Quantum mechanics with patterns of light: Progress in high dimensional and multidimensional entanglement with structured light

Quantum mechanics is a mature topic that dates back more than 100 years. Revolutionary in its day, it has become one of the most tested theories in Physics and a pillar of our modern world view. Equally impressive are the technological advances that have emerged from the application of the theory, including lasers and transistors. Despite these successes, the theory has always invoked debate because of seemingly counterintuitive predictions. Foremost among these are the predictions of “spooky action at a distance” as a result of quantum entanglement, the so-called EPR paradox,¹ as well as other paradoxes such as those of Hardy² and Leggett.³ Yet, quantum entanglement is a quintessential property of quantum mechanics and a much sought after resource for information processing and communication. But entanglement is notoriously fragile, decaying through decoherence when quantum systems interact with an environment, a process not so well understood. For this reason, photonic quantum states have great appeal,⁴ exploiting the weak interacting nature of photons, although it is generally accepted that composite approaches are likely to be needed for a complete system, e.g., fleeting photonic states may be ideal for long distance transmission but rather more difficult to harness for storing and retrieving quantum information.^5–7 For these and many other reasons, photonic quantum states have been extensively studied in the context of the second quantum revolution—that of harnessing quantum states in general, and entangled states in particular, for quantum inspired sensing, metrology, imaging, computation, and communication.^8–14 This is only fitting given that it was the nature of light itself that launched the first quantum revolution.

Early photonic entanglement experiments considered polarization as the degree of freedom of the photon to exploit, partly because the theory itself was well developed for two-level quantum systems and partly because the optical tools for polarization control were readily available and sophisticated. The first violation of a Bell-type test was performed using polarization as the entangled degree of freedom,^15–17 and even today, the much mooted quantum network appears closer to reality with polarization, with seminal demonstrations reaching distances of up to 1200 km via free-space satellite-based distribution¹⁸ and over 100 km over conventional optical fiber,¹⁹ critical distances to close the network loop while facilitating easy integration into classical networks for fast and secure communication.^20–22 

Given the very advanced control of polarization-based quantum experiments with “qubits” (two dimensional quantum states), it is moot to ask why bother with spatial modes of light, so-called structured light, the topic of this review. Spatial modes of light offer a route to accessing high dimensional Hilbert spaces with dimensionality well beyond the two dimensional limit of polarization. Such “qudits” (high dimensional quantum states) have known advantages over their qubit analogs,^23–28 summarized as follows:

1. A large encoding alphabet to realize $I=log2(d)$ bits per photon of information. Clearly, d can be as large as the experimental parameters allow, and so one can encode significantly more information per photon than the 1 bit/photon available with qubits. This means that fewer photons need to be sent as qudits in order to transmit the same information as a qubit channel.

2. A limited cloning fidelity that scales inversely with dimensions, with an upper bound given by $F=12+[1/(1+d)]$⁠. For qubits, the cloning fidelity can be as high as $F=56≈83%$⁠, whereas for a very high dimensional state, we find that $F→50%$⁠. This means that it is far more difficult for an eavesdropper to hack a high dimensional state than it would be to hack a qubit state.

3. As a consequence of (2), there are security benefits such as an increased error threshold for quantum key distribution (QKD). The error threshold is below 13% for qubits but up to 50% for qudits, with the exact values depending on the protocol and states used.

4. Larger violations of Bell-type inequalities, limited to $S<22$ for qubits but $S>22$ and scales with the dimensions for qudits. Violation of local realism is not only important for fundamental studies of quantum mechanics but also crucial in certain QKD protocols, indicating the presence of an eavesdropper. Qudits therefore hold the promise of QKD transmission even when a qubit channel would cease to be secure.

Now, given the advantages, one asks how best to realize such qudit states. There has been tremendous effort in doing so, for example, using time-frequency entangled photons,^29,30 path entanglement,^31,32 and atoms/ions,^33–35 to name but a few, but often restricted to multiple two-level systems, which we refer to as multidimensional, rather than true high dimensional states. An alternative avenue to realizing multiple photon and high dimensional entanglement is to exploit the orthonormal and complete basis provided by certain classes of spatial modes of light, the so-called structured light³⁶ in the form of quantum states, the subject of this review. This exciting field emerged only recently following seminal work using the orbital angular momentum (OAM) of light as a basis.³⁷ Since then, the field has exploded in activity and depth with several noteworthy reviews.^38–42 

In this review, we concentrate on recent progress in the field, first covering the basics of what constitutes a spatial mode toolkit insofar as quantum experiments are concerned. We then delve into the concepts of high dimensional Hilbert spaces spanned by spatial modes and highlight the progress made in pushing the dimensionality boundary. We start with scalar structured light in two and high dimensions before moving on to vectorial structured light as multiple two dimensional entangled states and high dimensional single photon states. We review hybrid entanglement that combines spatial modes with polarization and consider some example applications of such spatial mode entanglement. A very recent development has been the push to combine an increased dimension with multiple photons (more than two), for example, to realize teleportation and entanglement swapping with spatial modes of light. Finally, we conclude with some remarks on the challenges in using high dimensions with spatial modes of light, pointing out the open and critical questions that remain yet unanswered.

In this section, we outline in tutorial style how one would go about building the core toolkit required to understand and perform spatial mode entanglement experiments, which we illustrate in Fig. 1. The easy part is the creation step: the workhorse of many quantum optics laboratories, spontaneous parametric downconversion (SPDC), gives you high dimensional entanglement for free.^43,44 To understand this, consider the linear momentum conservation laws embedded in the phase matching conditions of the crystal: a pump photon with momentum $ℏkp$ is incident on a crystal, with the output of the SPDC process two photons, the signal (denoted as state A) and the idler (denoted as state B), of momenta $ℏks$ and $ℏki$ such that $kp=ks+ki$ and $ωp=ωs+ωi$⁠. The phase matching conditions look very much like the core criteria for entanglement. The resulting SPDC biphoton state in linear momentum space can be written as

