[] [0]0 2011201100000Review Paper

toroidal

11institutetext: Optoelectronics Research Centre, University of Southampton,
Southampton SO17 1BJ, United Kingdom 22institutetext: Centro de
Investigaciones en Óptica, A. C., Loma del Bosque 115, Col. Lomas del
Campestre, 37150, León, Gto., México 33institutetext: Wang Da-Heng
Collaborative Innovation Center for Quantum Manipulation and Control, Harbin
University of Science and Technology, Harbin 150080, China

# Nonseparable states of light: From quantum to classical

Yijie Shen and Carmelo Rosales-Guzmán 11**2233**<EMAIL_ADDRESS>(Y.S);
<EMAIL_ADDRESS>(C.R-G);

###### Abstract

Controlling the various degrees-of-freedom (DoFs) of structured light at both
quantum and classical levels is of paramount importance in optics. It is a
conventional paradigm to treat diverse DoFs separately in light shaping.
While, the more general case of nonseparable states of light, in which two or
more DoFs are coupled in a nonseparable way, has become topical recently.
Importantly, classical nonseparable states of light are mathematically
analogue to quantum entangled states. Such similarity has hatched attractive
studies in structured light, e.g. the spin-orbit coupling in vector beams.
However, nonseparable classical states of light are still treated in a
fragmented fashion, while its forms are not limited by vector beams and its
potential is certainly not fully exploited. For instance, exotic space-time
coupled pulses nontrivial light shaping towards ultrafast time scales, and
ray-wave geometric beams provide new dimensions in optical manipulations. Here
we provide a bird’s eye view on the rapidly growing but incoherent body of
work on general nonseparable states involving various DoFs of light and
introduce a unified framework for their classification and tailoring,
providing a perspective on new opportunities for both fundamental science and
applications.

###### keywords:

Nonseparable state, structured light, quantum-analogue system, entanglement.

##  1 Introduction

Modern optics can be divided in several branches but all can be classified as
either quantum or classical optics, featuring salient fundamental divides [1,
2]. For instance, a classical state can be measured with full certainty
without being affected, at the quantum level, however, a measurement of a
quantum state introduces uncertainty as it, counterintuitively, changes its
properties. What is the origin of such counterintuitive phenomena? The main
clue comes from nonlocality and nonseparability in quantum entanglement [3,
4]. Nonlocality usually refers to the interaction among particles, acting at a
distance, popularized by Einstein’s description as “spooky action at a
distance”. Nonseparability means the state of any particle in a system cannot
be separated. The nonlocality and nonseparability of entanglement seem
counterintuitive but have been proved by rigorous experimental verification
[5, 6, 7].  In quantum optics, entangled systems of multiple photons with
different degrees of freedom (DoFs) have been demonstrated, where the
measurement of a DoF on one photon affects the states of others [8, 9, 10]. In
classical optics, every effect is exact locally realistic without nonlocality,
but the nonseparability can be constructed by local DoFs of light to emulate
quantum-like nonseparable states.

Figure 1: Separable and nonseparable states of light. The fundamental
parameters as DoFs to control in structured light include: amplitude,
polarizaiton, spatial mode, temporal envelope, phase, wave vector, and
frequency. While the paradigm is to shape light structure with separable
control of various DoFs. The more general light control is in its nonseparable
states, such as the vector beams with nonseparability of spatial mode and
polarization, space-time nonseparable structured pulse, and momentum-
polarization nonseparable ray-wave structured light.

Importantly, nonseparability, can also be achieved between the multiple
intrinsic DoFs of a single photon, which is known as local entanglement [11,
12, 13, 14]. This is precisely the type of entanglement that can be used to
describe  the nonseparability of classical light,  that is controversially
termed “classical entanglement” [15, 16, 17, 18, 19, 20]. Typical examples are
spin-orbit vector beams, nonseparable in the spatial and polarization DoFs
[21]. This existing parallelism between classical and quantum nonseparable
states enable the applicability of many of the quantum methodologies to
characterize some of the properties of quantum-like structured light fields,
for instance, the topological phase and Bell-like inequality for spin-orbit
laser modes [22, 23]. Hereafter we will refer to such nonseparable classical
light fields as  entanglement-like states and define them as the nonseparable
sum of product states of more than two DoFs of light, which exist in different
vector spaces. Such definition has been widely endorsed and achieved with
different DoFs in spatial, temporal, and frequency domains [11, 12, 13, 14,
15, 16, 17, 18, 19, 20]. Akin to quantum entangled states, such nonseparable
states of any two or more DoFs gives rise to a Hilbert space that is the
tensor product of the spaces corresponding to the individual DoFs. Hence,
although the nonlocal action does not exist in classical optics,
nonseparability is a common mathematical relationship to both  quantum
entangled and  classical nonseparable states of light. Based on this
similarity, the methodology of quantum optics can be nicely adapted to
characterize some of the properties of quantum-like structured light and is
also useful in many applications [24, 25, 26, 27].

In mathematics, a function is nonseparable if it cannot be factorized as a
product of two or more independent terms. For example, the function
$F(x,y)=af_{1}(x)g_{1}(y)+bf_{1}(x)g_{2}(y)=[f_{1}(x)][ag_{1}(y)+bg_{2}(y)]$
is separable, conversely, the function
$G(x,y)=af_{1}(x)g_{1}(y)+bf_{2}(x)g_{2}(y)$ is non separable, here $a$ and
$b$ are constant terms, $f_{1}$ and $f_{2}$ ($g_{1}$ and $g_{2}$) are
orthogonal functions.  Based on this definition, a generalized nonseparable
state with $m$ DoFs can be written using Dirac’s notation as,

$\displaystyle\left|\psi\right\rangle=$
$\displaystyle\sum_{i=1}^{n}\alpha_{i}\bigotimes_{j=1}^{m}\left|d_{j}^{(i)}\right\rangle$
$\displaystyle=$
$\displaystyle{{\alpha}_{1}}\left|d_{1}^{(1)}\right\rangle\left|d_{2}^{(1)}\right\rangle\cdots\left|d_{m}^{(1)}\right\rangle+{{\alpha}_{2}}\left|d_{1}^{(2)}\right\rangle\left|d_{2}^{(2)}\right\rangle\cdots\left|d_{m}^{(2)}\right\rangle$
$\displaystyle+\cdots+{{\alpha}_{n}}\left|d_{1}^{(n)}\right\rangle\left|d_{2}^{(n)}\right\rangle\cdots\left|d_{m}^{(n)}\right\rangle,$
(1)

where the coefficients $\alpha_{i}$ ($i=1,2,\cdots,n$) are complex nonzero
values that obey the relation
$\sum_{i=1}^{n}\alpha_{i}\alpha_{i}^{\dagger}=1$, the ket
$\left|d_{j}^{(i)}\right\rangle$ ($j=1,2,\cdots,m$) denotes the $j$-th DoF,
and, importantly, any DoF substate $\left|d_{j}^{(i)}\right\rangle$ cannot be
factorized (the values $d_{j}^{(i)}$ for $i=1,2,\cdots,n$ cannot be all the
same). For the sake of clarity, the following definitions  from quantum
mechanics are given:

Definition 1: Dimensionality is defined by the number of linearly independent
eigenstates of a DoF. For example, the polarization state of light is
2-dimensional, because it has two eigenstates, that can be horizontal or
vertical polarization, noted as $|H\rangle$ and $|V\rangle$. While, orbital
angular momentum (OAM) state of light, $|\ell\rangle$ ($\ell=0,\pm 1,\pm
2,\cdots$), theoretically allow infinite dimensionality, where the eigenvalues
can be arbitrary integers.

