# A robust basis for multi-bit optical communication with vectorial light

Keshaan Singh  Isaac Nape  Wagner Tavares Buono  Angela Dudley  Andrew Forbes
<EMAIL_ADDRESS>School of Physics, University of the Witwatersrand,
Private Bag 3, Johannesburg 2050, South Africa

###### Abstract

Increasing the information capacity of communication channels is a pressing
need, driven by growing data demands and the consequent impending data crunch
with existing modulation schemes. In this regard, mode division multiplexing
(MDM), where the spatial modes of light form the encoding basis, has enormous
potential and appeal, but is impeded by modal noise due to imperfect channels.
Here we overcome this challenge by breaking the existing MDM paradigm of using
the modes themselves as a discrete basis, instead exploiting the polarization
inhomogeneity (vectorness) of vectorial light as our information carrier. We
show that this encoding basis can be partitioned and detected almost at will,
and measured in a channel independent fashion, a fact we confirm
experimentally using atmospheric turbulence as a highly perturbing channel
example. Our approach replaces conventional amplitude modulation with a novel
modal alternative for potentially orders of magnitude channel information
enhancement, yet is robust to fading even through noisy channels, offering a
new paradigm to exploiting the spatial mode basis for optical communication.

††preprint: APS/123-QED Figure 1: Modal vectorness for multi-bit encoding.
Conceptual visualization of a sequence of vector beams with tuneable
vectorness (denoted as $\mathcal{C}$). Each $\mathcal{C}$ value is binned into
the user-defined basis allowing for the received $\mathcal{C}$ to map to a
bit-string of length determined by the basis choice. After propagating through
an aberrating channel (e.g. atmospheric turbulence, underwater, single-mode
fiber or even transparent cellular material) the intensity and polarization
profiles of the beams are distorted, however the $\mathcal{C}$ is left
unaffected.

Optical communication has been an integral part of human society since
recorded time, initially visual and thus free-space based, with wire-based
solutions only emerging 200 years ago, from copper wire to modern day fibre
optics. Today, our optical communication solutions are rapidly reaching their
capacity limit Richardson (2010); Richardson et al. (2013), requiring new
degrees of freedom for packing information into light. Here so-called
structured light comes to the fore Forbes et al. (2021), where light’s spatial
degree of freedom is used in space division multiplexing (SDM) Li et al.
(2014) and mode division multiplexing (MDM) Berdagué and Facq (1982), for more
channels and more capacity per channel, with significant advances over recent
years Trichili et al. (2019, 2020). Topical among the multiplexing techniques
is the use of orbital angular momentum (OAM) modes, with excellent reviews
available on the topic Padgett (2017); Willner et al. (2015, 2021). For
instance, in free-space reaching petabits-per-second data rates in the
laboratory Wang et al. (2014) and 80 gigabits per second over 260 metres Zhao
et al. (2016) while advances in optical fibre have shown 1.6 Tbit/s OAM
communication over kilometer lengths with a custom ring core fiber Bozinivic
et al. (2013). In both these channels, modal cross-talk is a limiting factor.

Atmospheric turbulence induced modal scattering in free-space links is a
particularly severe example Anguita et al. (2008); Ren et al. (2016); Krenn et
al. (2014); Rodenburg et al. (2012); Zhang et al. (2020); Malik et al. (2012);
Chen et al. (2016); Tyler and Boyd (2009); Cox et al. (2020), impeding
classical and quantum communication links with structured light to only
hundreds of meters Zhao et al. (2016). To mitigate this, there has been active
research in searching for robust states of structured light in turbulence,
including Bessel-Gaussian Mphuthi et al. (2018, 2019); Lukin (2014); Bao-Suan
and Ji-Xiong (2009); Zhu et al. (2008); Nelson et al. (2014); Ahmed et al.
(2016); Cheng et al. (2016); Doster and Watnik (2016); Watkins et al. (2020);
Vetter et al. (2019); Yuan et al. (2017), Hermite-Gaussian Cox et al. (2019);
Ndagano et al. (2017a); Restuccia et al. (2016); Ndagano et al. (2017b),
Laguerre-Gaussian Trichili et al. (2016); Zhao et al. (2015); Zhou et al.
(2019); Xie et al. (2016); Li et al. (2017) and Ince-Gaussian Gu et al. (2019)
beams, but none have shown to be robust in laboratory and real-world
experiments. This can be explained best from the perspective of OAM: the
atmosphere itself can transfer OAM to and from the beam, so that all beams
will be affected equivalently regardless of their initial structure Klug et
al. (2021). A recent development has been the creation of vectorial
combinations of OAM, for vector vortex light Zhan (2009); Rosales-Guzmán et
al. (2018) with inhomogeneous polarization patterns, and these too have found
applications in optical communication, both classical Milione et al. (2015);
Zhu et al. (2021) and quantum Ndagano et al. (2017c); Sit et al. (2017); Nape
et al. (2018); Ndagano et al. (2018). Unfortunately they have been found not
to be resilient in turbulence Cai et al. (2008); Ji-Xiong et al. (2010); Wang
and Pu (2008); Cox et al. (2016a); Ndagano et al. (2017d), a fact that has
been generalised to many different channels, revealing that the vectorial
nature of such light remains invariant even as the spatial pattern itself
alters Nape et al. (2022). It is this “altering” or distorting of the pattern
that impedes MDM, since its very premise is to “recognise” the original
pattern in its undistorted form.

Here we present a novel approach to exploiting MDM in optical communication,
foregoing the discrete modal basis of MDM and instead exploiting the
polarisation inhomogeneity (vectorness) of vectorial light as an encoding
basis in a manner that does not require the mode itself to be recognised: a
modal basis without the penalty of a modal detection. Similar to amplitude, it
spans 0 (scalar light) to 1 (fully vectorial) but with the pertinent benefit
that the full range can be used as a high-dimensional alphabet, rather than
the two dimensional (on/off) amplitude alphabet. Because the vectorness is an
invariant quantity, it remains intact even in the presence of aberrations,
where all parties will receive the same information regardless of their
particular channel conditions. This implies the potential for optical
communication free of modal-induced noise without the need for adaptive optics
or digital corrective procedures. We demonstrate our encoding scheme on a
highly-aberrated channel of dynamically changing atmospheric turbulence,
chosen as an extreme example of a noisy channel, showing correction-free data
transmission under a wide range of conditions, with the dimensionality of the
encoding primarily limited by detector noise. Our approach replaces
conventional amplitude modulation with a modal alternative for potentially
orders of magnitude channel information enhancement, offering a new approach
to exploiting structured light for optical communication.

## RESULTS

Figure 2: Experimental demonstration. a) Diagram showing the setup used to
generate vector fields with tune-able concurrence (EMITTER) and measure the
concurrence (RECEIVER) after propagating through heated air (CHANNEL) - [Li \-
lens, HWPi \- half-wave plate, WP - Wollaston prism, DMDi \- digital micro-
mirror device, LP - linear polarizer, CCD - charge-coupled device]. b)
Exemplar multiplexed binary amplitude holograms used to holographically
control the concurrence of generated beams via weighting parameter $a$ \-
associated bit strings are displayed. c) Exemplar Stokes intensities which are
integrated over the dashed circles in order to calculate the concurrence
[$P_{H}$ \- horizontal, $P_{V}$ \- horizontal, $P_{D}$ \- diagonal and $P_{R}$
\- right-circular polarization integrated powers].

