2}(\beta)},$
(48)

which are actually dependent on $\delta$, $t_{0}$, $\varphi$, and $\Gamma$.
Similar to the case of the PQS1, the PQS2 also produces an undesired term
(i.e., the term on the second line of Eq. (46)) of which mode 2 is a two-
photon state. This term gives rise to an increase in the heralding probability
[due to the second term in the RHS of Eq. (LABEL:eq:ProbHy2)] at the cost of
decreasing the corresponding fidelity [due to the second term in the
denominator of the RHS of Eq. (48))].

Figure 5: (a), (b) Success probability
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ and fidelity
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ given in Eqs. (LABEL:eq:ProbHy1)
and (45), respectively, as functions of the input amplitude $\delta$ and the
transmissivity $t$ for the preparation of the hybrid entanglement
$|\Phi^{\mathrm{DV\text{-}CV}}\rangle$ in Eq. 39 using the PQS1 developed in
Subsec. III.1. (c), (d) Success probability
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ and fidelity
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ given in Eqs. (LABEL:eq:ProbHy2)
and (48), respectively, as functions of the input amplitude $\delta$ and the
squeezing parameter $|\Gamma|$ for the preparation of the same entanglement
but using the PQS2 developed in Subsec. III.2. In plotting these figures
$\phi=0$ and $t_{0}=0.5$ were chosen.

To display our results graphically we choose $\varphi=0$ and $t_{0}=0.5$,
which give $\alpha=\beta=\delta$ and
$|\Xi\rangle_{12}=N_{0}(|\delta_{H}\rangle_{1}|\delta_{H}\rangle_{2}+|\\!-\delta_{V}\rangle_{1}|-\delta_{V}\rangle_{2})$).
With such parameters we plot $P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ and
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ as functions of the input
amplitude $\delta$ and the transmissitivity $t$ in Figs. 5a and 5b and
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ and
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ as functions of the input
amplitude $\delta$ and the squeezing parameter $|\Gamma|$ in Figs. 5c and 5d.
For the parameter ranges in Fig. 5, we observe that the PQS1 is superior to
the PQS2 in terms of the success probability but is inferior in terms of the
fidelity. Discretely, $P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ is of order
$10^{-2}-10^{-1}$, much larger than
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$, which is in the range
$10^{-4}-10^{-3}$. $F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ varies quite
largely from about 0.25 to above 0.97, whereas
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ remains very close to 1 for almost
the whole domain of the used parameters. These manifest the pros and cons of
the two PQS implementations.

Closer looking at $P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ in Fig. 5a and
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ in Fig. 5b reveals two apparently
contrast patterns in their variations with respect to $\delta$ and $t$:
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ decreases when increasing both
$\delta$ and $t$ while $F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ improves.
This is understandable, since from Eqs. (LABEL:eq:ProbHy1) and (45) we find
that $\lim_{t\to 1}P^{\mathrm{DV\text{-}CV}}_{PQS1}=0$ and $\lim_{t\to
1}F^{\mathrm{DV\text{-}CV}}_{PQS1}=1$, indicating that the state-preparation
fidelity can be made arbitrarily high by adjusting the transmissitivity $t$ to
close to 1 but with a price of an unrealistically low heralding probability.
We also note that increasing $\delta$, roughly speaking, leads to a decrease
in $f_{0}^{2}(\beta)\equiv f_{0}^{2}(\delta)$ and $f_{1}^{2}(\beta)\equiv
f_{1}^{2}(\delta)$ and an increase in
$f_{1}^{2}(\beta)/f_{0}^{2}(\beta)\equiv\delta^{2}$. The former directly links
to a reduction of $P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$ (see Eq.
(LABEL:eq:ProbHy1)); the latter is equivalent to lessening the contribution of
the unwanted term in the output state in Eq. (43) and thus enhancing
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS1}}$. As for
$P^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ and
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$, the first quantity increases with
$|\Gamma|$ and for a given $|\Gamma|$ is maximized by a particular value of
$\delta$ which is numerically found to be near 1. The second quantity changes
very slowly with $\delta$ and $|\Gamma|$ and stays close to 1, which is not
surprising if we look at its expression in Eq. (48). For
$|\Gamma|=10^{-2}-10^{-1}\ll 1$ the second term in the denominator of
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}$ is negligible, making
$F^{\mathrm{DV\text{-}CV}}_{\mathrm{PQS2}}\simeq 1$.