Definition 2: Multi-partite entanglement refers to the entanglement achieved
between multiple nonseparable systems in a same DoF. For example, the state
given by two photons entangled in the polarization DoF described by
$|\psi\rangle=\frac{1}{\sqrt{2}}(|H\rangle_{A}|H\rangle_{B}+|V\rangle_{A}|V\rangle_{B})$,
is a bipartite entangled state.

Definition 3: Hyper-entanglement refers to the entanglement achieved between
multiple nonseparable systems in multiple DoFs, e.g., the polarization-OAM
hyper-entangled photon pair,
$|\psi\rangle=\frac{1}{{2}}(|H\rangle_{A}|H\rangle_{B}+|V\rangle_{A}|V\rangle_{B})(|\ell\rangle_{A}|\ell\rangle_{B}+|-\ell\rangle_{A}|-\ell\rangle_{B})$
[28]. Further, if each system has an independent DoF, then it is called
hybrid-entanglement, e.g., the polarization-OAM entangled state,
$|\psi\rangle=\frac{1}{\sqrt{2}}(|H\rangle_{A}|\ell\rangle_{B}+|V\rangle_{A}|-\ell\rangle_{B})$
[29].

Quantum entangled states are usually constructed by different subsystems
nonseparable in one or more DoFs and classified according to the number of
subsystems or the number of DoFs. However, in classical domain without
nonlocality, it is impossible to have nonseparability between different
systems, the kind of nonseparability available is local and happens between
different DoFs of the same system, as expressed by Eq. (1).

Figure 2: Typical quantum-analogue models of classical light. The spin
operator has two eigenvalues ($\sigma=\pm 1$) with spin-up and -down
eigenstates, the vector representations of which are the same as the Jones
vectors of left- and right-handed circular polarizations; the OAM operator has
possible infinite eigenvalues ($\ell=0,\pm 1,\pm 2,\cdots$) with eigenstates
of Hilbert factor $\exp(i\ell\theta$), which represents the vortex waves with
helical isophase surfaces.

Notably, light has various intrinsic parameters or DoF [30], such as,
amplitude, polarization, spatial mode, temporal envelope, phase, wave vector,
and frequency, see Fig. 1. A certain amount of parameters can be controlled as
multiple DoFs when they can be treated independently in a optical system.
However, the paradigm of light shaping is to modulate various DoFs separately.
The more general light control is in its nonseparable states. For example,
vector beams are the solutions of optical modes with nonseparability in their
spatial mode and polarization DoF [31]. The existence of space-time
nonseparable solution of localized waves, that cannot be expressed as products
of spatial and temporal functions, has been also proved [32]. The spectral
vector beams are examples of wavelength-polarization nonseparable states of
light [33].

Here we review the rapidly growing topic of nonseparable states of classical
light and some of its analogies to quantum entangled states. Although quantum
entanglement is fundamentally different from classical light, nonseparability
is a common property to both. Based on this analogy, it makes sense to adapt
the tools that have been already developed in quantum mechanics to analyze
quantum-like nonseparable states of classical light and demonstrate novel
classical structures that can mimic certain properties of quantum entanglement
[34, 25]. Based on a quantum-inspired methodology, we provide a unified
framework for the classification of nonseparable structured light, guiding the
new forms of nonseparable structured light to be controlled and providing new
opportunities for both fundamental science and applications.

##  2 Analogies between quantum and classical states

In this section we revisit some of the mathematical and physical similarities
between quantum and classical optics as the foundation to extend these to
other complex nonseparable states. To start with, it is worth mentioning that
classical analogies have been proposed for a wide range of quantum states
[35]. As a first example, the quantum spin angular momentum (SAM) state
corresponds to the circular polarization state of light [36]. Thus, an
arbitrary polarization state can be expressed as a superposition of the two
eigenstates. As a second example, the quantum state associated to the orbital
angular momentum (OAM) also has its classical analog, the vortex beam carrying
a helical phase [37] (Fig. 2) and a well defined amount of OAM per photon.
Notably, any arbitrary phase distribution of light can be decomposed into
various OAM eigenstates.

Besides the above mentioned classical-quantum  analogies between polarization
and SAM as well as vortex beam and OAM, there exist an even larger
generalization of classical optical modes resembling quantum states. Actually,
the basic equations for quantum states and classical modes usually share a
same mathematical formation (See Supporting Information for more detailed
derivations of the formal analogies). For instance, the light modes
propagating in free space satisfy the basic equation, i.e. paraxial wave
equation, as a form of Schrödinger equation just with $z$ variable replacing
the time [38].

$\nabla^{2}_{\mathbf{r}}u(\mathbf{r},z)=-2ik\frac{\partial
u(\mathbf{r},z)}{\partial z},$ (2)

where $\nabla^{2}_{\mathbf{r}}=\partial^{2}/\partial
x^{2}+\partial^{2}/\partial y^{2}$, $(\mathbf{r},z)=(x,y,z)$, the mode is
assumed to be propagating along $z$-axis, and $u(\mathbf{r},z)$ represents the
envelope function related to the electric field by
$\mathbf{E}(\mathbf{r},z,t)=\hat{\mathbf{e}}u(\mathbf{r},z)e^{-i\omega
t}e^{-ikz}$, where $\hat{\mathbf{e}}$ is polarization vector and $k=\omega/c$
is wavenumber. Therefore, the mathematical methods in quantum mechanics to
solve Schrödinger equation can be transferred to solve classical mode with a
proper postulate.

Postulate [25]: A transverse mode $u(x,y)$ of paraxial light is represented by
a vector $|u\rangle$ in an infinite-dimensional Hilbert space isomorphic to
$\mathcal{L}^{2}(\mathbb{R}^{2})$, the square modulus of which is associated
to its transverse intensity $|u|^{2}=\langle
u|u\rangle\propto||\mathbf{E}||^{2}$. In contrast, for a quantum state it
represents its probability density.

According to the quantum mechanics operator algebra, we can define the
position and momentum operators, $\hat{x}=x$, $\hat{y}=y$ and
$\hat{p}_{\mathbf{r}}=-i\nabla_{\mathbf{r}}$, respectively, to transform Eq.
(2) in the equivalent form [39],

$\frac{\text{d}}{\text{d}z}|u(z)\rangle=-\frac{i}{2k}\hat{p}^{2}|u(0)\rangle\Rightarrow|u(z)\rangle=\hat{U}(z)|u(0)\rangle,$
(3)

where the propagation operator $\hat{U}(z)=\exp{(-\frac{i}{2k}\hat{p}^{2}z)}$
describes the transverse mode propagation along $z$-axis [40] and corresponds
to the time-dependent evolution of a particle state in quantum mechanics.
Using such operators we can define the ladder operators as [39]:

$\hat{a}_{x}=\frac{1}{\sqrt{2k}}(k\hat{x}+i\hat{p}_{x}),\quad\hat{a}_{y}^{\dagger}=\frac{1}{\sqrt{2k}}(k\hat{y}+i\hat{p}_{y}),$
(4)

which are of the same form of the ladder operator for Fock states or number
states of photons [41]. From such operators, high-order transverse modes
$|u_{n,m}\rangle$ can be obtained, by applying the raising operator to the
fundamental Gaussian mode $|u_{0,0}\rangle$ [39],

$|u_{n,m}\rangle=\frac{(\hat{a}_{x}^{\dagger})^{n}}{\sqrt{n!}}\frac{(\hat{a}_{y}^{\dagger})^{m}}{\sqrt{m!}}|u_{0,0}\rangle.$
(5)

The eigenfuncation of this result refers to a Hermite–Gaussian (HG) modes, the
conventional laser mode solved in Cartesian coordinate system. Furthermore,
the Laguerre–Gaussian (LG) modes in circular coordinate and general
Hermite–Laguerre–Gaussian (HLG) modes can be easily obtained by applying
additional SU(2) transformation to the ladder operators [42, 43, 44].