Here, the coefficients c_n are the spectral power distribution of the series, whose distribution depends on the conditions in the experiment, i.e., $|cn|2$ is the probability of finding photon A in state $|kn⟩$ and photon B in state $|kp−kn⟩$⁠. The expression in Eq. (1) shows some of the possibilities that might arise from this conservation law, and it is clear that the resulting state is nonseparable: a measurement of photon A in a particular direction (momentum) and energy results in the other photon collapsing into a correlated state, e.g., if photon A is measured as $|k1⟩$⁠, then photon B will be measured as $|kp−k1⟩$⁠. While the summation is infinite, there will be practical bounds set by the experimental conditions. For example, the allowed spectrum of states is dictated by factors such as the wavelength filters used in the experiment, typically $Δλ≈5−10$ nm about the central wavelength of the downconverted light, the size of the pump light at the crystal, and the mode overlap at the detectors (see later). In a type I SPDC experiment, the downconverted photons are produced with the same polarization, orthogonal to the pump. One can engineer the arrangement of the wavevectors as one likes, for example, to be aligned along the propagation of the pump or to diverge from the direction of the pump. Photons of the same wavelength are emitted on concentric cones about the pump axis and can be tuned by either the angle of the crystal or crystal temperature, depending on the type of crystal used. In a collinear arrangement, the wavevectors of the SPDC photons are aligned and the downconverted light appears as shown in Fig. 1. In a type II process, the polarization states of the SPDC photons are orthogonal to one another, and so this is often used to produce polarization entangled photon pairs: the SPDC light has in general two concentric rings with the entangled states sampled from the intersection of the two. The process can be engineered (by crystal choice) to be degenerate so that within the allowed bandwidth, the SPDC photons have the same wavelength.

It is always possible to express the high dimensional entanglement in another basis, and here, we wish to use the spatial modes of light as a basis. The requirement is that the mode set forms a complete, orthogonal, and orthonormal set. Examples would include the Laguerre–Gaussian (⁠$LGp,ℓ$⁠) modes of radial order p and azimuthal order ℓ, the Hermite–Gaussian (⁠$HGn,m$⁠) modes of orders n and m (associated with zeros in the field along the x and y directions, respectively), Bessel–Gaussian (⁠$BGkr,ℓ$⁠) modes with a continuous radial wavevector, k_r, and so on.⁴⁵ A popular choice in entanglement experiments with spatial modes is that of modes carrying OAM. It has been understood for just over 25 years now that light with a helical wavefront can carry OAM.⁴⁶ Thus, if we contract the mode set to reveal the OAM content, perhaps by setting p = 0 in the Laguerre–Gaussian basis, then we may write an OAM vortex mode as

where $Aℓ(r)$ is the radial enveloping function whose shape depends on the type of spatial mode used, e.g., Laguerre–Gaussian or Bessel–Gaussian (see discussion later), and also in general on ℓ. Equation (1) can be expressed in the spatial mode basis as

where we have applied the fact that OAM is conserved down to the single photon so that the signal and idler OAM must add to that of the pump: $ℓpℏ=ℓsℏ+ℓiℏ$⁠. Very often, the pump photon is Gaussian (⁠$ℓp=0$⁠) so that the SPDC state reduces to a Schmidt basis

This is still a high dimensional entangled state, with a dimensionality that depends on the number of spatial modes involved in the summation. This number changes from basis to basis, and so rather counter-intuitively, one can adjust the dimensionality of the entanglement by selecting the basis and even the parameters within a basis such as the phase and scale of the mode.^47,48

Selecting the basis manifests as a choice of measurement, which brings us to the second key part of the toolkit: how to make a pattern sensitive detector to perform the measurement step with spatial modes of light. We do have such a “pattern sensitive detector” in the form of a single mode fiber: it only allows a Gaussian to pass through and gives a “click” at the physical detector at the other end. But this is only good for one pattern, whereas we need a detector for any pattern. The idea is to exploit the reciprocity of light in laser beam shaping solutions. Say that a hologram is designed to convert a Gaussian mode into a pattern of type, M_A. Then, by reciprocity, passing M_A backward through the hologram will return the pattern to a Gaussian mode. Going backward implies coding the conjugate of the original pattern, $MA*$⁠. If a single mode fiber is placed after the hologram, then only the Gaussian mode will be collected, and so a click will result only if the input is M_A. Ideally, if the input is any other pattern, then the result would be zero. Thus, a beam shaping element, perhaps a computer generated hologram, and a single mode fiber together form the core components of a spatial mode detector,^49,50 as illustrated graphically in Fig. 2.

The detection probability, $|η|2$⁠, in the SPDC experiment is then calculated from the overlap integral,

where $f(x)$ describes the crystal, pumped by a mode M_p, $g(x)$ describes the single mode fibers (SMFs), and the projections are done into modes M_A and M_B for SPDC photons A and B, respectively. For example, in the thin crystal approximation (when the crystal is much thinner than the Rayleigh length of the pump mode), $f(x)≈1$⁠; for single mode fiber collection, the fiber functions are simply the Gaussian mode that they accept, $g(x)∝exp (−(r/w0)2)$⁠, and usually, the pump is also Gaussian: $Mp(x)∝exp (−(r/wp)2)$⁠. Continuing with OAM as an example, the modes to project into would be $MA=exp (iℓAϕ)$ and $MB=exp (iℓBϕ)$ (here, we consider only the azimuthal part as the SMFs already contain the radial Gaussian components of each mode), resulting in

Only when $ℓA=−ℓB$⁠, would the detectors result in coincidences.

The first quantum entanglement experiment with spatial modes of light exploited correlations in OAM.³⁷ In this work, the Zeilinger group used SPDC as an entanglement source and hard-coded computer generated holograms to perform projective measurements in the OAM basis. They exploited the reciprocity of light discussed earlier: the group used fork holograms, traditionally used to create classical OAM states, as detection devices. First, the work put to rest some confusion in the community as to whether or not OAM was conserved down to the single photon level. The strong antidiagonal correlations confirmed that indeed the signal and idler photon OAM add to that of the pump (⁠$ℏ$ terms dropped): $ℓp=ℓs+ℓi$⁠. This is now referred to as the spiral spectrum, with its full-width at half-maximum (the spiral bandwidth), an indication of how many dimensions are available in the experiment. Next, to confirm entanglement, the group performed a version of ghost imaging. One arm was projected into a superposition state of $|0⟩+|1⟩$ by moving the fork hologram off-axis, while the other arm was scanned to measure the resulting intensity structure. If the state was entangled, then a shifted vortex would be observed with peaks on either side of it, exactly as is seen classically.⁵¹ The observation of a shifted vortex with asymmetric intensity peaks rather than a blob of light with no dip confirmed that the state was indeed of the form $|0⟩+|1⟩$⁠, a coherent superposition and not an incoherent mixture of modes. This was an ingenious test of entanglement at a time when little of today's spatial mode toolkit existed.

