# Deterministic Distribution of Orbital Angular Momentum Multiplexed
Continuous-variable Entanglement and Quantum Steering

Li Zeng  The State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan 030006, China  Rong Ma  The State Key Laboratory of Quantum Optics and
Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University,
Taiyuan 030006, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan 030006, China  College of physics and electronic
engineering, Shanxi University, Taiyuan 030006, China  Hong Wen  The State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-
Electronics, Shanxi University, Taiyuan 030006, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Meihong Wang  The State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan 030006, China  Jun Liu  The State Key Laboratory of Quantum Optics and
Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University,
Taiyuan 030006, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan 030006, China  Zhongzhong Qin  The State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-
Electronics, Shanxi University, Taiyuan 030006, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
e-mail<EMAIL_ADDRESS>Xiaolong Su  The State Key Laboratory of Quantum
Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi
University, Taiyuan 030006, China  Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan 030006, China

###### Abstract

Orbital angular momentum (OAM) multiplexing provides an efficient method to
improve data-carrying capacity in various quantum communication protocols. It
is a precondition to distribute OAM multiplexed quantum resources in quantum
channels for implementing quantum communication. However, quantum steering of
OAM multiplexed optical fields and the effect of channel noise on OAM
multiplexed quantum resources remain unclear. Here, we generate OAM
multiplexed continuous-variable (CV) entangled states and distribute them in
lossy or noisy channels. We show that the decoherence property of entanglement
and quantum steering of the OAM multiplexed states carrying topological
charges $l=1$ and $l=2$ are the same as that of the Gaussian mode with $l=0$
in lossy and noisy channels. The sudden death of entanglement and quantum
steering of high-order OAM multiplexed states is observed in the presence of
excess noise. Our results demonstrate the feasibility to realize high data-
carrying capacity quantum information processing by utilizing OAM multiplexed
CV entangled states.

††journal: pr

##  1 Introduction

Einstein-Podolsky-Rosen (EPR) entanglement plays a crucial role in quantum
information processing, such as quantum communication, quantum computation,
and quantum precision measurement [1, 2, 3, 4, 5]. Besides entanglement,
quantum steering, which stands between entanglement [1] and Bell nonlocality
[6] in the hierarchy of quantum correlations [7], has been identified as a
useful quantum resource. Different from entanglement and Bell nonlocality,
quantum steering shows unique asymmetry or even one-way characteristics [8, 9,
10, 11, 12, 13, 14, 15, 16], and thus allows asymmetric quantum information
processing. For example, quantum steering enables one-side device independent
quantum key distribution [17, 18, 19].

Multiplexing provides an efficient method to enhance the data-carrying
capability in both classical and quantum communication systems by combining
multiple channels into a single channel. By utilizing different degrees of
freedom (DOFs) of light, such as wavelength [20, 21], polarization [22],
temporal [23, 24, 25] or spatial [26, 27] modes, different types of
multiplexing can be realized. Orbital angular momentum (OAM) of light [28] has
also been found to be an attractive DOF to realize multiplexing due to its
infinite range of possibly achievable topological charges [29, 30]. OAM has
found applications in discrete-variable quantum information processing, such
as high dimensional OAM entanglement generation [31], and 18-qubit
entanglement with six photons’ three DOFs including OAM [32].

Figure 1: (a) Experimental setup for the generation and distribution of OAM
multiplexed CV quantum entanglement and steering in a lossy or noisy channel.
Pr: probe beam; Conj: conjugate beam; LO-l,P and LOl,C: local oscillators of
Pr and Conj fields; AM: amplitude modulator; PM: phase modulator; GL: Glan-
laser polarizer; GT: Glan-Thompson polarizer; PBS: polarization beam splitter;
HWP: half-wave plate; VPP: vortex phase plate; M: mirror; BS: 50:50 beam
splitter; BHD1, BHD2: balanced homodyne detectors; SA: spectrum analyzer. (b)
Double-$\Lambda$ energy level structure for the FWM process in cesium vapor
cell. $\Delta$: one-photon detuning.

Four-wave mixing (FWM) process in warm alkali vapor cell has found a wide
range of applications [33, 34, 35, 36, 37, 38, 39]. Especially, spatial-multi-
mode advantage of the FWM process, attributed to its cavity-free
configuration, makes it an ideal optical parametric amplifier to generate
entangled images [35] and reconfigurable multipartite entanglement [36].
Quantum correlated twin beams carrying OAM were generated based on the FWM
process in rubidium vapor [37]. OAM multiplexed bipartite and multipartite CV
entangled states have been generated based on the FWM process [40, 41, 42].
Furthermore, OAM multiplexed deterministic all-optical quantum teleportation
has also been demonstrated by utilizing OAM multiplexed bipartite CV entangled
state generated from the FWM process [43]. To enhance the data-carrying
capacity in quantum communication based on OAM multiplexed CV entangled
states, it is essential to distribute them in lossy and noisy quantum channels
towards practical applications. The distribution of weak coherent field and
single photons carrying OAM in fiber, free space, and underwater have been
experimentally investigated [44, 45, 46]. However, it remains unclear whether
the quantum entanglement and steering of OAM multiplexed CV entangled states
are more sensitive to loss and noise than commonly used Gaussian mode with
$l=0$.

