# Quantum communication networks with optical vortices

Şerban Suciu  Horia Hulubei National Institute of Physics and Nuclear
Engineering, 077125 Bucharest–Măgurele, Romania  Faculty of Physics,
University of Bucharest, P.O.Box MG-11, Bucharest-Magurele, Romania  George
Andrei Bulzan  National Institute for Research and Development in
Microtechnologies IMT, Bucharest 077190, Romania  Faculty of Physics,
University of Bucharest, P.O.Box MG-11, Bucharest-Magurele, Romania  Tudor-
Alexandru Isdrailă  Horia Hulubei National Institute of Physics and Nuclear
Engineering, 077125 Bucharest–Măgurele, Romania  Department of Physics,
Tamkang University, 151 Yingzhuan Rd., Tamsui Dist., New Taipei City 251301,
Taiwan  Alexandra Maria Pălici  Horia Hulubei National Institute of Physics
and Nuclear Engineering, 077125 Bucharest–Măgurele, Romania  Dept. Photonics
Engineering, Technical University of Denmark (DTU), DK-2800 Kgs. Lyngby,
Denmark  Stefan Ataman  Extreme Light Infrastructure-Nuclear Physics (ELI-NP),
Horia Hulubei National Institute for Physics and Nuclear Engineering, 30
Reactorului Street, 077125 Bucharest–Măgurele, Romania  Cristian Kusko
National Institute for Research and Development in Microtechnologies IMT,
Bucharest 077190, Romania  Radu Ionicioiu  Horia Hulubei National Institute of
Physics and Nuclear Engineering, 077125 Bucharest–Măgurele, Romania

###### Abstract

Quantum communications bring a paradigm change in internet security by using
quantum resources to establish secure keys between parties. Present-day
quantum communications networks are mainly point-to-point and use trusted
nodes and key management systems to relay the keys. Future quantum networks,
including the quantum internet, will have complex topologies in which groups
of users are connected and communicate with each-other. Here we investigate
several architectures for quantum communication networks. We show that
photonic orbital angular momentum (OAM) can be used to route quantum
information between different nodes. Starting from a simple, point-to-point
network, we will gradually develop more complex architectures: point-to-
multipoint, fully-connected and entanglement-distribution networks. As a
particularly important result, we show that an $n$-node, fully-connected
network can be constructed with a single OAM sorter and $n-1$ OAM values. Our
results pave the way to construct complex quantum communication networks with
minimal resources.

keywords: quantum networks, OAM, QKD

##  I Introduction

Quantum computers pose a threat to present-day internet security, due to their
ability to efficiently break public-key cryptography. One way to mitigate this
quantum apocalypse is to deploy large-scale quantum communication networks.
Current quantum communications are usually point-to-point, using trusted nodes
and key management systems to establish secret keys between remote nodes.

Future quantum networks, including the quantum internet, will require to both
handle complex network topologies [1] and secure such networks [2]. These
networks will need to connect users situated in different locations and/or
domains. Consequently, in such networks it will be important to route a
quantum state $|\psi\rangle_{q}$ between different locations.

Most of the information we exchange everyday is encoded in photons and carried
by optical fibres. The data capacity of a single optical fiber depends on the
spectral bandwidth over which low-loss signal transmission can be achieved, on
the one hand, and on our ability to use this bandwidth through suitable coding
and decoding schemes, on the other.

Due to the constant increase of worldwide data traffic, nonlinear effects [3]
impose limits on the capacity of optical fibers. To address this capacity-
crunch, space division multiplexing (SDM) using multi-core [4, 5] and
multimode [6] fibers have beed developed. In the quest for larger data
capacity, another solution is to use an extra degree of freedom, different
from wavelength [7, 8].

A good candidate for the extra degree of freedom is the orbital angular
momentum (OAM) of the photon [9, 10, 11]. The phase front of an OAM beam is
helical, with quantized angular momentum $l\hbar$, $l\in\mathbb{Z}$. Photons
carrying OAM have been used for different applications, such as object
identification [12], enhanced phase sensitivity [13], imaging [14, 15] and
metrology [16, 17]. Classical and quantum communication with OAM states have
both been demonstrated in fiber [8, 18, 19], including experimental mode-
division multiplexing [20].

Long-distance, high-dimensional QKD using OAM in both optical fibers [8] and
free-space [21] have recently enjoyed a renewed interest. This is due to
several benefits brought by high-dimensional systems: reduced overall
complexity of a quantum circuit via $d$-level gates [22, 23], increased raw-
key rates [24, 25], robustness to noise [26, 27, 28] and hacking attacks [29].
Hybrid states of OAM and polarization have also been used in QKD protocols, in
both fiber and free-space [30, 31, 32]. Furthermore, recent advances have
lowered the resource requirements for point-to-multipoint architectures [1]
and have also enhanced quantum digital signatures protocols [2]. OAM
multiplexing can offer an alternative to the development of wireless
communications [33], because unlike wavelength-division multiplexing (WDM), it
can generate orthogonal channels [33] in a line-on-site channel environment.

Due to this increased interest in both classical and quantum applications of
OAM, dedicated optical fibers [34] and multiplexing and demultiplexing
techniques [35, 36, 37, 38] have been maturing recently. Thus new methods to
route information encoded in OAM are needed. In contrast to wavelength, it is
relatively easy to change OAM using passive optical elements like spiral
phase-plates (SPPs) [39]. This makes OAM an attractive degree-of-freedom for
network routing.

In this paper we discuss several architectures for quantum communication
networks which use OAM for routing quantum states $|\psi\rangle_{q}$ around
the network. The paper is structured as follows: in Section II we describe the
quantum sorter [37], which is the main element in OAM
multiplexing/demultiplexing (MUX/DEMUX). In Section III we show OAM
implementations of several topologies for quantum communications networks:
point-to-point, point-to-multipoint, fully connected, and fully-connected
entanglement-distribution networks with a central network provider. Finally,
we conclude the article in Section IV.

##  II Quantum sorter

A central element of all the networks discussed here is the $d$-dimensional
quantum sorter $U_{d}$, and its inverse $U_{d}^{\dagger}$ [37]. A short
description is provided in Appendix A; $U_{d}$ ($U_{d}^{\dagger}$) is a
unitary operator which acts as a demultiplexer (multiplexer). In quantum
information parlance, the sorter $U_{d}$ is a controlled-$X_{d}$ gate between
the observable to be sorted and the path degree of freedom.

The sorter is universal, i.e., it can (de)multiplex any internal degree of
freedom: wavelength, spin, radial angular momentum, OAM etc, and has a
theoretical efficiency of 100% [37]. A definite advantage of the sorter is
that it can be implemented with linear optics. In Appendix A we provide a
physical intuition behind the sorter. Experimentally, sorting photons
according to their radial number has been realized in Refs. [40, 41]. The same
sorting mechanism has been recently applied to a novel method of mass
spectrometry [42]. Standard telecom networks use wavelength as the extra DoF
for multiplexing/demultiplexing. In this article we focus on the OAM degree-
of-freedom as a tool for MUX/DEMUX. The action of the sorter $U_{d}$ and its
inverse $U_{d}^{\dagger}$ is:

$\displaystyle U_{d}\ (DEMUX):\ |l\rangle_{\text{oam}}|k\rangle_{\text{path}}$
$\displaystyle\xrightarrow{U_{d}}$
$\displaystyle|l\rangle_{\text{oam}}|k\oplus l\rangle_{\text{path}}$
$\displaystyle U_{d}^{\dagger}\ (MUX):\
|l\rangle_{\text{oam}}|k\rangle_{\text{path}}$
$\displaystyle\xrightarrow{U_{d}^{\dagger}}$
$\displaystyle|l\rangle_{\text{oam}}|k\ominus l\rangle_{\text{path}}$ (1)

with $\oplus/\ominus$ addition/subtraction $\mathrm{mod}\,d$. Here both OAM
and path DoFs are qudits, i.e., $d$-dimensional quantum systems. Thus, if
photons with different OAM $l$ are incident on port 0 of the $U_{d}$ gate
(DEMUX), they will exit on output $l$, i.e., they will be sorted on different
outputs according to their OAM value. The $U_{d}^{\dagger}$ gate (MUX) works
in reverse.

The addition and subtraction $\mathrm{mod}\,d$ result in a cyclic property
which can be better understood for $l=\pm d$. In this case $|l\rangle_{oam}$
will be sorted on path $|0\rangle_{\text{path}}$, like $l=0$. This cyclic
property will play a crucial role in the design of various network
architectures for routing quantum states using OAM.

The cyclic property is also used in the construction of the $X_{d}$ gate [43].
The $X_{d}$ gate is a basic building block for qudit tomography [44, 45] and
for general qudit protocols. Another application of the quantum sorter is in
the generation of high-dimensional entangled states between an observable and
the path DoF. Hybrid quantum gates are a hot topic under active development
[46, 47, 48]. Accessing a larger alphabet allows us to encode more
information, resulting in higher channel capacity and better robustness to
noise.

##  III OAM-assisted quantum communication networks

In this section we start with a simple architecture and then gradually build
more complex networks. All networks discussed here can be used for QKD, either
in prepare-and-measure (BB84) or in entanglement-based protocols (E91, BBM92).
The only difference is in the equipment available to users. The networks can
also be used to route quantum information as part of a larger protocol. What
we denote as ”senders” and ”receivers” can represent anything from sources and
detectors to other networks or protocols. The scale can also vary form
waveguides in computer chips to optical fibers between cities or ground-to-
satellite links.

For simplicity, in the following we use only positive OAM values. One can
substitute the OAM
$|l_{\text{max}}-n\rangle_{\text{oam}}\mapsto|-n-1\rangle_{\text{oam}}$, where
$l_{\text{max}}$ is the largest OAM used in the protocol, and
$n\in\\{0,\;1,\;...,\;\lfloor\frac{l_{\text{max}}}{2}\rfloor\\}$ thus halving
the maximum OAM values required.