The approach in the aforementioned seminal work was advanced by replacing the hard-coded diffractive elements with dynamic spatial light modulators (SLMs)⁵² (see Fig. 3 and Ref. 53), opening the way for quantum measurements to be performed in real-time and in an automated manner, including in multiple dimensions,^54,55 dynamic Bell tests⁵⁶ and precise Quantum State Tomography (QST).⁵⁷ The latter studies in two dimensions used parallels between the measurement steps in polarization to infer what might be needed to probe entanglement with spatial modes. Using OAM as an example, one can create a Bloch Sphere for OAM analogous to the Poincaré Sphere for polarization.⁵⁸ Following this, the idea is to map the orthogonal and superposition states in one basis to that of the other, e.g., $|L⟩≡|ℓ⟩, |R⟩≡|−ℓ⟩$⁠, $|L⟩+exp (iθ)|R⟩≡|ℓ⟩+exp (iθ)|−ℓ⟩$⁠, and so on. One then simply reads off the necessary projection in OAM from the equivalent one in polarization. This is illustrated in Fig. 4 for Bell-type tests and QST projections; the reader is referred to a recent tutorial on the topic of QST with spatial modes of light⁵⁹ for more details. Later, this two dimensional measurement approach was extended to high dimensional QST,^60–62 and then, high dimensional Bell tests were performed on d = 3 states,^63–65 eventually up to d = 12.⁶⁶ These advances followed from theoretical work on Bell-type inequalities beyond qubits,⁶⁷ where each photon is projected onto a carefully engineered superposition state in order to maximally violate the inequality. Extensions of this idea used fractional OAM to create the superpositions, again showing a strong violation of the inequality.^68–70 

While there has been tremendous effort in studying entanglement with OAM, other spatial modes have received considerably less attention. The reasons are no doubt partly historical and partly because OAM is intuitive and easy to measure, requiring phase-only holograms, no amplitude modulation, and no inherent scale parameter control. One extension has been the demonstration of entangled Bessel modes.⁷¹ The switch from vortex OAM modes to Bessel OAM modes allowed the radial wavevector, k_r, of the Bessel function to be used as an extra degree of freedom with which to control the OAM spectrum.⁷² Moreover, k_r control could be implemented with phase-only axicons and binary phase functions for easy coding on SLMs. As k_r was increased, the spiral spectrum flattened and the bandwidth increased—an early demonstration of controlling the spiral spectrum through radial mode control. An intriguing question at the time was whether the known classical properties of spatial modes could be transferred to the quantum realm. It was known that the nondiffracting nature of classical light would be transferred to that of a single heralded photon,⁷³ but would this be true for a biphoton high dimensional quantum state? In particular, would self-healing Bessel beams also result in self-healing quantum states when expressed in the Bessel basis? The answer was yes. By obstructing a quantum state with an opaque disk and then later projecting into a Bessel mode, the entangled state could be recovered from the noise, even in high dimensions,⁷⁴ shown in Fig. 5. Thus, spatial mode entanglement gives not only access to higher information capacity and security but also additional benefits derived from the very nature of the basis itself.

Following this, several other spatial modes have been used in entanglement experiments, including Ince–Gaussian modes,⁷⁵ Hermite–Gaussian modes,⁷⁶ Walsh modes,⁷⁷ and the radial p modes in the Laguerre–Gaussian basis.^78–82 A core requirement for QSTs is the creation of various mutually unbiased bases, which too have been created for various mode sets, including OAM.⁸³ The findings have been that, as expected, the OAM alone is not a complete basis, and thus, there are “missing” dimensions when comparing theory with experiments. In general, both indices of the mode set are necessary in order to correctly describe an optical mode on the transverse plane.

All the aforementioned work concentrated on the measurement step for postselection of the state. By this approach, it is possible to tune the dimensionality in a standard SPDC experiment using SLMs and crystal parameters, with typical laboratory values on the order of d = 20.⁴⁸ Engineering of a particular high dimensional quantum state without postselection is a subject still in its infancy. A few approaches have been successfully demonstrated thus far: the first relies on shaping the pump light to engineer the spatial mode spectrum seen in the downconverted photons. Here, the objective is to shape M_p to engineer the spectrum through the adjustment of the overlap in Eq. (5). This was first done with a flipped Gaussian pump, approximating a Hermite Gaussian mode, to shape the downconverted spectrum away from the traditional antidiagonal⁸⁴ and later extended to creating Bell states,⁸⁵ controlling a three dimensional state by an SLM controlled pump,⁸⁶ production of high dimensional Bell states up to d = 5,⁸⁷ and direct transfer of the pump state to the downconverted state,^88,89 all of which can easily be understood and simulated with classical light using the Klyshko model.^90,91

The second approach utilizes quantum interference at a beam splitter to engineer high dimensional states based on their symmetry. This was successfully shown with OAM up to d = 6.⁹² The physics of the Hong-Ou-Mandel (HOM) effect for an incoming entangled state dictates that only antisymmetric states result in coincidences (one photon in each path), while all symmetric states result in no coincidences (two photons in one path). By using dove prisms prior to the beam-splitter, the authors could engineer an OAM dependent phase to control the state symmetry and hence which states pass through the filter.

Yet, another approach is to use path information, e.g., by mapping the position to OAM with mode sorters for d = 3 states,⁹³ using spatial entanglement in multicore fiber up to d = 4,⁹⁴ and recently exploiting entanglement by path identity,^95,96 as illustrated in Fig. 6. The core idea of the latter approach is to create superpositions of photon pairs from different origins (a range of SPDC sources) and then arrange the crystals in such a way that the emerging photons have identical paths. Although each entangled pair is initially produced in a simple Gaussian state (⁠$|0⟩|0⟩$⁠), judiciously selected and positioned optical elements between the crystals can be used to transform the final state to some engineered superposition, including GHZ and W states. The experiment confirmed the approach, demonstrating states up to d = 3.⁹⁶ 

So far, we have only exploited the amplitude and phase of structured light in states based on scalar modes, while vector combinations hold tremendous promise for realizing the best of both worlds. Thus, while spatial mode developments have benefited from the precise control of an individual degree of freedom (DoF), more exotic forms of entangled photonic states have surfaced, interfacing various DoFs and in some cases using all available DoFs.^97–99 Among these exotic quantum states, hybrid entanglement emerged. Hybrid entanglement involves the entanglement of two spatially separated particles, each in a unique DoF.