Here, we present the deterministic distribution of OAM multiplexed CV quantum
entanglement and steering in lossy and noisy channels. In the experiment, the
OAM multiplexed entangled fields are generated deterministically based on the
FWM process in warm cesium vapor and distributed deterministically in quantum
channels. We show that the CV entangled states carrying topological charges
$l=1$ and $l=2$ are as robust against loss as Gaussian mode with $l=0$. Sudden
death of entanglement and quantum steering of high-order OAM multiplexed CV
entangled state is observed in the presence of noise. Our results pave the way
for applying OAM multiplexed CV entanglement and quantum steering in high
data-carrying capacity quantum communication.

##  2 The principle and experimental setup

Figure 1(a) shows the schematic of experimental setup, and Fig. 1(b) shows the
double-$\Lambda$ energy level structure used for the FWM process, which is
formed from the 133Cs $D_{1}$ line with an excited level
($6P_{1/2},F^{\prime}=4$) and two ground levels ($6S_{1/2},F=3$ and $F=4$).
The pump beam is about 1.6 GHz blue detuned from $6S_{1/2},F=3\rightarrow
6P_{1/2},F^{\prime}=4$ transition, and the probe beam is 9.2 GHz red shifted
relative to the pump beam. The pump and probe beams are combined by a Glan-
laser (GL) polarizer and then cross each other in the center of the cesium
vapor cell at an angle of 6 mrad [47]. The gain of the FWM process is around 3
with a pump power of 240 mW and a probe power of 3 $\mu$W. By injecting the
probe beam carrying topological charge $l$ of OAM mode, conjugate beam
carrying topological charge $-l$ of OAM mode is generated on the other side of
the pump, which satisfies OAM conservation in the FWM process. The topological
charge of OAM mode $l=1$ or $l=2$ is added to the probe beam by passing it
through a vortex phase plate (VPP). The pump beam is filtered out by using a
Glan-Thompson (GT) polarizer with an extinction ratio of $10^{5}$:1 after the
vapor cell.

The Conj field is kept by Alice, while the Pr field is distributed to a remote
quantum node owned by Bob through a lossy or noisy channel. The lossy channel
is simulated by a half-wave plate (HWP) and a polarization beam splitter
(PBS). The noisy channel is modeled by combining the Pr field with an
auxiliary beam at a PBS followed by a HWP and a PBS. The auxiliary beam
carries the same frequency and topological charge with the Pr field, and is
modulated by an amplitude modulator and a phase modulator with white noise
[16]. To characterize the OAM multiplexed CV entangled state, its covariance
matrix (CM) is experimentally measured by utilizing two sets of balanced
homodyne detectors (BHDs). In order to extract the CV quadrature information
carried by the OAM mode with a topological charge $l$, local oscillator (LO)
with opposite topological charge $-l$ is required. In our experiment, the
spatially mode-matched LO beams used in the BHDs are obtained from a second
set of FWM process which is around 5 mm above the first set of FWM process in
the same vapor cell [35]. More details of the experimental parameters can be
found in Appendix B.

Figure 2:  (a), (b) Beam patterns of the OAM multiplexed CV entanglement for
$l=1$ and $l=2$ in a lossy channel and corresponding transmitted patterns
through a tilted lens. (c) Dependence of PPT values of the OAM multiplexed CV
entanglement on transmission efficiency $\eta$ for $l=0$, $l=1$ and $l=2$ in
lossy channels. The PPT value is 0.46$\pm$0.01 at $\eta=1$. Curves and data
points show theoretical predictions and experimental results, respectively.
Error bars of experimental data represent one standard deviation and are
obtained based on the statistics of the measured data.

The Hamiltonian of the OAM multiplexed FWM process can be expressed as [40]:

$\hat{H}=\sum_{l}i\hbar\gamma_{l}\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}+h.c.$
(1)

where $\gamma_{l}$, which is defined as the interaction strength of each OAM
pair, is proportional to the pump power. $\hat{a}^{\dagger}_{l,P}$ and
$\hat{a}^{\dagger}_{-l,C}$ are the creation operators related to OAM modes of
the Pr and Conj fields, respectively. Since the pump beam does not carry OAM
($l=0$), the topological charges of the Pr and Conj fields are opposite due to
OAM conservation in the FWM process. The output state of the OAM multiplexed
FWM process is:

$\ket{\Psi}_{out}=\ket{\Psi}_{-l}\otimes\cdots\otimes\ket{\Psi}_{0}\otimes\cdots\otimes\ket{\Psi}_{l}$
(2)

where $\ket{\Psi}_{l}=\ket{\psi_{l,P},\psi_{-l,C}}$ presents a series of
independent OAM multiplexed CV EPR entangled state generated in the FWM
process. $\ket{\psi_{l,P}}$ and $\ket{\psi_{-l,C}}$ represent Pr field
carrying topological charge $l$ and Conj field carrying topological charge
$-l$ , respectively.