###  III.1 Point-to-point architecture

Point-to-point networks are a simple case in which pairs of users are
connected by their own quantum channel. In practice this results in a messy
and convoluted network of cables. To reduce the number of cables needed,
especially for long-distance communication, individual signals are in practice
multiplexed into the same channel. For example, different labs from two cities
can share the same channel for intercity transmission.

Figure 1: A. Logical network for two pairs connected in a point-to-point
topology. B. Two point-to-point pairs using a common channel. Signals from
each pair are multiplexed into the common (long-range) channel and
demultiplexed at the destination. The information can be recovered and
separated because each sender-receiver pair has allocated a unique OAM value.

In Figure 1 two pairs Alice-Charlie and Bob-Diana share a single long-range
quantum channel (instead of dedicated channels for each pair). Each pair has
assigned a unique OAM value. The pairs are indexed by consecutive numbers
which represents their assigned OAM, input port at the multiplexer and output
port at the demultiplexer.

For example, if Bob wants to send a quantum state $|\psi\rangle_{q}$ to Diana,
he uses input port $|1\rangle_{\text{path}}$ with OAM
$|1\rangle_{\text{oam}}$. This state is input into the multiplexer which
redirects it to port $|0\rangle_{\text{path}}$ of the long-range channel:

$|\psi\rangle_{q}|1\rangle_{\text{oam}}|1\rangle_{\text{path}}\xrightarrow{MUX}|\psi\rangle_{q}|1\rangle_{\text{oam}}|0\rangle_{\text{path}}$
(2)

Diana recovers $|\psi\rangle_{q}$ on output port $|1\rangle_{\text{path}}$ of
the demultiplexer at her end:

$|\psi\rangle_{q}|1\rangle_{\text{oam}}|0\rangle_{\text{path}}\xrightarrow{DEMUX}|\psi\rangle_{q}|1\rangle_{\text{oam}}|1\rangle_{\text{path}}$
(3)

The quantum state $|\psi\rangle_{q}$ can be any internal degree of freedom
(different from OAM). Usually we use polarization to encode the quantum state
$|\psi\rangle_{q}=|\psi\rangle_{\text{pol}}=\alpha|H\rangle+\beta|V\rangle$.
In Appendix B we discuss an example of an OAM-assisted BB84 protocol in
polarization.

Since the multiplexer and demultiplexer are modelled by a unitary operation,
the protocol also works in reverse. We can reverse the direction in Figure 1
such that Charlie and Diana are now the senders and everything works
similarly. This is true for all communication protocols discussed here.

###  III.2 Point-to-multipoint architecture

A point-to-multipoint architecture is a natural extension from the point-to-
point one. Instead of linking pairs of users, a point-to-multipoint network
links a group of users with one or more other groups. However, members of the
same group cannot communicate with each-other. The logical network is a
bipartite graph.

The simple setup with one multiplexer and one demultiplexer works in this
case, but only if the numbers of senders and receivers are coprime (see
Appendix C for a proof). Expanding on the previous example, different labs
from two cities can now share not only a transmission line, but also choose to
which lab from the other city to send data.

Figure 2: A. Logical network for two groups connected in a point-to-multipoint
topology. B. Two groups share a single, long-range channel to communicate with
members of the other group. To ensure that each pair has assigned a unique OAM
value, the dimensions of multiplexer and demultiplexer must be coprime; here
$d_{s}=2$ and $d_{r}=3$.

Figure 2 shows an example for two senders and three receivers. Alice (Bob) can
communicate with any receiver (Charlie, Diana, Eve) using an even (odd) OAM
value.

For the general case, suppose we have a set of $d_{s}$ senders and $d_{r}$
receivers, with $d_{s}$, $d_{r}$ relatively coprime. In this case any sender-
receiver pair has associated a unique OAM, thus the receiver can distinguish
between different senders. This value is determined by solving a system of
congruence relations:

$\displaystyle l_{sr}\equiv$ $\displaystyle s\;(\mathrm{mod}\;d_{s})$
$\displaystyle l_{sr}\equiv$ $\displaystyle r\;(\mathrm{mod}\;d_{r})$

where $d_{s}$ and $d_{r}$ are, respectively, the dimension of the multiplexer
(sender) and demultiplexer (receiver). In order for a sender
$s\in\\{0,\ldots,d_{s}-1\\}$ to communicate with a receiver
$r\in\\{0,\ldots,d_{r}-1\\}$, they use the OAM value $l_{sr}$ given by

$l_{sr}=pd_{s}+s=qd_{r}+r$ (4)

see Appendix C. The total number of OAM values is $d_{s}d_{r}$.

Although in practice we can always choose the dimensions of the multiplexer
and demultiplexer to be coprime (e.g., by embedding them into a larger set),
this can be an issue for more complex networks. The coprimality constraint can
be eliminated by modifying the demultiplexer as in Figure 3 (B). We call this
a group demultiplexer since it splits an input channel into $d_{s}\cdot d_{r}$
outputs and then groups them back together into $d_{r}$ channels.

Figure 3: A. Logical network for two groups connected in a point-to-multipoint
topology. B. Schematic for a group demultiplexer. C. General point-to-
multipoint protocol for an arbitrary number of senders and receivers. On the
receiver side, the demultiplexer has been replaced by a group demultiplexer
(dashed outline). With this change we eliminate the coprimality condition and
the OAM assignment is simplified. D. General point-to-multipoint for multiple
groups. Since the multiplexer is a unitary transformation, it has the same
number of input and output ports. By using the other available outputs, one
group can communicate to several other groups (situated at different
locations).

In Figure 3 (C) we use the group demultiplexer to create a more general
network. Now any sender $s$ can transmit to any receiver $r$ by an appropriate
OAM value $l_{sr}$:

$l_{sr}=s+rd_{s}$ (5)

A step-by-step analysis of this protocol is given in Appendix D; here we only
give the main result:

$|\psi\rangle_{q}|l_{sr}\rangle_{\text{oam}}|s\rangle_{\text{path}}\xrightarrow{\text{network}}|\psi\rangle_{q}|l_{sr}\rangle_{\text{oam}}|r\rangle_{\text{path}}$
(6)

This ensures that quantum information, encoded in the state
$|\psi\rangle_{q}$, is routed along the network from sender $s$ to receiver
$r$.

In the following schemes a group demultiplexer can be replaced by a simple
demultiplexer, provided that: (i) the number of senders and receivers are co-
prime, and (ii) the OAM values satisfy the congruence relations discussed
above. Also, any network can work in reverse, i.e., receivers become senders,
multiplexers become demultiplexers (and vice-versa) and group demultiplexers
become group multiplexers.

In point-to-multipoint networks, a group communicates with multiple other
groups. A useful use-case scenario is a network connecting multiple cities:
labs in one city communicate to labs in multiple cities. However, one network
connects just one city with others. This creates a physical star-network
topology, the logical network topology remains the same. Each sender forms a
logical star-network topology with all receivers, yet as a group the point-to-
multipoint logic is unchanged.

In Figure 3 (D) we split de group demultiplexer at the receivers end into two.
In large networks it will be useful to put a group multiplexer at the senders
and have simple demultiplexers at the receivers. This helps to reduce the
costs, since using group multiplexers scales as $d_{s}\cdot d_{r}$.

Compared to the previous protocol, we now use other outputs of the multiplexer
to communicate with different groups. Everything remains the same, except from
an offset of the OAM value, which depends on the receiver group:

$l_{sgr}=s+rd_{s}\ominus g$ (7)

where $g$ is the group number (i.e., the output port of the multiplexer).

Consider the example in Figure 3 (D), where Bob intends to communicate to Eve;
we have sender $s=1$ transmitting to receiver $r=0$ from group $g=1$, with a
multiplexer of size $d_{s}=2$. Thus their OAM value is $0$. Notice that this
OAM value is no longer unique, since Alice uses the same OAM to communicate to
Charlie.

This architecture helps to reduce the OAM bandwidth, i.e., the number of OAM
values required. Both schemes in Figure 3, have $d_{s}=2$ senders and
$d_{r}=4$ receivers. However, in case C we need $d_{s}d_{r}=6$ OAM values,
whereas in case D we need only $d_{r}=4$ values. In both cases any sender can
communicate with any receiver.

A variation of this architecture is to use a group multiplexer at the senders
side and only demultiplexers at the receivers side. In this case the OAM value
is

$l_{sgr}=r+sd_{r}\ominus gd_{s}$ (8)

###  III.3 Fully-connected networks

Finally, we generalize the previous schemes to a fully-connected network, in
which any two users can communicate with each other. In this case all nodes
are both senders and receivers. In the previous point-to-multipoint protocol,
this will work for a reasonable numbers of users, as the size of the group
demultiplexer scales as $O(n^{2})$.

Figure 4: A. Logical network for a fully connected network topology. B. A
fully-connected, $n$-node quantum network; each node is both a sender and a
receiver. A single $n$-dimensional sorter (MUX/DEMUX) routes information
between all the nodes. C. Circulators on the node side (a), and DEMUX side
(b), allow each node to be both a transmitter Tx and a receiver Rx. Each node
is connected to the (central) DEMUX by a single fiber, and thus to rest of the
network.

Surprisingly however, a single MUX/DEMUX is enough to create a fully-connected
network with $n$ nodes, see Figure 4. Here the senders can view the OAM value
as an indexing list: $0$ for themselves, $1$ for the next user (mod $d$), $2$
for the second over (mod $d$) and so on. Now each node is both a sender and a
receiver.