In the sections to follow, we cover the basic techniques for generating and detecting hybrid entangled polarization and spatial mode states both in their quantum and classical manifestations, focussing mainly on their manipulation in high dimensional state spaces.

To begin, let us consider two photons that are spatially separated and defined in the spatial and polarization DoFs, as illustrated in Fig. 7. The biphoton state can be expressed as

where a and b are the normalized probability amplitudes; while the orthogonal modes $|uR,L⟩∈H∞$ represent the spatial modes on an infinite dimensional Hilbert space (⁠$H∞$⁠), $|R⟩$ and $|L⟩$ are the canonical right- and left-circular polarization states defined on the qubit space, $H2$⁠, otherwise known as the Poincaré Sphere. When $a=b=12$⁠, the biphoton is said to be maximally entangled and the individual subsystems cannot be written as a separable product.

The concept of using independent DoFs with spatially separated photons was initially proposed by Zukowski and Zeilinger in 1991 for the purpose of enhancing Bell inequality violations for nonlocality tests on entangled photons.¹⁰⁰ Prior to their proposal, Bell inequality violations were demonstrated using polarization qubits (see Ref. 17 for example). Their adaptation allowed photon A's path to be controlled by photon B's polarization. A decade later, the concept was physically implemented by Walborn¹⁰¹ and later by Ma.¹⁰² The idea is to interchange the path and polarization: the polarization eigenmodes of photon B determine the physical path that photon A occupies. This was illustrated in Walborn's quantum eraser experiment:¹⁰¹ a polarizer filtered the state, say $|R⟩(|L⟩)$ or a superposition (⁠$|R⟩+|L⟩$⁠), causing photon A to traverse a single path $|uR⟩(|uL⟩)$ or to occupy both paths simultaneously $(|uL⟩A+|uR⟩A)$⁠, thereby leading to path interference. Intriguingly, this meant that the particle or wave-nature of one photon could be controlled with an independent DoF of a distant photon.

The field advanced rapidly after the introduction of spin–orbit (SO) coupling optics, interfacing polarization (spin), and OAM (orbit) at the single photon level.¹⁰³ In the following year, the same group reported hybrid entanglement of spatially separated photons using polarization entangled photons (type II phase matching) and converting one of the pairs into the OAM DoF using SO optics,¹⁰⁴ repeated later in a similar approach using type I phase matching.¹⁰⁵ The difference was that the photons were initially defined in the OAM DoF and required an OAM to spin mapping. Accordingly, the spatial modes were now associated with integer topological charges to produce an OAM hybrid state given as

Following these advances the term “hybrid entanglement” was colloquially used to describe systems expressed in the form of Eq. (7). Hybrid entanglement between transverse spatial modes carrying orbital angular momentum and polarization is a promising tool for quantum information processing¹⁰⁶ and facilitated by easy generation using standard SPDC processes together with SO coupling optics.^105,107 The reader is referred to Ref. 108 for a comprehensive review on the historical developments.

In those studies, A and B still refer to two photons, each entangled in one DoF. It is also possible to interpret Eq. (8) as one photon with local entanglement in two DoFs, A and B. Interestingly, the nonseparability of the hybrid state in Eq. (8) is not unique to quantum mechanics, with local entanglement appearing in many classical systems, including classical light.^109,110 This is sometimes referred to as classically entangled light. There are, however, subtle differences between entanglement in spatially separated multiparticle systems and in single photons and classical beams:¹¹¹ entanglement between spatially separated systems (intersystem entanglement) is richer than entanglement between different DoFs of a single system (intrasystem entanglement). Both the biphoton and single photon forms of hybrid entangled states have been used for quantum information processing in recent years, including high dimensional state preparation of a wide variety of hybrid entangled states using hyperentangled photon pairs,¹¹² while several studies have considered the quantumlike behavior of local entanglement in classical fields, demonstrating a full suite of quantum tools applied successfully to characterize classically entangled light.¹¹³ 

In optical fields, the local entanglement can be identified in modes having complex electric fields with spatially varying polarization vectors, for example, the radially and azimuthally polarized fields shown in Fig. 8 with Laguerre–Gaussian and Bessel–Gaussian profiles. These modes are commonly referred to as cylindrical vector vortex beams and are solutions to the paraxial Helmholtz equation for electromagnetic fields.¹¹⁴ To describe such modes mathematically, we use the Jones formalism for polarization and associate the circular polarization eigenstates with column vectors $|L⟩→êL=(1,0)⊤$ and $|R⟩→êR=(0,1)⊤$ and subsequently define the vector field as

Here, $uL,R(x,y)$ represents the traverse electric fields in each of the orthogonal polarization components, while a and b represent the normalized weightings. It is convenient to adapt Eq. (9) to the bra–ket notation,

and we see that we are back to the initial hybrid state but now only expressing local entanglement: the state $Φ$ possesses local correlations between the spatial and polarization DoFs, which in the OAM basis may be expressed as

where the distinct topological charges $ℓ1,2$ (⁠$⟨ℓ1|ℓ2⟩=0$⁠) characterize the azimuthal phase of each transverse mode marked with a unique polarization. Projections onto linear polarization states collapse the spatial DoF onto the superposition state $(|ℓ1⟩+exp (iα)|ℓ2⟩)/2$⁠, where α is the intermodal phase. Physically, α can be changed by rotating a polarizer, resulting in a variation of the intensity distribution of the spatial modes. For example, as shown in Fig. 8 for $ℓ1,2=±1$ LG and BG profiles, the orientation of the spatial modes (petals) changes with the corresponding polarization projection, and therefore, the measured polarization uniquely determines the observed spatial mode. Conversely, for separable (scalar) modes, only a fluctuation in intensity with no change in the distribution would be observed.