A new encoding scheme. Our technique involves exploiting the spatial mode
basis using vector fields of the form

$|\Psi\rangle=\sqrt{a}|\psi_{+}\rangle|H\rangle+\sqrt{1-a}|\psi_{-}\rangle|V\rangle\,,$
(1)

where $|\psi_{\pm}\rangle$ represents any pair of orthogonal spatial modes,
$|H(V)\rangle$ are the horizontal(vertical) linear polarization Jones vectors
and the real amplitude determined by $a$ ensures that the total power in the
field remains constant for $a\in[0,1]$. Such beams are ubiquitous, and in
particular are the natural modes of both optical fibre and free space when
$|\psi_{\pm}\rangle$ are chosen appropriately Rosales-Guzmán et al. (2018). It
has been shown that fields of this type have a degree of non-separability in
their spatial and polarization degrees-of-freedom (DoFs) which can be
quantified using a quantum inspired metric (the concurrence), termed the
vector quality factor McLaren et al. (2015); Ndagano et al. (2016), and which
we will refer to as the vectorness for short. The concurrence, $\mathcal{C}$,
for the field in Eqn 1 is given by Wootters (1998)

$\mathcal{C}=2\sqrt{\langle\psi_{+}|\psi_{+}\rangle\langle\psi_{-}|\psi_{-}\rangle-|\langle\psi_{+}|\psi_{-}\rangle|^{2}}\,,$
(2)

which reduces to $\mathcal{C}=2|\sqrt{a(1-a)}|$. We see that the weighting
parameter $a$ allows $\mathcal{C}$ to be varied monotonically from a minimum
value of $\mathcal{C}=0$, which represents completely scalar fields
(homogeneously polarized), to a maximum of $\mathcal{C}=1$ which represents
fields with maximally non-separable spatial and polarization DoFs
(inhomogeneously polarized), with the required amplitude modulation trivially
found from

$a=\frac{1}{2}(1\pm\sqrt{1-\mathcal{C}^{2}})\,.$ (3)

Note that both $C$ and $a$ span from 0 to 1, but the invariance of $C$ allows
us to use it as a multi-bit encoding basis. This is because all measures of
$C$ will agree on the value regardless of the detector type (and we will show,
regardless of the channel conditions), a property inherited from its quantum
origin. This is in stark contrast to a measurement of the amplitude, $a$,
directly, where disparate detector efficiencies and channel noise/loss mean no
universal agreement on $a$, thus reducing the full multi-bit bandwidth
spanning 0 through to 1, to just 0 or 1 (one bit) for light or no light. To
use the vectorness as measured by $\mathcal{C}$ as an encoding basis we can
assign unique information to values separated by $\Delta C$, for $N\approx
1/\Delta C$ as the number of elements in the basis (we will return later to
how small $\Delta C$ can be). This allows us to transmit $d=\log_{2}N$ bits
per on/off pulse, rather than the one bit with the traditional non-modal
amplitude approach. We illustrate this concept in Fig. 1, where initial modes
(IN) are passed through an aberrating channel and emerge distorted (OUT).
Although the spatial mode structure appears scrambled and would have high
modal cross-talk, the vectorness remains intact with no cross-talk, and can
therefore be used as a multi-bit encoding basis. Here concurrence values of
$\mathcal{C}=\\{0,0.33,0.66,1\\}$ are used with $N=4$ as an example. Although
in the remainder of this paper we will use turbulence as our example channel,
it could be optical fibre, underwater or cellular media too, as illustrated in
Fig. 1.

Invariance of the basis to turbulence. Now that we have defined how we
construct a communication basis from the concurrence of classical vector
beams, we can investigate how this basis responds to propagation through the
atmosphere where spatially varying air densities due to both pressure and
temperature variations induce a spatially varying refractive index according
to the Gladstone-Dale law. We select turbulence as an example only, chosen
because it represents a particularly dynamic and extreme distorting medium. We
can express the total phase change, $\Phi(\vec{r}_{T})$, through a width of
turbulent medium using the thin screen approximation Herman and Strugala
(1990). This phase acts only on the spatial DoF via the operator in the
position basis $\hat{P}_{\Phi}=\int
e^{i\Phi(\vec{r}_{T})}|\vec{r}_{T}\rangle\langle\vec{r}_{T}|d\vec{r}_{T}$. We
also choose to allow propagation to the far-field for phase-only distortions
to become phase and intensity fluctuations, modelled by using a Fourier
transform operator $\mathcal{F}$. We thus express our field after propagating
into the far-field through turbulence as

$|\phi_{\pm}\rangle=\mathcal{F}\hat{P}_{\Phi}|\psi_{\pm}\rangle\,.$ (4)

We can now take note of an important property of this transformation, namely,
that it is unitary and therefore preserves inner products such that
$\langle\phi_{\pm}|\phi_{\pm}\rangle=\langle\psi_{\pm}|\psi_{\pm}\rangle\ $
and $\langle\phi_{\mp}|\phi_{\pm}\rangle=\langle\psi_{\mp}|\psi_{\pm}\rangle\
$. From Eqn 2 we see immediately that this implies the invariance of the
vectorness (concurrence) to this channel, and obviously extended to other
similar transformations. The result of this is illustrated in Fig. 1, where
the vectorness and the bit-strings they are mapped to are unchanged through
the channel, even though the intensity and polarization profiles are severely
distorted. A more detailed derivation of this theory is given in the
Supplementary Information. This theoretical concept reveals the potential of
vectorness as a robust multi-bit information carrier.

Practical demonstration. In order to verify the effectiveness of the suggested
encoding scheme, we utilized the experimental setup seen in Fig. 2, with a
more detailed breakdown given in the Methods section. The vector beams were
generated using an interferometer consisting of a Wollaston prism (WP), an
imaging system and a digital-micromirror device (DMD1), where the independent
complex modulation of horizontally and vertically polarized components of a
diagonally polarized (approximate) plane wave was facilitated by two
multiplexed binary amplitude holograms Rosales-Guzmán et al. (2020). In order
to control $\mathcal{C}$ the relative efficiencies of the holograms were tuned
according to Eqn. 3 by varying the amplitude, $a$, with exemplar multiplexed
holograms for a $N=4$ concurrence basis shown in Fig. 2.b. The vectorial beam
was passed through a dynamically aberrated channel created by a heater set to
a steady state temperature of $T\approx 185^{\circ}$C to induce turbulence in
the air along $\approx 200$ mm of beam path. The resulting beam was then
passed to the detection system, a custom built single shot Stokes polarimetry
arrangement, composed of a second DMD2 encoded with multiplexed binary
gratings which diffracted $4$ copies of the vector field into the $\pm 1$
diffraction orders. The beams were allowed to propagate to the far field for
the phase distortion to manifest as both amplitude and phase distortion. The
four copies were then imaged onto the sensor of a CCD camera - initially a
low-cost FLIR Chameleon was used and later replaced by a higher quality FLIR
Grasshopper to highlight the role of detector sensitivity. Prior to the CCD,
each of the four copies were passed through different combinations of quarter-
wave and half-wave plates and a common linear polarizer in order to capture
horizontally ($I_{H}$), vertically ($I_{V}$), diagonally ($I_{D}$) and right-
circularly ($I_{R}$) polarized Stokes intensity measurements from which the
concurrence was calculated. Furthermore, the Stokes measurements form an over-
complete set of polarisation projections, making our decoding step independent
of the polarisation basis that the modes were prepared in, i.e., the sender or
the channel can alter the polarization basis without any effect on the
outcome. Figure 2.c shows exemplar Stokes intensities integrated over the
dashed region to obtain associated powers $P_{H,V,D,R}$, from which the
concurrence could be determined (see Methods).

Figure 3: Discreetized bases, crosstalk and multi-bit information transfer. a)
Received concurrence of an input value varied in 16 discreet steps (insets
show partial results for 8 and 32 steps) through still (solid line) and
turbulent (dashed line) air. b) Crosstalk matrices (with associated fidelity)
for the emitted and received concurrence through still (top) and turbulent
(bottom) air for bases with $N=8,16$ and $32$ (left to right). c) 3 Bit (8
level) image transmitted through still (top) and turbulent (bottom) air - red
pixels highlight errors (inset shows the ground truth image) while the values
report the image fidelity. Figure 4: Noise, statistical variation and bit-
errors. a) Mean received concurrence (over 50 points) per element in the
$N=16$ basis in both still (circles) and turbulent (triangles) air, error bars
are visible in the inset. b) Distribution of received $\mathcal{C}$ values
about their target for two example basis elements both without (top) and with
(bottom) turbulence - values report the mean deviations across the $N=16$
basis. c) Measured standard deviation for each element in the $N=16$ basis
through still and turbulent air, where the large overlap illustrates the
turbulence invariance. d) Theoretically predicted BER as a function of the
statistical spread $\delta\mathcal{C}$ and basis choice $N$, the white solid
line represents the points where $N\delta\mathcal{C}=1$ and white dashed lines
represent the deviations of our receiver under different channel conditions -
$\delta\mathcal{C}_{\text{still}}$ and
$\tilde{\delta\mathcal{C}}_{\text{turb}}$.

