# Topological rejection of noise by non-local quantum skyrmions

Pedro Ornelas  School of Physics, University of the Witwatersrand, Private Bag
3, Wits 2050, South Africa  Isaac Nape  School of Physics, University of the
Witwatersrand, Private Bag 3, Wits 2050, South Africa  Robert De Mello Koch
School of Science, Huzhou University, Huzhou 313000, China  Mandelstam
Institute for Theoretical Physics, School of Physics,University of the
Witwatersrand, Private Bag 3, Wits 2050, South Africa  Andrew
Forbes111Corresponding author. e-mail<EMAIL_ADDRESS>School of
Physics, University of the Witwatersrand, Private Bag 3, Wits 2050, South
Africa

###### Abstract

An open challenge in the context of quantum information processing and
communication is improving the robustness of quantum information to
environmental contributions of noise, a severe hindrance in real-world
scenarios. Here we create quantum skyrmionic topologies across two degrees of
freedom in two entangled photons and show that the topological invariant
remains resilient even as the purity of the state decreases due to
environmental noise, collapsing only once the state is maximally mixed. We
discuss how such noise perturbations can be considered a smooth deformation of
the state, and provide a full theoretical treatment to support the
experimental findings. Our approach can be viewed as a digitization of quantum
information, holding exciting promise for intrinsic quantum resilience through
topology.

Quantum entanglement is an essential and viable resource for future quantum
technologies. It has proven its worth in applications as diverse as secure
quantum communication gisin2002quantum ; scarani2009security ; ma2007quantum ,
even over large distances in satellite based networks liao2017satellite ;
bedington2017progress , quantum computing thomas2022efficient ;
raussendorf2001one ; briegel2009measurement , quantum imaging of complex
objects pepe2016correlation ; sephton2023revealing , quantum metrology
giovannetti2011advances  and lithography boto2000quantum ; kok2001quantum .
However, it is well known that noise deteriorates entanglement, for instance,
due to noisy detectors, stray light (e.g., daylight), lost photons or other
irreversible environmental effects, diminishing our ability to certify,
identify or exploit the full benefits of quantum entanglement for information
processing kumar2003effect ; nielsen2001quantum ; lloyd1997capacity ;
liang2013quantum . High dimensional ($d>2$ dimensions) entangled states are
able to increase the noise threshold to which the usual entanglement witnesses
certify entanglement ecker2019overcoming ; peterfreund2021multidimensional ;
almeida2007noise ; qu2022robust ; tsokeng2018dynamics ; qu2022retrieving ,
reaching $p_{c}=1/(1+d)$ before the state is separable, but comes with the
cost of limited state purity zhu2021high  and complex quantum state analysis
nape2023quantum . An emerging trend is to exploit quantum topological
photonics yan2021quantum , which has already realised stable quantum emitters
mehrabad2020chiral ; dai2022topologically ; mittal2018topological , robust
transport of entanglement through quantum circuits barik2018topological ;
blanco2018topological  and entanglement storage parmee2022optical , but here
the topology is not in the quantum state but in carefully structured
topological matter. All these schemes seek entanglement persistence in non-
ideal systems, but this does not guarantee robustness of the information
carried by that entanglement. Can quantum information be preserved in the
presence of noise even if the entanglement is fragile?

Here we propose and demonstrate noise resilience of quantum information by
harnessing a nontrivial topology that emerges from the entanglement of bi-
photon quantum states. To realise this we create a nonlocal skyrmionic
topology as a shared property of two entangled photons and show that the
topology is invariant to external noise, which we simulate by a generic noise
model that mixes a target pure state with a maximally mixed state. Such a
model encompasses various decoherence channels that may originate from shot
noise, thermal noise, ambient light, depolarisation channels, to name but a
few. We provide an intuitive explanation, supported by a rigorous theoretical
treatment, as to why such noise can be considered a smooth deformation of the
state (an operation that does not modify its topology). We present
experimental results across various topologies and for a wide range of noise
levels, confirming the topological invariance. Our work opens a new path to
quantum information processing and communication in noisy quantum systems and
channels without the need for compensation or mitigation strategies.

Figure 1: To probe the resilience of skyrmionic quantum topology to noise, we
(a) create a bi-photon entangled state, with photon A in the spatial degree of
freedom and photon B in the polarization degree of freedom, for a non-local
skyrmionic topology as a shared emergent property of the two photons, shown as
a stereographic projection. When the quantum topology is exposed to white
noise sources (torches), theory predicts (b) that as the noise increases,
quantified by a decrease in quantum contrast ($Q_{c}$), so the purity of the
state decreases while the skyrmion number remains invariant. (c) In the
geometric picture of photon B’s Hilbert space as expressed by Stokes vectors
($S_{x},S_{y},S_{z}$), the change of purity is a sphere of diminishing size,
which for even mixed states can always be renormalized while leaving the
topology unaltered. Once the state is maximally-mixed, the Hilbert space of
photon B has zero radius, no entanglement and thus only a trivial topology
($N=0$).

As our aim is to study quantum skyrmionic topology in the presence of noise,
we begin by constructing a bi-photon entangled state of the form
$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{\ell_{1}}_{A}\ket{P_{1}}_{B}+\ket{\ell_{2}}_{A}\ket{P_{2}}_{B}\right)$,
where $\ell_{1}$ and $\ell_{2}$ denote orbital angular momentum (OAM) of
$\ell_{1}\hbar$ and $\ell_{2}\hbar$ per photon, respectively and
$\ket{P_{1}},\ket{P_{2}}$ are orthogonal polarization states, as depicted
graphically in Fig. 1 (a). When $|\ell_{1}|\neq|\ell_{2}|$ the bi-photon
correlations form the desired skyrmionic mapping from
$\mathcal{R}^{2}\to\mathcal{S}^{2}$, so that the entanglement defines a
topology ornelas2024non . The topological invariant, the skyrmion number
($N$), characterizes the number of times $\mathcal{R}^{2}$ (stereographically
projected Hilbert space of photon A) wraps $\mathcal{S}^{2}$ (Hilbert space of
photon B), i.e., the number of times photon B’s collapsed states wrap its own
Hilbert space after completing a full set of spatial position measurements on
photon A. This mapping is evident when expressing the OAM states of photon A
in the position basis, using
$|\ell\rangle=\int_{\mathcal{R}^{2}}|\text{LG}_{\ell}\left(\mathbf{r}\right)|e^{i\ell\phi}|\mathbf{r}\rangle$
where $\text{LG}_{\ell}\left(\mathbf{r}\right)$ are the Laguerre-Gaussian
fields and $|\mathbf{r}\rangle$ are position states. This recasts our state to

