Topological rejection of noise by quantum skyrmions: Supplementary Information

Pedro Ornelas,1 Isaac Nape,1 Robert de Mello Koch,2, 3 and Andrew Forbes1

1School of Physics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
2School of Science, Huzhou University, Huzhou 313000, China

3Mandelstam Institute for Theoretical Physics, School of Physics,
University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

I. 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. S1 (a)), can in general be written in the
form

|Ψ⟩ = 1√
2

(
|ℓ1⟩A |P1⟩B + eiδ |ℓ2⟩A |P2⟩B

)
, (S.1)

where ℓ1 and ℓ2 denote OAM of ℓ1ℏ and ℓ2ℏ per photon,
respectively, and |P1⟩ , |P2⟩ are orthogonal polarization
states, while δ allows for a relative phase between the two
components of the state vector. Photon A’s states live on
the Hilbert space HA spanned by the states {|ℓ1⟩ , |ℓ2⟩}
and photon B’s states live on the Hilbert space HB

spanned by the states {|P1⟩ , |P2⟩}. |Ψ⟩ belongs to the
Hilbert space H = HA⊗HB . For states with |ℓ1| ≠ |ℓ2| it
has been shown that the correlations between these pho-
tons form the desired skyrmionic mapping from S2 → S2

(equivalent to the mapping R2 → S2 through a stereo-
graphic projection of the spatial S2 to the real plane R2)
as depicted in Fig. S1 (b) [1]. Expressing photon A in
position basis, using |ℓ⟩ =

∫
R2 |LGℓ (r) |eiℓϕ|r⟩d2r where

LGℓ (r) are the Laguerre-Gaussian fields and |r⟩ are po-
sition states, we find

|Ψ⟩ =
∫
R2

|r⟩A (a(rA) |P1⟩B + b(rA) |P2⟩B) d
2rA, (S.2)

where a(rA) ≡ |LGℓ1 (rA) |, b(rA) ≡ eiΘ(ϕA)|LGℓ2 (rA) |,
Θ(ϕA) = ∆ℓϕA+δ and ∆ℓ = ℓ2−ℓ1. In this formulation,
it can be deduced that a spatial measurement on photon
A, collapses photon B into a particular polarization state
as shown in Fig. S1 (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 po-
larization state as depicted in Fig. S1 (c). This non-local
topology can be depicted compactly by combining the
spatial sphere and polarization states together as shown
in Fig. S1 (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 be cal-
culated using the equation N = 1

4π

∫
R2 Σz(x, y)dxdy [2]

where Σz(x, y) = ϵpqrSp
∂Sq

∂x
∂Sr

∂y and ϵpqr is the Levi-

Civita symbol. The integral in the Skyrmion number
calculation computes the surface area of photon B’s pa-
rameter space covered by the mapping defined by the

wavefunction given in Eqn. S.2. Thus, since the parame-
ter space is a unit sphere, dividing by the surface area of
the sphere (4π), we find the wrapping number N of the
non-local topology.

The quantum Stokes parameters, Si, are given by
the expectation values of the Pauli matrices, which
are calculated by taking the diagonal matrix element
at position r for photon A and the partial trace over
photon B, such that Si = TrB(⟨|r⟩A A⟨r| ⊗ σB,i⟩) =
TrB (σB,i A⟨r|Ψ⟩⟨Ψ|r⟩A). From this it can be shown
that the Skyrmion number depends on the difference
between ℓ1 and ℓ2, according to N = m∆ℓ with m =
sign (|ℓ1| − |ℓ2|)[2–5]. The value of ∆ℓ known as the vor-
ticity controls the magnitude of the Skyrmion number
and can be used to switch between Skyrmions (m = 1)
and anti-Skyrmions (m = −1). It should be noted that
while we are studying states with integer topological
numbers of mappings between spheres, the existence of
meron structures with half-integer topological numbers
have been studied in classical vectorial fields [6, 7] al-
beit these structures are strictly not skyrmions as they
do not arise from maps between spheres. Furthermore,
while the chosen polarization basis and the helicity, δ,
play no role in determining the topology of the state it
can be used to switch between different textures. Con-
ventionally, Skyrmions are defined in the |R⟩ , |L⟩ polar-
ization basis, Sz = S3, with Bimerons defined under an
arbitrary choice of polarization basis [5] (see Ref. [8] for
a discussion on the topic). However, we are free to per-
form a rotation of the Poincaré sphere which does not
alter the topology of our states such that we can instead
consider Skyrmions defined in the |H⟩ , |V ⟩ polarization
basis, Sz = S1, following the convention we used in our
previous work [1]. Furthermore, we can also switch be-
tween different textures by changing δ, for example if
N = 1 and δ = 0, π we have a Neél-type Skyrmion, and
if δ = ±π

2 we have a Bloch-type Skyrmion. We note that
the state could have been expressed in a different basis,
such as momentum. However, this would incur a signif-
icant computational penalty, as the Fourier transforms
required implies that our Skyrmion number calculation
would become a complicated double convolution of the
Fourier transform of the Stokes parameter. Therefore
the continuous position basis is not only more intuitive,
it is computationally less cumbersome.