All Gaussian properties of the CV entangled state $\ket{\Psi}_{l}$ can be
determined by the covariance matrix $\sigma_{AB}$ with the matrix element
$\sigma_{ij}=\langle\hat{\xi}_{i}\hat{\xi}_{j}+\hat{\xi}_{j}\hat{\xi}_{i}\rangle/2-\langle\hat{\xi}_{i}\rangle\langle\hat{\xi}_{j}\rangle$,
where
$\hat{\xi}\equiv(\hat{X}_{-l,C},\hat{Y}_{-l,C},\hat{X}_{l,P},\hat{Y}_{l,P})^{T}$,
$\hat{X}=\hat{a}+\hat{a}^{{\dagger}}$ and
$\hat{Y}=(\hat{a}-\hat{a}^{{\dagger}})/i$ represent amplitude and phase
quadratures of the Conj and Pr fields, respectively. The covariance matrix of
the OAM multiplexed entangled state after distribution in a lossy or noisy
channel is as following

$\sigma_{AB,\delta,\eta}=\left(\begin{array}[]{cccc}V_{a}&0&V_{c}&0\\\
0&V_{a}&0&-V_{c}\\\ V_{c}&0&V_{b}&0\\\ 0&-V_{c}&0&V_{b}\end{array}\right)$ (3)

with $V_{a}=\frac{V+V^{\prime}}{2}$,
$V_{b}=\eta\frac{V+V^{\prime}}{2}+(1-\eta)(1+\delta)$ and
$V_{c}=\sqrt{\eta}\frac{V^{\prime}-V}{2}$. $V$ and $V^{\prime}$ represent the
variances of squeezed and anti-squeezed quadratures of the optical mode,
respectively. The derivation of the covariance matrix can be found in Appendix
A. For pure squeezed states, $VV^{\prime}=1$. While $VV^{\prime}>1$ after pure
squeezed states suffer from loss or noise. $\eta$ and $\delta$ represent
transmission efficiency and excess noise of the quantum channel, respectively.
We have $\delta=0$ in a lossy channel, while we have $\delta>0$ in a noisy
channel. The submatrices $\sigma_{A}=V_{a}I$ and $\sigma_{B}=V_{b}I$, with $I$
as the identity matrix, are the covariance matrices corresponding to the
states of Alice’s and Bob’s subsystems, respectively. The CV entangled state
is a symmetric state and an asymmetric state when $\sigma_{A}=\sigma_{B}$ and
$\sigma_{A}\neq\sigma_{B}$, respectively.

Figure 3:  Dependence of PPT values of the OAM multiplexed CV entanglement on
transmission efficiency $\eta$ for $l=0$, $l=1$ and $l=2$ in noisy channels.
Three different amounts of excess noise $\delta=0.15$ (black), $\delta=0.5$
(red), and $\delta=1$ (blue) are compared. The light blue plane shows the
boundary for sudden death of entanglement where PPT value equals to 1. The
three vertical dashed lines indicate corresponding transmission efficiencies
where entanglement starts to disappear. Curves and data points show
theoretical predictions and experimental results, respectively. Error bars of
experimental data represent one standard deviation and are obtained based on
the statistics of the measured data. Figure 4:  Quantum steerabilities of OAM
multiplexed CV entangled state distributed in a lossy (a) or noisy (b)
channel. The excess noise shown in (b) is $\delta=0.15$. Solid and dashed
curves show theoretical predictions of $\mathcal{G}^{A\rightarrow B}$ and
$\mathcal{G}^{B\rightarrow A}$, respectively. Data points show experimental
results. Error bars of experimental data represent one standard deviation and
are obtained based on the statistics of the measured data.

The Peres-Horodecki criterion of positivity under partial transpose (PPT)
criterion is a sufficient and necessary criterion to characterize the
entanglement of CV bipartite entanglement [48]. If the smallest symplectic
eigenvalue $\nu$ of the partially transposed covariance matrix is smaller than
1, bipartite entanglement exists. Otherwise, it’s a separable state.
Furthermore, smaller $\nu$ represents stronger entanglement.