$\displaystyle\ket{\Psi}=\int_{\mathcal{R}^{2}}\ket{\mathbf{r}_{A}}\left(a(\mathbf{r}_{A})\ket{P_{1}}_{B}+b(\mathbf{r}_{A})\ket{P_{2}}_{B}\right)d^{2}r_{A},$
(1)

where
$a(\mathbf{r}_{A})\equiv|\text{LG}_{\ell_{1}}\left(\mathbf{r}_{A}\right)|$ and
$b(\mathbf{r}_{A})\equiv\exp(\Delta\ell\phi_{A})|\text{LG}_{\ell_{2}}\left(\mathbf{r}_{A}\right)|$
and $\Delta\ell=\ell_{2}-\ell_{1}$. The skyrmion number of such a state is
then calculated from gao2020paraxial

$N=\frac{1}{4\pi}\int_{\mathcal{R}^{2}}\epsilon_{pqr}S_{p}\frac{\partial
S_{q}}{\partial x}\frac{\partial S_{r}}{\partial y}dxdy\,,$ (2)

where $\epsilon_{pqr}$ is the Levi-Civita symbol and it is assumed that the
sphere has been normalized such that $\Sigma_{i=1}^{3}S_{i}^{2}=1$, i.e., a
mapping to the unit sphere. To find the necessary quantum Stokes parameters,
$S_{i}$, one computes the expectation values of the Pauli matrices, calculated
by taking the diagonal matrix element at position $\bar{r}$ for photon A and
the partial trace over photon B, such that
$S_{i}=\text{Tr}_{B}(\langle|\bar{r}\rangle_{A}\langle\bar{r}|_{A}\otimes\sigma_{B,i}\rangle)=\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$.

A common way to simulate real-world noisy conditions is to use a white
(isotropic) noise model, by mixing the initially pure state with a maximally
mixed state dur1999quantum , covering a wide range of random noise sources
e.g., from the source (multi-photon events), the channel (background and lost
photons), and the detector (dark counts). To keep the analysis generic we will
only use the purity of the state as our measure of the noise, and not
knowledge of the noise source itself. Without loss of generality we restrict
ourselves to the original OAM Hilbert space of photon A so that the density
matrix can be written as

${}_{A}\langle\bar{r}|\rho|\bar{r}\rangle_{A}=p{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{4}\mathbb{1}_{4},$
(3)

and $\mathbb{1}_{4}$ is the $4\\!\times\\!4$ identity matrix. The purity of
this state is given by $\gamma=\text{Tr}(\rho^{2})=p^{2}+\frac{1-p^{2}}{4}$
with $p\in[0,1]$, e.g., a pure state ($|\Psi\rangle\langle\Psi|$) for $p=1$
and a maximally mixed state ($\frac{1}{4}\mathbb{1}_{4}$) for $p=0$. To
calculate the skyrmion number we must determine the effect of the noise on the
quantum Stokes parameters, whose action is to produce new parameters,
$S_{i}^{\prime}$. Using the linearity of the trace operation and the fact that
the Pauli matrices are traceless, we obtain the result

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$ $\displaystyle
p\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$
(6) $\displaystyle+\frac{1-p}{2}\text{Tr}_{B}\left(\sigma_{B,i}\right)$
$\displaystyle=$ $\displaystyle
p\,\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$
(8) $\displaystyle=$ $\displaystyle pS_{i}$ (10)

where $S_{i}$ are the quantum Stokes parameters for the pure state. Given that
the skyrmion number calculation is defined for maps to spheres of unit radius
i.e., $\Sigma_{i=1}^{3}S_{i}^{2}=1$, the purity $p$ factors out of the
calculation when performing normalization. The conclusion is that since the
integrand of Eq. 2 is unchanged, so too is the skyrmion number. Hence the
topology of the state remains unchanged so long as a small portion of the pure
state survives ($p>0$ or $\gamma>\frac{1}{4}$). See the SI for a more detailed
derivation of this main result. This invariance is shown graphically in Fig. 1
(b) where the skyrmion number remains constant as the purity decreases with
increasing noise, here quantified by the quantum contrast, $Q_{c}$. The
skyrmion number only changes once the purity drops to $0.25$, signalling a
maximally-mixed state with no entanglement. As our topology is derived from
the quantum wavefunction itself, no entanglement naturally implies no topology
(i.e., a trivial topology with $N=0$).

This noise rejection mechanism can be understood intuitively if we consider
the spectral decomposition of the Pauli observables, given by
$\sigma_{B,i}=\lambda^{+}_{i}P^{+}_{i}+\lambda^{-}_{i}P^{-}_{i}$ where
$P^{\pm}_{i}=P_{i}^{\pm}=|\lambda^{\pm}_{i}\rangle\langle\lambda^{\pm}_{i}|$
for positive and negative eigenvalues $\lambda^{\pm}=\pm 1$, thus

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\text{Tr}_{B}\left(P^{+}_{i}\left[p\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{2}\mathbb{1_{2}}\right]\right)$
(14)
$\displaystyle-\text{Tr}_{B}\left(P^{-}_{i}\left[p\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{2}\mathbb{1_{2}}\right]\right),$

which exposes a subtle point: the noise rejection mechanism stems from
subtracting measured values of the projections $P^{\pm}_{i}$, which each
contain the same noise contribution,
$\frac{1-p}{2}\text{Tr}_{B}\left(P^{\pm}_{i}\right)=\frac{1-p}{2}$.
Normalizing each of the $S_{i}$ parameters by
$S_{0}^{{}^{\prime}2}=\text{Tr}_{B}\left({}_{A}\langle\bar{r}|\rho|\bar{r}\rangle_{A}\right)$
reveals the smooth deformation that takes place under the addition of this
noise to the system. With this normalization it can be shown that
$\sum_{i}^{3}S_{i}^{{}^{\prime}2}=\frac{4p^{2}}{(p+1)^{2}}\leq 1$ which
reveals that photon B’s state is mapped to a state on a shell in the interior
of its Hilbert space (a unit two sphere), with a radius dictated by the purity
of the state, as depicted in Fig. 1 (c). This does not change the topology:
$\mathcal{R}^{2}$ still maps to a sphere $\mathcal{S}^{2}$, albeit one of a
smaller radius, with the same skyrmion number. When the state is maximally
mixed, $\gamma=\frac{1}{4}$, the calculated skyrmion number vanishes as the
radius of the shell has reduced to zero. In this singular limit
$\mathcal{R}^{2}$ maps to a single point, depicted in Fig. 1 (c), consistent
with the fact that when there is no entanglement, there can be no non-trivial
topology ornelas2024non . The topology remains robust to the noise so long as
some entanglement persists.