Table 1:  OAM assignments for the fully-connected network in Figure 4. Each row is obtained from the one above by a circular right-shift. sender receiver | A | B | C | D
---|---|---|---|---
A | $|0\rangle$ | $|1\rangle$ | $|2\rangle$ | $|3\rangle$
B | $|3\rangle$ | $|0\rangle$ | $|1\rangle$ | $|2\rangle$
C | $|2\rangle$ | $|3\rangle$ | $|0\rangle$ | $|1\rangle$
D | $|1\rangle$ | $|2\rangle$ | $|3\rangle$ | $|0\rangle$

Thus a fully-connected network with $n$ users requires only a single
$n$-dimensional quantum sorter (acting as a MUX/DEMUX) and $n$ OAM values. In
fact, since the OAM value $\ell=0$ is used to connect a node to itself, we
need only $n-1$ OAM values.

At first sight it looks like each node needs two different channels to connect
to the network, one for sending and one for receiving. However, using two
circulators, one at the user side and the other at the MUX/DEMUX side, a node
can use a single channel for both sending and receiving, see Figure 4 (C).

This design is the most general possible, connecting all network nodes. It can
be used as a fully-connected network in prepare-and-measure QKD. The strength
of this approach can be seen in Table 2, since the fully-connected network has
the lowest resource requirements.

Table 2: Resource scaling for different network architectures. The fully connected network requires one quantum sorter, whereas all the other architectures require at least two. Network architecture | Resource scaling
---|---
Point-to-point | 1$\times U_{d_{s}}$; 1$\times U_{d_{r}}$
Point-to-multipoint (general) | 1$\times U_{d}$; $\left(\frac{d}{2}+1\right)\times U_{2}$
Point-to-multipoint (groups) | 1$\times U_{d_{s}}$; $d_{s}\times U_{d_{r}}$
Fully Connected | 1$\times U_{d}$
Entanglement distrib. (active) | 1$\times U_{d}$; $\left(\frac{d}{2}+1\right)\times U_{2}$
Entanglement distrib. (passive) | 3$\times U_{d}$; $2d\times$ SPP($i$)

###  III.4 Entanglement-distribution networks

So far we have discussed networks for prepare-and-measure QKD protocols, such
as BB84. Another important class of QKD protocols are entanglement-based ones,
e.g., E91 or BBM92. Although entanglement-based protocols are more secure than
prepare-and-measure ones, they are also more difficult to implement, as they
require to distribute entanglement between nodes.

Multi-user entanglement-distribution networks have been experimentally
demonstrated for wavelength multiplexing [49, 50]. Similar passive-switching
networks with a central node can be designed for OAM.

Entanglement-distribution networks can be actively- or passively switched. In
an active network, the central node (the source) generates pairs of
polarization-entangled photons. An active switch then assigns the correct OAM
values $r$ and $s$ to the two photons and then distributes the photons to the
corresponding nodes $r$ and $s$.

For example, the actively-switched entanglement distribution scheme in Ref.
[49] can be translated directly in the OAM domain with a general point-to-
multipoint network, as in Figure 5 (A). Here ”Alice” and ”Bob” are two
entangled photons that are distributed based on their assigned OAM value.

Figure 5: A. Fully-connected network for entanglement distribution using an
active central provider. If in the general point to multipoint network one
group sends entangled pairs (or multipartite entangled states) to the other,
the receiving group becomes a fully-connected network for entanglement-based
QKD. B. Fully-connected network for entanglement distribution from a passive
central provider. In this scheme the senders are replaced by a source of
entangled photons which are then randomly assigned OAM numbers, thus
generating random pairs of users which share an entangled state.

Notice that even though the network is not fully connected in the prepare-and-
measure regime, it becomes so for entanglement distribution. It is fully
connected, i.e., any two users can share an entangled pair. A passive-
switching network for entanglement distribution has been experimentally
demonstrated in Ref. [50].

A passively-switched network works in a similar way to the choice of
measurement basis in BB84, where a beam-splitter chooses randomly the basis.
In the case of a passive OAM network, two Fourier gates $F_{d}$ (which
generalize the beamsplitter for $d>2$) put the signal and idler photon into
two distinct multi-path interferometers, one for signal and one for idler, see
Figure 5 (B). For the signal photon we have:

$F_{d}|0\rangle_{\text{path}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle_{\text{path}}$
(9)

Each path $i$ of the interferometer has an $i$-th order spiral phase plate
(SPP) and changes the OAM
$|0\rangle_{\text{oam}}{\rightarrow}|i\rangle_{\text{oam}}$

$\text{SPP}^{i}|0\rangle_{\text{oam}}|i\rangle_{\text{path}}=|i\rangle_{\text{oam}}|i\rangle_{\text{path}}$
(10)

The paths (channels) are then multiplexed into a common exit path.
Subsequently, the two photons are input into path 0 (the signal) and,
respectively 1 (the idler) of a final OAM demultiplexer, which distributes the
two photons to the final users, Figure 5 (B). The final quantum state of the
two photons is (for simplicity we omit the polarization part):

$\frac{1}{d}\sum_{i=0}^{d-1}\sum_{j=0}^{d-1}|i,j\ominus
1\rangle_{\text{oam}}|i,j\rangle_{\text{path}}$ (11)

In this case entanglement distribution between nodes is done randomly,
according to the OAM values of signal and idler (via post-selection). Similar
to other passively-switched networks [50], the pairs of nodes (randomly)
receiving the entangled pair are identified by coincidences in their
detectors.

###  III.5 Non-ideal case

So far we have discussed the ideal, noiseless case. We now briefly analyse the
effect of noise. There are two types of losses: (i) in the quantum channel
(optical fibres, free-space, underwater etc); (ii) in the quantum sorters.
Losses in the quantum channel depend on the specific losses in the fibers
(e.g., Raman absorption), atmospheric turbulence, water turbidity etc . Since
these are common to all quantum communication protocols using the same type of
channel, we will not discuss them here.

Losses due to a non-ideal mass sorter have been discussed in Ref. [42],
including a discussion about decoherence. The OAM sorter and mass sorter have
the same quantum network: they are both equivalent to a controlled-$X_{d}$
gate $C(X_{d})$. Thus the conclusion of the previous analysis also holds for
the OAM sorter. Specifically, for $d=3$ the probability of sorting correctly
is $>96\%$ even for phase errors as high as $2\pi/15$, representing 20% of the
relevant phase for the system, see Fig.6 of Ref. [42].

##  IV Discussion and conclusions

The development of quantum communication networks and the advent of the future
quantum internet is contingent on the ability to route quantum information in
networks with complex topologies. In the scenario investigated here, a set of
users send quantum states $|\psi\rangle_{q}$, entangled or not, to another set
of receivers. Similar to the classical case, the scarcity of certain
resources, like long-distance optical cables, means that signals between
different nodes use a common communication channel. This implies that we need
to multiplex/demultiplex quantum signals from/to different users (nodes). In
order to achieve this, here we use OAM to route the quantum state
$|\psi\rangle_{q}$ between different users.

In this article we have discussed several network architectures for pairwise
communication between multiple parties. Starting from a simple, one-to-one
network, we then developed one-to-many and fully-connected networks for
distributing quantum states. We have shown that a fully-connected network with
$n$ nodes can be achieved with minimal resources: a single quantum sorter
acting as a MUX/DEMUX, connected to all $n$ nodes. Moreover, this fully-
connected network requires only $n-1$ OAM values. Finally, we have developed a
novel entanglement-distribution protocol which has several advantages compared
to the current wavelength-based networks.

The central element of all the networks discussed here is the quantum sorter
which acts as a MUX/DEMUX. The quantum sorter has a cyclic property which was
used extensively in building a multitude of network architectures. These
networks can be used to distribute quantum states between nodes, both in
prepare-and-measure protocols (BB84) and in entanglement-based ones (E91,
BBM92).

The protocols described here can be implemented either in optical fibers or in
free-space. As future applications, we envisage our protocols to be used in a
wide-range of communication tasks, such as terrestrial networks (intra- and
inter-city), satellite-to-satellite or satellite-to-ground.

###### Acknowledgements.

The authors acknowledge support from a grant of the Romanian Ministry of
Research and Innovation, PCCDI-UEFISCDI, project number PN-
III-P1-1.2-PCCDI-2017-0338/79PCCDI/2018, within PNCDI III. R.I. acknowledges
support from PN 19060101/2019-2022. S.A. acknowledges the support of contract
PN 23 21 01 05 funded by the Romanian Ministry of Research, Innovation and
Digitalization and by the Extreme Light Infrastructure Nuclear Physics Phase
II, a project co-financed by the Romanian Government and the European Union
through the European Regional Development Fund and the Competitiveness
Operational Programme (No. 1/07.07.2016, COP, ID 1334). G.A.B. and C.K.: this
research was funded by the Ministry of Research, Innovation and Digitization,
CNCS-UEFISCDI via project LACAS grant number PN-III-P2-2.1-PED-2021-1233.
T.A.I. is supported by National Science and Technology Council, Taiwan, under
Grant no. NSTC 112-2112-M-032-008-MY3, 112-2811-M-032-002-MY3 and
111-2923-M-032-002-MY5.