In addition, recent advances in the field of structured light have evinced
novel kinds of quantum-analogue classical states [35, 19, 20]. As a result,
quantum analogies are not limited to free-space light beams. For instance, the
Schrödinger equation and related processing methods were also extended to
study the classical light behavior in fibers [45, 46], nonlinear media [47,
48], complex waveguide systems [49, 50]. More typical examples and details
will be reviewed in the following sections. Therefore, it is still highly
appealing to explore more intriguing quantum-analogue formation in various
kinds of light fields.

##  3  Two-DoF nonseparable states

Hereinafter, we introduce a unified framework for a comprehensive
classification of nonseparable states of light. We start from the classical
nonseparable states with two DoFs, such as space-polarization or space-time
and end with a generalized description.

Figure 3: Structured light analogs of typical quantum states. a,b, A
circularly polarized vector vortex beam is represented as the tensor product
of a OAM state and SAM state (a), and a vector vortex beam is represented by a
Bell state as the formation of OAM-SAM qubit entanglement (b) [18]. c-e, The
evolution of  quantum-like entangled elliptical mode can be mapped on a
Poincaré sphere (c) and a Bell state of such mode is experimentally measured
with polarization distributions (d) through the Stokes parameters (e) [51].
f,g, The multi-path ray-wave vortex beams with right- and left-handed twisted
structures have the same formation of the quantum coherent state with coherent
phase parameters of $\pi$ and 0 [52]. h-j, The evolution of ray-wave
nonseparable mode can be mapped on a Poincaré sphere (h) [43], where the pole
represents the circular twisted geometric mode carrying OAM (i), see the phase
and spatial distributions inserted, and the equator the planer geometric mode
(j). k, An  state with an oscillatory degree of nonseparability is constructed
by a longitudinal-variant vector beam, that is superposed by two vortex beams
with opposite propagation and orthogonal polarization [53]. l, The flying
doughnut pulse with focused single-cycle structure and isodiffracting spectrum
fulfills the formation of space-spectrum  quantum-like entangled state [54].

###  3.1 Space-polarization nonseparable states

In quantum mechanics, a qubit is the basic unit to encode information in
bipartite systems, which is written as [55]:

$\displaystyle\left|{{\psi}}\right\rangle=\alpha\left|0\right\rangle\left|1\right\rangle+\beta
e^{i\varphi}\left|1\right\rangle\left|0\right\rangle$ (6)

where $\alpha^{2}+\beta^{2}=1$ and $\varphi$ is the phase delay between the
two eigenstates. The maximally entangled states (leading to the strongest
violations of Bell’s inequality) in a bipartite qubit system are known as Bell
states, and is the simplest case of entanglement of two particles [8]:

$\displaystyle\left|{{\psi}^{\pm}}\right\rangle=\frac{\left|0\right\rangle\left|1\right\rangle\pm\left|1\right\rangle\left|0\right\rangle}{\sqrt{2}}$
(7)

In a  nonseparable light field, only local entanglement can be simulated using
the intrinsic DoFs of a classical light field. For example, by selecting the
OAM (spatial mode) and the SAM (polarization) DoF of light we obtain space-
polarization nonseparable light modes with spatially variant polarization [18,
19, 20], as shown in Figs. 3a,b. Such qubit state can be expressed
mathematically as:

$\left|{{\psi}}\right\rangle=\cos\theta|\ell\rangle|R\rangle+\sin\theta
e^{i\varphi}|-\ell\rangle|L\rangle$ (8)

where $\theta$ is a parameter that allows a smooth transition between scalar
($\cos\theta=0$ or $\sin\theta=0$) and vector modes, ${\varphi}$ is an
intermodal phase, $|\ell\rangle$ is an azimuthal index known as the
topological charge and $|R\rangle$, $|L\rangle$ are the states of right- and
left-handed circular polarizations (RCP and LCP). Such vector beams can be
mapped on the surface of a higher-order Poincaré sphere (HOPS) [56], revealing
the quantum-like spin-to-orbital conversion (SOC). In this representation,
there is a unique HOPS for each value of $\ell$, furthermore, it can be
generalized to a hybrid-order Poincaré sphere by allowing modes with different
topological charges $\ell_{1}$ and $\ell_{2}$ [57]. An even larger
generalization can be done by using both, the azimuthal and radial indices
that fully describe the transverse profile of LG modes. The two polarization
states can also be selected as any pair of elliptical orthogonal polarized
states [58]. Moreover, the spatial DoF is not restricted to the set of LG
modes but other sets of solutions to the paraxial wave equation can also be
used. This is the case of the Ince-Gaussian [59, 60], Mathieu- [61] or
parabolic-Gaussian[62] beams, which are solutions to the paraxial wave
equation in elliptical and parabolic coordinates, respectively. In recent
time, these and other families of modes have been employed to generate
quantum-like entangled beams with elliptical shapes, giving rise to new
families of Bell-like vector novel types of vector beams [63, 64, 65, 66], see
Figs. 3c-e.

Space-polarization nonseparable states can be generated in various ways, for
example using liquid crystal q-plates, which rely on the conversion of SAM to
OAM [67, 68, 69, 70]. A q-plate converts a circularly polarized Gaussian beam
into a vortex beam of topological charge $\ell=\pm 2q$ and reverts its
polarization direction, $q$ is a parameter related to the number of cyclic
rotations ($q$) around the center of the plate. The sign of the topological
charge is determined by the handedness of the input polarization state, see
Fig. 4a. Hence, to generate a  quantum-like entangled mode, a linearly
polarized beam is used as input, which is a superposition of RCP and LCP
components. Another optical device recently proposed and with the capability
to convert arbitrary elliptical polarization states to arbitrary output states
of total angular momentum is the J-plate [71, 58]. The principle behind the
J-plate relies on the use of subwavelength gratings with periods smaller that
the wavelength of the input light, as shown in Fig. 4b. In recent years
metamaterial approaches have been introduced that allow to realize vectorial
states of increasing complexity, such as the metalens shown in Fig. 4c, which
combine ondemand focusing to create multiple complex vector vortex states
simultaneously [72]. In order to increase the compactness of the generating
device as well as the purity of the generated modes, novel approaches have
proposed the insertion of a metamaterial into the laser cavity [73], (Fig. 4g)
to directly generate high-purity nonseparable state with control over SOC even
at extremely large topological charges [74, 75].

Although compact, the metasurface as solid-state component is hard to
reconfigure and does not allow modulation of light. As a recently emerged
method, digital holography provides flexible, reconfigurable and programmable
ways not only to generate arbitrary vector modes but also to access a higher
dimensional polarization-OAM space. The first device that was used to this end
was the liquid crystal Spatial Light Modulator (SLM), that even though it can
only modulate linearly polarized states, it can tailor in an almost arbitrary
way the spatial shape of light [76, 77, 78]. Hence, in order to generate
quantum-like entangled modes, the spatial DoF of each polarization component
is manipulated independently after which they are recombined along a common
propagation axis. To this end, several techniques have been proposed, based on
one or two SLMs incuding the use of temporal sequences where a laser beam is
bounced twice over a single SLM [79, 80, 81, 82, 83], as shown in Fig. 4d.
Another device that is gaining popularity in recent years due to its many
advantages, such as high refresh rates, polarization-independence, amongst
others, is the Digital Micromirror Device (DMD) [84, 85, 86, 87], as shown in
Fig. 4e. Notably, the polarization-independence property of DMDs allows the
modulation of any polarization state, a property that has been widely
exploited in recent times for the generation of  quantum-like entangled modes
with arbitrary spatial shape and polarization distributions [88, 89].