Figure 2: (a) Experimental setup for the generation, preparation and detection
of non-local skyrmionic quantum states, with details given in the main text
and Supplementary Information. The desired hybrid states are prepared via a
spatial-to-polarization coupling (SPC) technique (shown as an inset). A
tuneable isotropic noise source (NS) is introduced in the form of ambient
lights. To confirm the system performance we show (b) quantum state tomography
(QST) results (top) along with the reconstructed experimental density matrices
(middle) and quantum Stokes parameters (bottom) for the ideal example state,
$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{H}_{B}+\ket{1}_{A}\ket{V}_{B}\right)$.
To calibrate the noise, we show (c) purity, $\gamma$, as a function of quantum
contrast, $Q_{c}$ with theory (dashed lines) and experiment (points) in good
agreement. Abbreviations: Barium Borate (BBO), band-pass filter (BPF), 50:50
beam splitter (BS), mirror (M), lens (L), spatial-to-polarization coupling
(SPC), spatial light modulator (SLM), avalanche photodiode (APD), noise source
(NS), coincidence counter (CC), half-wave plate (HWP), quarter-wave plate
(QWP), linear polarizer (LP).

To verify the findings we use the experiment depicted in Fig. 2(a), with full
details given in the SI. Entangled OAM biphotons of wavelength 710 nm were
generated through spontaneous parametric down-conversion (SPDC) in a 3 mm long
Type-I Barium Borate (BBO) nonlinear crystal pumped by a 355 nm wavelength
collimated Gaussian beam. To achieve our topologically non-trivial states we
used the spatial-to-polarization coupling (SPC) state preparation stage shown
as the inset to Fig. 2(a), exchanging the OAM-OAM correlations between photons
A and B for OAM-Polarization correlations. To do so, Photon A was left
unaltered while photon B’s spatial information was coupled to its polarization
using a double bounce off a spatial light modulator (SLM B). The incoming
horizontally polarized photons were modulated in the first pass and the
vertically polarized photons in the second (by splitting the SLM screen in two
parts), with the quarter-wave plate (QWP) serving to accommodate the fact that
the SLM can only modulate horizontally polarized light. For non-trivial
topologies the holograms displayed on SLM B were set such that
$|\ell_{1}|-|\ell_{2}|\neq 0$. In the detection stage, photon A was directed
to a spatial light modulator (SLM A) for spatial projective measurements while
photon B was passed through a HWP orientated to $45^{\circ}$ and a linear
polarizer orientated at $90^{\circ}$ for polarization projective measurements.
Both photons were collected by single-mode fibres (SMFs) coupled to an
avalanche photon detector (APD) and measured in coincidence. Six projective
measurements on each photon allowed for an overcomplete quantum state
tomography (QST) with 36 entries, which was used to reconstruct the quantum
state (with and without noise), with full details given in the SI. Initially,
the experiment was run without noise to certify the creation and detection of
the example state
$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{H}_{B}+\ket{1}_{A}\ket{V}_{B}\right)$.
Fig. 2 (b) shows the experimental QST for the state along with the
reconstructed density matrix and quantum Stokes parameters. The data is in
agreement with the results expected from the desired pure state, with a purity
of $\gamma=0.80$ and skyrmion number $N=0.96\pm 0.01$ (theory: $N=1$). To
introduce noise we used an incoherent light source of variable intensity,
which is known to be a good approximation to white noise ecker2019overcoming .
The contribution of the white light source to the final state was determined
from the average quantum contrast, $Q_{c}$, which estimated the signal-to-
noise (SNR) ratio in our system, reaching values as high as 32.3 and as low as
2.24 for low and high levels of noise, respectively. This impacts directly on
the purity of the state via the relation zhu2021high

$\gamma=\frac{1}{4}\left[3\left(\frac{Q_{c}-1}{Q_{c}+1}\right)^{2}+1\right],$
(15)

which we used as a measure of the impact of the noise, ranging from our target
pure state (no noise) to a maximally mixed (separable) state (high noise). In
Fig. 2 (c) the experimentally obtained purity is plotted against average
quantum contrast and is in excellent agreement with theory. This demonstrates
that precise control over the experimental purity was achieved through control
over the quantum contrast of the system. The use of the purity of the state in
the experimental analysis negates the need for any knowledge of the noise
source. Finally, for each noise level we measured our topological invariant
using the reconstructed density matrix.

The experiment was run for seven different initial states with varying
topology, $N\in\\{-3,-2,-1,0,1,2,3\\}$, each for a wide range of isotropic
noise levels, and in each case the quantum Stokes parameters inferred from a
quantum state tomography and the resulting Skyrmion number calculated. Example
data is shown in Fig. 3 for the states
$\rho=p\ket{\Psi}\bra{\Psi}+\frac{1-p}{4}\mathbb{1}_{4}$ where
$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{H}_{B}+\ket{3}_{A}\ket{V}_{B}\right)$
in the presence of varying isotropic noise levels from pure (point 1,
$\gamma=0.80$) to partially-mixed (point 2, $\gamma=0.45$) and then maximally
mixed (point 3, $\gamma=0.25$). As expected the QST data shown in Fig. 3(a)
reveals a characteristic saturation of each projection measurement with an
increase in isotropic noise. This is accompanied by a clear transition of the
density matrix describing a pure state (entangled) to that describing a
maximally-mixed state (no entanglement) as the purity is decreased, which can
be seen in Fig. 3(b). The skyrmion number for these three states (1-3) was
$N=2.99\pm 0.01$, $N=2.99\pm 0.01$ and $N=0.01\pm 0.01$, respectively
revealing that the skyrmion number remains intact until the entanglement
vanishes altogether. This can be seen in the trends of Fig. 3(c), with
theoretical (dashed lines) and experimental data (points) for the skyrmion
number as a function of purity in excellent agreement across all topologies.
The zoomed inset shows experimental data near the classical limit,
$\gamma=0.25$, where it is clear that a topological transition from $N\neq 0$
to $N=0$ occurs because without entanglement there can be no topology.