###  IV.2 DV-DV polarization Bell pair

Figure 6: (a), (b) Success probability
$P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}$ and fidelity
$F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}$ given in Appendix A as functions
of the input amplitude $\delta$ and the transmissivity $t$ for the preparation
of the PBP $|\Phi^{\mathrm{DV\text{-}DV}}\rangle$ in Eq. 40 using the PQS1
developed in Subsec. III.1. (c), (d) Success probability
$P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}$ and fidelity
$F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}$ given in Appendix A as functions
of the input amplitude $\delta$ and the squeezing parameter $|\Gamma|$ for the
preparation of the same entanglement but using the PQS2 developed in Subsec.
III.2. In plotting these figures $\phi=0$ and $t_{0}=0.5$ were chosen.

Given the output state in Eq. (43) or (46), that is close to the hybrid
entangled state
$(|1_{H}\rangle_{2}|\alpha_{H}\rangle_{1}-e^{i\varphi}|1_{V}\rangle_{2}|\\!-\alpha_{V}\rangle_{1})/\sqrt{2}$,
we continue truncating irrelevant components in mode 1 to get the PBP
$|\Phi^{\mathrm{DV\text{-}DV}}\rangle$ in Eq. 40 via the PQS1 as well as the
PQS2. Detailed calculations for such processes are provided in Appendix A and
here we highlight the main results only. Figure 6 shows the performance for
the preparation of the PBP using the two PQSs. The patterns for the success
probabilities and the fidelities in Fig. 6 are substantially similar to those
in Fig. 5, except the success probabilities are now lower by several orders
due to double truncation. We again see an evident trade-off in the
performances of the two QPS implementations. The QPS1 yields a good heralding
probability but a modest fidelity, while the QPS2 features at a very low
success probability and a close-to-unit fidelity.

##  V Discussion

Figure 7: (a) Reduction of the number of photodetectors in the PQS1
implementation. In the left, two spatially separated photons in modes $1$ and
$2$ of the same horizontal $(H)$ polarization are detected by two detectors D1
and D2. In the right, employing a HWP, a delay circle, a mirror, and a PBS
makes the two photons detected by only one detector D but at adjustably
separated arrival times. (b) Similar to (a) but for the PQS2 implementation
with two detected photons initially having orthogonal polarizations.

The results in the Sec. IV clearly show the advantages and disadvantages of
each of the two PQS implementations in the entanglement production. The PQS1
is beneficial when it comes to the heralding probability but possesses a
moderate fidelity. The PQS2, in contrast, performs with a low success
probability but yields a very high fidelity. Concerning the consumed
resources, the PQS1 requires two ancilla photons of orthogonal polarizations
and four photon-number-resolving (PNR) detectors. The PQS2 differently
involves one SPDC crystal accompanied by optical pumping to stimulate
squeezing and two PNR detectors. Notably, owing to the interplay between
polarization and spatial DoFs Kok and Lovett (2010) we can reduce the number
of PNR detectors in both the PQS implementations. Concretely, instead of
detecting two spatially separated photons by two detectors as in Figs. 2 and 3
we exploit delay circles, mirrors, HWPs, and PBSs, as illustrated in Fig. 7,
to detect the photons by only one detector at the same spatial mode but at
different arrival times. By this the number of needed PNR detectors in the
PQS1 implementation is decreased from 4 to 2, whereas that in the PQS2
implementation becomes 1.