###  3.2 Space-time nonseparable states

In addition to qubit Bell states, high-dimensional Bell states have became
increasingly topical [90], an $n$-dimensional case of which is given by:

$|\psi\rangle=\frac{1}{\sqrt{n}}\left(|0\rangle|0\rangle+|1\rangle|1\rangle+\cdots+|n-1\rangle|n-1\rangle\right)$
(9)

In quantum optics, there is a characteristic example of such Bell states, the
OAM-entangled photon pair,
$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{\ell=1}^{n}|\ell\rangle|-\ell\rangle$
[91]. But in the classical case, it is not possible to realize such states
based on the polarization DoF, as it is limited in two-dimensions. The
alternative route to this end is the selection of the two DoFs to form
spatiotemporal structured light pulse, that is using space and time (or
equivalently frequency or wavelength). Space-time non-separable pulses have a
long history dating back 4 decades, where they were introduced in the context
of non-diffracting waves [32].  The method of termed “electromagnetic
directed-energy pulse trains” was proposed as a well-endorsed way to solve
general finite-energy space-time nonseparable pulses [92], special cases of
which were subsequently studied by Hellwarth and Nouchi, who found closed-form
expressions that describe focused single-cycle pulse solutions to the
homogeneous Maxwell’s equations. This family of pulses includes both linearly
polarized pulses, termed “focused pancakes” [93], as well as pulses of
toroidal symmetry, termed “flying doughnuts” [94], which were experimentally
generated very recently [95]. The crucial characteristic induced by the space-
time nonseparable pulse is the isodiffraction nature, whereby the spatial
distribution of different spectral components in the transverse plane does not
suffer distortion upon propagation. [96, 97], as demonstrated by the map of
colored lines in Fig. 3l. Recently, a quantum-analogue method to characterize
space-time (or -frequency) nonseparability was proposed, highlighting the link
between isodiffracting pulses and high-dimensional Bell states. Particularly,
spatiotemporal pulses can typically be described by a bipartite state with the
space and time DoFs, but only specific type of space-time nonseparable pulses
(e.g. isodiffracting pulses) can be the Bell state with maximal
nonseparability [54]:

$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|r_{i}\rangle|t_{i}\rangle$ (10)

or equivalently,

$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|r_{i}\rangle|\omega_{i}\rangle$
(11)

where the spectral states $|\omega_{i}\rangle$ are (monochromatic) states of
light of defined frequency $\omega_{i}$ and with defined radial position of
peak intensity, and spatial states  $|r_{i}\rangle$ are generally
polychromatic states of light located at the position with defined radial
ratio $r_{i}/r_{\text{max}}$ ($r_{\text{max}}$ is the radial position at which
the total intensity of the light field reaches its maximum). For common
pulses, the general state should be a general high-dimensional bipartite
state, while the isodiffracting case corresponds to the maximally entangled
state.

The exploration of the analogy between the space-time nonseparability and
quantum entanglement would inject new vitality to the characterization of
spatiotemporal structured pulses by using quantum-inspired methods such as
state tomography and quantum measures. Space-time nonseparability can be
employed for the construction of propagation-invariant spatiotemporal pulses
[98, 99, 100], as well as the above discussed isodiffracting single-cycle
pulses [54]. Recently, the state tomography method is also used to measure
more general kinds of classical wide-band structured beams or pulses [54].  In
addition, the study of space-time nonseparable states may not lbe imited in
ultrashort pulses, which can also be accessed by the construction of
propagation-dependent spatial mode, for instance, the polarization-variant
polarization or OAM mode recently realized by metasurface design [101, 102].

###  3.3 Ray-wave nonseparable states

In addition to the Bell-like states, there are also many other intriguing
bipartite quantum states available to be exploited. For example, the coherent
state is a superposition of various Fock states, quantum state whose dynamics
most closely resembles the oscillatory behavior of a classical harmonic
oscillator [103]. In two-mode SU(2) symmetric system, the normalized SU(2)
coherent state is given by [104]:

$\left|\phi\right\rangle=\frac{1}{{{2}^{{N}/{2}\;}}}\sum\limits_{K=0}^{N}{{{\left(\begin{matrix}N\\\
K\\\
\end{matrix}\right)}^{{1}/{2}\;}}{{\text{e}}^{\text{i}K\phi}}\left|K,N\right\rangle}$
(12)

where $K$ and $N$ are number of bosons in the first and second modes of a Fock
eigenstate $|K,N\rangle$ and $\phi$ is a phase parameter. As a salient
property of coherent state, its wavefunction distribution is always coupled
with the movement trajectory of the classical harmonic oscillator, which is
termed quantum-classical correspondence [105]. The original proposal of
coherent state involves the description of the photon stationary properties
for distinguishing the quantum and classical field [103], while in recent
years it stands as a powerful tool to create geometric structured light. The
SU(2) coherent state was originally used to characterize the statistic
properties of photon flows in quantum optics. It can be used to describe
structured light beams by replacing the Fock bases $|K,N\rangle$ by the
classical light eigenmodes, either in freespace [106, 107, 108], waveguide
[109], or microlaser systems [110, 111]. The resulting SU(2) geometric modes
harness the striking property of ray-wave duality, whereby the spatial
wavepacket is always coupled with a geometric ray trajectory, akin to the
quantum-classical correspondence of quantum coherent state. For example, the
light modes in a microcavity, with billiard-like trajectory bouncing back-and-
forth periodically within the boundary of cavity, can be represented by the
coherent state field, and controlled by the parameters of coherent state [110,
111]. Such modes hold promise in the study of nonlinearity, chaos and non-
Hermitian physics [112, 113, 114]. The coherent state field can also represent
various exotic freespace geometric modes coupled with multi-path ray
trajectory, moreover, the geometric ray trajectories can be tailored by
control of a simple solid-state laser cavity [115, 52]. We can use the  set of
states, $|K,N\rangle$, to represent a set of frequency-degenerate freespace
eigenmodes, e.g. HG, LG, and general HLG modes, see Figs. 3f,g as an example
of LG-based ray-wave geometric modes carrying OAM. Inspired by the OAM
evolution of conventional HLG eigenmodes, the ray-wave geometric modes also
feature an OAM evolution that can be mapped on a Poincaré-like sphere [43], as
shown in Fig. 3h. Due to the ray-wave duality structure, such geometric mode
opens additional propagation-variant 3D spatial structures controlled by
multiple parameters (see Figs. 3i,j) that the common beams do not have.

###  3.4  Other two-DoFs nonseparable states

In contrast to the Bell-state vector beams, the higher-dimensional states of
light provide access to a broader parameter space to manipulate customized
patterns of light. A number of recent works have demonstrated control of
higher dimensional beams based on the conventional vector beams. For example,
the OAM state can be represented by the standing-wave mode, rather than the
conventional traveling-wave LG mode. Thus it is formed by the superposition of
two traveling-wave modes propagating in opposite directions, resulting into  a
beating of the degree of nonseparability, i.e, a nonseparabl mode with an
oscillatory degree of nonseparability along the longitudinal direction [53],
as shown in Fig. 3k. This scheme was recently improved by using higher-order
OAM modes for complete control of SOC dynamics [116, 117]. The standing-wave
eigenmodes can also be used to construct coherent state of light resulting
into standing-wave ray-wave nonseparable structured light [118]. The concept
of vector beams were also transferred in spectral domain to create novel
wavelength/frequency-polarization nonseparable structures [33].  In addition
to the above-mentioned DoFs, structured light can possess many other kinds of
DoFs (as Fig. 1 shows), and the combination of them can be used to construct
more kinds of 2-DoF nonseparable states. For instance along this line,
pioneering works have demonsrated the generation of Bell states by combining
linear momentum and OAM, i.e. momentum-OAM nonseparable state [119].  In the
field of nonlinear optics, it has been demonstrated that the use of path and
polarization degrees of freedom allows nearly perfect optical switching
between different OAM operations, such as, OAM addition and doubling [120].