Multi-bit encoding. The key to our proposed technique is that $\mathcal{C}$
does not change due to beam distortions, so that the full range from 0 to 1
can be used in a user-defined number of steps ($N$) independent of how
perturbing the channel is. This sub-division of the available encoding space
is shown in Fig. 3.a for $N=16$ (with $N=\\{8,32\\}$ included as an inset). To
indicate the low cross-talk, the transmitted vector beam was allowed to “idle”
at each basis element (delineated into bins shown as shaded bars) while
repeated measurements were taken, confirming only small dynamic changes in
still and turbulent air alike. The robust nature of the encoding is quantified
by the cross-talk matrices in Fig. 3.b for up to $N=32$, corresponding to 32
vectorial modes, yet with low levels of cross-talk. Finally, we use the system
to transmit information over this dynamically changing turbulent channel with
no adjustment to the received data signal, resulting in the high fidelity
images shown in Fig. 3.c for still (top) and turbulent (bottom) air using
$N=8$. The consistency of the results under different channel conditions
validates the concept. The notable feature of this approach is that the number
of modes used can be tailored up to a maximum $N_{\text{max}}=1/\delta C$,
where $\delta C$ is the inherent noise in the system - calculated as the
standard deviation of repeated measurements at a given $\mathcal{C}$. Next, we
take a closer look at this to estimate the potential of the approach and show
that, as expected, $\delta C$ is limited primarily by detector noise and not
the condition of the channel itself.

Noise and information capacity. We see from the “idling” results of Fig. 3.a
that the noise of the still and turbulent air are comparable, suggesting that
detector noise is the primary cause of the statistical variation of a given
vectorness about the target, as anticipated by theory. This results in only
very small errors between what was emitted and what was received, as
illustrated in Fig. 4.a, for $N=16$ as an example, with the small scale of the
deviation shown in the enlarged inset. A statistical analysis of the
experimental noise, shown graphically in Fig. 4.b, reveals that
$\delta\mathcal{C}_{\text{still}}\approx 3.7\times 10^{-3}$ in still air (no
turbulence), which we take to represent the inherent noise of the detection
system. When instantaneous turbulent channel conditions are introduced we
observe only a slight increase in noise to
$\tilde{\delta\mathcal{C}}_{\text{turb}}\approx 4.6\times 10^{-3}$, which we
attribute to the signal-to-noise limit of our detector, i.e., defocusing
aberrations in turbulence leading to a signal below the noise threshold of the
detector (see Supplementary Information). It is notable that a spread in the
received $\mathcal{C}$ due to detector noise is akin to the spectral spread
observed in OAM due to channel aberrations. We confirm too that the noise is
not mode specific, as shown by the small difference in variance with and
without turbulence in Fig. 4.c. This validates the claim that the minimum
subdivision of the encoding space, or maximum number of modes (or bits), is
limited primarily by detector noise and not channel conditions, contrary to
traditional MDM schemes.

To probe the potential of this approach we consider how a given detector noise
limit ($\delta\mathcal{C}$) and subdivision choice ($N$) affect the resulting
bit-error-rate (BER), using our turbulence channel as an example. The BER is
the ratio of incorrectly received bits to total transmitted bits and
represents how crosstalk in the scheme affects the information transfer. The
results of this are illustrated graphically in Fig. 4.d. In order to quantify
the relationship between the cross-talk inducing $\delta\mathcal{C}$ and the
system fidelity we can analytically inspect the overlap of neighbouring basis
elements to reveal the BER for a given choice of basis (i.e., $N$) and
$\delta\mathcal{C}$

$\text{BER}(N,\delta\mathcal{C})=\bar{E}(N)\frac{\delta\mathcal{C}}{2}\sqrt{\frac{\pi}{2}}e^{-\frac{N^{-2}}{2\delta\mathcal{C}}}\text{erf}\left(\sqrt{\frac{2}{\delta\mathcal{C}}}\right)\,,$
(5)

where $\text{erf}()$ represents the error function and $\bar{E}(N)$ is the
mean bit-error per erroneously binned measurement for a given basis choice
(see Supplementary Information). If we inspect the BER for different channel
conditions and basis choices, as shown in Fig. 4.d, we notice how the reducing
$\delta\mathcal{C}$ allows for the use of denser choice of basis while
maintaining a low BER. The white curve in Fig. 4.d indicates the special cases
of $N\delta\mathcal{C}=1$, with the associated line acting as a limit for the
basis size under a given channel condition. It is however notable that this
overlap is significantly lower than the modal overlap experienced by MDM
systems, and that with a suitable detector acceptable BERs on the order of
$10^{-9}$ are plausible. These results indicate that the effectiveness of the
technique for high dimensional information transfer places the burden only on
the signal-to-noise ratio of the detector system. The notably poor diffraction
efficiency offered by DMDs Scholes et al. (2019) and the limited dynamic range
of CCDs means that our particular system suffers from a lower signal-to-noise
ratio (SNR) than alternatives, such as the sensitive photodiodes used in free-
space communication links. The white dashed lines in Fig. 4.d indicate the
mean measured $\delta\mathcal{C}$ for our system with and without turbulence.
The associated $\text{BER}(N=8)\approx 2\times 10^{-3}$ also agrees well with
the fidelities reported in Fig. 3.c.

Figure 5: Channel invariance comparison to OAM encoding. Crosstalk matrices
through a) ideal and b) aberrated channels using both OAM (with $N=11$) and
vectorness (with $N=11$ and $N=50$) encoding schemes. The turbulence phase
screen used to aberrate the beams is shown in the left panel of b). A
truncated 32 mode subset of the $N=50$ basis is highlighted showing improved
fidelity.

Comparative performance. The highlight of the proposed vectorness based
encoding scheme is that its invariance to the channel conditions makes it a
more robust choice in comparison to direct modal encoding schemes. To
illustrate this point we will use the OAM basis as a comparative example.
Additionally we changed the CCD detector from a 58.81 dB to a 60.62 dB dynamic
range model to further highlight the role of detector choice and to allow a
larger basis to be explored (more subdivisions). The setup of Fig. 2 (using
DMD2) allowed us to generate and perform spatial mode projections on scalar
Laguerre-Gaussian beams ($\mathrm{LG}_{p}^{l}$) of azimuthal order $l$ and
radial order $p$. The results in Fig. 5.a show the baseline case with no
turbulence for direct OAM encoding and indirect OAM encoding using the
vectorness. In the direct OAM case, we utilized an 11 mode basis spanning the
range of $l\in[-5,5]$ with $p=0$. Here the size of the OAM basis was
constrained by the system aperture, as increasing mode order results in ever
larger beams. Our modal vectorness scheme can be executed with low order modes
for the same size across all bit-levels, and so does not suffer from this
limitation. The detector noise limit in the vectorness scheme allowed us to
probe a $N=50$ basis, far larger than OAM encoding approaches which typically
range from $N=4$ to $N=34$ Wang et al. (2012a); Zhu et al. (2021). To the best
of our knowledge $N=50$ is the largest number of vectorial modes used in a
communication link experiment. The fidelities in Fig. 5.a are indicative of
experimental limitations (detector noise) and can be taken as the baseline
values.

Since we are interested in the effect of the channel, we replaced our dynamic
channel aberrations with a static turbulence phase screen with a fixed Fried
parameter of $r_{0}=1.5$ mm as displayed in the left panel of Fig. 5.b (see
Methods for the generation technique). The static screen allowed for a
controlled comparison of how the crosstalk in each of the schemes changes when
the channel perturbations are introduced. The 37.72 % decrease in fidelity
using OAM encoding is in stark contrast to the 0 % change observed using
vectorness encoding at the same basis size ($N=11$). When pushing the
dimensionality of the new encoding scheme further, the channel induced
crosstalk only increases marginally, in the order of 1 %.

There are two important observations one can make from the vectorness data in
Fig 5. Firstly, the baseline (no turbulence) indicates how the detector choice
limits the maximum subdivision number by its noise, setting a limit on how
small $\delta C$ can be. For our detector at $N=50$ the baseline fidelity is
85 %, but can be improved to 93 % by using a smaller number of modes
($N<N_{\mathrm{max}}$) in the available space - while also maintaining the
modal density. With better detector technology the maximum number of modes can
be significant, or the impact of noise made small by using a smaller encoding
alphabet. Secondly, a comparison of the baseline (no turbulence) to the
baseline with channel noise (with turbulence) makes crystal clear that the
channel noise is negligible in the new scheme, validating our central claim.
In no other modal approach is the channel noise so inconsequential, here
contributing in the $\approx 1$ % range.