Figure 3:  (a) QST results for the example state
$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{H}_{B}+\ket{3}_{A}\ket{V}_{B}\right)$
in the presence of varying noise levels from pure (point 1, $\gamma=0.80$) to
partially-mixed (point 2, $\gamma=0.45$) and then maximally-mixed (point 3,
$\gamma=0.25$). (b) The experimental density matrices extracted from the QST
of each state. (c) Skyrmion number as a function of purity, with experiment
(points) in excellent agreement with theory (dashed lines). The collapse of
the topology when the maximally-mixed state limit is reached is shown in the
zoomed-in inset.

Skyrmionic photonic topologies have been celebrated for their potential to act
as resilient information carriers shen2024optical , but numerical studies have
returned mixed outcomes liu2022disorder . This report offers, to the best of
our knowledge, the first experimental results on their resilience to noise
(decoherence channels) and highlights a subtle point that places prior work
into a new context: the key question is whether the quantum channel is map
preserving or not. We show that isotropic noise is map preserving - it can be
viewed as a smooth deformation of the state (map) - and thus preserves
topology even when the entanglement itself is decaying. Our approach of
harnessing the emergent topology that arises from entanglement for noise
rejection can be understood in the context of digital and analogue
information. Topology gives rise to a discrete signal that captures the
entanglement information. Just as for classical digital signals, the discrete
quantum signal is completely resilient to noise which is unable to flip the
signal between its discrete values. We believe that this noise rejection by
topology mechanism could be exploited in practical quantum information
processing and communication protocols, e.g., quantum communication under
ambient (daylight) conditions, or quantum computing on noisy photonic chips,
and could be used in conjunction with the entanglement preserving strategies
mentioned in the introduction.

In conclusion we have shown that the skyrmionic topology of non-local quantum
states is resilient to noise by virtue of a smooth deformation of the state
(map) that leaves the topology unchanged, collapsing only when the state is
maximally mixed and no entanglement remains. We note that our noise model is
very general, covering a plethora of real-world scenarios. The demonstrated
resilience comes without the need for error mitigation or state purification
techniques, thus offering a new approach to quantum information resilience in
realistic and non-ideal conditions.