In addition to the nonseparable states discussed above, vector beams have led
to the generation of optical skyrmions. Skyrmions are special states of
quasiparticles with a sophisticated space-vector topology, and in recent time,
its optical counterparts were introduced from the quantum and material physics
into optics, as a powerful tool to shape new topological states of light [121,
122, 123]. More recently, skyrmion topology also found its efficacy in
tailoring reconfigurable patterns in vector beams by fully exploiting the
salient particle-like topology of skyrmions [124, 125, 126]. Thus, more
skyrmionic nonseparable states in structured light are expected in the future.

Finally, another promising path to follow is the construction of novel
structured light fields based on other famous quantum states,  such as the
structured vortex modes of cat state [127] and squeezed state [128].
Therefore, it is highly probable that in the near future and with the
exploitation of unexplored quantum-analogue methods, other kinds of bipartite
or two-DoF higher-dimensional nonseparable light will be found.

Figure 4: Generation schemes of nonseparable states of light. a,b, Schematic
representation of the spin-orbit interaction performed by (a) a q-plate and
(b) a J-plate [58]. c, Multiple  nonseparable vector modes generated from a
tailored metalens [72]. d, Experimental setup to generate quantum-analogue
modes using SLM [80]. e, Schematic representation of the generation of
quantum-analogue modes using a digital micromirror device [104]. f, Generation
of quantum-analogue modes directly from a laser with intracavity metasurface
[73]. g, Generation of multipartite quantum-analogue states in more than two
DoFs from an anisotropic microchip laser [129]. h, Generation of quantum-
analogue space-time nonseparable pulse (flying donut pulse) from a tailored
metasurface with singular antenna array [130]. L: lens; M: mirror; A:
aperture; QWP: quarter-wave plate; HWP: half-wave plate; WP: Wollaston prism.

##  4  Multi-DoF nonseparable states

In previous sections, we outlined a roadmap to control  quantum-like two-DoF
nonseparable states of structured light, by manipulating two intrinsic DoFs of
light. Importantly, the use of higher-order modes,  such as spatial modes with
higher values of OAM ($\ell>2$), gives access to a larger dimension which
allow to increase the information capacity per photon and the error threshold
of optical communication systems. In regards to quantum entangled states, it
has been demonstrated that the increase in dimension can translate into an
increased robustness to noise, of potential applications to high-dimensional
quantum key distribution protocols [131]. Nonetheless, the mode order or OAM
value that can be achieve experimentally has a technological limit. A way to
overcome this limit is to exploit multiple DoFs of light to  emulate
multipartite entangled states.  For classical structured light, in recent
decades, the nonseparable states were mostly discussed in vector beams, i.e.
the space-polarization two-DoF nonseparable states, the construction of
structured light with more-than-two DoFs is still elusive. Therefore, it is an
emerging topic to find more-than-two intrinsic DoFs of a structured light beam
to construct on-demand multipartite nonseparable states. Based on the recent
advances of structured light and optical manipulation, multiple new intrinsic
DoFs of light have been proposed and controlled, that enables the construction
of  high-dimensional nonseparable states in multiple DoFs.

###  4.1 Ray-wave-polarization nonseparable states

The maximally entangled states that involve multiple subsystems are named
Greenberger–Horne–Zeilinger (GHZ) states.  In quantum entanglement, GHZ states
require one DoF but in no-less-than-three DoFs. For example the set of
tripartite GHZ states that can be constructed as [132]:

$\displaystyle\left|{{\Phi}^{\pm}}\right\rangle=\frac{\left|0\right\rangle\left|0\right\rangle\left|0\right\rangle\pm\left|1\right\rangle\left|1\right\rangle\left|1\right\rangle}{\sqrt{2}}$
(13)
$\displaystyle\left|\Psi_{1}^{\pm}\right\rangle=\frac{\left|1\right\rangle\left|0\right\rangle\left|0\right\rangle\pm\left|0\right\rangle\left|1\right\rangle\left|1\right\rangle}{\sqrt{2}}$
(14)
$\displaystyle\left|\Psi_{2}^{\pm}\right\rangle=\frac{\left|0\right\rangle\left|1\right\rangle\left|0\right\rangle\pm\left|1\right\rangle\left|0\right\rangle\left|1\right\rangle}{\sqrt{2}}$
(15)
$\displaystyle\left|\Psi_{3}^{\pm}\right\rangle=\frac{\left|0\right\rangle\left|0\right\rangle\left|1\right\rangle\pm\left|1\right\rangle\left|1\right\rangle\left|0\right\rangle}{\sqrt{2}}$
(16)

All the 8 states are maximally entangled states. In general, a complete set of
$m$-partite GHZ states include $2^{m}$ maximally entangled states. In order to
construct the corresponding classical analogs of the GHZ states, we should
find three different local DoFs to mimic the DoF in three subsystems. For
instance, the usage of mode indices of two orthogonal directions and
polarization were applied to construct a tripartite  GHZ-like state [19].
However, the two mode indices in a single beam cannot be tuned independently.
By using the ray-wave vector vortex beam, as discussed in Section 3.3, it is
possible to create more-than-two DoFs in a structured beam, such as trajectory
shape, coherent-state phase, OAM, astigmatic degree and polarization [133].
Also, an experimental generation of tripartite nonseparable state with three
of these new DoFs was reported, and, importantly, such modes can be generated
directly from a laser cavity by operating an anisotropic microchip laser
[129], as shown in Fig 4g. In this demonstration, the cavity lases on multiple
ray-like trajectories or geometric paths, exploiting a ray-wave duality, with
differing coherent state phases and independent tunable polarization. In
addition, the polarization control was achieved via a precise off-axis pumping
of an anisotropic crystal [134]. In this method, the creation of multiple DoFs
and a single GHZ state is successful, but it is hard to fully control the
high-dimensional nonseparable state. Recently, to overcome this bottleneck, it
was reported that a complete set of tripartite GHZ states with all the eight
maximally entangled states, as Eqs. 13-16, can be fully controlled in
$2^{3}$-dimensional Hilbert space [135], by combing the spatial light
modulation and laser control in multi-DoF ray-wave structured light. Moreover,
this method is scalable to generate the multipartite GHZ state with even more
controllable DoFs and higher dimensions.