2

FIG. S1: (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 projec-
tion, 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 em-
bedded 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. QUANTUM TOPOLOGICAL INVARIANCE
TO ISOTROPIC NOISE

We now consider subjecting our state, |Ψ⟩, to environ-
mental/ “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

ρ = p |Ψ⟩ ⟨Ψ|+ 1− p

d2
1d2 , (S.3)

where d = 2, p ∈ [0, 1] is a parameter controlling the
degree of purity of the state with p = 1 giving a pure

state (|Ψ⟩⟨Ψ|) and p = 0 giving a maximally mixed state
( 1
d2 1d2) and 1d2 , the d2 × d2 identity matrix, is the iden-

tity operator on H. The purity, γ, of a state is given by
Tr(ρ2). It follows that γ is related to p according to

γ = Tr

((
p |Ψ⟩ ⟨Ψ|+ 1− p

d2
1d2

)2
)

= Tr

(
p2 |Ψ⟩ ⟨Ψ|+ 2p

(1− p)

d2
|Ψ⟩ ⟨Ψ|+ (1− p)2

d4
1d2

)
= p2 + 2p

(1− p)

d2
+

(1− p)2

d2

= p2 +
1− p2

d2
. (S.4)

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

S′
i = TrB (σB,i A⟨r|ρ|r⟩A)

= TrB

(
σB,i

(
pA⟨r|Ψ⟩⟨Ψ|r⟩A

+
1− p

4
A⟨r|14|r⟩A

))
(S.5)

It should be noted that without loss of generality we
have chosen to restrict ourselves to the original 2D OAM
hilbert space of photon A, by projecting onto only that
hilbert space. This is because isotropic noise treats ev-
ery OAM state in the exact same way (mixing it with
the identity), thus we are justified in only considering
states within which we start with a non-zero signal con-
tribution. To compute the diagonal matrix element of
14, start by noting that we have

a(r) = ⟨r|ℓ1⟩ b(r) = ⟨r|ℓ2⟩ (S.6)

which follow by comparing (S.1) and (S.2). We note that
we have ignored the factor 1√

2
in (S.6) since to obtain cor-

rectly normalized quantum Stokes parameters, we must
choose

|a|2 + |b|2 = 1 (S.7)

at all r. Using

14 = |ℓ1⟩A|P1⟩B A⟨ℓ1|B⟨P1|+ |ℓ1⟩A|P2⟩B A⟨ℓ1|B⟨P2|

+|ℓ2⟩A|P1⟩B A⟨ℓ2|B⟨P1|+ |ℓ2⟩A|P2⟩B A⟨ℓ2|B⟨P2|

we easily find (use (S.6) and (S.7))

⟨r|14|r⟩ = 2|a|2|P1⟩B B⟨P1|+ 2|a|2|P2⟩B B⟨P2|

+2|b|2|P1⟩B B⟨P1|+ 2|b|2|P2⟩B B⟨P2|

= 212

where 12 is the identity operator on HB and a ≡ a(r)
and b ≡ b(r) . Consequently we have

S′
i = TrB

(
σB,i

(
pA⟨r|Ψ⟩⟨Ψ|r⟩A +

1− p

2
12

))
(S.8)



3

It is satisfying that after taking the diagonal matrix ele-
ment, the maximally mixed state in H has been replaced
by the maximally mixed state in HB . Using the linearity
of the trace operation and the fact that the Pauli matri-
ces are traceless, we have

S′
i = pTrB (σB,i A⟨r|Ψ⟩⟨Ψ|r⟩A)

+
1− p

2
TrB (σB,i)

= pTrB (σB,i A⟨r|Ψ⟩⟨Ψ|r⟩A)

= pSi (S.9)

where Si 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 pa-
rameters, Si, by a constant factor p. The Skyrmion num-
ber assumes normalization such that Σ3

iS
2
i = 1. After

correct normalization of the Stokes parameters this con-
stant p factor does not contribute demonstrating that
isotropic noise does not alter the topology. This demon-
strates that the skyrmionic topology of the state is in-
variant to isotropic noise as long as a portion of the pure
state survives i.e., p > 0 or equivalently γ > 1

d2 .