The heralded entanglement preparation proposed in this paper crucially depends
on the availability of the input CV entanglements in Eqs. (1) and (7). These
entangled states as demonstrated in Fig. 1 are produced from a coherent state
and a cat state. The latter is of great importance in foundations of the
quantum theory Schrödinger (1935) and quantum applications Ralph  _et al._
(2003); Lund  _et al._ (2008) but in general is troublesome to prepare Mikheev
_et al._ (2019). Thankfully, in practice there are states that can be used as
cat states with high fidelities. Namely, kitten states, i.e., cat states with
small amplitudes (up to 1), can be very well approximated by a deterministic
squeezed vacuum Israel  _et al._ (2019). Large-amplitude cat states (larger
than $2$) can be prepared probabilistically via breeding of kitten states
Sychev  _et al._ (2017) or generalized photon subtraction Takase  _et al._
(2021). Moreover, single-photon inputs required in the PQS1 can be supplied
deterministically with high quality by semiconductor quantum-dot emitters
Senellart  _et al._ (2017); Tomm  _et al._ (2021); Li  _et al._ (2021).
Another salient ingredient in our entanglement generation scheme as well as in
many other quantum optical protocols is PNR detectors. Superconducting
transition edge sensors PNR detectors Lita  _et al._ (2008) have recently been
shown to operate with an efficiency exceeding $95\%$ Thekkadath  _et al._
(2020); Arrazola  _et al._ (2021). Another possibilities to discriminate
photon numbers include multiplexing of single-photon detectors Israel  _et
al._ (2019); Jönsson and Björk (2019); Provazník  _et al._ (2020); Nehra  _et
al._ (2020) and fine analyses of output signal waveforms Endo  _et al._
(2021). We note that for a small-amplitude input state (i.e., the state
$|\Xi\rangle_{12}$ in Eq. 1 with small $\alpha$ and $\beta$) of which more-
than-one-photon contributions are small compared to those of the vacuum and
the single-photon state, non-PNR detectors such as single-photon counting
modules (SPCMs) Wagenknecht  _et al._ (2010) or single-photon avalanche
photodiodes (SPADs) Nehra  _et al._ (2020) might suffice for our scheme.

We compare the entanglement preparation scheme proposed here with the existing
ones in the literature Kwon and Jeong (2015); Li  _et al._ (2018); Gouzien
_et al._ (2020); Le  _et al._ (2021); Barz  _et al._ (2010); Wagenknecht  _et
al._ (2010); Śliwa and Banaszek (2003); Li  _et al._ (2021). Our scheme to
generate the DV-CV hybrid entangled state in Eq. (39), in comparisons with
those in Refs. Kwon and Jeong (2015); Li  _et al._ (2018); Gouzien  _et al._
(2020); Le  _et al._ (2021), does not require a PBP input and uses a lower
number of PNR detectors, thus relaxing initial overheads. Also, the success
probability and the prepared-state fidelity of our scheme at a proper choice
of relevant parameters can be made considerably higher than those in Refs.
Kwon and Jeong (2015); Li  _et al._ (2018); Gouzien  _et al._ (2020); Le  _et
al._ (2021). As for the DV-DV PBP preparation, we estimate the count rate in
our scheme and compare it to those in Refs. Barz  _et al._ (2010); Wagenknecht
_et al._ (2010); Li  _et al._ (2021). In particular, we assume that the input
entangled state $|\Xi\rangle_{12}$ prepared deterministically from a squeezed
vacuum and the input single photons emitted from a quantum-dot emitter are
supplied on-demand with repetition rate $6.4\,\mathrm{MHz}$ Li  _et al._
(2021). We then employ the PQS1, choose $\delta=0.8$ and $t=0.98$ at which
$P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}\sim 3.6\times 10^{-5}$ and
$F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}>0.9$ (see Figs. 6a and 6b), and
adapt multiplexing PNR detectors in Refs. Nehra  _et al._ (2020); Israel  _et
al._ (2019) that are compatible with MHz repetition rates to generate the PBP
at a count rate approximately given by $6.4\,\mathrm{MHz}\times 3.6\times
10^{-5}\approx 230\,\mathrm{Hz}$. Using the PQS2, we operate it at
$\delta=0.8$ and $|\Gamma|=0.07$ with
$P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}\sim 2\times 10^{-6}$ and
$F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}>0.98$ (see Figs. 6c and 6d) and
again assume that the entangled input $|\Xi\rangle_{12}$ is supplied from a
squeezed vacuum but with a higher repetition rate of $80\,\mathrm{MHz}$ Israel
_et al._ (2019), so that the PBP preparation will have a count rate roughly at
$80\,\mathrm{MHz}\times 2\times 10^{-6}=160\,\mathrm{Hz}$. The count rates
estimated above for preparing the PBP with fidelities $>0.9$ are several
orders higher than those reported in Refs. Barz  _et al._ (2010); Wagenknecht
_et al._ (2010); Li  _et al._ (2021). Additionally, the use of a relatively
small squeezing parameter $|\Gamma|$ in the PQS2 here eases up the requirement
for very strong pumping and reduces the effect of higher squeezing orders
which can lead to false detections as in Refs. Barz  _et al._ (2010);
Wagenknecht  _et al._ (2010).