###  4.2 Space-time-polarization nonseparable states

GHZ states are multipartite entangled states where the DoF is limited in
2-dimension. While, the general multi-DoF hybrid-entangled states can be
constructed by different higher-dimensional DoFs, for example of a 3-DoF
hyper-entangled states:

$\left|\psi\right\rangle=\frac{1}{2\sqrt{n}}\left(\sum\limits_{i=0}^{n-1}{\left|i\right\rangle\left|i\right\rangle\left|0\right\rangle}+\sum\limits_{i=0}^{n-1}{\left|i\right\rangle\left|i\right\rangle\left|1\right\rangle}\right)$
(17)

where the first and second DoFs are $n$-dimensional, while the third DoF is
2-dimensional. An intuitive way to reach this goal is to find additional DoF
in the structured light fulfilling high-dimensional Bell states. As discussed
in Section 3.2, the isodiffracting single-cycle pulses can be expressed as
high-dimensional Bell state with two DoFs (space and time). The class of
isodiffractin pulses includes pancake-like pulses [96, 93] and doughnut-like
pulses [94, 97]. The focused pancake pulse is linearly polarised, while the
doughnut-like pulse presents a can perform complex structure of cylindrically
polarized vector fields, including the azimuthal (for transverse electric
modes) [136] and radial polarized fields (for transverse magnetic mode) [137]
(the vector structure of transverse electric mode is depicted in Fig. 3l).
Such cylindrical vector nature just performs as a platform to hold space-
polarization nonseparable states [138], as the subsystem of the pulse. Thus
the FD pulse has further properties to extend space-time-polarization
nonseparable state:

$\left|\psi\right\rangle=\frac{1}{2\sqrt{2}}\left(\sum\limits_{i}{\left|{{r}_{i}}\right\rangle\left|{{t}_{i}}\right\rangle\left|H\right\rangle}+\sum\limits_{i}{\left|{{r}_{i}}\right\rangle\left|{{t}_{i}}\right\rangle\left|V\right\rangle}\right)$
(18)

or equivalently,

$\left|\psi\right\rangle=\frac{1}{2\sqrt{2}}\left(\sum\limits_{i}{\left|{{r}_{i}}\right\rangle\left|{{\omega}_{i}}\right\rangle\left|H\right\rangle}+\sum\limits_{i}{\left|{{r}_{i}}\right\rangle\left|{{\omega}_{i}}\right\rangle\left|V\right\rangle}\right)$
(19)

Moreover, taking advantage of recent advances in the metasurfaces [130], the
generation of such pulse was also proposed [139], and observed experimentally
in recent times [95], where the metamaterial converter can be constructed by a
cylindrically symmetric array of low Q-factor dipole resonators oriented
radially or azimuthally oriented electric dipoles and arranged in concentric
rings, as shown in Fig. 4h. Such pulses have been shown to efficiently couple
to toroidal multipoles and to non-radiating charge-current configurations
(anapoles) [140, 136, 141]. Recently, more new kinds of spatiotemporal vector
pulses were designed [142, 143, 144, 145], and the method for experimental
characterization of ultrashort vector pulses was proposed [146, 147], which
can be used for fully measuring the distribution of the three DoFs, i.e.
space, time, and polarization. Thus we can expect that the full control of
tripartite nonseparable spatiotemporal pulse is just around the corner.

###  4.3 Beyond  three-DoF nonseparable states

The use of multiple degrees of freedom at the single photon level has been
used to investigate the generation of entanglement between two noninteracting
qubits that share a common environment, which have been prepared in
independent individual states. Here, the two qubits where described by the
transverse an polarization degrees of freedom, while the common enviroment was
set by the photon path [148]. In a similar way, achieving entanglement in
high-dimensional quantum systems with one or more degrees of freedom is of
great relevance to increase the information capacity or to enable new quantum
information protocols [149]. Noteworthy, the classical analogs of multi-
partite high-dimensional quantum states have already been suggested but
controlling more than three DoF experimentally represents a real challenge.
Recently, the structured light model to create up-to-five-partite GHZ states
has been theoretically predicted [135], it requires the generation of more
complex ray-wave geometric mode of higher-order coherent state so as to access
two new DoF, center-OAM and sub-OAM, but has not been experimentally reported
yet. A model of astigmatic hybrid SU(2) vector vortex beams was proposed to
involve a new DoF, astigmatic degree, into ray-wave structured light [133],
but this DoF is still a candidate to be used to build nonseparable state.
There were successful methods to control multipartite nonseparable state by
designing a beam splitting system, where the increase of splitting paths
reveals the creation of more parties [150, 151, 152, 153]. However, in this
method, the DoFs would no longer be intrinsic to one paraxial beam and the
control would become increasingly complicated and problematic. Therefore, it
is still an open challenge, very much in its infancy, to find no-less-than
four DoFs that are easy to control and intrinsic to a single paraxial
structured beam.

Besides the quantum states selectively discussed above, there are still a
number of states available to be explored for generating quantum-analogue
structured light, such as the W state [154] and Laughlin state [155], which
all hold unique properties to be applied in classical light. Therefore, the
methodology of constructing more kinds of quantum-like nonseparable state
holds resource to realize multi-DoF higher-dimensional control of light for
spurring new applications and extending the frontier of fundamental science.

##  5 Applications of nonseparable states of light

Achieving applications of nonseparable states of light, fully or partially
relying on the existing parallelism between these and quantum entangled
states, is at the heart of future technologies. Importantly, these are paving
the road to applications that have demonstrated challenging with quantum
states but can be easily implemented with quantum-analogue states. This is the
case of various applications, where quantum inspired approaches started
opening a new era of improvement and performance. Although, the unique
properties of quantum-analogue modes of light have inspired many other
applications (see for example Refs. [31, 37]), here we will only focus on
those that directly exploits the nonseparability aspect.

###  5.1 Beam quality measurement

Since nonseparable light modes share the same mathematical formalism as
quantum entangled states, it is almost natural to exploit quantum mechanic
tools to quantitatively characterize the corresponding properties of classical
structured light. Importantly, Over the past century, an extended toolbox has
been developed to quantify the quality of a quantum entangled state, including
state tomography, density matrix, fidelity, linear entropy, concurrence, etc.
[156, 157]. In the meantime, the measures for evaluating various properties of
classical structured light is very limited, some original beam quality
measures are only related to the scalar propagation property. While, the
structured light has developed to harness multiple complex DoFs in itself, and
new quantum-analogue measure methods are acquired to meet its development. For
instance, the quantum state tomography measurement can be applied to vector
vortex beams [34], see Fig. 5a, to reconstruct the density matrix for
characterizing the vector vortex beams and calculating the fidelity to reflect
the quality to the ideal light state, akin to a measure of qubit [158]. We can
also apply the concurrence or entanglement of formation, a measure of
nonseparability of entangled state, to measure the quality of vector beams, as
a new continual quantification from pure scalar beam to vector beam [159, 160,
161]. Moreover, the quantum Bell measurement method and Bell’s inequality are
also available to use for revealing the spin-to-orbital coupling in a
classical beam [98, 158]. Besides the pure state systems, the quantum
mechanics of mixed state was also transferred to  complex classical beams for
exploring high-dimensional quantum-analogue system [162], which can also
realize the counter-intuitive quantum teleportation effect in classical beams
system [163, 164].  In a similar way, the concept of nonseparability has been
also used for spatiotemporal quantum-analogue pulses, e.g. the Schmidt number
or Schmidt rank was used to characterize the invariant propagation effect of a
classical pulse [100]. Since the quantum state tomography method was modified
to  high-dimensional state [162], we can apply analogous method to more
complex structured light, e.g. the space-time nonseparable pulses, and many
quantum concepts (fidelity, concurrence, linear entropy, etc.) were open to
quantitatively characterize the spatiotemporal propagation dynamics [54].

Figure 5: Applications of nonseparable states of light. a, State tomography
measurement is implemented to measure the quality of classical structured
light beams [34]. b, When an object traverses across a  quantum-like
nonseparable beam, its spatial and polarization DoFs become a function of the
instantaneous position of the object [165]. c, The rotational and longitudinal
speeds of an object moving in three dimensions can be measured simultaneously
using a nonseparable light mode. d, Schematic of the transmission of a grey
scale image using sixteen orthogonal nonseparable modes [166]. e, Schematic of
the transmission through a free-space turbulent channel of a quantum entangled
photon and a  quantum-like nonseparable mode [27]. f, Schematic of high-
dimensional long-distance quantum cryptography for encrypting colorful images
with structured light photon [167]. g, Schematic of the implementation of
quantum walks using classical structured light: the intensity (i) and phase
(ii) distributions of nonseparable superposition (iii) of OAM states and
polarization for a 4-step walk, and the probability distribution (iv) of
walker occupies positions after 50 steps of symmetrical walk [168].