## References

  * (1) N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Reviews of Modern Physics, vol. 74, no. 1, p. 145, 2002.
  * (2) V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Reviews of Modern Physics, vol. 81, no. 3, p. 1301, 2009.
  * (3) X. Ma, C.-H. F. Fung, and H.-K. Lo, “Quantum key distribution with entangled photon sources,” Physical Review A, vol. 76, no. 1, p. 012307, 2007.
  * (4) S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, et al., “Satellite-to-ground quantum key distribution,” Nature, vol. 549, no. 7670, pp. 43–47, 2017.
  * (5) R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Information, vol. 3, no. 1, p. 30, 2017.
  * (6) P. Thomas, L. Ruscio, O. Morin, and G. Rempe, “Efficient generation of entangled multiphoton graph states from a single atom,” Nature, vol. 608, no. 7924, pp. 677–681, 2022.
  * (7) R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Physical Review Letters, vol. 86, no. 22, p. 5188, 2001.
  * (8) H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nature Physics, vol. 5, no. 1, pp. 19–26, 2009.
  * (9) F. V. Pepe, F. Di Lena, A. Garuccio, G. Scarcelli, and M. D’Angelo, “Correlation plenoptic imaging with entangled photons,” Technologies, vol. 4, no. 2, p. 17, 2016.
  * (10) B. Sephton, I. Nape, C. Moodley, J. Francis, and A. Forbes, “Revealing the embedded phase in single-pixel quantum ghost imaging,” Optica, vol. 10, no. 2, pp. 286–291, 2023.
  * (11) V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature photonics, vol. 5, no. 4, pp. 222–229, 2011.
  * (12) A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Physical Review Letters, vol. 85, no. 13, p. 2733, 2000.
  * (13) P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Physical Review A, vol. 63, no. 6, p. 063407, 2001.
  * (14) D. Kumar and P. Pandey, “Effect of noise on quantum teleportation,” Physical Review A, vol. 68, no. 1, p. 012317, 2003.
  * (15) M. A. Nielsen and I. L. Chuang, “Quantum computation and quantum information,” Phys. Today, vol. 54, no. 2, p. 60, 2001.
  * (16) S. Lloyd, “Capacity of the noisy quantum channel,” Physical Review A, vol. 55, no. 3, p. 1613, 1997.
  * (17) H.-Q. Liang, J.-M. Liu, S.-S. Feng, and J.-G. Chen, “Quantum teleportation with partially entangled states via noisy channels,” Quantum Information Processing, vol. 12, no. 8, pp. 2671–2687, 2013.
  * (18) S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, et al., “Overcoming noise in entanglement distribution,” Physical Review X, vol. 9, no. 4, p. 041042, 2019.
  * (19) E. Peterfreund and M. Gavish, “Multidimensional scaling of noisy high dimensional data,” Applied and Computational Harmonic Analysis, vol. 51, pp. 333–373, 2021.
  * (20) M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Physical Review Letters, vol. 99, no. 4, p. 040403, 2007.
  * (21) R. Qu, Y. Wang, X. Zhang, S. Ru, F. Wang, H. Gao, F. Li, and P. Zhang, “Robust method for certifying genuine high-dimensional quantum steering with multimeasurement settings,” Optica, vol. 9, no. 5, pp. 473–478, 2022.
  * (22) A. T. Tsokeng, M. Tchoffo, and L. C. Fai, “Dynamics of entanglement and quantum states transitions in spin-qutrit systems under classical dephasing and the relevance of the initial state,” Journal of Physics Communications, vol. 2, no. 3, p. 035031, 2018.
  * (23) R. Qu, Y. Wang, M. An, F. Wang, Q. Quan, H. Li, H. Gao, F. Li, and P. Zhang, “Retrieving high-dimensional quantum steering from a noisy environment with n measurement settings,” Physical Review Letters, vol. 128, no. 24, p. 240402, 2022.
  * (24) F. Zhu, M. Tyler, N. H. Valencia, M. Malik, and J. Leach, “Is high-dimensional photonic entanglement robust to noise?,” AVS Quantum Science, vol. 3, no. 1, p. 011401, 2021.
  * (25) I. Nape, B. Sephton, P. Ornelas, C. Moodley, and A. Forbes, “Quantum structured light in high dimensions,” APL Photonics, vol. 8, no. 5, 2023.
  * (26) Q. Yan, X. Hu, Y. Fu, C. Lu, C. Fan, Q. Liu, X. Feng, Q. Sun, and Q. Gong, “Quantum topological photonics,” Advanced Optical Materials, vol. 9, no. 15, p. 2001739, 2021.
  * (27) M. J. Mehrabad, A. P. Foster, R. Dost, E. Clarke, P. K. Patil, A. M. Fox, M. S. Skolnick, and L. R. Wilson, “Chiral topological photonics with an embedded quantum emitter,” Optica, vol. 7, no. 12, pp. 1690–1696, 2020.
  * (28) T. Dai, Y. Ao, J. Bao, J. Mao, Y. Chi, Z. Fu, Y. You, X. Chen, C. Zhai, B. Tang, et al., “Topologically protected quantum entanglement emitters,” Nature Photonics, vol. 16, no. 3, pp. 248–257, 2022.
  * (29) S. Mittal, E. A. Goldschmidt, and M. Hafezi, “A topological source of quantum light,” Nature, vol. 561, no. 7724, pp. 502–506, 2018.
  * (30) S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science, vol. 359, no. 6376, pp. 666–668, 2018.
  * (31) A. Blanco-Redondo, B. Bell, D. Oren, B. J. Eggleton, and M. Segev, “Topological protection of biphoton states,” Science, vol. 362, no. 6414, pp. 568–571, 2018.
  * (32) C. D. Parmee, M. R. Dennis, and J. Ruostekoski, “Optical excitations of skyrmions, knotted solitons, and defects in atoms,” Communications Physics, vol. 5, no. 1, p. 54, 2022.
  * (33) P. Ornelas, I. Nape, R. de Mello Koch, and A. Forbes, “Non-local skyrmions as topologically resilient quantum entangled states of light,” Nature Photonics, pp. 1–9, 2024.
  * (34) S. Gao, F. C. Speirits, F. Castellucci, S. Franke-Arnold, S. M. Barnett, and J. B. Götte, “Paraxial skyrmionic beams,” Physical Review A, vol. 102, no. 5, p. 053513, 2020.
  * (35) W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeaters based on entanglement purification,” Physical Review A, vol. 59, no. 1, p. 169, 1999.
  * (36) Y. Shen, Q. Zhang, P. Shi, L. Du, X. Yuan, and A. V. Zayats, “Optical skyrmions and other topological quasiparticles of light,” Nature Photonics, vol. 18, no. 1, pp. 15–25, 2024.
  * (37) C. Liu, S. Zhang, S. A. Maier, and H. Ren, “Disorder-induced topological state transition in the optical skyrmion family,” Physical Review Letters, vol. 129, no. 26, p. 267401, 2022.
  * (38) H. Kuratsuji and S. Tsuchida, “Evolution of the stokes parameters, polarization singularities, and optical skyrmion,” Physical Review A, vol. 103, no. 8, p. 023514, 2021.
  * (39) Y. Shen, E. C. Martínez, and C. Rosales-Guzmán, “Generation of optical skyrmions with tunable topological textures,” ACS Photonics, vol. 9, no. 1, pp. 296–303, 2022.
  * (40) Y. Shen, “Topological bimeronic beams,” Optics Letters, vol. 46, no. 15, pp. 3737–3740, 2021.
  * (41) I. Nape, A. G. de Oliveira, D. Slabbert, N. Bornman, J. Francis, P. H. S. Ribeiro, and A. Forbes, “An all-digital approach for versatile hybrid entanglement generation,” Journal of Optics, vol. 24, p. 054003, mar 2022.

##  I Supplementary: Formulation of non-local quantum skyrmion

A pure non-local biphoton hybrid entangled state of two photons, A and B,
correlated in OAM-Polarization (as depicted in Fig. 4 (a)), can in general be
written in the form

$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{\ell_{1}}_{A}\ket{P_{1}}_{B}+e^{i\delta}\ket{\ell_{2}}_{A}\ket{P_{2}}_{B}\right),$
(16)

where $\ell_{1}$ and $\ell_{2}$ denote OAM of $\ell_{1}\hbar$ and
$\ell_{2}\hbar$ per photon, respectively, and $\ket{P_{1}},\ket{P_{2}}$ are
orthogonal polarization states, while $\delta$ allows for a relative phase
between the two components of the state vector. Photon A’s states live on the
Hilbert space ${\cal H}_{A}$ spanned by the states
$\\{\ket{\ell_{1}},\ket{\ell_{2}}\\}$ and photon B’s states live on the
Hilbert space ${\cal H}_{B}$ spanned by the states
$\\{\ket{P_{1}},\ket{P_{2}}\\}$. $|\Psi\rangle$ belongs to the Hilbert space
${\cal H}={\cal H}_{A}\otimes{\cal H}_{B}$. For states with
$|\ell_{1}|\neq|\ell_{2}|$ it has been shown that the correlations between
these photons form the desired skyrmionic mapping from
$\mathcal{S}^{2}\to\mathcal{S}^{2}$ (equivalent to the mapping
$\mathcal{R}^{2}\to\mathcal{S}^{2}$ through a stereographic projection of the
spatial $\mathcal{S}^{2}$ to the real plane $\mathcal{R}^{2}$) as depicted in
Fig. 4 (b) ornelas2024non . Expressing photon A in position basis, using
$|\ell\rangle=\int_{\mathcal{R}^{2}}|\text{LG}_{\ell}\left(\mathbf{r}\right)|e^{i\ell\phi}|\mathbf{r}\rangle$
where $\text{LG}_{\ell}\left(\mathbf{r}\right)$ are the Laguerre-Gaussian
fields and $|\mathbf{r}\rangle$ are position states, we find