Quantum steering for bipartite Gaussian states of CV systems can be quantified
by [49]

$\mathcal{G}^{A\rightarrow
B}(\sigma_{AB})=\mbox{$\max\big{\\{}0,\,\frac{1}{2}\ln{\frac{\det\sigma_{A}}{\det\sigma_{AB}}}\big{\\}}$},$
(4)

where $\mathcal{G}^{A\rightarrow B}(\sigma_{AB})>0$ represents that Alice has
the ability to steer Bob’s state. Similarly, we have

$\mathcal{G}^{B\rightarrow
A}(\sigma_{AB})=\mbox{$\max\big{\\{}0,\,\frac{1}{2}\ln{\frac{\det\sigma_{B}}{\det\sigma_{AB}}}\big{\\}}$},$
(5)

which represents Bob’s ability to steer Alice’s state. From the expressions of
$\mathcal{G}^{A\rightarrow B}(\sigma_{AB})$ and $\mathcal{G}^{B\rightarrow
A}(\sigma_{AB})$, it can be seen that Alice and Bob have the same steerability
if $\det\sigma_{A}=\det\sigma_{B}$ is satisfied; i.e., the bipartite Gaussian
state is a symmetric state. If the state is an asymmetric state, the
steerabilities of Alice and Bob will be different.

##  3 Results

To verify the OAM property of the optical fields, we measure the spatial beam
patterns of quantum states $\ket{\Psi}_{1}$ and $\ket{\Psi}_{2}$ transmitted
through a lossy channel, which are shown in the top rows of Fig. 2(a) and
2(b), respectively. It is obvious that the Pr and Conj fields are both
Laguerre-Gaussian beams. To infer their topological charges, they are passed
through a tilted lens and imaged on a camera. As shown in the bottom rows of
Fig. 2(a) and 2(b), the number of dark stripes gives the number of the
topological charge and the direction gives its sign [50]. As the transmission
efficiency of the Pr field decreases, its optical intensity also decreases,
while its topological charge remains unchanged. Additional beam patterns of
the Pr and Conj fields can be found in Appendix D.

The covariance matrices of the OAM multiplexed entangled states are
reconstructed by measuring the noise variances of the amplitude and phase
quadratures of the Conj and Pr fields $\Delta^{2}\hat{X}_{-l,C}$,
$\Delta^{2}\hat{Y}_{-l,C}$, $\Delta^{2}\hat{X}_{l,P}$, and
$\Delta^{2}\hat{Y}_{l,P}$, as well as their correlation variances of amplitude
and phase quadratures $\Delta^{2}(\hat{X}_{l,P}-\hat{X}_{-l,C})$, and
$\Delta^{2}(\hat{P}_{l,P}+\hat{P}_{-l,C})$, respectively. Details about the
measurement of the covariance matrices can be found in Appendix C. Based on
the covariance matrix of each OAM multiplexed entangled state at different
loss and noise levels, its quantum entanglement and quantum steering
characteristics are evaluated experimentally.

Fig. 2(c) shows the dependence of PPT values of the CV bipartite entangled
state carrying different topological charges on the transmission efficiency of
the Pr field. The correlation and anti-correlation levels of the initial CV
entangled states carrying topological charges $l=0$, $l=1$, and $l=2$ are all
around $-3.3$ dB and 6.1 dB, which correspond to $V=0.47$ and
$V^{\prime}=4.11$, respectively. The entanglement between the Pr and Conj
fields degrades as the transmission efficiency decreases. However, the
entanglement is robust against loss, i.e., it always exists until the
transmission efficiency reaches 0. It is obvious that the CV bipartite
entangled state carrying topological charges $l=1$, $l=2$ are as robust to
loss as their Gaussian counterpart $l=0$.

Figure 3 shows the dependence of PPT values of the CV bipartite entangled
state carrying different topological charges in noisy channels. Compared with
the results in Fig. 2(c), the entanglement disappears at a certain
transmission efficiency of the Pr field in the presence of excess noise, which
demonstrates the sudden death of CV quantum entanglement. Furthermore, the
higher the excess noise is, the sooner entanglement disappears. The
transmission efficiencies where entanglement starts to disappear are
$\eta=0.10,0.28$ and 0.44, respectively, for the excess noise
$\delta=0.15,0.5$ and 1 in the units of shot-noise level (SNL). We show that
OAM multiplexed CV entangled states carrying high order topological charges
$l=1$, $l=2$ exhibit the same decoherence tendency as their Gaussian
counterpart $l=0$ in noisy channels.

The dependence of steerabilities $\mathcal{G}^{A\rightarrow B}$ and
$\mathcal{G}^{B\rightarrow A}$ on the transmission efficiency $\eta$ and
topological charge $l$ in lossy and noisy channels are shown in Fig. 4 (a) and
Fig. 4 (b), respectively. In a lossy channel, the steerabilities for both
directions always decrease when the transmission efficiency decreases. One-way
steering is observed in the region of $0<\eta<0.72$ for OAM multiplexed CV
entangled state carrying different topological charges $l=0$, $l=1$, and
$l=2$. In a noisy channel, where the excess noise $\delta=0.15$ (in the units
of SNL) exists, the steerabilities $\mathcal{G}^{A\rightarrow B}$ and
$\mathcal{G}^{B\rightarrow A}$ are lower than those in the lossy channel [13,
51]. Futhermore, Alice loses its steerability in the region of $0<\eta<0.45$,
while Bob loses its steerability in the region of $0<\eta<0.75$, which
confirms sudden death of quantum steering in a noisy channel [51]. It is worth
noting that the CV entangled state carrying topological charges $l=1$, $l=2$
has the same steerabilities as their counterpart $l=0$.