###  5.2 Metrology and sensing

Optical metrology is a broad area of science and technology that encompasses
all measuring techniques that use light as the main sensing tool, allowing for
non-contact measurements at high speeds and with high precision. These can be
used, for example, to measure the physical properties of solids and fluids,
such as, their geometrical properties chemical composition, temperature,
pressure, velocities, amongst other. In this regard, the nonseparability of
quantum-analogue light beams provide with an alternative tool for the
development of novel metrology techniques. This was demonstrated in a recent
technique that exploits the nonseparability of such modes to perform real-time
sensing of position and velocity of opaque objects moving in two dimensions
[165]. The key idea relies in the fact that when an object traverses across a
nonseparable beam, its space-polarization non-separability becomes a function
of the instantaneous position of the object, see Fig. 5b. Importantly, the
modulation of the spatial degree of freedom is correlated with the global
polarization state of the  nonseparable mode for an extraction of the required
information, directly from the changes in polarization. Along the same line,
and also taking advantage of the nonseparability of such modes, a recent
technique demonstrated the ability to measure simultaneously the longitudinal
and angular speed of objects moving along spiral trajectories using a single
interrogating beam [169]. To this end, the moving target is illuminated with a
nonseparable mode constituted by a Gaussian and a vortex mode with orthogonal
polarization, see Fig. 5c. The speed of the object is then determined from the
light scattered back from the objects. Key to this is the fact that the
information contained in the back-scattered light can be unambiguously
separated upon detection. In this way, while the longitudinal speed is
measured with the Gaussian component of the  nonseparable mode, the rotational
one is determined with the vortex mode. The nonseparability of quantum-
analogue light modes can be also used to determine the optical properties of
objects. This was demonstrated in a recent technique that proposed a real-time
Stokes polarimetry to measure the temperature of a temperature-dependent
birefringent crystal [170]. Stokes polarimetry enables the reconstruction of
polarization through a minimum of four intensity measurements. In this way, a
real-time reconstruction of polarization enables the possibility to track in
over time the evolution of a given polarization-dependent property. In the
experiment, a radially polarized beam was sent through a birefringent crystal
while its temperature was increased causing a rotation of the polarization
distribution. In this way, the angle of rotation of the state of polarization
directly indicates the temperature of the crystal. In a similar work, the
unique properties of  nonseparable modes were proposed as a two-DoF
polarimetry scheme, a classical analogue of two-photon polarimetry. This with
the aim of enhancing the performance of Mueller matrix metrology [171]. In its
conventional form, Mueller matrix techniques allows to determine certain
properties of an object by analyzing the polarization of light transmitted
through or scattered from it. For this, the object must be sequentially
illuminated with at least four different polarization states. In the new
version implemented with quantum-analogue modes, the object is illuminated
with a single light beam containing all polarization components, such as a
radially polarized mode, in contrast to the conventional way that requires
four beams. Afterwards, the light scattered or transmitted is analyzed
simultaneously in both DoFs, polarization and spatial. Importantly, in a
similar way to conventional Mueller matrix polarimetry, the polarization
degree of freedom is used to prove the object while the spatial one is used to
post-select the polarization state. This technique allows a real-time
determination of the Mueller matrix allowing for fast tracking of the optical
properties of the object.  In addition to the space-polarization nonseparable
vector modes, it is also an emerging direction to design other kinds of
classical nonseparable state to enable novel metrology. For instance, the
spectral vector mode as wavelength-polarization nonseparable state was
designed and enabled a high-speed spectroscopic metrology [33].

###  5.3 Communications

The realization that current communication systems will inevitably face a
bandwidth capacity crunch, has risen an increasing interest in the development
of alternative methods to encode and transmit information. Current techniques
already exploit most of the physical properties of light, wavelength-division,
time-division, quadrature amplitude modulation and polarization-division. Even
though these techniques have significantly improved the transmission capacity
of optical systems, most of them have reach their maximum capacity and will
not be able to satisfy the exponentially-increasing global capacity demand.
Importantly, mode division multiplexing provides a potential solution to this
problem. Here information is encoded in the high-dimensional orthogonal space
of the spatial modes of light, the shape of light. Due to their orthogonality,
each mode can be used as a communication channel with the ability to carry
independent data stream, increasing in this way the overall capacity in
proportion to the number of modes. Despite all the advances along this
research line (see for example [172]), one of the major problems concerning
both, free space and optical fibers communication is modal cross talk [173].
In free space, it is believed that  space-polarization nonseparable modes
perform better in the presence of atmospheric turbulence than common scalar
modes [174, 175, 176], even though some studies contradict this believe [177].
The amount of research devoted to both, classical and quantum communication
using  quantum-like modes is enormous [178]. Pioneering work along this
direction demonstrated, in a proof-of principle experiment, the transmission
of information in free-space using four orthogonal  space-polarisation
nonseparable modes, multiplexed and transmitted as one single light mode using
mode division multiplexing. In this demonstration, each mode, generated with
q-plates was able to transmit information at 20-Gbit/s [179]. Similar
techniques have demonstrated the transmission of  such light modes along
optical fibers, which represents the core of our current communication systems
[180, 181]. An example of which is the transmission of data over 2 km using a
fiber-based system that used vector modes generated from a photonic integrated
device [182, 183]. In addition to mode division multiplexing,  nonseparable
modes can also be used as an “alphabet” with which to encode and transmit
information. In this approach, each vector mode of the  multidimensional basis
is used as a symbol of the alphabet. In a first attempt to demonstrate this,
four vector beams were encoded and successfully decoded with very low cross
talk (lower 9%) using a Mach–Zehnder interferometer [184]. An increase in the
number of vector modes was achieved by Wang et al. who took advantage of the
flexibility of SLMs to generate a basis containing 16 vector modes [166]. With
this, the authors were able to achieve hexadecimal encoding and used as an
example the encoding and decoding of a $64\times 64$ pixels black and white
image, which was sent pixel by pixel. To this end, the grey value of each
pixel is first converted to a hexadecimal number and then assigned to one of
the 16  nonseparable modes, see Fig. 5d. Similar experiments have demonstrated
the realization of multi-ary encoding/decoding in a high-dimensional space
formed by multiple possible states of hybrid vector beams [185]. Recently,
multi-DoF nonseparable ray-wave structured light was also exploited to improve
the capacity, speed, and dimensionality in communication system [186].

###  5.4  Information error corrections

The similarities between quantum and classical states of light can be
exploited in the implementation of secure communications using the fundamental
laws of quantum mechanics in what is known as Quantum Key Distribution (QKD).
The secure transmission of information involves the implementation of a secure
communication channel for the distribution of a cryptographic key. It is in
this step where quantum mechanics provides an unconditional secure
distribution of the cryptographic key by alerting from the presence of an
intruder in the communication channel, based on fundamental laws of quantum
mechanics, such as the non-cloning theorem and the Heisenberg’s uncertainty
principle. Along this line, high-dimensional QKD protocols based on the
spatial mode of light have become popular as they can potentially increase the
transmissions distances as well as the information capacity [172, 178].
Unfortunately,  if we do not account for the perturbations in the quantum
channel, they can produce entanglement decay which, paradoxically, can only be
repaired once the quantum link is working. A solution to this problem was
proposed recently by taking full advantage of the similarity between quantum
and spin orbit nonseparable states of light [27]. In this proposal it was
demonstrated that the quantum entanglement decay of a photon pair is identical
to that of  space-polarisation nonseparable multi-photon states. This in the
case where the channel acts on one of the two quantum entangled photons and
similarly on one degree of freedom of the  quantum-like nonseparable mode, see
Fig. 5e. In this way, a full characterization of a quantum channel can be
performed using quantum-analogue light modes instead of single photons. This
discovery provides the basis to perform precise quantum error correction in
short and long-haul optical communication. Moreover, even though it was
demonstrated for a free-space channel perturbed by atmospheric turbulence, it
is expected to also work in optical fibers.  Incidentally, in a recent article
it was demonstrated that entanglement itself can also be used to measure the
transmission matrix of the quantum channel, which in turn allows the recovery
of quantum correlations that otherwise are lost [187].

