Non-local skyrmions as topologically resilient quantum entangled states of light

It may state two separate claims as one.

The reduced state of photon A is an incoherent mixture of OAM, and the reduced state of photon B is unpolarised.

“The reduced state of photon A is an incoherent mixture of OAM, while that of photon B is unpolarised, so that individually neither has any salient topological structure.”

The quote only partly states this claim.

The experimentally generated state for l1=1, l2=0, γ=0 revealed a fidelity of F=95.0% when compared with the expected pure theoretical state.

“When compared with the expected pure theoretical state, the experimentally generated state revealed a fidelity of F = 95.0%.”

It may state two separate claims as one.

The experimental quantum skyrme number for the first realisation of a non-local quantum entangled skyrmion is N_exp = 0.972, close to the theoretical value N_th = 1.

“An experimental quantum skyrme number of N exp = 0.972 which is very close to the theoretical value of N th = 1, for the first realisation of a non-local quantum entangled skyrmion.”

It may state two separate claims as one.

The non-local quantum skyrme number N can be tailored as desired and can be any non-zero integer.

“While the topological structure of each individual photon is always trivial and with zero skyrme number, the non-local skyrme number (we call it the quantum skyrme number) denoted by N , can be tailored as desired, with the skyrmion existing in the entanglement between the photons themselves.”

• Also stated in: § Discussion and Conclusion · plain

  “The topologically non-trivial twophoton entangled state has a skyrme number that can be any non-zero integer, and this topological structure is not derived from either local photon state, but rather, from the entanglement between them.”

It may state two separate claims as one.

The experimental probability density of finding a photon in space is given by the spatial structure of photon A, and the experimentally reconstructed skyrmion is a spin textured field with a defined topology.

“The experimental probability density of finding a photon in space, given by the spatial structure of photon A, and the experimentally reconstructed skyrmion as a spin textured field with a defined topology.”

It may state two separate claims as one.

When l2 = -l1, no quantum skyrmion is produced and N = 0.

“Notice that when ℓ 2 = -ℓ 1 i.e., opposite twists to the OAM components, as would be created in typical spinorbit hybrid entanglement experiments [39], no quantum skyrmion is produced and N = 0.”

• cites [39] Blanco-Redondo, A., Bell, B., Oren, D., Eggleton, B. J. & Segev, M. Topological protection of biphoton states. Science 362, 568–571 (2018). sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Probably not a substantial claim.

A quantum state tomography was performed on the selected entangled state |Ψ⟩ = 1/√2(|1_A⟩|H_B⟩ + |0_A⟩|V_B⟩) by spatial measurements on photon A and polarisation measurements on photon B, with coincidences collected for all outcomes.

“A quantum state tomography was performed on the selected entangled state, |Ψ = 1 √ 2 (|1 A|H B + |0 A|V B ) by spatial measurements on photon A (columns) and polarisation measurements on photon B (rows), with coincidences collected for all outcomes shown as false colour from low (black) to high (white).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

An example QST and associated density matrix was derived from data captured for the state |Ψ⟩ = |3_A⟩|H_B⟩ + |0_A⟩|V_B⟩.

“An example of a generated QST and associated density matrix is shown in figure 9, derived from data captured for the state |Ψ = |3 A |H B + |0 A |V B .”

It may state two separate claims as one.

Entangled states where |l1| = |l2| have N = 0 and map to rings on S^2 with close to zero coverage and are topologically equivalent.

“Entangled States where |ℓ1| = |ℓ2| have N = 0 and map to rings on S 2 with close to zero coverage and are topologically equivalent.”

It may state two separate claims as one.

In Figure 4a, the topologically trivial class is mapped to a ring, and its orientation can easily be discerned; the experimental reconstruction deviates from the undistorted prediction.

“since the topologically trivial class is mapped to a ring, as shown in Figure 4a, its orientation can easily be discerned.Indeed, in Figure 4a we can see the deviation from the undistorted prediction of the ring (solid line) and the experimental reconstruction (data points), the latter always with some inherent perturbation.”

Probably not a substantial claim.

Experimental skyrme numbers for the example states were measured.