III. 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 de-
composition of the Pauli matrices, σB,i = λ+

i P
+
i +λ−

i P
−
i

where P±
i = |λ±

i ⟩⟨λ
±
i | and the eigenvalues are λ±

i = ±1.
Substituting this into Eqn. S.9 we find

S′
i = TrB

(
P+
i

[
p A⟨r|Ψ⟩⟨Ψ|r⟩A +

1− p

2
12

])

−TrB

(
P−
i

[
p A⟨r|Ψ⟩⟨Ψ|r⟩A +

1− p

2
12

])
= (I+i,pure + I+i,noise)− (I−i,pure + I−i,noise),

(S.10)

where I±i,pure = TrB
(
P±
i p A⟨r|Ψ⟩⟨Ψ|r⟩A

)
and I±i,noise =

TrB
(
P±
i

1−p
2 12

)
. Experimentally, we measure the quan-

tities I+i,exp = I+i,pure+I+i,noise and I−i,exp = I−i,pure+I−i,noise.
Since the eigenvalues of the Pauli matrices are non-
degenerate we have that Tr

(
P+
i

)
= Tr

(
P−
i

)
= 1. There-

fore I+i,noise = I−i,noise, so that S′
i = I+i,exp − I−i,exp =

I+i,pure − I+i,pure. This shows that each pair of projec-
tive measurements required to calculate each Pauli ob-
servable, receive identical noise contributions, which thus
cancels in their difference, explaining the noise rejection.

IV. EXPERIMENT

A schematic of the experiment conducted is shown in
Fig. S2. 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 pho-
tons were correlated in OAM, sharing the non-separable
state, |Ψ⟩ = Σℓcℓ|ℓ⟩A ⊗ | − ℓ⟩B , where the coefficients,
cℓ, determined the weightings for each subspace that was
spanned by the OAM eigenstates, |±ℓ⟩, 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

|Ψ⟩ = 1√
2
(|ℓ1⟩A|H⟩B + |ℓ2⟩A|V ⟩B) , (S.11)

through the use of a digital spatial-to-polarization cou-
pling (SPC) approach [9] where the spatial information of
photon B was coupled to polarization. To achieve the de-
sired arbitrary state given in Eq. S.11, 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 contri-
butions from states which will not couple into the fibre,

i.e we only consider |m⟩ = |0⟩) |H, ℓ
′

1⟩ + |V, ℓ′2⟩
SLM−−−→

|H, ℓ
′

1 + ℓ1⟩ + |V, ℓ′2⟩
QWP−−−→ |L, ℓ′1 + ℓ1⟩ + |R, ℓ

′

2⟩
M−→

|R, ℓ
′

1 + ℓ1⟩ + |L, ℓ′2⟩
QWP−−−→ |V, ℓ′1 + ℓ1⟩ + |H, ℓ

′

2⟩
SLM−−−→

|V, ℓ′1 + ℓ1⟩ + |H, ℓ
′

2 + ℓ2⟩, where coupling into the fibre

ensures that ℓ1 = −ℓ
′

1 and ℓ2 = −ℓ
′

2 thereby erasing
the OAM information from photon B as it becomes a
separable DOF in photon B. It is clear then that the de-
sired OAM subspace, {ℓ1, ℓ2} is selected by displaying
those modes on the SLM. Following the state prepara-
tion, spatially separated projective measurements were
performed on both photons and they were observed in
coincidence allowing for the construction of a full quan-
tum state tomography of the biphoton state. Photon A
was detected through a coupled detection system consist-
ing 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◦ and a linear
polarizer (LP) orientated at 90◦.
To identify the highest-order topological state achiev-

able in the experiment, we must first identify the highest
OAM state achievable. We conducted an OAM spiral
bandwidth measurement, which yielded a Schmidt mode
number ofK = 11. This result indicates that the highest-
order OAM state that can be reliably measured in the ex-
periment is ℓ = ±5. However, due to losses introduced by



4

the optics in the SPC technique this was further reduced
to ℓ = ±3. We have provided experimental evidence for
the resilience of states with topological number up to
N = 3 (using OAM basis states |0⟩, |ℓ⟩, ℓ ̸= 0 with OAM
values from -3 to 3), which means that we were able to
produce 6 states with distinct topological features. That
being said it would be possible within the current con-
figuration to exceed |N | = 3 as we can use the OAM
values ℓ = −2, 3 and ℓ = 2,−3 to reach an upper value of
|N | = 5. Beyond this higher order topological numbers
can be achieved by improving the efficiency of the SPDC
process, the SPC and the detection of higher-order OAM
values.