##  4 Conclusion

The distribution of OAM multiplexed CV entanglement and quantum steering in
quantum channels with homogeneous loss and noise, such as fiber channels, are
experimentally simulated in our work. There are also other quantum channels
with inhomogeneous loss and noise, such as atmospheric turbulence and
diffraction. Recently, it has been shown that other optical fields carrying
OAM, such as vector beams, are turbulence-resilient in atmospheric turbulence
[52]. Thus it is worthwhile to investigate the turbulence-resilient
characteristics of OAM multiplexed CV quantum entanglement and steering, which
have the potential to substantially improve the quantum communication distance
and fidelity.

In summary, we experimentally demonstrate quantum steering of OAM multiplexed
optical fields and investigate the distribution of OAM multiplexed CV
entanglement and quantum steering in quantum channels. We show that the
decoherence property of CV entanglement and quantum steering of the OAM
multiplexed optical fields carrying topological charges $l=1$ and $l=2$ are
the same as that of the counterpart Gaussian mode with $l=0$ in lossy and
noisy channels. The sudden death of entanglement and quantum steering of high-
order OAM multiplexed optical fields is observed in the presence of excess
noise. Our results demonstrate the feasibility to improve the quantum
communication capacity in practical quantum channels by utilizing OAM
multiplexed CV entanglement and quantum steering.

Figure 5: Detailed experimental schematic for distributing OAM multiplexed CV
entanglement in a noisy channel. The lossy channel is realized by blocking the
auxiliary beam. D-shaped mirrors are utilized to combine or separate light
beams with small distances. HWP: half-wave plate; PBS: polarization beam
splitter; EOM: electro-optic modulator; VPP: vortex phase plate; GL: Glan-
laser polarizer; GT: Glan-Thompson polarizer; Pr: probe beam; Conj: conjugate
beam; AM: amplitude modulator; PM: phase modulator; M: mirror; DM: D-shaped
mirror; PZT: piezoelectric ceramics; BS: 50:50 beam splitter; BHD: balanced
homodyne detector; SA: spectrum analyzer. Figure 6: The measured quantum
correlation noises for initially generated OAM multiplexed CV entangled states
carrying topological charge $l=0$ (a), $l=1$ (b) and $l=2$ (c), respectively.
Brown curve at 0 dB shows the SNL. Other six curves show the noise variances
of $\Delta^{2}\hat{X}_{-l,C}$, $\Delta^{2}\hat{Y}_{-l,C}$,
$\Delta^{2}\hat{X}_{l,P}$, $\Delta^{2}\hat{Y}_{l,P}$, as well as noise
variances of their joint amplitude or phase quadrature
$\Delta^{2}(\hat{X}_{l,P}-\hat{X}_{-l,C})$, and
$\Delta^{2}(\hat{P}_{l,P}+\hat{P}_{-l,C})$. These six curves are all
normalized to the same SNL. All the measurements are performed at 1.2 MHz. The
electronic noise of the BHDs and background noise from leaked pump fields are
subtracted from the SNL and signals, respectively. Figure 7: The images of OAM
modes of the Pr beam and Conj beam generated from FWM process. (a) $l=0$; (b)
$l=1$; (c) $l=2$. From left to right: Conj beam, leaked pump beam, and Pr
beam.

## APPENDIX A: Theoretical model

The Hamiltonian of the orbital-angular momentum (OAM) multiplexed four-wave
mixing (FWM) process can be expressed as:

$\hat{H}=\sum_{l}i\hbar
k_{l}\hat{a}_{0,Pump}\hat{a}_{0,Pump}\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}e^{i\theta}+h.c.$
(6)

where $\hat{a}_{0,Pump}$, $\hat{a}^{\dagger}_{l,P}$, and
$\hat{a}^{\dagger}_{-l,C}$ are the annihilation operator of the pump field,
the creation operators related to OAM modes of the Pr and Conj fields,
respectively. $\theta$ is the phase of the pump field. The first term
represents the process in which two pump photons are converted into a Pr
photon and a Conj photon. $k_{l}$ is interaction strength of each OAM pair.
Since the pump beam does not carry OAM ($l=0$), the topological charges of the
Pr and Conj fields are opposite due to OAM conservation in the FWM process.
The second term h.c. is the Hermitian conjugate of the first term, and
represents the reversed process in which a Pr photon and a Conj photon are
converted to two pump photons.