## References

  * Fröhlich  _et al._ [2013] B. Fröhlich, J. F. Dynes, M. Lucamarini, A. W. Sharpe, Z. Yuan, and A. J. Shields, A quantum access network, Nature 501, 69 (2013).
  * Yin  _et al._ [2022] H.-L. Yin, Y. Fu, C.-L. Li, C.-X. Weng, B.-H. Li, J. Gu, Y.-S. Lu, S. Huang, and Z.-B. Chen, Experimental quantum secure network with digital signatures and encryption, National Science Review 10, nwac228 (2022).
  * Richardson [2010] D. J. Richardson, Filling the light pipe, Science 330, 327 (2010).
  * Sakaguchi  _et al._ [2012] J. Sakaguchi, B. J. Puttnam, W. Klaus, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, K. Imamura, H. Inaba, K. Mukasa, _et al._ , 305 Tb/s space division multiplexed transmission using homogeneous 19-core fiber, Journal of Lightwave Technology 31, 554 (2012).
  * Sakaguchi  _et al._ [2019] J. Sakaguchi, W. Klaus, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Nakanishi, T. Taru, T. Takahata, and T. Kobayashi, 228-spatial-channel bi-directional data communication system enabled by 39-core 3-mode fiber, Journal of Lightwave Technology 37, 1756 (2019).
  * Ryf  _et al._ [2012] R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, _et al._ , Mode-division multiplexing over 96 km of few-mode fiber using coherent 6$\times$ 6 MIMO processing, Journal of Lightwave Technology 30, 521 (2012).
  * Huang  _et al._ [2013] H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. Tur, B. Erkmen, K. Birnbaum, _et al._ , 100 Tbit/s free-space data link using orbital angular momentum mode division multiplexing combined with wavelength division multiplexing, in  _2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC)_ (IEEE, 2013) pp. 1–3.
  * Bozinovic  _et al._ [2013] N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers, Science 340, 1545 (2013).
  * Molina-Terriza  _et al._ [2001] G. Molina-Terriza, J. P. Torres, and L. Torner, Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum, Phys. Rev. Lett. 88, 013601 (2001).
  * Andrews [2011] D. L. Andrews,  _Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces_ (Academic Press, 2011).
  * Zhou  _et al._ [2019] Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, Using all transverse degrees of freedom in quantum communications based on a generic mode sorter, Opt. Express 27, 10383 (2019).
  * Uribe-Patarroyo  _et al._ [2013] N. Uribe-Patarroyo, A. Fraine, D. S. Simon, O. Minaeva, and A. V. Sergienko, Object Identification Using Correlated Orbital Angular Momentum States, Phys. Rev. Lett. 110, 043601 (2013).
  * D’Ambrosio  _et al._ [2013] V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, Photonic polarization gears for ultra-sensitive angular measurements, Nature Communications 4, 2432 (2013).
  * Tamburini  _et al._ [2006] F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Overcoming the Rayleigh Criterion Limit with Optical Vortices, Phys. Rev. Lett. 97, 163903 (2006).
  * Pălici  _et al._ [2022] A. M. Pălici, T.-A. Isdrailă, S. Ataman, and R. Ionicioiu, Interaction-free imaging of multipixel objects, Phys. Rev. A 105, 013529 (2022).
  * Lavery  _et al._ [2013] M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, Detection of a Spinning Object Using Light’s Orbital Angular Momentum, Science 341, 537 (2013).
  * Cvijetic  _et al._ [2015] N. Cvijetic, G. Milione, E. Ip, and T. Wang, Detecting lateral motion using light’s orbital angular momentum, Scientific Reports 5, 15422 (2015).
  * Ingerslev  _et al._ [2018] K. Ingerslev, P. Gregg, M. Galili, F. Da Ros, H. Hu, F. Bao, M. A. U. Castaneda, P. Kristensen, A. Rubano, L. Marrucci, _et al._ , 12 mode, WDM, MIMO-free orbital angular momentum transmission, Optics Express 26, 20225 (2018).
  * Sit  _et al._ [2018] A. Sit, R. Fickler, F. Alsaiari, F. Bouchard, H. Larocque, P. Gregg, L. Yan, R. W. Boyd, S. Ramachandran, and E. Karimi, Quantum cryptography with structured photons through a vortex fiber, Optics Letters 43, 4108 (2018).
  * Zhang  _et al._ [2020] J. Zhang, J. Liu, L. Shen, L. Zhang, J. Luo, J. Liu, and S. Yu, Mode-division multiplexed transmission of wavelength-division multiplexing signals over a 100-km single-span orbital angular momentum fiber, Photon. Res. 8, 1236 (2020).
  * Sit  _et al._ [2017] A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, High-dimensional intracity quantum cryptography with structured photons, Optica 4, 1006 (2017).
  * Nikolaeva  _et al._ [2022] A. S. Nikolaeva, E. O. Kiktenko, and A. K. Fedorov, Efficient realization of quantum algorithms with qudits (2022), arXiv:2111.04384 [quant-ph] .
  * Ralph  _et al._ [2007] T. C. Ralph, K. J. Resch, and A. Gilchrist, Efficient Toffoli gates using qudits, Phys. Rev. A 75, 022313 (2007).
  * Sheridan and Scarani [2010] L. Sheridan and V. Scarani, Security proof for quantum key distribution using qudit systems, Phys. Rev. A 82, 030301 (2010).
  * Cerf  _et al._ [2002] N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Security of quantum key distribution using $\mathit{d}$-level systems, Phys. Rev. Lett. 88, 127902 (2002).
  * Ecker  _et al._ [2019] S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, Overcoming Noise in Entanglement Distribution, Phys. Rev. X 9, 041042 (2019).
  * Liu and Fan [2009] Z. Liu and H. Fan, Decay of multiqudit entanglement, Phys. Rev. A 79, 064305 (2009).
  * Bechmann-Pasquinucci and Tittel [2000] H. Bechmann-Pasquinucci and W. Tittel, Quantum cryptography using larger alphabets, Phys. Rev. A 61, 062308 (2000).
  * Bouchard  _et al._ [2017] F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, High-dimensional quantum cloning and applications to quantum hacking, Science Advances 3, e1601915 (2017).
  * Cozzolino  _et al._ [2019] D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, Air-core fiber distribution of hybrid vector vortex-polarization entangled states, Advanced Photonics 1, 046005 (2019).
  * D’Ambrosio  _et al._ [2012] V. D’Ambrosio, E. Nagali, S. P. Walborn, S. S. Leandro Aolita, L. Marrucci, and F. Sciarrino, Complete experimental toolbox for alignment-free quantum communication, Nature Communications 3 (2012).
  * Vallone  _et al._ [2014] G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons, Phys. Rev. Lett. 113, 060503 (2014).
  * Lee  _et al._ [2017] D. Lee, H. Sasaki, H. Fukumoto, K. Hiraga, and T. Nakagawa, Orbital Angular Momentum (OAM) Multiplexing: An Enabler of a New Era of Wireless Communications, IEICE Transactions on Communications E100.B, 1044 (2017).
  * Gregg  _et al._ [2015] P. Gregg, P. Kristensen, and S. Ramachandran, Conservation of orbital angular momentum in air-core optical fibers, Optica 2, 267 (2015).
  * Leach  _et al._ [2002] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88, 257901 (2002).
  * O’Sullivan  _et al._ [2012] M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, Near-perfect sorting of orbital angular momentum and angular position states of light, Opt. Express 20, 24444 (2012).
  * Ionicioiu [2016] R. Ionicioiu, Sorting quantum systems efficiently, Scientific Reports 6, 25356 (2016).
  * Fu  _et al._ [2018] D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, Realization of a scalable Laguerre-Gaussian mode sorter based on a robust radial mode sorter, Opt. Express 26, 33057 (2018).
  * Zhou  _et al._ [2022] Z. Zhou, P. Li, J. Ma, S. Zhang, and Y. Gu, Generation and Detection of Optical Vortices with Multiple Cascaded Spiral Phase Plates, Photonics 9, 354 (2022).
  * Zhou  _et al._ [2017] Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. Hashemi Rafsanjani, A. E. Willner, and R. W. Boyd, Sorting photons by radial quantum number, Phys. Rev. Lett. 119, 263602 (2017).
  * Gu  _et al._ [2018] X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, Gouy phase radial mode sorter for light: Concepts and experiments, Phys. Rev. Lett. 120, 103601 (2018).
  * Ionicioiu [2023] R. Ionicioiu, Interferometric mass spectrometry, Journal of the American Society for Mass Spectrometry 34, 1160 (2023).
  * Isdrailă  _et al._ [2019] T.-A. Isdrailă, C. Kusko, and R. Ionicioiu, Cyclic permutations for qudits in d dimensions, Scientific Reports 9, 6337 (2019).
  * Asadian  _et al._ [2016] A. Asadian, P. Erker, M. Huber, and C. Klöckl, Heisenberg-weyl observables: Bloch vectors in phase space, Phys. Rev. A 94, 010301 (2016).
  * Pălici  _et al._ [2020] A. M. Pălici, T.-A. Isdrailă, S. Ataman, and R. Ionicioiu, OAM tomography with Heisenberg-Weyl observables, Quantum Science and Technology 5, 045004 (2020).
  * Daboul  _et al._ [2003] J. Daboul, X. Wang, and B. C. Sanders, Quantum gates on hybrid qudits, Journal of Physics A: Mathematical and General 36, 2525 (2003).
  * Khan and Perkowski [2006] F. S. Khan and M. Perkowski, Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition, Theoretical Computer Science 367, 336 (2006).
  * Fickler  _et al._ [2014] R. Fickler, R. Lapkiewicz, M. Huber, M. P. J. Lavery, M. J. Padgett, and A. Zeilinger, Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information, Nature Communications 5, 4502 (2014).
  * Wengerowsky  _et al._ [2018] S. Wengerowsky, S. K. Joshi, F. Steinlechner, H. Hubel, and R. Ursin, An entanglement-based wavelength-multiplexed quantum communication network, Nature 564, 225 (2018).
  * Joshi  _et al._ [2020] S. K. Joshi, D. Aktas, S. Wengerowsky, M. Loncaric, S. P. Neumann, T. S. Bo Liu, G. C. Lorenzo, Z. Samec, L. Kling, A. Qiu, M. Razavi, M. Stipcevic, J. G. Rarity, and R. Ursin, A trusted-node-free eight-user metropolitan quantum communication network, Science Advances 6, eaba0959 (2020).