Lastly, to vary the degree of isotropic noise in the ex-
periment, a variable white light source was introduced
whilst keeping the configuration of the rest of the ex-
periment the same as discussed above. The white light
source was characterized (See Fig. 4 in the main text) on
the basis of how it affected the quantum contrast or state
purity of the state (discussed in more detail in the next
section). Therefore, we expect our model and experiment
to remain agnostic to the origin of the noise whether that
be the entanglement source, the detectors or the channel.

V. PURITY AND QUANTUM CONTRAST

The purity of the states generated within the experi-
ment can directly be controlled by controlling the quan-
tum contrast. The discussion that follows is based on the
supplementary material given in reference [10].

Using the result given by Eq. S.4 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 Qc of
the state, which gives the ratio between the accidental
and two-photon coincidences. This allows a relation be-
tween the probability p and Qc as follows

Qc =
1− p+ pd

1− p
=⇒ p =

Qc − 1

Qc − 1 + d
(S.12)

where d is the dimensionality of the state. Substituting
Eq. S.4 and solving for γ then gives

γ =
d
(
Q2

c − 2Qc + 2
)
+ 2(Qc − 1)

d(d+Qc − 1)2
. (S.13)

Experimentally a full quantum state tomography is per-
formed, consisting of 36 projective measurements (for

d = 2) each with its own Qc. When computing the Qc

for a state, the average Qc across all 36 measurements
is used. Furthermore, the Qc is directly computed from
the accidental coincidences and two photon coincidences
according to

Qc =
1

T

C

AB
(S.14)

where C is the two photon coincidences and Nacc = TAB
is the accidental coincidences with T being the coinci-
dence window and A,B being the number of single pho-
ton detection events detected by the APDs in the optical
path of photon A and B, respectively. With this in mind,
the Qc value used in Eq. S.13 is given by

Qc =
1

36T

6∑
i,j=1

Ci.j

Ai.jBi.j
(S.15)

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

VI. CONCURRENCE AND FIDELITY

Beyond purity, quantities such as Concurrence and Fi-
delity also serve as entanglement witnesses for our states.

The fidelity was used to analytically compare our mea-
sured partially mixed states, ρ against an initial pure
state ρT = |Ψ⟩⟨Ψ|

F =

(
Tr

(√√
ρT ρ

√
ρT

))2

, (S.16)

The fidelity is 0 if the states are not identical or 1
when they are identical up to a global phase. However,
since ρ is a partially mixed state, we find that when ρ
becomes completely mixed, that is ρ = 1

41d2 , then F = 1
4 .

The concurrence was used to measure the degree of
entanglement between the hybrid entangled photons. It
was measured from

C(ρ) = max{0, λ1 − λ2 − λ3 − λ4}, (S.17)

where λi are eigenvalues of the operator R =
Tr
(√√

ρρ̃
√
ρ
)
in descending order and ρ̃ = σy⊗σyρ

∗σy⊗
σy. The concurrence ranges from 0 for separable and
completely mixed states to 1 for entangled states.

[1] 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.

[2] 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.



5

FIG. S2: 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).

[3] 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.

[4] Y. Shen, E. C. Mart́ınez, and C. Rosales-Guzmán, “Gen-
eration of optical skyrmions with tunable topological tex-
tures,” ACS Photonics, vol. 9, no. 1, pp. 296–303, 2022.

[5] Y. Shen, “Topological bimeronic beams,” Optics Letters,
vol. 46, no. 15, pp. 3737–3740, 2021.

[6] D. Marco, I. Herrera, S. Brasselet, and M. A. Alonso,
“Propagation-invariant optical meron lattices,” ACS
Photonics, 2024.

[7] K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Sin-
gular polarimetry: Evolution of polarization singulari-
ties in electromagnetic waves propagating in a weakly

anisotropic medium,” Optics express, vol. 16, no. 2,
pp. 695–709, 2008.

[8] J. Chen, A. Forbes, and C.-W. Qiu, “More than just
a name? from magnetic to optical skyrmions and the
topology of light,” Light: Science & Applications, vol. 14,
no. 1, p. 28, 2025.

[9] 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.

[10] 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.


	Formulation of non-local quantum skyrmion
	Quantum topological invariance to isotropic noise
	Rejection mechanism underlying projective measurements
	Experiment
	Purity and Quantum contrast
	Concurrence and Fidelity
	References