Simultaneous entanglement swapping of multiple orbital angular momentum states of light

High-bit-rate long-distance quantum communication is a proposed technology for future communication networks and relies on high-dimensional quantum entanglement as a core resource. While it is known that spatial modes of light provide an avenue for high-dimensional entanglement, the ability to transport such quantum states robustly over long distances remains challenging. To overcome this, entanglement swapping may be used to generate remote quantum correlations between particles that have not interacted; this is the core ingredient of a quantum repeater, akin to repeaters in optical fibre networks. Here we demonstrate entanglement swapping of multiple orbital angular momentum states of light. Our approach does not distinguish between different anti-symmetric states, and thus entanglement swapping occurs for several thousand pairs of spatial light modes simultaneously. This work represents the first step towards a quantum network for high-dimensional entangled states and provides a test bed for fundamental tests of quantum science.


Supplementary Note 1: Entanglement swapping in high dimensions
The state of the two photon pairs produced by spontaneous parametric downconversion (SPDC) in paths A, B and C, D, respectively, can be written as where c n represents complex coefficients, we use the notation n := −n and denote symmetric and antisymmetric Bell states with a plus and a minus sign, respectively. The photons in path B and path C are subjected to a 50 : 50 beam splitter which superposes the two beams. The inversion of the helicity of the OAM modes, → − , upon reflection in the beam spitter is compensated by two additional reflections employing a mirror in path B before and another one in path C behind the beam splitter. The action of the beam splitter, combined with both mirrors, is thus characterised by the transformation rules where we denote the original before the beam splitter and the reflected path thereafter by the same letter; accordingly a photon in the input port B exits in path B upon reflection and in path C upon transmission. Using these transformation rules on the input state in Supplementary Equation (1), and under the condition that each output path of the beam splitter contains a single photon, we obtain a state after the beam splitter that only consists of antisymmetric photon pairs. For d-dimensions, one can express it as where K is a normalization constant that compensates for the loss of the terms with two photons in the same output path of the beam splitter and d = 1 + 2N . In the case where we only consider = ±1, ±2, we have Simultaneous detection of a single photon in each of the two output ports of a symmetric beam splitter causes a projection onto the antisymmetric component of the input state. The dimension of the corresponding antisymmetric state space is given by the number of ways in which the OAM values of the input space can be combined into pairs of 's to form an antisymmetric state Ψ − . For example, four OAM values would give six antisymmetric basis states, featuring in the BC-components of the state in Supplementary Equation (4). On the other hand, the antisymmetric basis states also feature as components of the photon pair in AD, because the OAM must sum to zero in each term of the state. In general, considering d OAM levels, we can produce a state of the form given in Supplementary Equation (4) with our setup consisting of d(d − 1)/2 antisymmetric basis states that involve both photon pairs.
Note that the state expressed in Supplementary Equation (4) represents the Schmidt decomposition of an entangled state, i.e., it has the form where the Schmidt bases are photon pairs in AD and BC, respectively. So, apart from the entanglement among the different pairs, there is also the maximal entanglement within the pairs between the single photons. The detection of the photons in paths B and C without measurement of their OAM values results in a statistical mixture of the antisymmetric states as in Supplementary Equation (4). By tracing over the OAM degrees of freedom of the photons in paths B and C, we obtain for d dimensions Restricted to = ±1, ±2, the result reduces to The projection onto the antisymmetric space of the photons in B and C transfers entanglement between the systems in A and B to the remote systems in A and D, which were not entangled before. This constitutes entanglement swapping.

Supplementary Note 2: Pure final state
We note that, by using a filter in paths A and D that projects onto any two-dimensional subspace with OAM values { , }, one obtains an antisymmetric state |Ψ − , AD , which is maximally entangled. Such a filter in front of the detectors in paths B and C could be used to prepare a particular antisymmetric state remotely in paths A and D. A similar procedure could be exploited for various purposes of quantum communication between three or four parties, such as secure bit commitment or QKD protocols.
By means of particular filters for photons in BC, it is also possible to obtain a pure state with a multitude of entangled levels instead of a mixture in AD. For example, projecting on a superposition of singlet states Ψ − nn in BC results, as shown below, in a superposition of such states in AD. According to Supplementary Equation (4), the state of photons in BC after the beamsplitter reads with |n ≡ Ψ − nn and α n ≡ Kc 2 n , where components Ψ − n,m with different OAM values n = m are not mentioned explicitly. A filter in BC projecting onto the state |x ≡ ( N n=1 |n )/ √ N leads to The resulting state of the photons in AD, N n=1α n |n ≡ α n Ψ − nn (with normalised coefficientsα n = c 2 n / n |c 2 n | 2 ), is a pure entangled state of Schmidt rank N . Such a filter could be realised, e.g., by parametric up-conversion of the photon pair in AD to a photon of double the frequency (the inverse process to SPDC) and subsequent measurement of its OAM, conditioning on the OAM value = 0.

Supplementary Note 3: Background subtraction
In our experiment, we use spatial light modulators (SLMs) and single-mode fibres to detect photons in paths A and D and multi-mode fibres to detect photons in paths B and C. The use of multi-mode fibres is so that we detect within the 4-dimensional space spanning the modes { = −2, = −1, = +1, = +2} in an unrestricted fashion, i.e. we do not select out a subspace in paths B and C. The use of additional SLMs and single-mode fibres, rather than multi-mode fibres, would enable a choice of which particular space we detect in, but it would also remove the ability to observe multiple subspaces at once to obtain a high-dimensional state.
A consequence of the multi-mode fibres is, however, that the rate of the single-photon detection events at detectors B and C is significantly higher than that recorded at detectors A and D. Supplementary Table I provides representative single-channel and coincidence count rates recorded for the = ±1 and = ±2 subspaces. The B and C single-channel rates recorded with the multi-mode fibres are on average ≈4 times higher than those recorded at A and D. This trend is observed across all the subspaces that we investigate.
The large single-channel rates resulting from the multi-mode fibres contribute to unwanted 4-way coincidence counts that do not participate in any entanglement swapping. However, these unwanted 4-way coincidences that arise from uncorrelated photon detection events (Eq. (12) in the main paper) can be calculated and subtracted off the measured 4-way counts. We can use either the raw counts or the background-subtracted counts to calculate density matrices from which fidelities and concurrences are extracted. Supplementary Figure 3 and Supplementary Table II provide evidence of the impact of the background subtraction on the quality of our entanglement swapping. As can be seen, a higher fidelity and concurrence (0.80 ± 0.10 and 0.68 ± 0.18) is observed for the density matrix generated using the background-subtracted data as compared to that using the raw counts (0.54 ± 0.08 and 0.09 ± 0.14).
This increase in quality when background subtraction is applied is to be expected. If, as mentioned above, we project into the = ±1 subspace using SLMs and single-mode fibres in paths B and C, the raw count rates would only differ from the background-subtracted rates by ≈ 1%. This in turn would result in a very small difference between the two different density matrices, and high fidelities and concurrences would be observed in both cases.