##  Appendix A Quantum OAM sorter

The OAM sorter $U_{d}$ used here is an example of the more general universal
quantum sorter introduced in [37]. The sorter is a Mach-Zehnder interferometer
with $d$ paths and with different phase-shifts in each arm. For path $k$, the
phase-shifts are given by Dove prisms rotated with angles $\alpha_{k}=k\pi/d$,
$k=0,\ldots,d-1$, see Figure 6. The $F_{d},F_{d}^{\dagger}$ are discrete
Fourier gates acting only on the path degree of freedom; they are equivalent
to multi-mode couplers with appropriate phases:

$F_{d}|k\rangle_{\text{path}}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\omega^{kj}|j\rangle_{\text{path}}$
(12)

with $\omega=e^{2\pi i/d}$ a root of unity of order $d$. We also have
$F_{d}^{4}=I,F_{d}^{\dagger}=F_{d}^{3}$ and
$F_{d}^{2}|k\rangle=|-k\rangle=|d-k\rangle$.

Due to constructive interference, a particle with OAM $|j\rangle$ entering the
interferometer on input $0$, will exit with unit probability on path $j$:
$|j\rangle_{\mathrm{oam}}|0\rangle_{\mathrm{path}}\rightarrow|j\rangle_{\mathrm{oam}}|j\rangle_{\mathrm{path}}$.

Figure 6: The OAM sorter $U_{d}$ is a Mach-Zehnder interferometer with $d$
paths and with different phase-shifts $\alpha_{k}$ on each path. A photon with
OAM $|j\rangle$ enters on path $0$ and exits through path $|j\rangle$ with
unit probability, i.e., is sorted according to its OAM value.

This might seem quite abstract. We now give a physical intuition of how the
sorter works. Consider first the simplest case of sorting two states, $d=2$.
In this case the sorter is a Mach-Zehnder interferometer with a Dove prism in
each arm; the prisms are rotated relative to each other by $\pi/2$ [35]. Thus,
an incoming photon with $\ell=0$ will get a relative phase-shift (between the
two arms) $\delta\varphi=0$ and will exit, with unit probability, through
output port 0 (constructive interference). On the other hand, an incoming
photon with $\ell=1$ will get a relative phase-shift $\delta\varphi=\pi/2$ and
will exit through output port 1, again with unit probability (destructive
interference).

The general case is similar. The sorter is now a Mach-Zehnder interferometer
with $d$ paths and its role is to induce state-dependent relative phases
between the arms. For example, a photon in the state $\ell=0$ will have
relative phases $(0,0,\ldots,0)$ and will exit with unit probability on exit
0. A photon with $\ell=1$ will have relative phases
$(0,\pi/d,\ldots,(d-1)\pi/d)$ and will exit with unit probability on exit 1,
and so on. Thus the relative phases experienced by a photon depend on the OAM
value $\ell=k$. In turn, the relative phases determine, through
constructive/destructive interference, the output $k$ through which the photon
will exit.

##  Appendix B BB84

The OAM-assisted BB84 protocol in polarization encoding can be implemented
with the network shown in Figure 1. In this case there are $d$ senders and
receivers. Sender $i$ sends to receiver $i$ the qubit
$|\psi_{i}\rangle_{q}:=\alpha_{i}|H\rangle+\beta_{i}|V\rangle$, encoded in
polarization. Each pair (sender $i$, receiver $i$) has allocated an OAM value
$i\in\\{0,\ldots,d-1\\}$, thus the initial state at sender $i$ is

$|\psi_{i}\rangle_{q}|i\rangle_{\text{oam}}|i\rangle_{\text{path}}$ (13)

Since the polarization qubit is unchanged by the MUX/DEMUX network, we will
thus omit it for simplicity. The action of the device is

$|i\rangle_{\text{oam}}|i\rangle_{\text{path}}\xrightarrow{U_{d}^{\dagger}}|i\rangle_{\text{oam}}|0\rangle_{\text{path}}\xrightarrow{U_{d}}|i\rangle_{\text{oam}}|i\rangle_{\text{path}}$
(14)

In order to measure the polarization state, each receiver has the standard
BB84 setup: a beam-splitter (BS), a half-wave plate (HWP), two polarizing
beam-splitters (PBS) and 4 single-photon detectors (SPDs).

The protocol also works for the networks discussed in Section III, if the
sender chooses an appropriate OAM value. In Figure 3, the sender $s$ choses an
OAM value equal to $r$ in order to send to receiver $r$ (see Appendix D):

$|\psi_{sr}\rangle_{q}|r\rangle_{\text{oam}}|s\rangle_{\text{path}}$ (15)

In Figure 2, the sender $s$ communicates with receiver $r$:

$|\psi_{sr}\rangle_{q}|sqd_{r}+rpd_{s}\rangle_{\text{oam}}|s\rangle_{\text{path}}$
(16)

where $p$ and $q$ are the Bezout coefficients of the identity
$pd_{s}+qd_{r}=1$ and $d_{s}$ and $d_{r}$ are the number of senders and
receivers respectively (see Appendix C).

##  Appendix C Co-prime case

For $d_{s}$ senders and $d_{r}$ receivers we can prove that any sender $s$ can
transmit to any receiver $r$ if they share an OAM state
$|l_{sr}\rangle_{\text{oam}}$ satisfying the conditions

$\displaystyle l_{sr}=$ $\displaystyle pd_{s}+s$ $\displaystyle l_{sr}=$
$\displaystyle qd_{r}+r$

with $s\in\\{0,\ldots,d_{s}-1\\}$, $r\in\\{0,\ldots,d_{r}-1\\}$ and $p$, $q$
integers. This is equivalent to the following congruence relations

$\displaystyle l_{sr}\equiv$ $\displaystyle s\;(\mathrm{mod}\;d_{s})$
$\displaystyle l_{sr}\equiv$ $\displaystyle r\;(\mathrm{mod}\;d_{r})$ (17)

From the Chinese remainder theorem we know that $d_{s}$ and $d_{r}$ need to be
co-prime and only one $l_{sr}\in\\{0,\ldots,d_{s}d_{r}-1\\}$ satisfies these
conditions for fixed $s$ and $r$. This means that we can design a network with
only a $d_{s}$-dimensional MUX $U_{d_{s}}^{\dagger}$ and a $d_{r}$-dimensional
DEMUX $U_{d_{r}}$, see Figure 2.

Given a sender $s$ and a receiver $r$, we can find their assigned OAM state
$|l_{sr}\rangle_{\text{oam}}$ by solving the congruence relations (17). This
gives $l_{sr}=sqd_{r}+rpd_{s}$, where $p$ and $q$ are the Bezout coefficients
of the identity $pd_{s}+qd_{r}=1$, which are calculated using the extended
Euclidean algorithm.

The protocol requires the number of senders and the number of receivers to be
coprime. In practice we can always satisfy the co-primality condition by
embedding the number of senders and/or receivers into larger sets with co-
prime cardinality.

##  Appendix D General point-to-multipoint OAM network

Our goal is to establish a general pairwise communication protocol between
$d_{s}$ senders and $d_{r}$ receivers, such that any sender $s$ can
communicate with any receiver $r$. The protocol must be free from the co-
primality condition discussed in Appendix C. In this case there are
$d_{s}d_{r}$ pairs (sender $s$, receiver $r$), with
$s\in\\{0,\ldots,d_{s}-1\\}$ and $r\in\\{0,\ldots,d_{r}-1\\}$.

We group the OAM values as in Table 3, where each line shows the OAMs
available to sender $s$, and each column the OAMs received by receiver $r$.

Table 3:  OAM correspondence chart for $d_{s}$ senders and $d_{r}$ receivers. sender receiver | $0$ | $1$ | … | ${d_{r}-1}$
---|---|---|---|---
$0$ | $|0\rangle$ | $|d_{s}\rangle$ | … | $|(d_{r}-1)d_{s}\rangle$
$1$ | $|1\rangle$ | $|d_{s}+1\rangle$ | … | $|(d_{r}-1)d_{s}+1\rangle$
… | … | … | … | …
${d_{s}-1}$ | $|d_{s}-1\rangle$ | $|2d_{s}-1\rangle$ | … | $|d_{r}d_{s}-1\rangle$

Notice that the element on row $s$, column $r$ is $|s+rd_{s}\rangle$, thus we
can assign each pair (sender $s$, receiver $r$) a unique OAM state
$|l_{sr}\rangle_{\text{oam}}$, with $\;l_{sr}=s+rd_{s}$, requiring
$d_{s}d_{r}$ OAM states in total.

Example: For two senders ($d_{s}=2$) and three receivers $(d_{r}=3)$, if
sender $0$ ($s=0$) wants to communicate with receiver $1$ ($r=1$), the
appropriate OAM state is
$|l_{sr}\rangle_{\text{oam}}=|s+rd_{s}\rangle_{\text{oam}}=|0+1\cdot
2\rangle_{\text{oam}}=|2\rangle_{\text{oam}}$, see Table 4.