The pump field is much stronger than the Pr and Conj fields in the FWM
process, so it can be regarded as classical field. By combing the intensity of
the pump field $|\alpha_{Pump}|^{2}$ with $k_{l}$, i.e.
$|\alpha_{Pump}|^{2}k_{l}=\gamma_{l}$, and taking $\theta=0$, the Hamiltonian
can be simplified as [40]:

$\hat{H}=\sum_{l}i\hbar\gamma_{l}\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}+h.c.$
(7)

The output state of the OAM multiplexed FWM process is as following:

$\ket{\Psi}_{out}=\ket{\Psi}_{-l}\otimes\cdots\otimes\ket{\Psi}_{0}\otimes\cdots\otimes\ket{\Psi}_{l}$
(8)

where
$\ket{\Psi}_{l}=\hat{S}(\gamma_{l})\ket{vac}=\ket{\psi_{l,P},\psi_{-l,C}}$
means the EPR entangled state with Pr and Conj fields carrying topological
charge $l$ and $-l$, respectively. $\ket{vac}$ presents vacuum state, and
$\hat{S}(\gamma_{l})=e^{\gamma_{l}(\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}-\hat{a}_{l,P}\hat{a}_{-l,C})}$
presents two-mode squeezing operator for the vacuum state carrying OAM
topological charge $l$. It is obvious that a series of CV EPR entangled states
carrying independent topological charges are generated in the FWM process,
i.e., OAM multiplexing is realized.

All Gaussian properties of the CV Gaussian entangled state $\ket{\Psi}_{l}$
can be determined by its covariance matrix $\sigma_{AB}$, with the matrix
element
$\sigma_{ij}=\langle\hat{\xi}_{i}\hat{\xi}_{j}+\hat{\xi}_{j}\hat{\xi}_{i}\rangle/2-\langle\hat{\xi}_{i}\rangle\langle\hat{\xi}_{j}\rangle$,
where
$\hat{\xi}\equiv(\hat{X}_{-l,C},\hat{Y}_{-l,C},\hat{X}_{l,P},\hat{Y}_{l,P})^{T}$,
$\hat{X}=\hat{a}+\hat{a}^{{\dagger}}$ and
$\hat{Y}=(\hat{a}-\hat{a}^{{\dagger}})/i$ represent amplitude and phase
quadratures of the Conj and Pr fields.

The covariance matrix of CV bipartite entangled state can be written as:

$\sigma_{AB}=\left(\begin{array}[]{cccc}\frac{V+V^{\prime}}{2}&0&\frac{V^{\prime}-V}{2}&0\\\
0&\frac{V+V^{\prime}}{2}&0&\frac{V-V^{\prime}}{2}\\\
\frac{V^{\prime}-V}{2}&0&\frac{V+V^{\prime}}{2}&0\\\
0&\frac{V-V^{\prime}}{2}&0&\frac{V+V^{\prime}}{2}\end{array}\right)=\left(\begin{array}[]{cc}\frac{V+V^{\prime}}{2}I&\frac{V^{\prime}-V}{2}Z\\\
\frac{V^{\prime}-V}{2}Z&\frac{V+V^{\prime}}{2}I\end{array}\right)$ (9)

where $V$ and $V^{\prime}$ represent the variances of squeezed and anti-
squeezed quadratures of the optical mode, respectively. $I$ and $Z$ are the
Pauli matrices:

$I=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\quad
Z=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$ (10)

Then we consider the distribution of CV entangled state $\ket{\Psi}_{l}$ in a
lossy and noisy channel. Let $\hat{a}_{l,P}$ and $\hat{a}_{-l,C}$ represent
the annihilation operators of the Pr and Conj fields, respectively. After the
Pr field $\hat{a}_{l,P}$ is distributed in a lossy channel, it becomes
$\hat{a}^{\prime}_{l,P}=\sqrt{\eta}\hat{a}_{l,P}+\sqrt{1-\eta}\hat{\mu}$,
where $\hat{\mu}$ represents vacuum state with variance of 1. Similarly, the
Pr field becomes
$\hat{a}^{\prime}_{l,P}=\sqrt{\eta}\hat{a}_{l,P}+\sqrt{1-\eta}(\hat{\epsilon}+\hat{\mu})$
after it is distributed in a noisy channel with excess noise
$\Delta^{2}(\hat{X}_{\epsilon})=\Delta^{2}(\hat{Y}_{\epsilon})=\delta$ [13].
$\delta=0$ means that there is no excess noise, and only loss exists in the
channel. $\delta>0$ means that there exists excess noise in the channel. So
the covariance matrix of the CV entangled state $\ket{\Psi}_{l}$ after
distribution in the lossy or noisy channel is as following:

$\sigma_{AB,\delta,\eta}=\left(\begin{array}[]{cccc}V_{a}&0&V_{c}&0\\\
0&V_{a}&0&-V_{c}\\\ V_{c}&0&V_{b}&0\\\
0&-V_{c}&0&V_{b}\end{array}\right)=\left(\begin{array}[]{cc}V_{a}I&V_{c}Z\\\
V_{c}Z&V_{b}I\end{array}\right)$ (11)

with $V_{a}=\frac{V+V^{\prime}}{2}$,
$V_{b}=\eta\frac{V+V^{\prime}}{2}+(1-\eta)(1+\delta)$ and
$V_{c}=\sqrt{\eta}\frac{V^{\prime}-V}{2}$.