“Experimental skyrme numbers for example states.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The insets show the probability of detection as measured and calculated from photon A.

“The insets show the probability of detection as measured (Exp) and calculated (Th) from photon A.”

Probably not a substantial claim.

Table I reports the measured experimental skyrme number, fidelity, and concurrence for each target Skyrmion topology {N, φ}.

“TABLE I: Table of experimental data.The Skyrmion topology, N , and type φ is given in the form {N, φ} along with the associated measured values for the experimental skyrme number, Nexp, the fidelities of each reconstructed density matrix, F , and their concurrences.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Experimental fidelities for the example states were measured when compared to theoretical maximally entangled pure states.

“Experimental fidelities for the example states when compared to theoretical maximally entangled pure states.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The coverage on the Poincaré Sphere was reconstructed, with experimental data shown as blue dots.

“Reconstruction of the coverage on the Poincaré Sphere, with experimental data shown as blue dots.”

It may state two separate claims as one.

As entanglement asymptotically approaches vanishing, the topology abruptly changes from non-trivial to trivial, showing that skyrmionic topology is a shared property of the entangled photons and necessarily vanishes once there is no entanglement.

“We see from Figure 5a that as we asymptotically approach vanishing entanglement, so the topology abruptly changes from non-trivial (N = 0) to trivial (N = 0), emphasising that the skyrmionic topology is a shared property of the entangled photons, which necessarily vanishes once there is no entanglement.”

It may state two separate claims as one.

For N=1 and N=3, the quantum skyrme number remains intact in the presence of decaying entanglement and falls off to zero only when the entanglement itself vanishes.

“The results for N = 1 and N = 3 reveal a new form of topological resilience, with the quantum skyrme number intact in the presence of decaying entanglement, satisfying (1) and (3), falling off to zero only when the entanglement itself vanishes, satisfying (2).”

It may state two separate claims as one.

The non-local skyrmionic configuration is formed in the entanglement between the two photons, with photon A carrying the spatial degree of freedom expressed on an OAM Bloch Sphere and photon B carrying the polarization degree of freedom expressed on the polarisation Bloch Sphere.

“The non-local skyrmionic configuration is formed in the entanglement between the two photons, with b, photon A carrying the spatial degree of freedom (DoF) expressed on an OAM Bloch Sphere, and c, photon B carrying the polarization DoF expressed on the polarisation Bloch Sphere.”

It may state two separate claims as one.

The paper demonstrated experimental control over the topology of two-photon entangled states showing exotic topologies N ∈ {-3, -1, 0, 1, 3} and textures (anti-, Neel- and Bloch types).

“We demonstrated experimental control over the topology of our two-photon entangled states showing exotic topologies (N ∈ {-3, -1, 0, 1, 3}) and textures (anti-, Neel-and Bloch types).”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Example states traverse the skyrmionic landscape from anti-skyrmions (N < 0) through non-skyrmionic states (N = 0) to Bloch-type (N = 1, γ = π/2), Neel-type (N = 1, γ = 0), and higher-order skyrmions (N > 1).

“Example states are shown graphically and labelled with numbers 1 through 10, traversing the skyrmionic landscape from anti-skyrmions (N < 0) through non-skyrmionic states (N = 0) to Bloch-type (N = 1, γ = π/2), Neel-type (N = 1, γ = 0) and higherorder skyrmions N > 1.”

It may state two separate claims as one.

For N = 3, the non-local field has three segments, each with a full coverage of S^2, shown as red, green and blue experimental dots; the entire state maps to S^2 with three windings for a coverage of 299%, close to 300%.

“For N = 3, the non-local field has three segments, each with a full coverage of S 2 , shown as red, green and blue experimental dots.The entire state maps to S 2 with three windings for a coverage of 299%, close to 300%, illustrated in the composite sphere (right) with all three experimental data sets.”

Probably not a substantial claim. It may state two separate claims as one.

A stereographic mapping of the experimental data shows spatial information and probability of detection, with the skyrmionic state depicted by the direction of the arrows and their location, a direct outcome from the joint state of photons A and B.