###  5.5 Computation and others

Another advance of quantum effect is shown in the usage of computation, the
quantum computation has proved its extremely higher speed and efficiency than
the classical computation [188]. Since we can create various quantum-analogue
states in classical light, it also inspires us to apply quantum computation
algorithms and methods in classical optical computation system. A typical
algorithm is the quantum walk [189]. Quantum walks are motivated by the
widespread use of classical random walks in the design of randomized
algorithms for the usages of data retrieval and searching. The classical walks
allow only one path in a retrieval process, while quantum walks promise to
retrieve multiple paths simultaneously benefited by the quantum uncertainty
nature, perfectly overcoming the classical drawback. Therefore, for some
oracular problems, quantum walks provide an exponential speedup over any
classical algorithm. Recently, complex structured light was used to implement
a quantum walk and largely enhance its ability of data correction [190, 191].
It was also proposed that the use of  quantum-like modes allow to emulate the
evolution of a quantum walk in real time [192], which was also recently
realized experimentally [168], see Fig. 5g. The classical quantum walk
technique was also extended to higher-dimensional topological state for more
powerful optical computations [193].  Spin-orbit nonseparable states have been
also used to illustrate some of the concepts of game theory in its quantum
version, more precisely, to implement the quantum version of the prisoners
dilemma [194]. In addition, artificial intelligence and machine learning
algorithms were recently used for identifying multi-DoF nonseparable states of
light [195, 196], which enable the further improvement of related
applications. Finally, the use of nononseparable modes are taking relevance as
an academic tool to simulate quantum key distribution protocols, see for
example [197, 198] and references therein.

In short, in this review we highlighted many of the analogies that have been
posted over the years between nonseparable light fields and quantum states but
is very likely that we missed some other that are being proposed as we write
this review. We also discussed a variety of applications, some of which are
inspired in the similarities between nonseparable fields and quantum states.
Importantly, there are still some quantum effects which have not found their
parallel with nonseparable states, some of which might never find it, these is
the case, for example, of quantum imaging [199] or quantum storage [200].

##  6 Conclusions and perspectives

In this review we outlined the many similarities that nonseparable states of
classical light hold with quantum entangled states. To this end, we
systematically constructed the theoretical framework of nonseparable states
[20, 25, 135]. Such  entanglement-like states are represented mathematically
as a nonseparable superposition of the various DoFs of light, which can be
classified akin to the classification of entangled states with different
partite numbers and dimensionalities[34]. This similarity performs as a bridge
connecting the quantum and classical worlds and allows to apply the quantum
tools to describe many of the properties of  classical nonseparable states of
light [158, 161, 98]. Building on a large body of work on vector vortex modes,
bipartite nonseparable states in polarisation and spatial mode, here we
generalize the family of multipartite nonseparable state of light with diverse
DoFs. With the recent technological advance, the family of nonseparable states
of light is no longer limited to conventional vector beams, with an increasing
numbers of intriguing higher-dimensional structured light modes (e.g. ray-wave
structured light and space-time nonseparable pulse) enabling the simulation of
an increasing number of higher-dimensional quantum states, which allows the
transfer of more quantum tools to classical fields [54]. Therefore, we
anticipate that in the following years, many other examples of quantum-
analogue non-separable states of light will emerge.

Besides the techniques for the generation of quantum-analogue states of light,
it is also important to develop techniques for the characterization of such
states of light using quantum-inspired methodology. Pioneering work along this
line, has demonstrated that many concepts, such as fidelity, concurrence, and
linear entropy from quantum mechanics, which are used to measures of the
degree of entanglement, can be applied also to  space-polarisation
nonseparable states to measure the degree of coupling between the spatial and
polarization DoFs in vector modes [160, 159]. Similarly, measures have been
proposed to characterize space-time nonseparable structures, such as flying
doughnuts [54]. The techniques for the characterisation of quantum-analogue
modes are still in its infancy and it is still challenging to adapt similar
measurements to other DoFs. Nonetheless, the mathematical nature of
entanglement does not distinguish between classical or quantum states, the
only limitation is technological. Therefore it is very likely that
technological advancements will provide the required tools to extend the
concept of nonseparability to other novel DoFs, such as self-torque [201],
astigmatic degree [133], skyrmion number [124], supertoroidal order [123], in
the near future.

From the applications perspective, classically nonseparable states of light
have pioneered many of them, some of which are directly inspired by their
similarity with quantum entangled states. Such applications, which have
demonstrated challenging at the single photon level, are easier to develop
with  quantum-like classical nonseparable states and will form the core of
future technological advances. Nonetheless, the vast majority of applications
revolve around the use of vector modes,  nonseparable in their spatial and
polarisation DoFs, exploiting either their non-separability nature or their
non-homogeneous transverse polarisation distribution. The development of
applications of  nonseparable states in other DoFs has not been fully explored
and will definitely set the basis for future technological advances.

Hence, even though  many analogies between the quantum and classical worlds
have been established through the concept of nonseparability, only few DoFs
have been fully exploited to explore such analogies from both, the fundamental
and applications perspective. The zoology and methodology proposed in this
article would surely guide new nonseparable states of light with more DoFs to
be discovered. It is therefore highly probable that in the years to come
quantum-analogue nonseparable states of light will merge as one of the main
branches in modern optics.

The authors thank Dr. Nikitas Papasimakis for useful discussions. CRG
acknowledges support from the National Natural Science Foundation of China
(61975047 and 11934013) and the High-Level Talents Project of Heilongjiang
Province (Grant No. 2020GSP12).

YSYijie Shenis a senior research fellow in Optoelectronics Research Centre
(ORC), University of Southampton, UK, also working as a Marie S.-Curie
Research Fellow funded by MULTIPLY Marie S.-Curie Postdoctoral Fellowships. He
received the Ph.D. degree in optical engineering at the Department of
Precision Instrument, Tsinghua University, Beijing, China, in 2019. He
received the B.S. degree in mechanical engineering and automation from South
China University of Technology, Guangzhou, China, in 2015. During Mar. 2019 to
Jun. 2019, he was invited as a visiting researcher in School of Physics:
Structured Light Laboratory, University of the Witwatersrand (Wits
University), Johannesburg, South Africa, and also invited as a visiting
researcher in National Laser Centre, Council for Scientific and Industrial
Research (CSIR), Pretoria, South Africa. His research interests include
structured light, ultrafast nonlinear optics, quantum optics, metamaterial and
nanophotonics.

CRGCarmelo Rosales-GuzmánIs a principal investigator at Centro de
Investigaciones en Optica (CIO, Mexico) since 2020 and an associate professor
at the University of Science and Technology of Harbin (China) since 2018. He
was a postdoctoral research fellow at the structured light laboratory of the
University of the Witwatersrand (South Africa) from 2015 to 2018. He obtained
his PhD degree in 2015 with the highest distinction cum laude from ICFO-The
institute of photonics sciences (Barcelona, Spain). His main research interest
involves the generation, characterisation and applications of structured
light.

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