$\displaystyle\ket{\Psi}=\int_{\mathcal{R}^{2}}\ket{\mathbf{r}_{A}}\left(a(\mathbf{r}_{A})\ket{P_{1}}_{B}+b(\mathbf{r}_{A})e^{i\Theta(\phi_{A})}\ket{P_{2}}_{B}\right)d^{2}r_{A},$
(17)

where
$a(\mathbf{r}_{A})\equiv|\text{LG}_{\ell_{1}}\left(\mathbf{r}_{A}\right)|$,
$b(\mathbf{r}_{A})\equiv|\text{LG}_{\ell_{2}}\left(\mathbf{r}_{A}\right)|$,
$\Theta(\phi_{A})=\Delta\ell\phi_{A}+\gamma$ and
$\Delta\ell=\ell_{2}-\ell_{1}$. In this formulation, it is can be deduced that
a spatial measurement on photon A, collapses photon B into a particular
polarization state as shown in Fig. 4 (b). For a non-trivial skyrmionic
topology, we have that a full set of spatial measurements on photon A
collapses photon B into every possible polarization state as depicted in Fig.
4 (c). This non-local topology can be depicted compactly by combining the
spatial sphere and polarization states together as shown in Fig. 4 (c) where
the tail of each polarization state vector for photon B is adjacent to its
correlated position in photon A.

The Skyrmion number of these states can then calculated using the equation
$N=\frac{1}{4\pi}\int_{\mathcal{R}^{2}}\Sigma_{z}(x,y)dxdy$ gao2020paraxial
where $\Sigma_{z}(x,y)=\epsilon_{pqr}S_{p}\frac{\partial S_{q}}{\partial
x}\frac{\partial S_{r}}{\partial y}$ and $\epsilon_{pqr}$ is the Levi-Civita
symbol. The integral in the Skyrmion number calculation computes the surface
area of photon B’s parameter space covered by the mapping defined by the
wavefunction given in Eqn. 17. Thus, since the parameter space is a unit
sphere, dividing by the surface area of the sphere ($4\pi$), we find the
wrapping number $N$ of the non-local topology.

The quantum Stokes parameters, $S_{i}$, are given by the expectation values of
the Pauli matrices, which are calculated by taking the diagonal matrix element
at position $\bar{r}$ for photon A and the partial trace over photon B, such
that
$S_{i}=\text{Tr}_{B}(\langle|\bar{r}\rangle_{A}\langle\bar{r}|_{A}\otimes\sigma_{B,i}\rangle)=\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$.
From this it can be shown that the Skyrmion number depends on the difference
between $\ell_{1}$ and $\ell_{2}$, according to $N=m\Delta\ell$ with
$m=\text{sign}\left(|\ell_{1}|-|\ell_{2}|\right)$. gao2020paraxial ;
kuratsuji2021evolution ; shen2022generation ; shen2021topological

Figure 4: (a) Hybrid entangled state of photons A and B which share OAM-
polarization correlations. (b) Such a state also yields position-polarization
correlations where a position measurement on photon A yields in coincidence a
polarization state collapse for photon B. Through a stereographic projection,
this can also be seen as a mapping between a spatial sphere and a polarization
sphere. (c) Observation of photon B’s state in coincidence with spatial
measurements performed on photon A reveals the non-local skyrmionic structure
embedded within the entangled state. The topological structure can be
compactly represented by combining the spatial sphere of photon A and the
polarization state vectors of photon B whose tail is adjacent to its
correlated position in photon A.

##  II Supplementary: Quantum topological invariance to isotropic noise

We now consider subjecting our state, $|\Psi\rangle$, to environmental/
“white” noise, according to the isotropic model where the purity of the state
is degraded by mixing it with a maximally mixed state according to

$\rho=p\ket{\Psi}\bra{\Psi}+\frac{1-p}{d^{2}}\mathbb{1}_{d^{2}},$ (18)

where $d=2$, $p\in[0,1]$ is a parameter controlling the degree of purity of
the state with $p=1$ giving a pure state ($|\Psi\rangle\langle\Psi|$) and
$p=0$ giving a maximally mixed state ($\frac{1}{d^{2}}\mathbb{1}_{d^{2}}$) and
$\mathbb{1}_{d^{2}}$, the $d^{2}\,\times\,d^{2}$ identity matrix, is the
identity operator on ${\cal H}$. The purity, $\gamma$, of a state is given by
$\text{Tr}(\rho^{2})$. It follows that $\gamma$ is related to $p$ according to

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\left(p\ket{\Psi}\bra{\Psi}+\frac{1-p}{d^{2}}\mathbb{1}_{d^{2}}\right)^{2}\right)$
(19) $\displaystyle=$
$\displaystyle\text{Tr}\left(p^{2}\ket{\Psi}\bra{\Psi}+2p\frac{(1-p)}{d^{2}}\ket{\Psi}\bra{\Psi}+\frac{(1-p)^{2}}{d^{4}}\mathbb{1}_{d^{2}}\right)$
$\displaystyle=$ $\displaystyle
p^{2}+2p\frac{(1-p)}{d^{2}}+\frac{(1-p)^{2}}{d^{2}}$ $\displaystyle=$
$\displaystyle p^{2}+\frac{1-p^{2}}{d^{2}}.$

To calculate the Skyrmion number we must consider how the quantum Stokes
parameters are affected by the isotropic noise with the model given above,

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\text{Tr}_{B}\left(\sigma_{B,i}\,{}_{A}\langle\bar{r}|\rho|\bar{r}\rangle_{A}\right)$
(20) $\displaystyle=$
$\displaystyle\text{Tr}_{B}\Big{(}\sigma_{B,i}\big{(}p\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}$
(24)
$\displaystyle\qquad+\frac{1-p}{4}\,{}_{A}\langle\bar{r}|\mathbb{1}_{4}|\bar{r}\rangle_{A}\big{)}\Big{)}$