##  VI Conclusion

We presented a general scheme to prepare $n$-partite polarization entangled
states for an arbitrary $n\geq 2$ and considered two specific examples,
namely, the hybrid DV-CV entangled state between polarized single photons and
polarized coherent states in Eq. 39 and the DV PBP between polarized single
photons in Eq. 40. Our scheme involves neither postselection nor destruction
of photonsKwiat  _et al._ (1995); Kwiat and Weinfurter (1998). Different from
the existing heralded schemes Zou  _et al._ (2002a, b); Pittman  _et al._
(2003); Kwon and Jeong (2015); Li  _et al._ (2018); Gouzien  _et al._ (2020);
Le  _et al._ (2021); Barz  _et al._ (2010); Wagenknecht  _et al._ (2010);
Śliwa and Banaszek (2003); Li  _et al._ (2021), we harness the connections
among CV, hybrid DV-CV, and DV entanglements and propose a truncation
technique, i.e., the PQS, to map a given CV entanglement into the desired
entangled states in a heralded fashion. The needed input state is an entangled
coherent state in Eq. 1 or Eq. (7) which can be supplied by modern quantum
technologies following the schematic setup in Fig. 1. We designed two
different PQS implementations, the PQS1 in Fig. 2 and the PQS2 in Fig. 3. The
PQS1 employs a single-photon source and linear-optics devices while the PQS2
uses a two-mode squeezer which is a commonplace optical nonlinear element. The
former is advantageous in success probability and the later excels in
fidelity, as shown by detailed performance analysis of the entanglement
preparation in Sec. IV. Both the PQS1 and the PQS2 could be implemented within
the current optical toolbox. Effects of imperfect input states, inefficient
PNR detectors, nonideal device operations, and decoherence from surrounding
environments are beyond the scope of the present paper but will be the subject
of a subsequent work. Also, applications of the PQSs presented here to other
quantum information tasks such as noiseless linear amplification Winnel  _et
al._ (2020); He  _et al._ (2021) and studies of the transition from CV
entanglement to DV entanglement in the context of the nonclassicality and
classical simultability Andersen  _et al._ (2015); Rundle  _et al._ (2020);
García-Álvarez  _et al._ (2020) are worth pursuing.

###### Acknowledgements.

In this work D.T.L. is supported by the Australian Research Council Centre of
Excellence for Engineered Quantum Systems (EQUS, CE170100009), and N.B.A. is
supported by the National Foundation for Science and Technology Development
(NAFOSTED) under Project No. 103.01-2019.313.

##  Appendix A  Success probability and fidelity for the preparation of the
polarization Bell pair

###  A.1  Using the PQS1

We rewrite the output in Eq. (43) as

$|\alpha_{H}\rangle_{1}(g_{1}|1_{H}\rangle+g_{0}|0\rangle)_{2}+|\\!-\alpha_{V}\rangle_{1}(-g_{1}e^{i\varphi}|1_{V}\rangle+g_{0}e^{i\varphi}|0\rangle)_{2},$
(49)

where

$g_{1}=\sqrt{(1-t)t}N_{0}f_{1}(\beta),\hskip
14.22636ptg_{0}=(1-t)N_{0}f_{0}(\beta).$ (50)

We use the PQS1 in Subsec. III.1 to truncate non-single-photon components in
mode $1$ of this state and obtain the output