## DISCUSSION AND CONCLUSION

The proposed scheme and the results of our demonstration reveal its potential
for multi-bit encoding using a new modal version of amplitude modulation,
enabling a $d$-fold increase in information density using existing amplitude
modulation technology for superior data transmission rates. Unlike MDM
systems, our source of noise is primarily detector based, while being almost
completely invariant to channel aberrations, negating the need for adaptive
error-correction, which we showed using atmospheric turbulence as an extreme
example. This is illustrated by using 50 spatial modes in a turbulent channel
with a cross-talk that is comparable to the no-turbulence case, differing by
less then 2 %. For comparison, our crosstalk observed between 11 OAM modes was
37.72 % but zero for the vectorness approach, while other studies with 4
orthogonal vector OAM modes have reported $\approx 20$% crosstalk even in weak
turbulence conditions Cox et al. (2016b), while crosstalk between 7
neighbouring scalar OAM modes reached $13.2$% even with corrective measures
Zhou et al. (2021). Our observed BER using 8 modes, on the order of $2\times
10^{-3}$, is comparable to that achieved in error-corrected MDM systems Wang
et al. (2012b), even though we have deployed only a proof-of-principle version
of the experiment with rudimentary detectors. Our direct comparison to OAM
encoding revealed the superior channel invariance while also demonstrating how
a larger basis (e.g., up to 50) can be achieved with the same system aperture
which constrained the OAM basis to 11 modes - this is since the increase of
basis size in OAM encoding requires larger mode orders which require larger
apertures, while our approach can scale the number of modes up to the noise
floor of the detector.

We point out that while the particular vector beams we used were composed of
OAM endowed $\mathrm{LG}_{p}^{l}$ modes, the core feature of our scheme is the
utilization of vector beams (our modal basis) in a manner that does not
require the spatial modes nor orthogonal polarisation components to be
detected or recognised. Instead, only their vectorness is detected, an
invariant quantity that is found by an integrated modal signal. It is a modal
approach without the penalty of detecting modes. In this regard, while our
proof-of-principle experiment used a CCD camera for detection, fast and
sensitive photo-detectors are all that is required for a real-world
implementation. Because our channel-invariant basis is derived from spatial
modes without actually detecting them and is independent of the polarisation
basis that they were prepared in, it brings with it some significant benefits
over traditional MDM schemes that use the modes themselves as the basis. For
instance, (i) system misalignment is mitigated since the integrated detection
(a power) is spatially invariant, (ii) the modes can be selected with low
order to reduce divergence Willner and Liu (2021) (since more information does
not require higher mode numbers), (iii) like amplitude modulation, our scheme
does not prohibit other enhancement schemes such as wavelength division
multiplexing (WDM) to further improve information density, including even MDM
if modal correction is applied, and further, (iv) unlike alternative vectorial
techniques Milione et al. (2015); Zhu et al. (2021), the nature of our
detection scheme is over-complete even in the polarisation basis, and
therefore means that the sender and receiver do not have to agree on the
measurement basis, allowing the scheme to work even in polarization scrambling
media such as stressed optical fibre. These benefits are provided by the
vectorness, a physical quantity which displays resilience to transformations
which generally hinder modal communication systems. The trade-off lies in the
scaling of our scheme, in order to increase $d$ by $1$, $N$ has to double.
This places the burden of performance on the power and amplitude modulation
resolution of the emitter and the SNR of the receiver - all factors which
already receive considerable attention in free space optical communication
systems.

To conclude, we have presented how the vectorness of vectorial light can be
used as a new modal version of amplitude modulation, exploiting spatial modes
in a manner that makes them channel-invariant, with the number of modes used
limited primarily by the sensitivity of the detectors used. We have
demonstrated high fidelity, correction-free, multi-bit information transfer to
verify our technique, even through dynamic turbulence, an extreme example of a
communications channel. Our approach can be extended to other channels too,
such as optical fibre and under-water, since the invariance property will hold
in all such channels. We believe this approach will open up a new avenue for
high-bandwidth optical communication, with the immediate benefits of MDM but
without the modal cross-talk challenges. The vectorness presents a new scheme
for information encoding, which if coupled with other communication techniques
such as WDM and SDM, can be used to push the boundaries of optical
communication.

## METHODS

Experimental implementation. In order to demonstrate the effective high
dimensional information transfer using concurrence as the communication basis,
we utilized the setup shown in Fig. 1.b. A Gaussian beam produced by a HeNe
laser (wavelength 633 nm) was expanded and collimated using lenses L1 and L2
respectively. The plane of polarization was converted to $45^{\circ}$ using a
half-wave plate (HWP1) before passing through a Wollaston prism (WP) which
separated the horizontally and vertically polarized components of the expanded
beam at an angle of $\approx 1^{\circ}$. The plane at the WP was imaged onto
the screen of a digital micro-mirror device (DMD1) using a $4f_{1}$ imaging
system. DMD1 was addressed using two multiplexed binary holograms of the form

$\displaystyle H_{A/B}(\vec{r})$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\text{sign}[\cos(\varphi_{A/B}(\vec{r}_{T})+2\pi(\vec{g}^{A/B}_{T}\cdot\vec{r}_{T}))$
$\displaystyle-\cos(A_{A/B}(\vec{r}_{T}))],$ (6)

facilitating the modulation of the complex field
$U_{A/B}=A_{A/B}e^{i\varphi_{A/B}}$, where
$\vec{g}^{A/B}_{T}=(g_{x}^{A/B},g_{y}^{A/B})$ are grating frequencies Lee
(1979). In order to control the relative amplitudes of the resulting beams,
complimentary weighted random matrices were multiplied to $H_{A/B}$ according
to the method outlined in Ref. Hu et al. (2021). Examples of the multiplexed
gratings corresponding the different concurrence values are shown in Fig. 1.c.
By selecting $\vec{g}^{A/B}$ appropriately, the $+1$ diffraction orders of
independently modulated, orthogonally polarized components were spatially
overlapped - creating our vector beam $|\Psi\rangle$ Rosales-Guzmán et al.
(2020). For our case $|\psi_{\pm}\rangle\equiv LG_{0}^{\pm 1}$ were chosen,
where $LG_{p}^{l}$ is the Laguerre-Gaussian mode with azimuthal(radial) index
$l(p)$. The combined diffraction order was isolated using an aperture placed
at the focal plane of a $4f_{2}$ imaging system - which imaged $|\Psi\rangle$
onto a second DMD2. DMD2 was addressed by simple binary diffraction gratings
which were multiplexed in order to produce four copies of $|\Psi\rangle$,
these copies were allowed to propagate into the far field through a 200 mm
length of air - heated by a plate at 185∘C. Additionally 3 of the four paths
were each passed through a HWP2,3 (fast axes at 45 and 22.5∘, respectively) or
a quater-wave plate (QWP - fast axis at 45∘). Lenses L3 and L4 were used to
demagnify the four beams onto a CCD camera, while filtering using a linear
polarizer (LP - transmission axis at 0∘). The intensities projected onto the
CCD correspond to the horizontal, vertical, diagonal and right-circular
polarized components - which allowed for the determination of $\mathcal{C}$
Singh et al. (2020). Simulated examples of the far field Stokes intensities
showing the region of integration (dashed circles) are given in Fig. 1.d.

Concurrence measurement. The Stokes measurements were used to calculate $C$
according to Selyem et al. (2019)

$\displaystyle\mathcal{C}=$
$\displaystyle\sqrt{1-\frac{(P_{H}-P_{V})^{2}+(2P_{D}-P_{H}-P_{V})^{2}+(2P_{R}-P_{H}-P_{V})^{2}}{(P_{H}+P_{V})^{2}}}\,,$
(7)

where $P_{i}=\int\int I_{i}dr_{T}$ represent the powers obtained from the
transversely integrated Stokes intensities, the dashed circle in Fig. 2.c
indicate the necessary region of integration.