“A stereographic mapping of the experimental data showing spatial information (coverage) and probability of detection (colour), with the skyrmionic state depicted by the direction of the arrows and their location, a direct outcome from the joint state of photons A and B.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Examples of experimental stereographic projections were obtained.

“Examples of experimental stereographic projections.”

It may state two separate claims as one.

For N = -1, the wavefunction has a characteristic hyperbolic texture; for Neel and Bloch type skyrmions, characteristic hedgehog and spiral textures are seen.

“For N = -1, we see the characteristic hyperbolic texture embedded into the wavefunction, and for the Neel and Bloch type skyrmions we see the characteristic hedgehog and spiral textures.”

It may state two separate claims as one.

α = 0.5 when M = 0, and α = 1 when M = 1.

“Since the induced loss was to the vertically polarized term in equation 26, we have that α = 0.5(1) when M = 0(1).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Measured quantum Stokes projections were obtained directly from the tomography.

“Measured quantum Stokes projections directly from the tomography.”

It may state two separate claims as one.

From the simplified expression, N ∈ Z and N = m∆l, with m = 0 if |l1| = |l2|, m = 1 if |l1| > |l2|, and m = -1 if |l1| < |l2|.

“From this expression, one can see that N ∈ Z as expected and N = m∆ℓ with m =      0, |ℓ 1 | = |ℓ 2 | 1, |ℓ 1 | > |ℓ 2 | -1, |ℓ 1 | < |ℓ 2 |.”

It may state two separate claims as one.

By varying the grating depth, the degree of entanglement can be varied, with maximal entanglement when the grating is switched off and no entanglement when the grating is switched on with maximum depth.

“This establishes a relation between α and the grating depth, more specifically by varying the grating depth, we are able to vary the degree of entanglement, with maximal entanglement obtained when the grating is switched off, and no entanglement when the grating is switched on with a maximum depth.”

• Also stated in: § VIII. SUPPLEMENTARY: EXPERIMENTAL DATA · plain

  “To control the degree of entanglement, a grating is placed on one of the halves of the SLM in the SPC conversion with the grating depth controlling the level of entanglement, from no grating leading to maximal entanglement and maximum grating depth leading to no entanglement.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For N = 1, the coverage is close to 100%.

“In b the coverage is close to 100% for N = 1.”

It may state two separate claims as one.

All entangled states with N = 0 map to rings on S2 and are topologically equivalent, as confirmed by results in Figure 4a.

“All entangled states with N = 0 map to rings on S 2 and hence are topologically equivalent, with the results in Figure 4a confirming the mapping.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Example skyrmion topologies with varying textures are shown as experimental and theoretical reconstructions.

“Example skyrmion topologies with varying textures shown as experimental (Exp) and theoretical (Th) reconstructions.”

It may state two separate claims as one.

The entanglement between the photons is skyrmionic, as denoted by the vectorial arrows in space, which can be mapped stereographically to an abstract sphere that holds spatial information and the skyrmionic state.

“The entanglement between the photons is now skyrmionic, as denoted by the vectorial arrows in space, which can be mapped stereographically to an abstract sphere that holds spatial information (coverage) and the skyrmionic state (arrows and location).”

Probably not a substantial claim.

Magnification of the beam incident on the nonlinear crystal is required to flatten the spiral bandwidth of the crystal, thereby allowing access to higher-order OAM modes.

“Magnfication of the beam incident on the NC is required to flatten the spiral bandwidth of the NC thereby allowing access to higher order Orbital Angular Momentum (OAM) modes [44,45].”

• cites [44] Forbes, A. & Nape, I. Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light. AVS Quantum Sci. 1, 011701 (2019). sourceNot checked — no full text of the source
• cites [45] Hatcher, A. Algebraic Topology (Cambridge Univ. Press, 2002).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

A unitary rotation on the Stokes parameters has no effect on the density matrix and can be described as a change of coordinates that is a smooth deformation preserving the skyrme number, implying the topology remains unchanged.

“It can be shown that such a unitary operation has no effect on the density matrix and furthermore the rotation can be described as a change of coordinates (x, y, z) → (x , y , z ) which topologically speaking is a smooth deformation which we have shown preserves the skyrme number implying that the topology remains unchanged.”

It may state two separate claims as one.