Table 4:  Reference chart for $d_{s}=2$, $d_{r}=3$ and OAM DoF spanning from $|0\rangle$ to $|5\rangle$. sender receiver | 0 | 1 | 2
---|---|---|---
0 | $|0\rangle$ | $|2\rangle$ | $|4\rangle$
1 | $|1\rangle$ | $|3\rangle$ | $|5\rangle$

Figure 3 (C) shows the physical implementation of the general point-to-
multipoint network for two senders and four receivers. In general, sender $s$
starts with the OAM state $|s+rd_{s}\rangle_{\text{oam}}$ on path
$|s\rangle_{\text{path}}$. The state is then multiplexed into a single
transmission channel

$|s+rd_{s}\rangle_{\text{oam}}|s\rangle_{\text{path}}\xrightarrow{U_{d_{s}}^{\dagger}}|s+rd_{s}\rangle_{\text{oam}}|0\rangle_{\text{path}}$
(18)

The demultiplexer at the receiver’s end is much larger, splitting the channel
into $d_{s}d_{r}$ paths, one for each OAM. These are then grouped into $d_{r}$
groups of $d_{s}$ channels and multiplexed back together by $d_{r}$
multiplexers $U_{d_{s}}^{\dagger}$. For simplicity, we can take the
demultiplexer and the $d_{r}$ multiplexers to be a single device called a
group demultiplexer $G$, represented by a dashed outline in the figures (see
Appendix E). We mark the paths inside the device as local paths, therefore

$\displaystyle|s+rd_{s}\rangle_{\text{oam}}|0\rangle_{\text{path}}$
$\displaystyle\xrightarrow{U_{d_{s}d_{r}}}$
$\displaystyle|s+rd_{s}\rangle_{\text{oam}}|s+rd_{s}\rangle_{\text{local}}$
$\displaystyle|s+rd_{s}\rangle_{\text{oam}}|s+rd_{s}\rangle_{\text{local}}$
$\displaystyle\xrightarrow{U_{d_{s}}^{\dagger}}$
$\displaystyle|s+rd_{s}\rangle_{\text{oam}}|rd_{s}\rangle_{\text{local}}$

There are gaps of $d_{s}$ between output ports on the local path. We can map
back to the global path by dividing the output port by $d_{s}$
($|rd_{s}\rangle_{\text{local}}\mapsto|r\rangle_{\text{path}}$). The full
action of the group demultiplexer $G^{d_{r}}_{d_{s}}$ (makes $d_{r}$ groups of
dimension $d_{s}$) is then

$|s+rd_{s}\rangle_{\text{oam}}|0\rangle_{\text{path}}\xrightarrow{G^{d_{r}}_{d_{s}}}|s+rd_{s}\rangle_{\text{oam}}|r\rangle_{\text{path}}$
(19)

Finally, the action of the protocol is

$|s+rd_{s}\rangle_{\text{oam}}|s\rangle_{\text{path}}\xrightarrow{U_{d_{s}}^{\dagger},\;G^{d_{r}}_{d_{s}}}|s+rd_{s}\rangle_{\text{oam}}|r\rangle_{\text{path}}$
(20)

If sender $s$ wants, for example, to transmit a qubit to receiver $r$, they
encode the information in the polarization dof
$\alpha_{sr}|H\rangle+\beta_{sr}|V\rangle$ with OAM
$|s+rd_{s}\rangle_{\text{oam}}$ on path $|s\rangle_{\text{path}}$. The
receiver recovers the information encoded in the qubit via polarisation state
detection on path $|r\rangle_{\text{path}}$. This way we can ensure the
general pairwise quantum communication between several parties through a
single channel. Moreover, if the two groups of senders and receivers are the
same, the network becomes effectively a fully-connected network.

##  Appendix E Group demultiplexer

In Figure 3 (B), we introduce the group demultiplexer $G$. Imputing
consecutive OAM numbers on port $|0\rangle_{\text{path}}$ into a demultiplexer
$U_{d}$, we see that it distributes them on consecutive channels
$l\;\text{mod}\;d$ until it resets to output $0$ when $l$ reaches the next
multiple of $d$. A group demultiplexer $G^{d_{r}}_{d_{s}}$ in the same
situation outputs on the same channel $\lfloor\frac{l}{ds}\rfloor$ until it
moves to the next one when $l$ reaches the next multiple of $d_{s}$ as in
Table 5.

Table 5:  Outputs of a group demultiplexer. output OAM | $0$ | $1$ | … | ${d_{s}-1}$
---|---|---|---|---
$0$ | $|0\rangle$ | $|1\rangle$ | … | $|d_{s}-1\rangle$
$1$ | $|d_{s}\rangle$ | $|d_{s}+1\rangle$ | … | $|2d_{s}-1\rangle$
… | … | … | … | …
${d_{r}-1}$ | $|(d_{r}-1)d_{s}\rangle$ | $|(d_{r}-1)d_{s}+1\rangle$ | … | $|d_{r}d_{s}-1\rangle$

The above table is just the transpose of Table 3. Formally, we have

$|l\rangle_{\text{oam}}|k\rangle_{\text{path}}\xrightarrow{G^{d_{r}}_{d_{s}}}|l\rangle_{\text{oam}}|\lfloor\frac{k\oplus
l}{d_{s}}\rfloor\rangle_{\text{path}}$ (21)


[a,b]Maxim Kurkov

# Poisson gauge theory: a review

###### Abstract

In this paper we overview the Poisson gauge theory focusing on the most recent
developments. We discuss the general construction and its symplectic-geometric
interpretation. We consider explicit realisations of the formalism for all
non-commutativities of the Lie algebraic type. We discuss Seiberg-Witten maps
between Poisson gauge field-theoretical models.

##  1 From non-commutative geometry to Poisson gauge theory.

Various approaches to quantum gravity naturally yield non-commutative
geometry, what motivates studies on non-commutative quantum field theory [1].
An important class of field theories on non-commutative space-times is given
by non-commutative _gauge_ theories, see [2] for the latest review.

Let $\mathcal{M}$ be a flat manifold which represents the space-time. The
local coordinates on $\mathcal{M}$ are denoted through $x^{a}$, $a=0,...,n-1$.
The non-commutative structure of the space-time is encoded in the Kontsevich
star product111Of course, the star product is not the only way to treat the
non-commutative geometry, for example, there is the spectral approach, for a
review see [3].  of smooth functions on $\mathcal{M}$,

$f(x)\star g(x)=f(x)\cdot g(x)+\frac{{\rm
i}}{2}\\{f(x),g(x)\\}+...\,\,,\quad\quad\forall
f,g\in\mathcal{C}^{\infty}\big{(}\mathcal{M}\big{)}.$ (1)

In this formula the Poisson bracket is related to the Poisson bivector
$\Theta^{ab}$,

$\\{f(x),g(x)\\}={\Theta^{ab}(x)}\,\,\partial_{a}f(x)\,\,\partial_{b}g(x)\,,$
(2)

whilst the remaining terms, denoted through "…", contain second and higher
derivatives of $f$ an $g$.

According to the novel approach to non-commutative gauge theories, proposed in
[4], non-commutative deformations of the $U(1)$ gauge theory must exhibit the
following fundamental properties.

  * •

The infinitesimal gauge transformations should close the non-commutative
algebra,

$[\delta_{f},\delta_{g}]A_{a}(x)=\delta_{-{\rm
i}[f(x),g(x)]_{\star}}A_{a}(x)\,.$ (3)

  * •

The commutative limit has to reproduce the usual abelian gauge
transformations,

$\lim_{\Theta\to 0}\delta_{f}A_{a}(x)=\partial_{a}f(x)\,.$ (4)

Upon the semi-classical limit the star commutator,

$[f(x),g(x)]_{\star}=f(x)\star g(x)-g(x)\star f(x)={\rm
i}\\{f(x),g(x)\\}+...\,\,,$

tends to ${\rm i}\\{f(x),g(x)\\}$, and the full non-commutative algebra (3)
reduces to the Poisson gauge algebra,

$[\delta_{f},\delta_{g}]A_{a}(x)=\delta_{\\{f(x),g(x)\\}}A_{a}(x)\,.$ (5)

The corresponding gauge theory is called the _Poisson gauge theory_ [5].
Physically this theory provides a semi-classical approximation of the non-
commutative $U(1)$ gauge theory.

The rest of this paper is organised as follows. In Sec. 2 we describe the
general construction of Poisson gauge models. Sec. 3 is devoted to its
geometric interpretation. In Sec. 4 we present explicit formulae for non-
commutativities of the Lie algebraic type. In Sec. 5 we consider the
arbitrariness of the construction and its relation to Seiberg-Witten maps
between Poisson gauge models. We round up with the summary and concluding
remarks in Sec. 6.

##  2 General construction.

### a. Deformed gauge transformations.

For the canonical non-commutativity, i.e. when $\partial_{a}\Theta^{bc}=0$,
the required deformed gauge transformations can be easily constructed,

$\delta_{f}A_{a}(x)=\partial_{a}f(x)+\\{A_{a}(x),f(x)\\}\,.$ (6)

Though for non-constant Poisson bivectors $\Theta^{ab}$ this simple formula is
not compatible with the Poisson gauge algebra (5), it can be generalised in
the following way [6, 4],

$\delta_{f}A_{a}(x)=\gamma^{b}_{a}(x,A(x))\,\partial_{b}f(x)+\\{A_{a}(x),f(x)\\}\,.$
(7)

The matrix $\gamma(x,p)$, entering this relation, is a solution of the _first
master equation,_

$\gamma_{a}^{b}\partial^{a}_{p}\gamma^{d}_{c}-\gamma^{d}_{a}\partial_{p}^{a}\gamma^{b}_{c}+\Theta^{ba}\partial_{a}\gamma^{d}_{c}-\Theta^{da}\partial_{a}\gamma^{b}_{c}-\gamma^{a}_{c}\partial_{a}\Theta^{bd}=0,\quad\partial_{p}^{c}\equiv\frac{\partial}{\partial
p_{c}}\,,$ (8)

which tends to the identity matrix at the commutative limit,

$\qquad\lim_{\Theta\to 0}\gamma_{b}^{a}=\delta_{b}^{a}.$ (9)

One can check by direct calculation that these deformed gauge transformations,
indeed, close the required non-commutative Poisson gauge algebra (5) and
exhibit the correct commutative limit (4).