Now that we have described hybrid modes in the quantum and classical context, we can explore their description on Hilbert spaces. Previously, hybrid states with transverse spatial mode and polarization entanglement were restricted to a two dimensional state space. Quantum encoding beyond d = 2 dimensions is possible with the measurement and detection tools already developed. Moreover, the OAM eigenstates are defined on a large Hilbert space, i.e., $|ℓ⟩∈Hd$⁠, where d is the dimensionality of the space. This implies that there is an infinite set of two-dimensional spaces available in what we refer to as multidimensional. A key feature of hybrid states, which has not yet been fully exploited, is their potential for high dimensional encoding as single photons and for multiple two-dimensional encoding as entangled states.

While the hybrid entangled states are limited to multiple two-dimensional spaces, as single photons, a four dimensional alphabet can be constructed. To illustrate the potential benefit of increased encoding dimensions with single photon hybrid states, let us suppose that we are restricted to qubit (d = 2) subspaces of OAM modes, i.e., $H2=span{|ℓ1⟩,|ℓ2⟩}$⁠. By taking the tensor product of the OAM qubits and polarization qubits, a d = 4 subspace is spanned by the new basis ${|ℓ1⟩|L⟩,|ℓ1⟩|R⟩,|ℓ2⟩|L⟩,|ℓ2⟩|R⟩}$⁠, with each DoF contributing two dimensions. A convenient way to represent the new space is by the higher order Poincaré sphere (HOPS)¹¹⁵ illustrated in Fig. 9. The poles of the spheres map the circularly polarized OAM modes, and the equator maps the vector vortex modes.¹¹⁴ In general, there are infinitely many of these HOPSs since $ℓ1,2$ are boundless integers.

A four dimensional basis composed of nonseparable superposition of vector modes can be used to span the HOPS states (with global phases ignored),

Here, $ℓ1$ and $ℓ2$ represent the orthogonal spatial modes and A(r) represents the radial amplitude of each mode (for convenience, assumed to be the same). In the case of $ℓ1=−ℓ2$⁠, the basis states in Eq. (12) are known for their rotational invariance^116–118 since both the circular polarization and OAM states are invariant under arbitrary rotations about the z-axis owing to the mapping $|R(L)⟩→e±iζ|R(L)⟩$ and $|±ℓ⟩→e±iζ|±ℓ⟩$ for rotation angles ζ. For the modes in Eq. (12), the rotations result in a factorizable global phase, leaving the states unchanged—valuable for alignment free communication in space-based quantum networks.

Locally entangled hybrid modes are easily generated by a range of approaches including geometric phase elements tailored with liquid crystals,¹¹⁹ dielectric metasurfaces,¹²⁰ few mode fibers,¹²¹ interferometers,¹²² laser resonators,¹²³ and many more; see Ref. 124 for a recent review. The most commonly used methods for quantum information encoding are SO conversion schemes that exploit geometric phase control, with a typical geometric phase approach illustrated in Fig. 10. The development of SO coupling devices followed from the seminal work of Bhandari¹²⁵ and later with implementations using subwavelength diffraction gratings for transferring OAM to an input light field by introducing a space-variant Pancharatnam–Berry phase.^126,127 However, these devices were limited to the mid-infrared wavelength region due to manufacturing limitations at the time. Following these challenges, it was later that Marrucci et al.¹²⁸ proposed using birefringent liquid crystals in the form so-called q-plates. The coupling rules for input circular polarization states impinging on a q-plate can be summarized as

where q is the charge of the plate. As illustrated in Fig. 10, circular polarization states are inverted with an additional imparted OAM variation of $±2q$ depending on the handedness of the input polarization.

In Fig. 10, we illustrate the setup used for generating high dimensional hybrid modes using q-plates along with wave retarders. For example, the vector mode $|ψ⟩00$ is generated by removing the second wave-plate and orientating the half-wave plate at $θ1=0$⁠. Since $|H⟩|0⟩=12(|R⟩|0⟩+|L⟩|0⟩$⁠) is a superposition state, the SO coupling induced by the q-plate then yields the nonseparable state $12(|R⟩|ℓ⟩+ |L⟩|−ℓ⟩)$⁠. The scheme assumes that the input photon is horizontally polarized and has an initial OAM charge of zero, though there is no restriction on the input radial modes of the photons as illustrated in Fig. 10, showing examples of LG and BG profiles. Just as in the scalar field case, hybrid Bessel modes also allow QKD and state self-healing through obstacles.¹²⁹ It is also possible to create non-symmetric OAM pairs by using metasurfaces, shown graphically in Fig. 11.

More pertinent to this review is the creation of biphoton pairs that are hybrid entangled. To understand how these schemes work, consider the quantum state of the photon pair produced from type I SPDC,

where $|cℓ|2$ is the probability of finding photons A and B in the state $|±ℓ⟩$⁠. Both photons are initially horizontally polarized (⁠$|H⟩$⁠) and entangled in the OAM DoF. The hybrid entanglement between photons A and B is obtained by using geometric phase control to perform SO conversion in arm A, initially done with a q-plate.¹⁰³ Applying the transformation rules of the q-plate to photon A and postselecting on the $ℓ=2q$ OAM subspace of photon B results in

Equation (16) represents a maximally entangled Bell state where the polarization DoF of photon A is entangled with the OAM DoF of photon B. Further, since ℓ is boundless, various OAM subspaces can be accessed by simply selecting a SO coupling device with a different charge. The schemes to do so exploit SPDC as a resource for generating photons that are entangled in the OAM DoF. Typically, one photon is sent through a SO coupling device that maps the photon onto the polarization basis and is subsequently measured using polarization elements, while the second photon remains in the OAM DoF and is propagated to a spatial light modulator (SLM) for OAM projections. Subsequently, the photons are both coupled into single mode fibers (SMFs) and correlated in time using coincidence counters. Alternatively, one could start with type-II SPDC. In this case, the photons would initially be entangled in the polarization DoF and the SO coupling device would map one of the photons into the OAM DoF, resulting in a hybrid entangled state.

The standard QST for biphotons in one DoF (polarization or OAM) can easily be extended to perform measurements on two DoFs,^113,130 as illustrated in Fig. 4 and reviewed in Ref. 59. In the case of local entanglement, the projections are performed sequentially on the same photon (or classical field), while in the case of biphotons, each projection is performed on the appropriate photon, e.g., spatial mode projections on photon A and polarization projections on photon B.