Static turbulence screens. In order to probe the performance of our encoding
scheme along with alternatives under controllable conditions we implemented a
static turbulence phase screen, $\Phi(\vec{r}_{T})$ which was used to modulate
the phase of our beams during their generation (i.e. on DMD1). The screen was
generated using a fast-Fourier transform technique according to the following
expression

$\Phi(\vec{r}_{T})=\Re\left(\mathcal{F}^{-1}(M_{rand}\sqrt{\theta})\right)\,,$
(8)

where $\Re$ is the real-part, $M_{rand}$ is a complex Gaussian random matrix
centered at 0 with unit deviation and $\theta$ is the Kolmogorov-Weiner
spectrum - conventionally determined (in pixel coordinates $(i,j)$) by the
following expression

$\theta(i,j)=0.023\left(\frac{2D}{r_{0}}\right)^{\frac{5}{3}}(i^{2}+j^{2})^{-\frac{11}{3}}\,,$
(9)

where $D$ is the hard aperture of the screen and $r_{0}$ is the Fried
parameter (diameter over which the root-mean-squared wavefront aberration is
$1$ radian) Fried (1966). It has been found that Eqn. 9 does not accurately
model the statistics in the low spatial frequency region (which is notably
dominant in the Kolmogorov regime). In order to accurately model this region
of the spectrum we employed additional subharmonic sampling according to the
method outlined in Ref. Lane et al. (1992) \- when calculating our spectrum.
In our experiment we set $r_{0}=1.5$ mm.

Modal projection measurements. To facilitate a comparison to OAM based
encoding we generated scalar $U_{emitted}=LG_{p}^{l}$ modes by setting $a=0$
on DMD1. For simplicity only the OAM associated azimuthal index $l$ was varied
on the interval $[-5,5]$ to form the communication basis - while setting
$p=0$. To determine crosstalk between basis elements we utilized correlation
filters encoded on the second DMD2 along with the far-field propagation to
perform inner product measurements with each of the basis elements. The
correlation filters were implemented according to Eqn 6 with
$U_{A}=(LG_{0}^{l})^{*}$ \- resulting in the following intensity profile, to
be captured in the far-field by the CCD,

$I_{FF}=\left|\int
U_{emitted}(LG_{0}^{l})^{*}e^{i2\pi\vec{k}_{T}\cdot\vec{r}_{T}}d\vec{r}_{T}\right|^{2}\,,$
(10)

where it is clear that measuring the on axis intensity in the far-field (i.e
where $\vec{k}_{T}=0$) results in the absolute-squared inner product between
the emitted field and the given basis element. This procedure is referred to
as modal decomposition and a thorough tutorial is available in Ref. Pinnell et
al. (2020).

## Author contributions

The experiment was performed by K.S. and I.N. assisted with the theory. A.D.
and W.B. assisted with editing the manuscript and data analysis. All authors
contributed to writing the manuscript. A.F. conceived of the idea and
supervised the project.

## Competing Interests

The authors declare no competing interests.

## Data availability

The data that supports the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

Supplementary information: A robust basis for multi-bit optical communication
with vectorial light

##  I Theory

###  I.1 Concept and general approach

For the benefit of the reader, we will first introduce a general overview of
our approach, whereby the core concepts will be subsequently unpacked in
further theoretical detail. Our technique makes use of a family of structured
light beams Forbes et al. (2021) with coupled transverse spatial and
polarization components, called vector beams, that can be expressed as the
linear superposition state

$|\Psi\rangle=a_{1}|\psi_{1}\rangle|H\rangle+a_{2}|\psi_{2}\rangle|V\rangle\,,$
(S.1)

consisting of the spatial modes (not necessarily orthogonal),
$|\psi_{1,2}\rangle$, that are coupled to the linear horizontal ($|H\rangle$)
and vertical ($|V\rangle$) polarization states, respectively. The coefficients
are normalised so that $|a_{1}|^{2}+|a_{2}|^{2}=1$. Such beams can have
independent (factorizable) polarization and spatial components, and can
therefore be written as a separable product of polarization and spatial
degrees of freedom, often called a scalar beam. Scalar beams are associated
with uniform polarization field vectors. Contrarily, the polarization and
spatial components can be written as a non-separable product, analogous to
quantum entangled two photon states. In this case, the resulting beams have
non-uniform polarization field vectors and are called vector beams. In
general, the beams can have varying levels of inhomogeneity in the
polarization field vectors.

The degree of inhomogeneity of the field vectors can be directly related to
the degree of non-separbility between the polarization and spatial components
of the field. A measure that accurately quantifies this, is the vector quality
factor (VQF) or vectorness Ndagano et al. (2016) which is a measure that is
derived from concurrence Wootters and Zurek (1982), commonly used to quantify
how entangled (non-separable) independent subsystems are. It ranges from
$\mathcal{C}=0$ for completely scalar to $\mathcal{C}=1$ for completely non-
separable vector fields.

For channels (or propagation media) that are completely unitary, meaning that
only a change of basis in field components, i.e. $|\psi_{1,2}\rangle$, is
performed, local rotations of the field vectors are however, still observed
upon propagation, although the field retains the same degree of non-
separability Nape et al. (2018). For example, a one-sided channel that only
perturbs the spatial components evolves the initial field to the final state

$|\tilde{\Psi}\rangle=a_{1}|\phi_{1}\rangle|H\rangle+a_{2}|\phi_{2}\rangle|V\rangle\,,$
(S.2)

following the mapping $|\psi_{1,2}\rangle\rightarrow|\phi_{1,2}\rangle$. The
resulting concurrence, $\tilde{\mathcal{C}}=\mathcal{C}$, of the input and
output states are equivalent Nape et al. (2018).

When $a_{2}=\sqrt{1-a_{1}^{2}}$, and $|\psi_{1,2}\rangle$ are orthogonal, then
the concurrence is a monotonic function of the parameter $a_{1}$, and can
therefore be used as an encoding basis, where a desired concurrence is encoded
by controlling the inhomogeneity of the field via $a_{1}$. For simplicity in
the main text we choose $a_{1}=\sqrt{a}$. In the sections that follow, we
elucidate this concept further and show that we can design a correction-free
communication scheme with concurrence as our multi-bit basis. Next, we show
how the concurrence can be measured in a basis independent manner using
traditional Stokes polarimetry, a crucial step towards demonstrating our
scheme.

###  I.2 Basis independent non-separability measurement

The degree of non-separability (equivalently concurrence), between the
polarization and spatial components of a classical vector field can be
retrieved from the global Stokes parameters (i.e
$S_{i}=\int\tilde{S}_{i}(\vec{r}_{T})d^{2}r_{T}$, where
$\tilde{S}_{i}(\vec{r}_{T})$ are the transversely varying local Stokes
parameters) Selyem et al. (2019). Consider the field in Eq. (S.2), having
spatial components that are not necessarily orthogonal and setting
$a_{1}=a_{2}$. We can use the Gram-Schmidt orthonormalization procedure to
retrieve orthogonal normalized spatial fields $|\psi_{\pm}\rangle$ given by

$\displaystyle|\psi_{\pm}\rangle=\mathcal{A}_{\pm}\left(\frac{|\psi_{1}\rangle}{\sqrt{|\langle\psi_{1}|\psi_{1}\rangle|}}\pm\frac{|\psi_{2}\rangle}{\sqrt{|\langle\psi_{2}|\psi_{2}\rangle|}}\right)\,,$
(S.3) $\displaystyle\mathcal{A}_{\pm}=\left(2\pm
2\frac{\langle\psi_{1}|\psi_{2}\rangle}{\sqrt{|\langle\psi_{1}|\psi_{1}\rangle||\langle\psi_{2}|\psi_{2}\rangle|}}\right)^{-\frac{1}{2}}\,.$
(S.4)

This allows us to express $|\Psi\rangle$ in terms of orthonormal spatial field
and polarization components according to

$|\Psi\rangle=a|\psi_{+}\rangle|H\rangle+b|\psi_{-}\rangle|H\rangle+c|\psi_{+}\rangle|V\rangle+d|\psi_{-}\rangle|V\rangle\,.$
(S.5)

Under the assumptions of completely polarized (pure) states and expressing in
terms of orthonormal spatial components (i.e. a qubit system), the following
convenient expression for the concurrence can be used Wootters (1998)

$\displaystyle\mathcal{C}=2|ad-bc|,$ (S.6)
$\displaystyle\mathcal{C}=2\sqrt{\langle\psi_{1}|\psi_{1}\rangle\langle\psi_{2}|\psi_{2}\rangle-|\langle\psi_{1}|\psi_{2}\rangle|^{2}}\,.$
(S.7)

Equation S.7 follows from substitution of Eqs S.3 and S.4 into Eq. S.6. The
concurrence is conveniently retrieved in terms of the original unnormalized
arbitrary states.