To compute the diagonal matrix element of $\mathbb{1}_{4}$, start by noting
that we have

$a(\vec{r})\,\,=\,\,\frac{1}{\sqrt{2}}\langle\vec{r}|\ell_{1}\rangle\qquad\qquad
b(\vec{r})\,\,=\,\,\frac{1}{\sqrt{2}}\langle\vec{r}|\ell_{2}\rangle$ (25)

which follow by comparing (16) and (17). To obtain correctly normalized
quantum Stokes parameters, we must choose

$|a|^{2}+|b|^{2}=1$ (26)

at all $\vec{r}$. Using

$\displaystyle\mathbb{1}_{4}$ $\displaystyle=$
$\displaystyle|\ell_{1}\rangle_{A}|P_{1}\rangle_{B}\,{}_{A}\langle\ell_{1}|\,{}_{B}\langle
P_{1}|+|\ell_{1}\rangle_{A}|P_{2}\rangle_{B}\,{}_{A}\langle\ell_{1}|\,{}_{B}\langle
P_{2}|$
$\displaystyle+|\ell_{2}\rangle_{A}|P_{1}\rangle_{B}\,{}_{A}\langle\ell_{2}|\,{}_{B}\langle
P_{1}|+|\ell_{2}\rangle_{A}|P_{2}\rangle_{B}\,{}_{A}\langle\ell_{2}|\,{}_{B}\langle
P_{2}|$

we easily find (use (25) and (26))

$\displaystyle\langle\vec{r}|\mathbb{1}_{4}|\vec{r}\rangle$ $\displaystyle=$
$\displaystyle 2|a|^{2}|P_{1}\rangle_{B}\,{}_{B}\langle
P_{1}|+2|a|^{2}|P_{2}\rangle_{B}\,{}_{B}\langle P_{2}|$
$\displaystyle+2|b|^{2}|P_{1}\rangle_{B}\,{}_{B}\langle
P_{1}|+2|b|^{2}|P_{2}\rangle_{B}\,{}_{B}\langle P_{2}|$ $\displaystyle=$
$\displaystyle 2\mathbb{1}_{2}$

where $\mathbb{1}_{2}$ is the identity operator on ${\cal H}_{B}$.
Consequently we have

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\text{Tr}_{B}\Big{(}\sigma_{B,i}\big{(}p\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{2}\mathbb{1}_{2}\big{)}\Big{)}$
(33)

It is satisfying that after taking the diagonal matrix element, the maximally
mixed state in ${\cal H}$ has been replaced by the maximally mixed state in
${\cal H}_{B}$. Using the linearity of the trace operation and the fact that
the Pauli matrices are traceless, we have

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$ $\displaystyle
p\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$
(36) $\displaystyle+\frac{1-p}{2}\text{Tr}_{B}\left(\sigma_{B,i}\right)$
$\displaystyle=$ $\displaystyle
p\,\text{Tr}_{B}\left(\sigma_{B,i}\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$
(38) $\displaystyle=$ $\displaystyle pS_{i}$ (40)

where $S_{i}$ are the quantum Stokes parameters for the pure state without
isotropic noise. Therefore, the effect of isotropic noise on the state is to
multiply the Stokes parameters, $S_{i}$, by a constant factor $p$. The
Skyrmion number assumes normalization such that $\Sigma_{i}^{3}S_{i}^{2}=1$.
After correct normalization of the Stokes parameters this constant $p$ factor
does not contribute demonstrating that isotropic noise does not alter the
topology. This demonstrates that the skyrmionic topology of the state is
invariant to isotropic noise as long as a portion of the pure state survives
i.e., $p>0$ or equivalently $\gamma>\frac{1}{d^{2}}$.

##  III Supplementary: Rejection mechanism underlying projective measurements

To further clarify the result obtained in the previous section, we will now
argue that the invariance of the skyrmionic topology to isotropic noise can be
understood as noise rejection by projective measurements made on our state.
Towards this end we employ the spectral decomposition of the Pauli matrices,
$\sigma_{B,i}=\lambda^{+}_{i}P^{+}_{i}+\lambda^{-}_{i}P^{-}_{i}$ where
$P_{i}^{\pm}=|\lambda^{\pm}_{i}\rangle\langle\lambda^{\pm}_{i}|$ and the
eigenvalues are $\lambda^{\pm}_{i}=\pm 1$. Substituting this into Eqn. 40 we
find

$\displaystyle S_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\text{Tr}_{B}\left(P^{+}_{i}\left[p\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{2}\mathbb{1_{2}}\right]\right)$
(44)
$\displaystyle-\text{Tr}_{B}\left(P^{-}_{i}\left[p\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}+\frac{1-p}{2}\mathbb{1_{2}}\right]\right)$
$\displaystyle=$
$\displaystyle(I_{i,\text{pure}}^{+}+I^{+}_{i,\text{noise}})-(I_{i,\text{pure}}^{-}+I^{-}_{i,\text{noise}}),$
(46)

where
$I^{\pm}_{i,\text{pure}}=\text{Tr}_{B}\left(P^{\pm}_{i}p\,\,{}_{A}\langle\bar{r}|\Psi\rangle\langle\Psi|\bar{r}\rangle_{A}\right)$
and
$I^{\pm}_{i,\text{noise}}=\text{Tr}_{B}\left(P^{\pm}_{i}\frac{1-p}{2}\mathbb{1_{2}}\right)$.
Experimentally, we measure the quantities
$I_{i,\text{exp}}^{+}=I_{i,\text{pure}}^{+}+I^{+}_{i,\text{noise}}$ and
$I_{i,\text{exp}}^{-}=I_{i,\text{pure}}^{-}+I^{-}_{i,\text{noise}}$. Since the
eigenvalues of the Pauli matrices are non-degenerate we have that
$\text{Tr}\left(P^{+}_{i}\right)=\text{Tr}\left(P^{-}_{i}\right)=1$. Therefore
$I^{+}_{i,\text{noise}}=I^{-}_{i,\text{noise}}$, so that
$S_{i}^{\prime}=I_{i,\text{exp}}^{+}-I_{i,\text{exp}}^{-}=I_{i,\text{pure}}^{+}-I_{i,\text{pure}}^{+}$.
This shows that each pair of projective measurements required to calculate
each Pauli observable, receive identical noise contributions, which thus
cancels in their difference, explaining the noise rejection.