The aforementioned examples are all of the projective type, postselecting the state by a filter-based measurement. Deterministic measurements of arbitrary locally entangled modes are at present still a challenge and limit the capability of quantum information processing with an arbitrary number of modes. While these modes can be uniquely detected through modal decomposition with geometric phase optics,^131–133 the measurement at the single photon level cannot be achieved unambiguously. Milione et al. have shown that when using q-plates to prepare and detect vector vortex modes, one needs two oppositely charged q-plates to detect all the degenerate modes, resulting in losses of up to 50%.¹³² In the context of QKD, such a probabilistic detection would result in lower shift rates, canceling the benefits of the increased dimensionality. Nagali et al.¹⁰⁴ have shown that it may be possible to construct deterministic measurements for d = 4 dimensional basis states for all d + 1 mutual unbiased states with q-plates and dove prisms; however, this limits the subspace of the OAM DoF. Recently, refractive OAM mode sorters¹³⁴ were shown to have the potential of detecting up to 27 dimensions of OAM states.¹³⁵ Following this, Ndagano et al.¹³⁶ developed a deterministic scheme for multiple hybrid spaces in a lossless and deterministic way. To illustrate the method, consider states as defined in Eq. (12). Ndagano and co-workers showed how the states could be detected deterministically through a combination of geometric phase control and an interferometer. First, a polarization grating based on the geometric phase acts as a beam splitter for left- and right-circularly polarized photons, creating two paths

where the subscripts a and b refer to the polarization-marked paths. The photon paths, a and b, are interfered at a 50:50 beam splitter, resulting in

where the subscripts c and d refer to the output ports of the beam splitter and α is the dynamic phase difference between the two paths. Note that the polarization of the two paths is automatically reconciled in each of the output ports of the beam splitter due to the number of reflections for each input arm. At this point, the polarization information is contained in the path.

Finally, by passing each of the outputs in c and d through a mode sorter, the resulting modes can be coupled to a detector and measured deterministically. While it is trivial to measure such nonseparable vector states at the classical level with many photons, here each such state was detected with unit probability at the single photon level. For example, consider the modes $|00⟩$ and $|01⟩$ from Eq. (12), with θ = 0 and $θ=π$⁠,

Through this path interference, the vector modes are mapped as $|ψ⟩±ℓ,0→i|∓ℓ⟩$ in port d and $|ψ⟩±ℓ,π→−|±ℓ⟩c$ in port c. The modes are subsequently differentiated according to the OAM charge that they carry using OAM mode sorters. Thus, the combination of the path (c or d) and lateral location (⁠$+ℓ$ or $−ℓ$⁠) uniquely determines the original vector mode. In contrast to previous methods, the scheme can gain access to numerous OAM subspaces due to the versatility of the mode sorters. The challenges presented by traditional mode sorters are the alignment sensitivity and separation between the detected spots, both of which have been addressed in compact diffractive versions.¹³⁷ 

An interesting application of hybrid states is to abstract the notion of the path for fundamental tests of quantum mechanics. In fact, this was the very motivation for hybrid states in the first place. One such test is that of the quantum eraser. Briefly, a photon traverses two independent paths, resulting in characteristic interference fringes due to its wave nature. Interestingly, when the paths are marked (say with orthogonal polarizations), the interference fringes disappear, interpreted as particlelike behavior. However, performing a complimentary measurement of the path markers recovers the interference fringes even though the paths are distinguishable: the path information in the system has been erased. Traditional demonstrations of the quantum eraser experiment involved interferometers using actual physical path interference^101,138,139 but was recently generalized using hybrid states,¹⁴⁰ with the concept illustrated in Fig. 12.

The idea was to use OAM as an abstract path and polarization as a marker: a hybrid state. To mark the OAM information of photon B, the circular polarization of photon A was converted to linear polarization, resulting in the state

Now, the OAM “path” is marked by linear polarization states. When one path is selected in this way, no OAM interference pattern appeared. However, just as in the double slit case, a projection of the polarization of photon A onto a complimentary (diagonal) basis state collapsed photon B onto a superposition state of $|+ℓ⟩+|−ℓ⟩$⁠, yielding azimuthal fringes with a frequency of $2|ℓ|$⁠. The OAM path information is erased by the polarizer. The visibility of the fringes could be adjusted using the polarizer to allow a direct measure of abstract paths in quantum mechanics.

In addition to fundamental tests, hybrid states hold great promise for applications, notably quantum communication. In particular, hybrid modes have been exploited to realize high dimensional single photon QKD based on the “BB84” protocol¹⁴¹ in free-space^142,143 and fiber.¹⁴⁴ 

Suppose that Alice and Bob wish to share a secure key. Alice randomly prepares single photons in modes from two sets: a vector mode set, $|ψ⟩ℓ,θ$⁠, and a mutually unbiased scalar mode set, $|ϕ⟩ℓ,θ$⁠, defined as

$θ=[0,π]$ is the intramodal phase. For a given $|ℓ|$ OAM subspace, there exist four orthogonal modes in both the vector basis [Eq. (21)] and its mutually unbiased counterpart [Eq. (22)], such that $|⟨ϕ|ψ⟩|2=1/d$ with d = 4. The vector basis can be mapped onto the mode basis in Eq. (12) with the corresponding mutually unbiased basis (MUB) being

where D and A are the diagonal and antidiagonal polarization states. During the transmission, Alice randomly prepares her photon in a vector (scalar) mode state, while Bob randomly measures the photon with either a vector or scalar analyzer based on the detection scheme in Fig. 13(a). At the end of the transmission, Alice and Bob reconcile the prepare and measure bases and discard measurements in complementary bases.

A figure of merit for QKD channels is the quantum bit error rate (QBER), E, which characterizes the probability of obtaining errors in the key. Following this, the mutual information between Alice and Bob can be computed as²⁵ 

The rate at which Alice and Bob can generate a key, assuming practical implementations with BB84 states, can be determined from¹⁴⁵ 

where $Qμ$ is the photon yield for an average intensity μ, $Hd(·)$ is the high-dimensional Shannon entropy, and $fEC$ is a factor that accounts for error correction and is nominally $fEC=1.2$⁠. Moreover, $Δ=(1−P0−P1)/Qμ$ is the multiphoton rate, where P₀ and P₁ are the vacuum and single photon emission probabilities, respectively. The term $1−Δ$ accounts for photon splitting attacks¹⁴⁶ and can be reduced by using ideal photon sources where $Δ→0$⁠.