### b. Deformed field strength.

According to the general strategy of [4], the deformed field strength has to
transform in a covariant way,

$\delta_{f}{\cal F}_{ab}(x)=\\{{\cal F}_{ab}(x),f(x)\\}\,,$ (10)

and it has to reproduce the commutative limit correctly,

$\lim_{\Theta\to 0}{\cal
F}_{ab}(x)=\partial_{a}A_{b}(x)-\partial_{b}A_{a}(x)\,.$ (11)

Such a field strength has been obtained in [5] in the following form,

$\mathcal{F}_{ab}(x)=\rho_{a}^{c}(x,A(x))\,\rho_{b}^{d}(x,A(x))\,\big{(}\gamma_{c}^{l}(x,A(x))\,\partial_{l}A_{d}-\gamma_{d}^{l}(x,A(x))\,\partial_{l}A_{c}+\\{A_{c}\,,A_{d}\\}\big{)}\,.$
(12)

The matrix $\rho(x,p)$, which enters this formula, has to obey the _second
master equation_ ,

$\gamma^{c}_{d}\partial_{p}^{d}\rho_{a}^{b}+\rho_{a}^{d}\partial_{p}^{b}\gamma_{d}^{c}+\Theta^{cd}\partial_{d}\rho^{b}_{a}=0\,,$
(13)

and it has to recover the identity matrix at the commutative limit,

$\lim_{\Theta\to 0}\rho^{a}_{b}=\delta_{b}^{a}\,.$ (14)

### c. Gauge-covariant derivative.

For any field $\psi(x)$, which transforms in a covariant way,

$\delta_{f}\psi(x):=\\{f(x),\psi(x)\\}\,,$ (15)

the gauge-covariant derivative has to transform properly,

$\delta_{f}\left({\cal D}_{a}\psi(x)\right)=\\{{\cal D}_{a}\psi(x),f(x)\\}\,,$
(16)

and it has to exhibit a correct commutative limit,

$\lim_{{\Theta}\to 0}{\cal D}_{a}\psi(x)=\partial_{a}\psi(x)\,.$ (17)

A suitable gauge-covariant derivative has been constructed in [5] as follows,

${\cal
D}_{a}\psi(x)=\rho_{a}^{i}(x,A(x))\,\big{(}\gamma^{l}_{i}(x,A(x))\,\partial_{l}\psi(x)+\\{A_{a}(x),\psi(x)\\}\big{)}\,.$
(18)

### d. Gauge-covariant equations of motion.

The deformed covariant derivative (18) and the deformed field strength (12)
allow to write down the “natural" equations of motion (e.o.m.),

${\cal D}_{a}{\cal F}^{ab}=0\,,$ (19)

which are manifestly gauge-covariant,

$\delta_{f}\left({\cal D}_{a}{\cal F}^{ab}\right)=\\{{\cal D}_{a}{\cal
F}^{ab},f\\},$ (20)

and which recover the first pair of Maxwell’s equations in vacuum at the
commutative limit. The gauge-covariance condition (20) insures that the
infinitesimal gauge transformations (7) map solutions of the e.o.m. (19) onto
other solutions.

For the three-dimensional space-time, equipped with the $\mathfrak{su}(2)$
non-commutativity222Various aspects of the $\mathfrak{su}(2)$ non-
commutativity have been studied in [13, 14, 15]. , these “natural" equations
of motion are equivalent to the Euler-Lagrange equations, which come out from
the gauge-invariant classical action,

$S[A]=\int_{\mathcal{M}}{\rm
d}x\,\left(-\frac{1}{4}\mathcal{F}_{ab}\mathcal{F}^{ab}\right),$ (21)

see [16] for details. A more profound discussion on the gauge-covariant
equations of motion can be found in [17].

##  3 Symplectic geometric interpretation.

The symplectic geometric interpretetion of the Poisson gauge formalism,
designed in [16, 18], is based on the two pillars: the symplectic embedding,
and the set of constraints in the outcoming symplectic space. Extending the
Poisson structure from $\mathcal{M}$ to $T^{*}\mathcal{M}$,

$\\{x^{a},x^{b}\\}=\Theta^{ab}(x)\,,\qquad\\{x^{a},p_{b}\\}=\gamma^{a}_{b}(x,p)\,,\qquad\\{p_{a},p_{b}\\}=0\,,$
(22)

we construct the symplectic embedding. One can easily see that the first
master equation (8) for $\gamma$ is equivalent to the Jacobi identity on
$T^{*}\mathcal{M}$. By definition, the set of constraints in this symplectic
space reads,

$\Phi_{a}(x,p):=p_{a}-A_{a}(x),\quad\quad a=0,...,n-1\,.$ (23)

The constituents of the Poisson gauge formalism exhibit natural expressions in
terms of this symplectic geometric construction. In particular, the deformed
gauge transformation (6) can be represented through a simple Poisson bracket
on $T^{*}\mathcal{M}$,

$\delta_{f}A_{a}(x)=\\{f(x),\Phi_{a}(x,p)\\}_{\Phi(x,p)=0}\,.$ (24)

The deformed field strength (12) and the gauge-covariant derivative (18) can
also be represented in a similar manner,

${\cal
F}_{ab}(x)=\\{\Phi^{\prime}_{a}(x,p),\Phi^{\prime}_{b}(x,p)\\}_{\Phi^{\prime}(x,p)=0}\,,\qquad{\cal
D}_{a}\psi(x)=\\{\psi(x),\Phi^{\prime}_{a}(x,p)\\}_{\Phi^{\prime}(x,p)=0}\,,$
(25)

where a new set of the constraints is defined as follows,

$\Phi^{\prime}_{a}(x,p):=\rho_{a}^{b}(x,p)\,\Phi_{b}(x,p)\,.$ (26)

We remind that $\rho^{a}_{b}$ obeys the second master equation (13) and
exhibits the commutative limit (14). Since $\rho$ is a non-degenerate matrix,
the new and the old constraint surfaces coincide,

$\Phi^{\prime}_{a}(x,p)=0\,,\quad\Leftrightarrow\quad\Phi_{a}(x,p)=0\,.$ (27)

##  4 Lie algebraic non-commutativities and universal solutions.

Consider a class of Poisson bivectors, which are linear in coordinates,

$\Theta^{ab}=f^{ab}_{c}\,x^{c}\,.$

The constants $f^{ab}_{c}$ satisfy the Jacobi identity,

$f^{ec}_{a}f^{bd}_{c}+f^{bc}_{a}f^{de}_{c}+f^{dc}_{a}f^{eb}_{c}=0\,,$

therefore these constants can be seen as the structure constants of a Lie
algebra. For this purpose we say that the corresponding non-commutativities
are of the Lie algebraic type.

In [18] a special solution of the first master equation (8) has been presented
in terms of a single matrix-valued function, which is valid for _all_ Poisson
bivectors of the Lie algebraic type. Introducing the matrix333According to our
notations, for any matrix $B$ the upper index enumerates strings, whilst the
lower one enumerates columns. In particular for a product of two matrices $B$
and $C$ we write $[B\,C]^{i}_{j}=B^{i}_{k}\,C^{k}_{j}.$,

$[\hat{p}]^{b}_{c}=-{\rm i}f^{ab}_{c}p_{a},$

we can represent this universal solution as follows,

$\gamma(x,p)=G(\hat{p}),\quad\quad G(z):=\frac{{\rm
i}\,z}{2}+\frac{z}{2}\cot{\frac{z}{2}}\,.$ (28)

Also the second master equation (13) exhibits the universal solution for
$\rho$, expressed as a matrix valued function of the same matrix variable
$\hat{p}$. The explicit formula reads [16],

$\rho(x,p)=F(\hat{p}),\quad F(z)=\frac{e^{{\rm i}z}-1}{{\rm i}\,z}\,.$ (29)

These universal solutions for $\gamma$ and $\rho$ exhibit a simple connection
[16],

$\rho^{-1}=\gamma-{\rm i}\,\hat{p}.$ (30)

Substituting the universal solutions (28) and (29) in the formulae of Sec. 2,
we can build the Poisson gauge model completely: we know the deformed gauge
transformations (7), which close the Poisson gauge algebra (5), and we also
know the gauge-covariant equations of motion (19).

### Example.

In order to illustrate the formalism, we consider the $\kappa$-Minkowski non-
commutativity, which arises in various contexts, see e.g. [7, 8, 9, 10, 11,
12] and for the recent progress. The Poisson bivector, which corresponds to
the (generalised) $\kappa$-Minkowski non-commutativity, is defined in the
following way,

$\Theta^{ab}=2(\omega^{a}x^{b}-\omega^{b}x^{a})\,,$ (31)

where $\omega^{a}$, $a=0,...,n-1$, stand for the deformation parameters. The
corresponding structure constants read,

$f^{ab}_{c}=2(\omega^{a}\delta^{b}_{c}-\omega^{b}\delta^{a}_{c})\,.$ (32)

The universal solutions have been calculated in [16],

$\displaystyle\gamma^{a}_{b}(A)$ $\displaystyle=$ $\displaystyle(\omega\cdot
A)\left[1+\coth{(\omega\cdot A)}\right]\delta^{a}_{b}+\frac{1-(\omega\cdot
A)-(\omega\cdot A)\coth{(\omega\cdot A)}}{\omega\cdot A}\,\omega^{a}A_{b}\,,$
$\displaystyle\rho_{b}^{a}(A)$ $\displaystyle=$
$\displaystyle\frac{e^{2(\omega\cdot A)}-1}{2(\omega\cdot
A)}\,\,\delta^{a}_{b}+\frac{1+2(\omega\cdot A)-e^{2(\omega\cdot
A)}}{2(\omega\cdot A)^{2}}\,\,\omega^{a}A_{b}\,.$ (33)

These expressions are different from the ones, obtained in the previous
studies [19], using different techniques,

$\displaystyle\tilde{\gamma}^{a}_{b}(A)$ $\displaystyle=$
$\displaystyle\left[\sqrt{1+(\omega\cdot A)^{2}}+(\omega\cdot
A)\right]\delta^{a}_{b}-\omega^{a}\,A_{b}\,,$
$\displaystyle\tilde{\rho}^{a}_{b}(A)$ $\displaystyle=$
$\displaystyle\left[\sqrt{1+(\omega\cdot A)^{2}}+(\omega\cdot
A)\right]\,\delta_{b}^{a}-\frac{\sqrt{1+(\omega\cdot A)^{2}}+(\omega\cdot
A)}{\sqrt{1+(\omega\cdot A)^{2}}}\,\omega^{a}A_{b}\,.$ (34)

Therefore we have more than one Poisson gauge model, which corresponds to the
Poisson bivector (31). In the next section we take a closer look at the
arbitrariness of the construction.