## APPENDIX B: Details of the Experiment

The Ti:sapphire laser (Coherent MBR-110) is about 1.6 GHz blue detuned from
133Cs D1 line $6S_{1/2},F=3\rightarrow 6P_{1/2},F^{\prime}=4$ transition with
a total power of 1.2 W. As shown in Fig. 5, the laser beam is split into two
beams by a polarization beam splitter (PBS1). The horizontally polarized beam
is split into two beams by PBS2 and they serve as the pump beams for two sets
of four-wave mixing (FWM) processes in the same cesium vapor cell. The
vertically polarized beam, with a power of 30 mW, passes through a resonance
electro-optic modulator (EOM, Qubig GmbH PM - Cs) and three successive
temperature-stabilized etalons to realize 9.2 GHz frequency red shift [47].
The OAM of topological charge $l=1$ or $l=2$ is added to the frequency red-
shifted beam by passing it through a vortex phase plate (VPP, RPC Photonics).
Then the frequency red-shifted beam is split into three beams, among which the
first two beams serve as the probe beams of two FWM processes, and the third
beam serves as the auxiliary beam for noisy channel. The pump and probe beams
are combined in a Glan-laser (GL) polarizer and then cross each other in the
center of the cesium vapor cell (Traid Technology Inc.) at an angle of 6 mrad.
The vapor cell is 25 mm long and its temperature is stabilized at
$103^{\circ}$C.

The two sets of FWM processes are constructed in the same cesium vapor cell
with a hight difference of 5 mm. The bottom FWM process is used to generate
OAM multiplexed CV entangled state, while the top FWM process is used to
generate spatially matched local oscillators (LOs) with the Pr and Conj
fields. The pump power in the bottom FWM process for generating OAM
multiplexed CV entanglement is 240 mW. The probe gain of the bottom FWM
process is around 3, and the degree of initially generated CV entanglement is
around $-$3.3$\pm$0.1 dB. The pump power and seed probe power of the top FWM
processes are 450 mW and 100 $\mu$W, respectively, so that the shot-noise
level (SNL) is around 10 dB higher than the electronic noise of the homodyne
detector. The bottom FWM process is weakly seeded with a probe power of around
3 $\mu$W for relative phase locking of the Pr/Conj fields and their LOs in the
balanced homodyne detections.

The lossy channel is simulated by a half-wave plate (HWP10) and PBS6. The
noisy channel is modeled by combining the vertically polarized Pr field with
an horizontally polarized auxiliary beam at PBS5 followed by HWP10 and PBS6.
The auxiliary beam carries the same frequency and topological charge with the
Pr field, and is modulated by an amplitude modulator (AM) and a phase
modulator (PM) with excess noise at 1.2 MHz . The amount of excess noise is
adjusted by tuning the amplitude of the signal applied to the AM and PM, and
evaluated in the units of SNL. For example, excess noise $\delta=1$
corresponds to noise level that is 3 dB higher than the SNL. By tuning HWP10,
the lower transmission efficiency of the Pr field is, the higher excess noise
is coupled to the Pr field. In practical quantum communication protocols,
higher excess noise is coupled to quantum entangled state as the communication
distance increases, accompanying lower transmission efficiency. Therefore, our
experimental setting is similar to the realistic scenarios in practical noisy
quantum channel. To characterize the OAM multiplexed CV entangled state, its
covariance matrix is experimentally obtained by utilizing two sets of balanced
homodyne detectors (BHDs, Thorlabs PDB450A). The interference visibilities for
the two sets of BHDs are both around 99$\%$. The electrical gains of these two
BHDs are both $10^{5}$ V/A. The original photodiodes are replaced by high
quantum efficiency (QE) photodiodes with QE=98$\%$ at 895 nm (First sensor).
To measure amplitude quadrature $\hat{X}$ or phase quadrature $\hat{P}$ of the
Pr/Conj fields, the relative phases between them and their LOs are locked by
applying feedback signal from proportional–integral–derivative circuits and
high-voltage amplifiers to piezoelectric ceramics (PZT).