Such analysis was applied to self-healing Bessel–Gaussian vector modes with heralded single photons.¹²⁹ In this scheme, photons were shown to retain the amplitudes even after encountering an obstruction, showing the benefit of adapting the radial DoF of photons to noisy channels. Advancements in practical implementations of QKD using high dimensional single photon quantum states have seen seminal reports in noisy channels such as optical fiber¹⁴⁷ and free-space,¹⁴² the latter over a 300 m channel, shown in Fig. 14. A raw error rate of 14% was found with a secure key rate of 0.39 bits/photon, which with error correction could be improved substantially to an error rate of 11% and a secret key rate of 0.65 bits/photon. A valid QKD transmission was possible since the error threshold is 18% for d = 4 versus 11% for d = 2 using the same protocol.

Another exciting application is that of optimal cloning. Perfect cloning of unknown quantum states is forbidden by the no-cloning theorem.¹⁴⁸ However, there have been significant developments in achieving the best possible fidelity using universal quantum cloning machines with polarization and OAM qubits by exploiting the symmetrization method.^149–152 

The method is illustrated in Fig. 15(a). The target state to be copied, say $ρ=|ψ⟩⟨ψ|$ for $|ψ⟩∈Hd$⁠, where d is the dimension of the encoding Hilbert space, is interfered with a maximally mixed state $ρmix=Id/d$ using a 50/50 beam splitter. Two clones having identical states will exit the beam splitter coherently through the same port if they are indistinguishable owing to the HOM effect.¹⁵⁴ The cloning fidelity of this scheme scales with dimensionality as $F=12+[1/(1+d)]$⁠.

Recently, applications with high dimensional spatial modes have gained interest for increasing the cloning fidelity, demonstrated with OAM modes up to d = 7 dimensions.¹⁵⁵ Remarkably, hybrid photonic states were also shown to possess a cloning capability on a four-dimensional state space.¹⁵³ In the reported scheme, the authors exploited the simultaneous control of the spin and orbit DoFs, showing cloning fidelities of up to $70%$ for all mutual unbiased states. An illustration of this scheme is shown in Fig. 15(b). They achieved this by preparing hybrid modes encoded with the states of Eq. (12) (the target clone states) and in parallel randomly prepared photons encoded in a mixture of the hybrid modes (the ancillary photon). The photons were then interfered at a 50:50 beam splitter; after bunching, the photons were again separated in the path and detected using q-plates and wave-retarders.

The scheme in Ref. 153 can further exploit the benefit of the high dimensionality of OAM in the symmetrization method if the photons are encoded beyond a single OAM subspace. In general, the symmetrization procedure could be scaled up to N inputs to reach higher fidelity with increasing copies.¹⁵⁶ 

Pushing the dimensionality of quantum states has seen tremendous effort of late, and has included a shift from scalar spatial mode states to hybrid polarization-spatial mode states and more recently to pushing both the dimensionality and the photon number to realize protocols for teleportation, entanglement swapping, cluster states for quantum computing, GHZ and W states, and so on. Insofar as pushing the dimensionality is concerned for quantum computing, seminal advances include the demonstration of high dimensional quantum gates, showing a generalization of the Pauli X gate to d = 4 (Ref. 157) used to create the complete 16 Bell states in d = 4.¹⁵⁸ 

The first experiment to demonstrate multiple photons (more than two) entangled beyond qubits with spatial modes was performed by Malik et al.¹⁵⁹ The authors used two SPDC sources, each with high dimensionally entangled OAM biphotons, and then engineered the experiment to produce an asymmetric three photon state where two of the photons resided in a three-dimensional space and the third in two dimensions, given by

The workhorse of this experiment was an “OAM beam splitter” that sorts OAM modes based on their parity, illustrated in Fig. 16. An initial three dimensional state in the OAM subspaces ±1 and 0 from each crystal was passed through polarizing beam splitters to erase the path information. The OAM beam splitter was engineered to only result in coincidences when the two incoming photons (B and C) had the same OAM parity (both odd or both even). By measuring photon A in a superposition of 0 and –1, and heralding a photon in T, the three photon state could be produced.

Following this was the first demonstration of teleportation and entanglement swapping with OAM modes.¹⁶⁰ The experiment involved creating two pairs of entangled photons, $|ψ⟩AB$ and $|ψ⟩CD$⁠, and passing one photon from each pair to a beam splitter (say photons B and C). Only the antisymmetric states would result in coincidence, a consequence of which was that the initially independent photons A and D now became entangled. The four photon demonstration showed high fidelity teleportation and entanglement swapping in two dimensions and imperfect processes in high dimensions. To overcome this fundamental limit imposed by the use of linear optics, the use of additional photons in the experiment is required.¹⁶¹ 

Many advances have been reported since the seminal work on two photon entanglement in two dimensions with polarization¹⁷ and OAM,³⁷ summarized graphically in Fig. 17. Tremendous effort has been made to push the boundaries of entanglement in terms of photon number^162–169 and dimension.^37,63,170 advances include pushing the photon number up to 10,¹⁷¹ mixing three degrees of freedom with six photons for 18 qubit control,¹⁷² contrast inverted ghost imaging (in four dimensions) with entanglement swapped photons,¹⁷³ realizing three dimensional GHZ states,¹⁷⁴ and witness to a 100 × 100 dimensional entangled state.¹⁷⁵ The latter advance was possible due to a sophisticated dimensionality test using visibilities in multiple subspaces. Coupled to this have been advances in detection schemes, notably those of imaging. Full field analysis of spatial modes is crucial (recall the need for both radial and azimuthal modes to account for the missing dimensions in experiments) and can easily be implemented by imaging systems.¹⁷⁶ By measuring a multipixel full-field of view, the authors verified entanglement to beyond 2500 dimensions. Using compressive sensing and a double pixel camera, it was shown that spatially entangled fields could be scanned at a rate $d2/log (n)$ faster than simple raster scanning, realizing imaging at resolutions of more than thousand dimensions per detector for a channel capacity of 8.4 bits/photon, among the highest yet reported.¹⁷⁷ 

There has been a concerted drive to implement spatial mode entanglement in quantum information processing and communication applications. Foremost among these is the potential of increasing the information carrying potential and better security to realize high dimensional quantum key distribution. Laboratory demonstrations have included the first high dimensional test with entangled qutrits,⁶⁵ followed by carefully constructed MUBs to reach entanglement based QKD up to d = 5,¹⁷⁸ and later single photon preparation up to d = 7.¹⁷⁹ Mixing degrees of freedom has seen d = 4 QKD implemented with hybrid states,^129,142,180 recently reaching d = 8 using radial, azimuthal control, as well as polarization control.¹⁸¹ All these tests have been in free-space.