##  IV Supplementary: Experiment

A schematic of the experiment conducted is shown in Fig. 5. Entangled photon
pairs were generated through spontaneous parametric down-conversion (SPDC),
where a 355 nm wavelength, collimated Gaussian beam was sent through a 3 mm
long, Type-I Barium Borate (BBO) non-linear crystal (NC). A band-pass filter
(BPF) centred at 710 nm wavelength was used to filter out the unconverted pump
beam. From the SPDC process, the generated photons were correlated in OAM,
sharing the non-separable state,
$|\Psi\rangle=\Sigma_{\ell}c_{\ell}|\ell\rangle_{A}\otimes|-\ell\rangle_{B}$,
where the coefficients, $c_{\ell}$, determined the weightings for each
subspace that was spanned by the OAM eigenstates, $\ket{\pm\ell}$, for each
photon. The two entangled photons (photon A and B) were then spatially
separated using a 50:50 beam-splitter (BS). In order to prepare the desired
non-local skyrmionic state the initial OAM-OAM entangled state was mapped to
an arbitrary hybrid entangled state of the form

$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|\ell_{1}\rangle_{A}|H\rangle_{B}+|\ell_{2}\rangle_{A}|V\rangle_{B}\right),$
(48)

through the use of a digital spatial-to-polarization coupling (SPC) approach
Nape_2022  where the spatial information of photon B was coupled to
polarization. To achieve the desired arbitrary state given in Eq. 48, the OAM
DOF of photon B was coupled to orthogonal polarization states using a post-
selection of the desired OAM subspace with separate modulations by an SLM. The
transformations undergone by the state of photon B due to the entire SPC can
be broken down as follows (we ignore contributions from states which will not
couple into the fibre, i.e we only consider $\ket{m}=\ket{0}$)
$\ket{H,\ell^{{}^{\prime}}_{1}}+\ket{V,\ell^{{}^{\prime}}_{2}}\xrightarrow{\text{SLM}}\ket{H,\ell^{{}^{\prime}}_{1}+\ell_{1}}+\ket{V,\ell^{{}^{\prime}}_{2}}\xrightarrow{\text{QWP}}\ket{L,\ell^{{}^{\prime}}_{1}+\ell_{1}}+\ket{R,\ell^{{}^{\prime}}_{2}}\xrightarrow{\text{M}}\ket{R,\ell^{{}^{\prime}}_{1}+\ell_{1}}+\ket{L,\ell^{{}^{\prime}}_{2}}\xrightarrow{\text{QWP}}\ket{V,\ell^{{}^{\prime}}_{1}+\ell_{1}}+\ket{H,\ell^{{}^{\prime}}_{2}}\xrightarrow{\text{SLM}}\ket{V,\ell^{{}^{\prime}}_{1}+\ell_{1}}+\ket{H,\ell^{{}^{\prime}}_{2}+\ell_{2}}$,
where coupling into the fibre ensures that $\ell_{1}=-\ell^{{}^{\prime}}_{1}$
and $\ell_{2}=-\ell^{{}^{\prime}}_{2}$ thereby erasing the OAM information
from photon B as it becomes a separable DOF in photon B. It is clear then that
the desired OAM subspace, $\\{\ell_{1},\ell_{2}\\}$ is selected by displaying
those modes on the SLM. Following the state preparation, spatially separated
projective measurements were performed on both photons and they were observed
in coincidence allowing for the construction of a full quantum state
tomography of the biphoton state. Photon A was detected through a coupled
detection system consisting of an SLM and SMF coupled to an avalanche photon
detector (APD). Photon B was measured using a set of polarization optics, a
HWP orientated to $45^{\circ}$ and a linear polarizer (LP) orientated at
$90^{\circ}$.

Figure 5: Experimental setup for the generation and detection of non-local
skyrmionic states. Some abbreviations: mirror (M), non-linear crystal (NC),
lens (L), band-pass filter (BPF), 50:50 beam splitter (BS), half-wave plate
(HWP), spatial light modulator (SLM), quarter-wave plate (QWP), linear
polarizer (LP), avalanche photodiode (APD), coincidence counter (CC).

##  V Supplementary: Purity and Quantum contrast

The purity of the states generated within the experiment can directly be
controlled by controlling the quantum contrast. The discussion that follows is
based on the supplementary material given in reference zhu2021high .

Using the result given by Eq. 19 it is clear that the purity of a state is
dependent on the ratio between the signal and isotropic noise component of the
state, $p$. This can directly be attributed to the quantum contrast $Q_{c}$ of
the state, which gives the ratio between the accidental and two-photon
coincidences. This allows a relation between the probability $p$ and $Q_{c}$
as follows

$\displaystyle Q_{c}=\frac{1-p+pd}{1-p}\implies p=\frac{Q_{c}-1}{Q_{c}-1+d}$
(49)

where $d$ is the dimensionality of the state. Substituting Eq. 19 and solving
for $\gamma$ then gives

$\gamma=\frac{d\big{(}Q_{c}^{2}-2Q_{c}+2\big{)}+2(Q_{c}-1)}{d(d+Q_{c}-1)^{2}}.$
(50)

Experimentally a full quantum state tomography is performed, consisting of 36
projective measurements (for $d=2$) each with its own $Q_{c}$. When computing
the $Q_{c}$ for a state, the average $Q_{c}$ across all 36 measurements is
used. Furthermore, the $Q_{c}$ is directly computed from the accidental
coincidences and two photon coincidences according to

$\displaystyle Q_{c}=\frac{1}{T}\frac{C}{AB}$ (51)

where $C$ is the two photon coincidences and $N_{acc}=TAB$ is the accidental
coincidences with $T$ being the coincidence window and $A,B$ being the number
of single photon detection events detected by the APDs in the optical path of
photon A and B, respectively. With this in mind, the $Q_{c}$ value used in Eq.
50 is given by

$\displaystyle
Q_{c}=\frac{1}{36T}\sum_{i,j=1}^{6}\frac{C_{i.j}}{A_{i.j}B_{i.j}}$ (52)

where the indices $i,j$ reference a particular projective measurement used to
build the full QST.