We can now examine the inner products in the position basis (using
$\int|\vec{r}_{T}\rangle\langle\vec{r}_{T}|d^{2}r_{T}=\mathbb{I}$) where
Töppel et al. (2013)

$\displaystyle|\psi_{1(2)}(\vec{r}_{T})|e^{i\varphi_{1(2)}(\vec{r}_{T})}=\langle{\vec{r}_{T}|\psi_{1(2)}}\rangle\,,$
(S.8)
$\displaystyle\langle\psi_{1(2)}|\psi_{1(2)}\rangle=\int|\psi_{1(2)}(\vec{r}_{T})|^{2}d^{2}r_{T}\,,$
(S.9) $\displaystyle\&$
$\displaystyle\langle\psi_{1(2)}|\psi_{2(1)}\rangle=\int|\psi_{1(2)}(\vec{r}_{T})||\psi_{2(1)}(\vec{r}_{T})|e^{\pm
i\Delta\varphi}d^{2}r_{T}\,,$ (S.10)

where $|\psi_{1(2)}(\vec{r}_{T})|$ and $\varphi_{1(2)}(\vec{r}_{T})$ are the
respective amplitude and phase of the orthogonally polarized field components
in the given transverse plane. While
$\Delta\varphi=\varphi_{2}(\vec{r}_{T})-\varphi_{1}(\vec{r}_{T})$. Comparing
the above to the following well known expressions for the Stokes parameters

$\displaystyle\tilde{S_{0}}(\vec{r}_{T})=|\psi_{1}(\vec{r}_{T})|^{2}+|\psi_{2}(\vec{r}_{T})|^{2}\,,$
(S.11)
$\displaystyle\tilde{S_{1}}(\vec{r}_{T})=|\psi_{1}(\vec{r}_{T})|^{2}-|\psi_{2}(\vec{r}_{T})|^{2}\,,$
(S.12)
$\displaystyle\tilde{S_{2}}(\vec{r}_{T})=|\psi_{1}(\vec{r}_{T})||\psi_{2}(\vec{r}_{T})|\cos(\varphi_{2}(\vec{r}_{T})-\varphi_{1}(\vec{r}_{T}))\,,$
(S.13)
$\displaystyle\tilde{S_{3}}(\vec{r}_{T})=|\psi_{1}(\vec{r}_{T})||\psi_{2}(\vec{r}_{T})|\sin(\varphi_{2}(\vec{r}_{T})-\varphi_{1}(\vec{r}_{T}))\,,$
(S.14)

allows us to reduce the concurrence to an expression dependent only on the
global Stokes parameters

$\mathcal{C}=\sqrt{1-\frac{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}{S_{0}^{2}}}\,,$
(S.15)

where $S^{2}_{i}=\int|\tilde{S_{i}}(\vec{r}_{T})|^{2}d^{2}r_{T}$, for each
stokes parameter. Conventionally the Stokes parameters are determined by the
overcomplete set of 6 intensity measurements, we choose to use the minimum
number of four measurements of the horizontal ($I_{H}$), vertical ($I_{V}$),
diagonal ($I_{D}$) and right-circular ($I_{R}$) polarization intensity
projections to calculate the Stokes parameters as follows

$\displaystyle\tilde{S_{0}}=I_{H}+I_{V}\,,$ (S.16)
$\displaystyle\tilde{S_{1}}=I_{H}-I_{V}\,,$ (S.17)
$\displaystyle\tilde{S_{2}}=2I_{H}+\tilde{S_{0}}\,,$ (S.18)
$\displaystyle\tilde{S_{3}}=2I_{D}+\tilde{S_{0}}\,.$ (S.19)

Furthermore, since the global integration is linear the intensity projections
can be directly integrated - leading to the direct determination of the
concurrence according to

$\displaystyle\mathcal{C}=$
$\displaystyle\sqrt{1-\frac{(P_{H}-P_{V})^{2}+(2P_{D}-P_{H}-P_{V})^{2}+(2P_{R}-P_{H}-P_{V})^{2}}{(P_{H}+P_{V})^{2}}}\,,$
(S.20)

here $P_{i}=\int\int I_{i}dr_{T}$ represent the powers obtained from the
transversely integrated Stokes intensities.

###  I.3 Propagation through turbulent media

Now let us consider the effect of atmospheric turbulence, where spatially
varying air densities due to both pressure and temperature variations induce a
spatially varying refractive index according to the Gladstone-Dale law. We can
express the total phase change $\Phi(\vec{r}_{T})$ through a width of
turbulent medium using the thin screen approximation Herman and Strugala
(1990). Therefore the concurrence through the channel (including propagation
into the far field) is given by

$\displaystyle\tilde{\mathcal{C}}=2\sqrt{\langle\phi_{1}|\phi_{1}\rangle\langle\phi_{2}|\phi_{2}\rangle-|\langle\phi_{1}|\phi_{2}\rangle|^{2}},$
(S.21)
$\displaystyle|\phi_{1,2}\rangle=\mathcal{F}|\tilde{\psi}_{1/2}\rangle\,;\,\langle\phi_{1(2)}|=\langle\tilde{\psi}_{1(2)}|\mathcal{F}^{\dagger},$
(S.22)
$\displaystyle\langle\vec{r}_{T}|\tilde{\psi}_{1,2}\rangle=|\psi_{1,2}(\vec{r}_{T})|e^{i(\varphi_{1,2}(\vec{r}_{T})+\Phi(\vec{r}_{T}))}.$
(S.23)

Here the states $|\tilde{\psi}_{1,2}\rangle$ and $|\phi_{1,2}\rangle$
correspond to the aberrated and subsequently propagated optical fields,
respectively. The symbol, $\mathcal{F}$, denotes the Fourier transform
operator used here to map the optical modes to the far-field. Since
$\mathcal{F}$ is a unitary transformation (i.e.
$\mathcal{F}^{\dagger}\mathcal{F}=\mathcal{F}\mathcal{F}^{\dagger}\mathbb{I}$),
it preserves their inner products such that

$\tilde{\mathcal{C}}=2\sqrt{\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{2}|\tilde{\psi}_{2}\rangle-|\langle\tilde{\psi}_{1}|\tilde{\psi}_{2}\rangle|^{2}},$
(S.24)

where we get

$\displaystyle\langle\tilde{\psi}_{1,2}|\tilde{\psi}_{1,2}\rangle$
$\displaystyle=\int\left||\psi_{1,2}(\vec{r}_{T})|e^{i(\varphi_{1,2}(\vec{r}_{T})+\Phi(\vec{r}_{T}))}\right|^{2}d^{2}r_{T}$
$\displaystyle=\langle\psi_{1,2}|\psi_{1,2}\rangle,$ (S.25)
$\displaystyle\langle\tilde{\psi}_{1,2}|\tilde{\psi}_{2,1}\rangle$
$\displaystyle=\int|\psi_{1,2}(\vec{r}_{T})|^{2}e^{\pm(i\Delta\varphi+\Phi(\vec{r}_{T})-\Phi(\vec{r}_{T}))}d^{2}r_{T}$
$\displaystyle=\langle\psi_{1,2}|\psi_{2,1}\rangle,$ (S.26)

therefore $\tilde{\mathcal{C}}=\mathcal{C}$. This result represents a key
advantage presented by the spatially independent concurrence as a basis for
free-space communication - making it invariant to atmospheric turbulence. An
extremely noteworthy outcome of this is that any unitary (i.e. inner product
preserving) transformation will preserve the concurrence - therefore our
concept can be readily extended to thick medium models of atmospheric
turbulence, as well as stress induced birefringence in optical fiber.

