Quantum transport of high-dimensional spatial information with a nonlinear detector

Abstract

Information exchange between two distant parties, where information is shared without physically transporting it, is a crucial resource in future quantum networks. Doing so with high-dimensional states offers the promise of higher information capacity and improved resilience to noise, but progress to date has been limited. Here we demonstrate how a nonlinear parametric process allows for arbitrary high-dimensional state projections in the spatial degree of freedom, where a strong coherent field enhances the probability of the process. This allows us to experimentally realise quantum transport of high-dimensional spatial information facilitated by a quantum channel with a single entangled pair and a nonlinear spatial mode detector. Using sum frequency generation we upconvert one of the photons from an entangled pair resulting in high-dimensional spatial information transported to the other. We realise a d = 15 quantum channel for arbitrary photonic spatial modes which we demonstrate by faithfully transferring information encoded into orbital angular momentum, Hermite-Gaussian and arbitrary spatial mode superpositions, without requiring knowledge of the state to be sent. Our demonstration merges the nascent fields of nonlinear control of structured light with quantum processes, offering a new approach to harnessing high-dimensional quantum states, and may be extended to other degrees of freedom too.

Fig. 1. High-dimensional quantum transport enabled by nonlinear detection.

In our concept, information is encoded on a coherent source and overlapped with a single photon from an entangled pair in a nonlinear crystal for up-conversion by sum frequency generation, the latter acting as a nonlinear spatial mode detector. The bright source is necessary to achieve the efficiency required for nonlinear detection. Information and photons flow in opposite directions: one of Bob’s entangled photons is sent to Alice and has no information, while a measurement on the other in coincidence with the upconverted photon establishes the transport of information across the quantum link. Alice need not know this information for the process to work, while the nonlinearity allows the state to be arbitrary and unknown in dimension and basis.

Fig. 2. Realising a quantum transport channel.

a A pump photon (λ_p = 532 nm) undergoes spontaneous parametric downconversion (SPDC) in a nonlinear crystal (NLC₁), producing a pair of entangled photons (signal B and idler C), at wavelengths of λ_B = 1565 nm and λ_C = 808 nm, respectively. Photon B is directed to a spatial mode detector comprising a spatial light modulator (SLM_B) and a single mode fibre coupled avalanche photo-diode detector (APD). The state to be transferred is prepared as a coherent source A using SLM_A (λ_A = 1565 nm), and is overlapped in a second nonlinear crystal (NLC₂) with photon C, resulting in an upconverted photon D which is sent to a single mode fibre coupled APD. Photons B and D are measured in coincidence to find the joint probability of the prepared and measured states using the two SLMs. b The quantum transport channel’s theoretical modal bandwidth (K) as a function of the pump (w_p) and detected photons' (w₀ and w_D) radii, with experimental confirmation shown in c through e corresponding to parameter positions C, D, and E in b. K_th and K_ex are the theoretical and experimental quantum transport channel capacities, respectively. The cross-talk plots are shown as orbital angular momentum (OAM) modes prepared and transferred. The raw data is reported with no noise suppression or background subtraction, and considering the same pump power conditions in all three configurations.

Fig. 3. Quality of the quantum transport process.

Experimental fidelities (points) for our channel dimensions up to the maximum achievable channel capacity of K = 15 ± 1, all well above the classical limit (dashed line). The solid line forms a maximum fidelity for the measured transferred state. The inset shows the measured OAM modal spectrum of the optimised quantum transport channel with maximum coincidences of 320 per second for a 5 min integration time. The raw data is reported with no noise suppression or background subtraction.

Fig. 4. Visibilities and quantum state tomography.

a Measured coincidences (points) and fitted curve (solid) as a function of the phase angle (θ) of the corresponding detection analyser for the state $∣\varphi ⟩=∣\ell ⟩+\exp \left(i\theta \right)∣-\ell ⟩$, for three OAM subspaces of ℓ = ± 1, ± 2, and ± 3 (further details in the Supplementary Note 5). b The real (Re[ρ]) and imaginary (Im[ρ]) parts of the density matrix (ρ) for the qutrit state $∣\mathrm{\Psi }⟩=∣-1⟩+∣0⟩+∣1⟩$ as reconstructed by quantum state tomography. The inset shows the raw coincidences with maximum coincidences of 220 detected per second from the tomographic projections (full details in the Supplementary Notes 6 and 13). c Measurements for the quantum transport of a 4-dimensional state, constructed from the states ℓ = { ± 1, ± 3}. d Measurements showing the detection (solid bars) of all the prepared (transparent bars) OAM states comprising one of the MUB states. The raw data is reported with no noise suppression or background subtraction.

Fig. 5. Quantum transport in the Hermite-Gaussian basis.

Coincidence measurements for quantum transport of a a 3-dimensional and b a 9-dimensional HG_n,m state, constructed from the states (n, m) = {(0, 1), (1, 0), (1, 1)} and (n, m) = {(0, 0), (2, 0), (0, 2), (2, 2), (2, 4), (4, 2), (4, 4)}, respectively. The weights of the diagonal elements of the density operator of the transported state (solid bars) are in good agreement with the weights of the prepared state (transparent bars). The raw data is reported with no noise suppression or background subtraction.

Fig. 6. Summary of transferred states.

Similarities for transport of a 2,3,4 and 9-dimensional superposition states in the OAM (represented as $∣\phi ⟩$) and HG (represented as $∣\gamma ⟩$) bases shown and labelled to the left. Transferred states are $∣{\phi }_1⟩=∣0⟩+∣-1⟩$, $∣{\phi }_2⟩=∣-1⟩+∣1⟩$, $∣{\phi }_3⟩=∣0⟩-∣1⟩$, $∣{\phi }_4⟩=∣-2⟩+∣0⟩+∣2⟩$, $∣{\gamma }_1⟩=∣H{G}_{1,0}⟩+∣H{G}_{1,1}⟩+∣H{G}_{0,1}⟩$, $∣{\phi }_5⟩=∣-3⟩-i∣-1⟩+∣1⟩+i∣3⟩$, $∣{\gamma }_2⟩=∣H{G}_{0,0}⟩+∣H{G}_{1,0}⟩+∣H{G}_{1,1}⟩+∣H{G}_{0,1}⟩$ and $∣{\gamma }_3⟩=∣H{G}_{0,0}⟩+∣H{G}_{2,0}⟩+∣H{G}_{0,2}⟩+∣H{G}_{2,2}⟩+∣H{G}_{4,0}⟩+∣H{G}_{0,4}⟩+∣H{G}_{4,2}⟩+∣H{G}_{2,4}⟩+∣H{G}_{4,4}⟩$. The similarity of diagonal elements of the density matrix together with prior phase information confirms coherent transport up to d = 4 but not for d = 9, where only the diagonal elements are assessed. Raw data are reported without noise suppression or background subtraction.