$\displaystyle\sqrt{(1\\!-\\!t)t}f_{1}(\alpha)|1_{H}\rangle_{1}(g_{1}|1_{H}\rangle+g_{0}|0\rangle)_{2}$
$\displaystyle+$
$\displaystyle\sqrt{(1\\!-\\!t)t}f_{1}(-\alpha)|1_{V}\rangle_{1}(-g_{1}e^{i\varphi}|1_{V}\rangle+g_{0}e^{i\varphi}|0\rangle)_{2}$
$\displaystyle+$
$\displaystyle(1\\!-\\!t)f_{0}(\alpha)|0\rangle_{1}\big{[}g_{1}(|1_{H}\rangle-e^{i\varphi}|1_{V}\rangle)+g_{0}(1+e^{i\varphi})|0\rangle\big{]}_{2}.$

The success probability and the fidelity of such output compared to the
desired PBP $|\Phi^{\mathrm{DV\text{-}DV}}\rangle$ in Eq. 40 are

$\displaystyle P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}\\!$ $\displaystyle=$
$\displaystyle\\!\big{[}2(1-t)tf_{1}^{2}(\alpha)+2(1-t)^{2}f_{0}^{2}(\alpha)\big{]}(g_{1}^{2}+g_{0}^{2}),$
$\displaystyle F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS1}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{tf_{1}^{2}(\alpha)g_{1}^{2}}{\big{[}tf_{1}^{2}(\alpha)+(1-t)f_{0}^{2}(\alpha)\big{]}(g_{1}^{2}+g_{0}^{2})}.$
(53)

###  A.2  Using the PQS2

We rewrite the output in Eq. (46) as

$|\alpha_{H}\rangle_{1}(h_{1}|1_{H}\rangle+h_{0}|0\rangle)_{2}+|\\!-\alpha_{V}\rangle_{1}(-h_{1}e^{i\varphi}|1_{V}\rangle+h_{0}e^{i\varphi}|0\rangle)_{2},$
(54)

where

$h_{1}=K_{1}(-i\Gamma)N_{0}f_{1}(\beta),\hskip
5.69046pth_{0}=K_{0}(-i\Gamma)^{2}N_{0}f_{0}(\beta).$ (55)

We use the PQS2 in Subsec. III.2 to truncate non-single-photon components in
mode $1$ of this state and obtain the output

$\displaystyle
K_{1}(-i\Gamma)f_{1}(\alpha)|1_{H}\rangle_{1}(h_{1}|1_{H}\rangle+h_{0}|0\rangle)_{2}$
(56) $\displaystyle+$ $\displaystyle
K_{1}(-i\Gamma)f_{1}(-\alpha)|1_{V}\rangle_{1}(-h_{1}e^{i\varphi}|1_{V}\rangle+h_{0}e^{i\varphi}|0\rangle)_{2}$
$\displaystyle+$ $\displaystyle
K_{0}(-i\Gamma)^{2}f_{0}(\alpha)|1_{H},1_{V}\rangle_{1}$ $\displaystyle\times$
$\displaystyle\big{[}h_{1}(|1_{H}\rangle-e^{i\varphi}|1_{V}\rangle)+h_{0}(1+e^{i\varphi})|0\rangle\big{]}_{2}.$

The success probability and the fidelity of such output compared to the
desired PBP $|\Phi^{\mathrm{DV\text{-}DV}}\rangle$ in Eq. 40 are

$\displaystyle P^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}\\!$ $\displaystyle=$
$\displaystyle\\!\big{[}2K_{1}^{2}|\Gamma|^{2}f_{1}^{2}(\alpha)\\!+\\!2K_{0}^{2}|\Gamma|^{4}f_{0}^{2}(\alpha)\big{]}(|h_{1}|^{2}\\!+\\!|h_{0}|^{2}),$
$\displaystyle F^{\mathrm{DV\text{-}DV}}_{\mathrm{PQS2}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{K_{1}^{2}f_{1}^{2}(\alpha)|h_{1}|^{2}}{\big{[}K_{1}^{2}f_{1}^{2}(\alpha)+K_{0}^{2}|\Gamma|^{2}f_{0}^{2}(\alpha)\big{]}(|h_{1}|^{2}+|h_{0}|^{2})}.$

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