###  I.4 Forming a basis for communication

Naturally the concurrence is a continuously variable parameter, however in
order for us to utilize this property as a basis for information transfer we
will need to discretize it. For convenience we will form our vector beam using
orthonormal spatial components

$|\Psi\rangle=\sqrt{a}|\psi_{+}\rangle|H\rangle+\sqrt{1-a}|\psi_{-}\rangle|V\rangle\,,$
(S.27)

(i.e. a special case of Eq S.5) therefore by varying the coefficient $a$
according to

$a=\frac{1}{2}(1\pm\sqrt{1-\mathcal{C}^{2}}),$ (S.28)

we can achieve a linear variation of the concurrence (non-separability)
$\mathcal{C}\in[0,1]$. This means that by knowing the degree of non-
separability between the polarization and spatial components that we wish to
encode, $\mathcal{C}$, we can find a unique coefficient $a$ that translates it
into the vector field. For example, a scalar vector field corresponding to
$a=0$ encodes $\mathcal{C}=0$, while $a=1/2$ encodes $\mathcal{C}=1$.
Limitations imposed on the system through the detection (i.e. Stokes
measurement) apparatus will result in an upper limit to the number of unique
concurrence $N_{max}$ values which can be reliably detected. The dimension $d$
of our encoding scheme (i.e. the number of independently transferable bits per
beam) is then given by

$d=\log_{2}N\,,$ (S.29)

where $N$ is the chosen number of basis elements to be used.

##  II Mean bit error

Figure S.1: a) Diagram showing the bit errors introduced by erroneous
measurements of neighbouring concurrence values. Under the assumption that
these errors are distributed homogeneously through the basis the mean can be
calculated according to the expression in b).

The observed crosstalk experienced by the proposed system is generally
confined to the neighbouring concurrence values in the discreetized basis.
This means that the number of erroneous bits transmitted by an incorrectly
binned measurement is dependent on $N$. The mean number of erroneous bits
$\bar{E}$ is given by

$\bar{E}(N)=\sum\limits_{i=1}^{d}\frac{N}{2^{i}}\frac{i}{N-1}\,.$ (S.30)

A diagram illustrating the individual and mean errors for a $N=8$ system is
shown in Fig. S.1. The probability $P_{E}$ of an erroneously binned
measurement can be approximated using the overlap between neighbouring
gaussian distributions given by:

$\displaystyle P_{E}(N,\delta\mathcal{C})$
$\displaystyle=\int\limits_{0}^{1}e^{-\frac{(\mathcal{C}+(2N)^{-1})^{2}}{(\delta\mathcal{C})^{2}}}e^{-\frac{(\mathcal{C}-(2N)^{-1})^{2}}{(\delta\mathcal{C})^{2}}}d\mathcal{C}$
$\displaystyle=\frac{\delta\mathcal{C}}{2}\sqrt{\frac{\pi}{2}}e^{-\frac{N^{-2}}{2\delta\mathcal{C}}}\text{erf}\left(\sqrt{\frac{2}{\delta\mathcal{C}}}\right)\,.$
(S.31)

We can then multiply the mean error for a given basis choice by the error
probability for that choice in order to determine an effective bit error rate
(i.e. $BER(N,\delta\mathcal{C})=\bar{E}(N)P_{E}(N,\delta\mathcal{C})$).

##  III Impact of Noise

Figure S.2: Plot showing the impact of integrated noise subtraction of the
received concurrence value. The same Stokes intensity measurements were used
to calculate the received concurrence with and without integrated subtraction
where the correct (encoded) scaling is achieved with the appropriate
subtraction.

As illustrated in the main text the concurrence is invariant to phase
aberrations as they fall under the general umbrella of one-sided (polarization
independent) unitary transformations. Noise, however, does impact the
concurrence measurement independent of the source (electronic, detector or
environmental). The impact of the noise in the region where we expect zeros in
the Stokes intensities necessitates the subtraction of a background value from
the measurements. The noise subtraction can be done as a single integrated
value subtracted from all the powers $P_{i}$ and allows the measured
concurrence to reach zero (see Fig. S.2). It should be noted that the value of
this integrated noise will be dependent on the size of the integration region
and the detector shot noise. This subtraction can lead to information loss as
parts of the Stokes intensity measurements which fall below the background
level are also subtracted - therefore if an integrated Stokes intensity
$I_{i}$ measurement is expected to be below the background level a significant
increase in deviation for the encoded vectorness will be observed.

Figure S.3: Results of numerical simulations showing the variation of
$\delta\mathcal{C}$ (i.e. errorbars) with additive intensity noise (given as a
percentage of the maximum Stokes intensity). A vectorness value of
$\mathcal{C}=0.5$ was chosen as the target value and a single integrated noise
value - appropriate for each noise level - was subtracted.

While the single value noise subtraction provides a convenient way to extract
the true encoded vectorness values, spatial and temporal variations in the
noise will lead to the over-/under-compensation of the true instantaneous
noise conditions. In order to illustrate this we performed numerical
simulations where random additive noise was introduced into the far field of a
$\mathcal{C}=0.5$ beam. Each Stokes intensity was given a unique noise profile
and a single integrated value was subtracted from the integrated powers prior
to the vectorness calculation (as was done in the experiments) - 50
simulations were performed at each noise level to acquire the statistics. From
the results shown in Fig. S.3, it is clear that the mean value remains around
the target however the deviations $\delta\mathcal{C}$ increase with the noise
level as was observed in the experiments.