##  5 Arbitrariness and Seiberg-Witten maps.

Consider a given Poisson bivector $\Theta^{ab}(x)$. Let $\gamma(x,p)$ and
$\rho(x,p)$ be solutions of the master equations (8) and (13), which have the
commutative limits (9) and (14) respectively. For any invertible field
redefinition,

$A\to\tilde{A}(A),$ (35)

obeying the condition,

$\lim_{\Theta\to 0}\tilde{A}(A)=A,$ (36)

the quantities

$\tilde{\gamma}^{a}_{b}(x,\tilde{A})=\Bigg{(}\gamma_{c}^{a}(x,A)\cdot\frac{\partial\tilde{A}_{b}}{\partial
A_{c}}\Bigg{)}\Bigg{|}_{A=A(\tilde{A})},\quad\tilde{\rho}_{a}^{i}(x,\tilde{A})=\Bigg{(}\frac{\partial
A_{s}}{\partial\tilde{A}_{i}}\cdot\rho_{a}^{s}(x,A)\Bigg{)}\Bigg{|}_{A=A(\tilde{A})}\,,$
(37)

are again the solutions of the master equations, which have correct
commutative limits, see [5, 16] for details. Therefore, the matrices
$\tilde{\gamma}(x,p)$ and $\tilde{\rho}(x,p)$ define one more Poisson gauge
model for the same Poisson bivector $\Theta^{ab}(x)$.

The infinitesimal gauge transformation of the new model read,

$\tilde{\delta}_{f}\tilde{A}_{a}(x)=\tilde{\gamma}^{i}_{a}(x,\tilde{A}(x))\,\partial_{i}f(x)+{\\{\tilde{A}_{a}(x),f(x)\\}}\,.$
(38)

Upon the field redefinition the gauge orbits are mapped onto the gauge orbits
[5, 16],

$\tilde{A}\left(A+\delta_{f}A\right)=\tilde{A}(A)+\tilde{\delta}_{f}\tilde{A}(A)\,,$
(39)

thus the invertible field redefinition (35) defines a Seiberg-Witten map
between the two models.

### Example.

In the $\kappa$-Minkowski case, the universal solutions (33) and the solutions
(34), which have been previously obtained in [19], are connected through the
relation (37) and the following Seiberg-Witten map [16],

$\tilde{A}_{a}=\frac{\sinh(\omega\cdot A)}{\omega\cdot
A}\,A_{a}\,\quad\Leftrightarrow\quad
A_{a}=\frac{\mathrm{arcsinh}\,(\omega\cdot\tilde{A})}{\omega\cdot\tilde{A}}\,\tilde{A}_{a}\,.$

##  6 Summary and concluding remarks.

The main points are the following.

  * •

For a given Poisson bivector $\Theta^{ab}$, defining the non-commutativity,
the main constituents of the Poisson gauge formalism, viz the deformed gauge
transformations, the deformed field strength and the deformed gauge-covariant
derivative, are completely determined by the matrices $\gamma$ and $\rho$,
which solve the two master equations.

  * •

The Poisson gauge theory exhibits an elegant symplectic-geometric description
in terms of symplectic embeddings and constrains in the extended space.

  * •

There exist universal solutions of the master equations, which are valid for
_all_ non-commutativities of the Lie algebraic type.

  * •

Invertible field redefinitions give rise to new solutions of the master
equations. All the outcoming Poisson gauge models are connected with each
other through Seiberg-Witten maps.

Of course, our mini-review does not cover all aspects of Poisson gauge theory.
In particular, an important connection with the $L_{\infty}$-algebras has not
been discussed. The interested reader is referred to [18] and [20]. See also
[21, 22] for applications of the $L_{\infty}$-structures to generalised gauge
symmetries and non-commutative gravity.

In conclusion, we would like to outline a few open questions. First, one has
to generalise the present results, introducing the charged matter. Second, it
would be very interesting to go beyond the semi-classical approximation
towards the full non-commutative gauge algebra (3). Third, one may wonder what
are the space-time symmetries of Poisson gauge models. In particular, one may
study the fate of the discrete symmetries. In addition to the obvious
phenomenological interest444See for instance [23] and references therein. in
the discrete symmetries breaking, the preserved symmetries such as PT allow
the non standard quantum field theories that may play the important role in
the dark energy physics [24, 25]. Some aspects of the $PT$-symmetry in the
non-commutative geometric context have already been studied in [26].

#### Acknowledgments

The author is grateful to Vlad Kupriyanov and Patrizia Vitale for discussions
and collaboration on related papers.

## References

  * [1] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378 (2003), 207-299 doi:10.1016/S0370-1573(03)00059-0
  * [2] K. Hersent, P. Mathieu and J. C. Wallet, “Gauge theories on quantum spaces,” [arXiv:2210.11890 [hep-th]].
  * [3] A. Devastato, M. Kurkov and F. Lizzi, “Spectral Noncommutative Geometry, Standard Model and all that,” Int. J. Mod. Phys. A 34 (2019) no.19, 1930010 doi:10.1142/S0217751X19300102
  * [4] V. G. Kupriyanov and P. Vitale, “A novel approach to non-commutative gauge theory,” JHEP 08 (2020), 041 doi:10.1007/JHEP08(2020)041
  * [5] V. G. Kupriyanov, “Poisson gauge theory,” JHEP 09 (2021), 016 doi:10.1007/JHEP09(2021)016
  * [6] V. G. Kupriyanov, “Non-commutative deformation of Chern–Simons theory,” Eur. Phys. J. C 80 (2020) no.1, 42 doi:10.1140/epjc/s10052-019-7573-y
  * [7] P. Mathieu and J. C. Wallet, “Single Extra Dimension from $\kappa$-Poincar\’e and Gauge Invariance,” JHEP 03 (2021), 209 doi:10.1007/JHEP03(2021)209
  * [8] F. Lizzi and F. Mercati, “$\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory,” Phys. Rev. D 103 (2021), 126009 doi:10.1103/PhysRevD.103.126009
  * [9] P. Mathieu and J. C. Wallet, “Twisted BRST symmetry in gauge theories on the $\kappa$-Minkowski spacetime,” Phys. Rev. D 103 (2021) no.8, 086018 doi:10.1103/PhysRevD.103.086018
  * [10] S. Meljanac, Z. Škoda and S. Kresic-Juric, “Symmetric ordering and Weyl realizations for quantum Minkowski spaces,” J. Math. Phys. 63 (2022) no.12, 123508 doi:10.1063/5.0094443
  * [11] S. Meljanac and R. Štrajn, “Deformed Quantum Phase Spaces, Realizations, Star Products and Twists,” SIGMA 18 (2022), 022 doi:10.3842/SIGMA.2022.022
  * [12] H. Kilian, “Quantum properties of $U(1)$-like gauge theory on $\kappa$-Minkowski,” [arXiv:2302.03998 [hep-th]].
  * [13] P. Vitale and J. C. Wallet, “Noncommutative field theories on $R^{3}_{\lambda}$: Toward UV/IR mixing freedom,” JHEP 04 (2013), 115 doi:10.1007/JHEP04(2013)115
  * [14] P. Vitale, “Noncommutative field theory on $\mathbb{R}^{3}_{\lambda}$,” Fortsch. Phys. 62 (2014), 825-834 doi:10.1002/prop.201400037
  * [15] J. M. Gracia-Bondia, F. Lizzi, G. Marmo and P. Vitale, “Infinitely many star products to play with,” JHEP 04 (2002), 026 doi:10.1088/1126-6708/2002/04/026
  * [16] V. G. Kupriyanov, M. A. Kurkov and P. Vitale, “Poisson gauge models and Seiberg-Witten map,” JHEP 11 (2022), 062 doi:10.1007/JHEP11(2022)062
  * [17] V. G. Kupriyanov, M. A. Kurkov and P. Vitale, “Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics.", in preparation.
  * [18] V. G. Kupriyanov and R. J. Szabo, “Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry,” J. Phys. A 55 (2022) no.3, 035201 doi:10.1088/1751-8121/ac411c
  * [19] V. G. Kupriyanov, M. Kurkov and P. Vitale, “$\kappa$-Minkowski-deformation of U(1) gauge theory,” JHEP 01 (2021), 102 doi:10.1007/JHEP01(2021)102
  * [20] O. Abla, V. G. Kupriyanov and M. A. Kurkov, “On the L∞ structure of Poisson gauge theory,” J. Phys. A 55 (2022) no.38, 384006 doi:10.1088/1751-8121/ac87df
  * [21] L. Jonke, “Generalized symmetries as homotopy Lie algebras,” [arXiv:2211.08098 [hep-th]].
  * [22] R. J. Szabo, “The $L_{\infty}$-structure of noncommutative gravity,” PoS CORFU2021 (2022), 218 doi:10.22323/1.406.0218
  * [23] O. O. Novikov and A. Zakharova, “States localized on a boundary of the time-dependent parity-breaking medium,” [arXiv:2206.04026 [hep-th]].
  * [24] A. A. Andrianov, C. Lan and O. O. Novikov, “PT Symmetric Classical and Quantum Cosmology,” Springer Proc. Phys. 184 (2016), 29-44 doi:10.1007/978-3-319-31356-6_3
  * [25] O. O. Novikov, “Inhomogeneous modes in the PT-symmetric quantum cosmology,” Phys. Part. Nucl. Lett. 15 (2018) no.4, 353-356 doi:10.1134/S1547477118040179 [arXiv:1712.03939 [hep-th]].
  * [26] O. O. Novikov, “$\mathcal{PT}$-symmetric quantum field theory on the noncommutative spacetime,” Mod. Phys. Lett. A 35 (2019) no.05, 2050012 doi:10.1142/S0217732320500121