The space of continuous maps from S2 to S2 is partitioned into equivalence classes, where maps in the same class can be continuously deformed into each other and maps in distinct classes cannot.

“An important corollary of the existence of a topological invariant is that the space of continuous maps is partitioned into equivalence classes [40].Any two maps in the same class can be continuously deformed into each other, while this is not possible for maps in distinct classes [40].”

• cites [40] Parmee, C. D., Dennis, M. R. & Ruostekoski, J. Optical excitations of skyrmions, knotted solitons, and defects in atoms. Commun. Phys. 5, 54 (2022). sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Combining the stereographic projection and Poincaré vector representation schemes allows the entirety of the topological information to be compactly represented on the surface of a sphere.

“Combining these two representation schemes, we can compactly represent the entirety of the topological information on the surface of a sphere, an example of which is shown in Figure 10c.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

For any biphoton state with |N| > 1, the same coordinate transformation is valid, except that the coordinate transformation follows a slightly different rule depending on |l2| - |l1|.

“For any biphoton state with |N | > 1, the same coordinate transformation discussed above is valid, except that the coordinate tranformation follows a slightly different rule which depends on |ℓ2| -|ℓ1|.”

Probably not a substantial claim. It may state two separate claims as one.

Every map from S2 to S2 has an integer degree N, which is a topological invariant unchanged by continuous deformation.

“Every map from S 2 to S 2 has an integer degree, N , measuring how many times the first sphere wraps the second.This degree is a topological invariant: it is unchanged by continuous deformation of the map.”

Probably not a substantial claim. It may state two separate claims as one.

Extracting the Quantum Stokes Parameters from the entangled state allows description of a spatially dependent vector S = (Sx(r,Θ), Sy(r,Θ), Sz(r,Θ)) with local normalization S•S = 1, describing a vector field whose vectors point from the centre of the Poincaré sphere to its surface and denote particular polarization states.

“Extracting the Quantum Stokes Parameters from the entangled state, allows for the description of a spatially dependent vector S = (S x (r, Θ), S y (r, Θ), S z (r, Θ)) which carries the local normalization condition S • S = 1, thus describing a vector field with vectors pointing from the centre of the Poincaré sphere to its surface and therefore denoting particular polarization states.”

It may state two separate claims as one.

Only the topologically trivial N=0 class of two-photon wavefunctions can have vanishing entanglement; wavefunctions belonging to any other class are always entangled.

“There is a quantitative relationship between the two: it is only the topologically trivial N = 0 class of two photon wavefunctions that can have vanishing entanglement.Wavefunctions belonging to any other class are always entangled.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

One is free to scale r and Θ by arbitrary real constants a, b^2 > 0 to restructure the representation so as to emphasize regions of interest.

“Beyond the inherent compactness of this representation, one is free to scale r and Θ by arbitrary real constants a, b 2 > 0 which allow for a simple scheme to restructure the representation so as to emphasize regions of interest.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The small deviation in coverage occurs at the pole of the mapped sphere, a natural artefact of mapping from R2 to S2 where data at spatially infinite distances cannot be recovered.

“The small deviation can be seen at the pole of of the mapped sphere, a natural artefact of mapping from R 2 to S 2 , where data at spatially infinite distances cannot be recovered.”

It may state two separate claims as one.

Full control over the quantum wavefunction allows manipulation of the texture and precise topology of the non-local skyrmion, thereby giving access to a plethora of topological structures.

“Full control over the quantum wavefunction allows for the manipulation of the texture and precise topology of the non-local skyrmion thereby giving access to a plethora topological structures.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The validity of restructuring the stereographic representation by scaling follows from the topology being invariant to smooth deformations caused by coordinate changes, and any scaling can be reversed through a second inverse scaling.

“The validity of such a restructuring of the representation follows from the fact that the topology is invariant to smooth deformations caused by coordinate changes, and in fact, any scaling can be reversed through a second inverse scaling.”

It may state two separate claims as one.

Any topologically non-trivial class enjoys resilience to perturbation, while the topologically trivial class does not.

“Further, any topologically non-trivial (N = 0) class enjoys this resilience, while the topologically trivial class does not.”