Unfortunately, a disadvantage to free-space communication with spatial modes (classical and quantum) is turbulence induced modal cross-talk. In the case of entangled states, this leads to a decay in entanglement,^182–185 including high-dimensional entangled states which have been studied theoretically^186,187 and experimentally¹⁸⁸ in turbulence conditions. In all cases, the state decays quickly, even in moderate to weak conditions. But correction is possible. One approach is to probe the quantum channel with classically entangled light using vector vortex beams. It has been shown that the decay in nonseparability of these beams is identical to the decay of entanglement in quantum states, allowing for real-time quantum error correction of the perturbed state.¹⁸⁰ This has been demonstrated with a six-state QKD protocol through turbulence with two dimensional OAM states,¹⁸⁹ with theoretical suggestions to extend this concept to high dimensions.¹⁹⁰ An alternative approach is to use a passive distillation technique based on quantum interference. Here, the idea is to prepare all initial states as antisymmetric, noting that turbulence will then scatter the modes into symmetric states. But passing the perturbed state through a HOM filter will remove all symmetric states when the outcome is conditioned on coincidence. The result is a passive filter than converts noise into loss and has shown to raise the fidelity of the perturbed state back that of the original.¹⁹¹ Finally, one can deploy adaptive optics, which has been predicted to successfully preserve entanglement of high dimensional states even in strong turbulence.¹⁸⁷ 

Quantum communication with spatial mode entanglement is far less advanced in fiber than in free-space, primarily due to the fact that custom fiber is often needed to avoid modal coupling and state decoherence. So far, demonstrations have included QKD in custom vortex fiber,¹⁹² entanglement transport in multimode optical fiber limited to less than 1 m,^193,194 and spatial mode entanglement transport in especially designed multimode fiber over km ranges,^147,195,196 illustrated in Fig. 18. Using hybrid entanglement, it was shown that single mode fiber could be used to distribute multiple two-dimensional states over 250 m,¹⁹⁷ replacing high dimensional entanglement in order to transport such quantum states down conventional optical fiber networks.

In the Introduction, we began by listing the advantages of qudits in general and spatial mode qudits by implication. Now, we note that there are some open challenges with regard to high dimensional spatial mode entanglement. First, although the toolkit has the core elements, it does not in general support deterministic detection of the modes and their superpositions, with only one exception for a d = 4 basis of scalar and vector OAM modes.¹⁸⁰ In all cases where SLMs (or geometric phase equivalents) are used, which is in virtually all quantum experiments with spatial modes, the detector is a filter that can select only one mode (or superposition) at a time. This means that although we have access to high dimensions, we sift through the dimensions one at a time, losing all the advantage. Second, the toolkit for measuring and inferring a high dimensional state is still very inefficient.⁶⁰ The standard approach is based on an over-complete set of projective measurements by QST, which scales unfavorably with the dimension (approximately d⁴).⁵⁹ This means that the measurement process can run for weeks for very high dimensions. Next, inferring information on the state is usually done via the density matrix, which itself is a computationally intensive task to calculate from the QST data. There are approaches that overcome some of these limitations, for example, using compressive sensing,¹⁹⁸ which has been demonstrated up to d = 17, or optimal bases,¹⁹⁹ which do hold promise for a speed up in state determination. Finally, even with a high dimensional density matrix on hand, it is not a trivial task to infer the degree of entanglement, a largely open challenge with some recent prospects for a faster measurement up to d = 8.²⁰⁰ It would be helpful to have entanglement witnesses for high dimensions that did not require a full QST nor a density matrix, an area of active research both theoretically and experimentally.²⁰¹ 

Although photons are weakly interacting, spatial mode entanglement is highly fragile and easily distorted in the most common quantum channels, despite the fact that high dimensional states are more robust to noise.²⁸ This was highlighted in Ref. 142 where they transmit nonseparable modes using a 300 m long intracity link in Ottawa, reporting quantum bit error rates of up to 14%. In such practical free-space channels, turbulence causes phase distortions. Moreover, it has been proven that vector modes are not any better than scalar modes in turbulence,²⁰² and though self-healing modes are resilient to obstructions, they are not resilient to phase distortions from turbulence.²⁰³ As a result, extensions to larger distances will require correction schemes. However, process tomographies in high dimensions are largely nonexistent, and only a handful of studies have attempted to address the preservation of high dimensional spatial mode entanglement in noisy channels.

An intriguing new avenue to explore is the parallel that seems to have emerged between nonseparable classical states and quantum entangled states.^109,110 Although seminal tests have been restricted to two dimensions,¹⁸⁰ it is understood that high dimensional classically entangled states may also be created.²⁰⁴ This may open new ways to develop quantum tests in high dimensions, first with classical light before beginning the complex task of handling single photons.

Finally, the benefits of high dimensional states to noise have so far not been observed in the laboratory. The few models that suggest this benefit assume “white noise” as the source and not noise resulting from modal cross-talk, with the latter being the dominant source of noise in quantum communications with spatial modes. The predicted noise tolerance suggests that in some d dimensional system, there is a chance of finding k dimensional entanglement. What is the value of k and how does one find those subspaces? These questions have not yet been answered. Indeed, when losses and concomitant degraded fidelity are factored into the equation, it has been found (experimentally) that multiplexing two-dimensional states gave twice the performance of a single four dimensional QKD protocol insofar as the QKD key rate was concerned.¹⁴⁷ Thus, the toolkit for high dimensional spatial modes is still too empty, and the theoretical benefits are not yet fully realized in experiment. This may change in the future, and certainly, these open challenges are worthy of investigation.

In conclusion, high dimensional entangled states, both in their scalar and hybrid manifestations, hold promise for a host of new quantum applications to fuel the next quantum revolution, particularly in quantum communication. Despite many advances, the toolbox for handling high dimensional states, theoretically and experimentally, remains rather empty, with most studies restricting themselves to replicating two dimensional studies. There is an urgent need for further work in this field if the true benefits of the dimensionality are to be realized.