## References

  * Richardson (2010) D. J. Richardson, Science 30, 327 (2010).
  * Richardson et al. (2013) D. Richardson, J. Fini, and L. Nelson, Nature Photonics 7, 354 (2013).
  * Forbes et al. (2021) A. Forbes, M. de Oliveira, and M. R. Dennis, Nature Photonics 15, 253 (2021).
  * Li et al. (2014) G. Li, N. Bai, N. Zhao, and C. Xia, Advances in Optics and Photonics 6, 413 (2014).
  * Berdagué and Facq (1982) S. Berdagué and P. Facq, Appl. Opt. 21, 1950 (1982).
  * Trichili et al. (2019) A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, IEEE Communications Surveys & Tutorials 21, 3175 (2019).
  * Trichili et al. (2020) A. Trichili, M. A. Cox, B. S. Ooi, and M.-S. Alouini, JOSA B 37, A184 (2020).
  * Padgett (2017) M. J. Padgett, Optics Express 25, 11265 (2017).
  * Willner et al. (2015) A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, et al., Advances in Optics and Photonics 7, 66 (2015).
  * Willner et al. (2021) A. E. Willner, K. Pang, H. Song, K. Zou, and H. Zhou, Applied Physics Reviews 8, 041312 (2021).
  * Wang et al. (2014) J. Wang, S. Li, M. Luo, J. Liu, L. Zhu, C. Li, D. Xie, Q. Yang, S. Yu, J. Sun, et al., in _2014 The European Conference on Optical Communication_ (IEEE, 2014), pp. 1–3.
  * Zhao et al. (2016) Y. Zhao, J. Liu, J. Du, S. Li, Y. Luo, A. Wang, L. Zhu, and J. Wang, in _Optical fiber communication conference_ (Optical Society of America, 2016), pp. Th1H–3.
  * Bozinivic et al. (2013) N. Bozinivic, Y. Y., Y. Ren, M. Tur, P. Kristensn, H. Huang, Willner, and S. Ramachandran, Science 340 (2013).
  * Anguita et al. (2008) J. A. Anguita, M. A. Neifeld, and B. V. Vasic, Appl. Opt. 47, 2414 (2008).
  * Ren et al. (2016) Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, et al., Optics Letters 41, 622 (2016).
  * Krenn et al. (2014) M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New Journal of Physics 16, 113028 (2014).
  * Rodenburg et al. (2012) B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, Optics Letters 37, 3735 (2012).
  * Zhang et al. (2020) L. Zhang, F. Shen, B. Lan, and A. Tang, Journal of Optics 22, 075607 (2020).
  * Malik et al. (2012) M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. Lavery, M. J. Padgett, and R. W. Boyd, Optics express 20, 13195 (2012).
  * Chen et al. (2016) C. Chen, H. Yang, S. Tong, and Y. Lou, Optics express 24, 6959 (2016).
  * Tyler and Boyd (2009) G. A. Tyler and R. W. Boyd, Optics letters 34, 142 (2009).
  * Cox et al. (2020) M. A. Cox, N. Mphuthi, I. Nape, N. Mashaba, L. Cheng, and A. Forbes, IEEE Journal of Selected Topics in Quantum Electronics 27, 1 (2020).
  * Mphuthi et al. (2018) N. Mphuthi, R. Botha, and A. Forbes, JOSA A 35, 1021 (2018).
  * Mphuthi et al. (2019) N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, R. Botha, and A. Forbes, Applied Optics 58, 4258 (2019).
  * Lukin (2014) I. P. Lukin, Applied optics 53, 3287 (2014).
  * Bao-Suan and Ji-Xiong (2009) C. Bao-Suan and P. Ji-Xiong, Chinese Physics B 18, 1033 (2009).
  * Zhu et al. (2008) K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, Optics Express 16, 21315 (2008).
  * Nelson et al. (2014) W. Nelson, J. Palastro, C. Davis, and P. Sprangle, JOSA A 31, 603 (2014).
  * Ahmed et al. (2016) N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, et al., Scientific reports 6, 22082 (2016).
  * Cheng et al. (2016) M. Cheng, L. Guo, J. Li, and Y. Zhang, IEEE Photonics Journal 8, 1 (2016).
  * Doster and Watnik (2016) T. Doster and A. T. Watnik, Applied Optics 55, 10239 (2016).
  * Watkins et al. (2020) R. J. Watkins, K. Dai, G. White, W. Li, J. K. Miller, K. S. Morgan, and E. G. Johnson, Optics express 28, 924 (2020).
  * Vetter et al. (2019) C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, Laser & Photonics Reviews 13, 1900103 (2019).
  * Yuan et al. (2017) Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, Scientific reports 7, 1 (2017).
  * Cox et al. (2019) M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, Journal of Lightwave Technology 37, 3911 (2019), ISSN 0733-8724, eprint 1901.07203, URL https://ieeexplore.ieee.org/document/8668467/.
  * Ndagano et al. (2017a) B. Ndagano, N. Mphuthi, G. Milione, and A. Forbes, Optics Letters 42, 4175 (2017a), ISSN 0146-9592, URL https://www.osapublishing.org/abstract.cfm?URI=ol-42-20-4175.
  * Restuccia et al. (2016) S. Restuccia, D. Giovannini, G. Gibson, and M. Padgett, Optics Express 24, 27127 (2016), ISSN 1094-4087, URL https://www.osapublishing.org/abstract.cfm?URI=oe-24-24-27127.
  * Ndagano et al. (2017b) B. Ndagano, N. Mphuthi, G. Milione, and A. Forbes, Optics letters 42, 4175 (2017b).
  * Trichili et al. (2016) A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. Ben Salem, M. Zghal, and A. Forbes, Scientific Reports 6, 27674 (2016).
  * Zhao et al. (2015) N. Zhao, X. Li, G. Li, and J. M. Kahn, Nature photonics 9, 822 (2015).
  * Zhou et al. (2019) Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. Lavery, A. E. Willner, and R. W. Boyd, Optics express 27, 10383 (2019).
  * Xie et al. (2016) G. Xie, Y. Ren, Y. Yan, H. Huang, N. Ahmed, L. Li, Z. Zhao, C. Bao, M. Tur, S. Ashrafi, et al., Optics letters 41, 3447 (2016).
  * Li et al. (2017) L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, et al., JOSA B 34, 1 (2017).
  * Gu et al. (2019) X. Gu, L. Chen, and M. Krenn, ArXiv p. 1906.03581v1 (2019).
  * Klug et al. (2021) A. Klug, I. Nape, and A. Forbes, New Journal of Physics 23, 093012 (2021).
  * Zhan (2009) Q. Zhan, Advances in Optics and Photonics 1, 1 (2009).
  * Rosales-Guzmán et al. (2018) C. Rosales-Guzmán, B. Ndagano, and A. Forbes, Journal of Optics 20, 123001 (2018).
  * Milione et al. (2015) G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, et al., Optics Letters 40, 1980 (2015), URL http://ol.osa.org/abstract.cfm?URI=ol-40-9-1980.
  * Zhu et al. (2021) Z. Zhu, M. Janasik, A. Fyffe, D. Hay, Y. Zhou, B. Kantor, T. Winder, R. W. Boyd, G. Leuchs, and Z. Shi, Nature communications 12, 1 (2021).
  * Ndagano et al. (2017c) B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, Scientific reports 7, 1 (2017c).
  * Sit et al. (2017) A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, et al., Optica 4, 1006 (2017).
  * Nape et al. (2018) I. Nape, E. Otte, A. Vallés, C. Rosales-Guzmán, F. Cardano, C. Denz, and A. Forbes, Optics express 26, 26946 (2018).
  * Ndagano et al. (2018) B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, Journal of Lightwave Technology 36, 292 (2018).
  * Cai et al. (2008) Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Optics express 16, 7665 (2008).
  * Ji-Xiong et al. (2010) P. Ji-Xiong, W. Tao, L. Hui-Chuan, and L. Cheng-Liang, Chinese Physics B 19, 089201 (2010).
  * Wang and Pu (2008) T. Wang and J. Pu, Optics communications 281, 3617 (2008).
  * Cox et al. (2016a) M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, D. J. Versfeld, and A. Forbes, Optics Express 24, 18105 (2016a).
  * Ndagano et al. (2017d) B. Ndagano, N. Mphuthi, G. Milione, and A. Forbes, Optics Letters 42, 4175 (2017d), ISSN 0146-9592, URL https://www.osapublishing.org/abstract.cfm?URI=ol-42-20-4175.
  * Nape et al. (2022) I. Nape, K. Singh, A. Klug, W. Buono, C. Rosales-Guzman, A. McWilliam, S. Franke-Arnold, A. Kritzinger, P. Forbes, A. Dudley, et al., Nature Photonics pp. 1–9 (2022).
  * McLaren et al. (2015) M. McLaren, T. Konrad, and Forbes Andrew, Physical Review A 92, 23833 (2015).
  * Ndagano et al. (2016) B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, Optics Letters 41, 3407 (2016).
  * Wootters (1998) W. K. Wootters, Physical Review Letters 80, 2245 (1998).
  * Herman and Strugala (1990) B. J. Herman and L. A. Strugala, in _Propagation of High-Energy Laser Beams through the
Earth’s Atmosphere_ (International Society for Optics and Photonics, 1990),
vol. 1221, pp. 183–192.

  * Rosales-Guzmán et al. (2020) C. Rosales-Guzmán, X.-B. Hu, A. Selyem, P. Moreno-Acosta, S. Franke-Arnold, R. Ramos-Garcia, and A. Forbes, Scientific Reports 10, 1 (2020).
  * Scholes et al. (2019) S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, Optical Engineering 59, 041202 (2019).
  * Wang et al. (2012a) J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, et al., Nature photonics 6, 488 (2012a).
  * Cox et al. (2016b) M. A. Cox, C. Rosales-Guzmán, M. P. Lavery, D. J. Versfeld, and A. Forbes, Optics Express 24, 18105 (2016b).
  * Zhou et al. (2021) Y. Zhou, J. Zhao, B. Braverman, K. Pang, R. Zhang, A. E. Willner, Z. Shi, and R. W. Boyd, Physical Review Applied 15, 034011 (2021).
  * Wang et al. (2012b) J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, et al., Nature Photonics 6, 488 (2012b), ISSN 1749-4885, URL http://dx.doi.org/10.1038/nphoton.2012.138.
  * Willner and Liu (2021) A. E. Willner and C. Liu, Nanophotonics 10, 225 (2021).
  * Lee (1979) W.-H. Lee, Applied Optics 18, 3661 (1979).
  * Hu et al. (2021) X.-B. Hu, S.-Y. Ma, and C. Rosales-Guzmán, Journal of Optics 23, 044002 (2021).
  * Singh et al. (2020) K. Singh, N. Tabebordbar, A. Forbes, and A. Dudley, JOSA A 37, C33 (2020).
  * Selyem et al. (2019) A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, Physical Review A 100, 063842 (2019).
  * Fried (1966) D. L. Fried, Journal of the Optical Society of America 56, 1372 (1966), ISSN 0030-3941, URL http://www.osapublishing.org/viewmedia.cfm?uri=josa-56-10-1372&seq=0&html=true.
  * Lane et al. (1992) R. Lane, A. Glindemann, and J. Dainty, Waves in random media 2, 209 (1992).
  * Pinnell et al. (2020) J. Pinnell, I. Nape, B. Sephton, M. A. Cox, V. Rodríguez-Fajardo, and A. Forbes, JOSA A 37, C146 (2020).
  * Wootters and Zurek (1982) W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982).
  * Töppel et al. (2013) F. Töppel, M. Ornigotti, and A. Aiello, New Journal of Physics 15, 113059 (2013).