## APPENDIX C: Measurement of the Covariance matrix

To reconstruct covariance matrix of the CV quantum entangled state, we perform
6 different measurements on the output optical modes. These measurements
include the variances of the amplitude and phase quadratures of the Conj field
and Pr field $\Delta^{2}\hat{X}_{-l,C}$, $\Delta^{2}\hat{Y}_{-l,C}$,
$\Delta^{2}\hat{X}_{l,P}$, $\Delta^{2}\hat{Y}_{l,P}$, as well as noise
variances of their joint amplitude or phase quadrature
$\Delta^{2}(\hat{X}_{l,P}-\hat{X}_{-l,C})$, and
$\Delta^{2}(\hat{Y}_{l,P}+\hat{Y}_{-l,C})$, respectively.
$\Delta^{2}\hat{X}_{-l,C}$ and $\Delta^{2}\hat{Y}_{-l,C}$
($\Delta^{2}\hat{X}_{l,P}$ and $\Delta^{2}\hat{Y}_{l,P}$) are experimentally
measured by locking the relative phase of Conj (Pr) field and its
corresponding local oscillator of BHD1 (BHD2) at amplitude quadrature or phase
quadrature. $\Delta^{2}(\hat{X}_{l,P}-\hat{X}_{-l,C})$ and
$\Delta^{2}(\hat{Y}_{l,P}+\hat{Y}_{-l,C})$ are experimentally measured by
locking the relative phases of BHD1 and BHD2 at amplitude quadrature or phase
quadrature, and then subtracting or adding the photocurrents with a radio-
frequency subtractor or adder. The SNL is achieved by blocking the Pr and/or
Conj field, so that only the noise of vacuum is measured. The settings of the
spectrum analyzer (Agilent E4411B) are 30 kHz resolution bandwidth, 100 Hz
video bandwidth, and zero span at 1.2 MHz.

Fig. 6 shows the measured 6 noise variance levels of the CV entanglement of
OAM multiplexed CV entangled state before being distributed in lossy and noisy
channels ($\delta=0,\eta=1$). As shown in Fig. 6, the noise levels of
amplitude and phase quadratures are
$\Delta^{2}\hat{X}_{-l,C}=\Delta^{2}\hat{Y}_{-l,C}=\Delta^{2}\hat{X}_{l,P}=\Delta^{2}\hat{Y}_{l,P}$=3.6$\pm$0.1
dB, and the noise levels of correlated quadratures are
$\Delta^{2}(\hat{X}_{l,P}-\hat{X}_{-l,C})$=$\Delta^{2}(\hat{Y}_{l,P}+\hat{Y}_{-l,C})$=$-$3.3$\pm$0.1
dB. Our FWM process works in the amplification regime, so amplitude
quadratures and phase quadratures of the Pr/Conj fields show strong
correlations and anti-correlations, respectively. It is clear that the degrees
of entanglement for CV entangled states carrying topological charge $l=1$,
$l=2$ are close to that of their spatially Gaussian counterpart $l=0$.

With the measured 6 noise variances, the cross correlation matrix elements are
calculated via

$\displaystyle Cov(\hat{\xi}_{i},\hat{\xi}_{j})$
$\displaystyle=\frac{1}{2}\left[\Delta^{2}\left(\hat{\xi}_{i}+\hat{\xi}_{j}\right)-\Delta^{2}\hat{\xi}_{i}-\Delta^{2}\hat{\xi}_{j}\right],$
(12) $\displaystyle Cov(\hat{\xi}_{i},\hat{\xi}_{j})$
$\displaystyle=-\frac{1}{2}\left[\Delta^{2}\left(\hat{\xi}_{i}-\hat{\xi}_{j}\right)-\Delta^{2}\hat{\xi}_{i}-\Delta^{2}\hat{\xi}_{j}\right].$

In the experiment, we obtain all the covariance matrices of quantum states
with different transmission efficiencies and excess noise, and then calculate
the smallest symplectic eigenvalue $\nu$ of the partially transposed
covariance matrix, $\mathcal{G}^{A\rightarrow B}(\sigma_{AB})$ and
$\mathcal{G}^{B\rightarrow A}(\sigma_{AB})$ to verify whether quantum
entanglement and quantum steering exist.

## APPENDIX D: Supplemental beam patterns

Supplemental beam patterns of the OAM modes of the Pr and Conj beams generated
from the FWM process are shown in Fig. 7. Pr and Conj beams carrying different
topological charges $l=0$, $l=1$, and $l=2$ are generated by removing or
switching different VPPs, and their beam patterns are taken after the GT
polarizer with a camera. It is obvious that the dark hollow patterns of the
Laguerre-Gaussian beams are enlarged as the topological charge increase.
Furthermore, the beam patterns of the Pr and Conj fields are symmetric with
respect to the pump beam.

Funding National Natural Science Foundation of China (NSFC) (Grants No.
11974227, No. 11834010, No. 61905135, and No. 62005149); Fund for Shanxi “1331
Project" Key Subjects Construction; Research Project Supported by Shanxi
Scholarship Council of China (2021-003).

Disclosures. The authors declare no conflicts of interest.

†These authors contributed equally to this Letter.

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