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Multidimensional quantum entanglement with large-scale integrated optics
Jianwei Wang, Stefano Paesani, Yunhong Ding, Raffaele Santagati, Paul Skrzypczyk, Alexia Salavrakos, Jordi Tura, Remigiusz Augusiak, Laura Mančinska, Davide Bacco, Damien Bonneau, Joshua W. Silverstone, Qihuang Gong, Antonio Acín, Karsten Rottwitt, Leif K. Oxenløwe, Jeremy L. O'Brien, Anthony Laing, Mark G. Thompson
TL;DR
Scaling high-dimensional entanglement requires stable, scalable technology for coherently embedding photon sources and precisely controlling qudit states. This work demonstrates an integrated silicon-photonics platform for on-chip multidimensional quantum processing, achieving high-quality entangled qudits and experimentally demonstrating multidimensional randomness expansion and related applications.
Problem
Scaling entanglement from qubits to high-dimensional qudits remains challenging because large arrays of identical photon sources and precise state control require stable, scalable technology.
Method
The platform combines integrated photon sources, interferometric control, compressed-sensing tomography, and device-independent dimensionality certification to generate, manipulate, and analyze multidimensional entangled states.
Results
High-quality entangled qudit states were generated, with visibility above 0.90 in all 120 source-interference experiments and above 0.98 in more than 80% of cases; randomness efficiency Hmin/n > 1 was preserved for 4 ≤ d ≤ 14.
Takeaways & Limitations
Fully on-chip multidimensional quantum processing provides a platform for applications including high-rate device-independent randomness generation and future high-dimensional chip-based quantum networks.
Takeaways & Limitations
In this implementation, generation and measurement occur on the same integrated structure, so the source and parties are separated only by millimetres; chip-to-chip distribution is needed for wider deployment.
Abstract
from arXiv · showhide
The ability to control multidimensional quantum systems is key for the investigation of fundamental science and for the development of advanced quantum technologies. Here we demonstrate a multidimensional integrated quantum photonic platform able to robustly generate, control and analyze high-dimensional entanglement. We realize a programmable bipartite entangled system with dimension up to $15 \times 15$ on a large-scale silicon-photonics quantum circuit. The device integrates more than 550 photonic components on a single chip, including 16 identical photon-pair sources. We verify the high precision, generality and controllability of our multidimensional technology, and further exploit these abilities to demonstrate key quantum applications experimentally unexplored before, such as quantum randomness expansion and self-testing on multidimensional states. Our work provides a prominent experimental platform for the development of multidimensional quantum technologies.
Large-scale integrated quantum photonic circuit
The platform addresses the challenge of scaling coherent high-dimensional entanglement generation and control by integrating sources, interferometers, and measurements on silicon. It verifies multidimensional states and correlations while enabling randomness expansion and self-testing.
- Large-scale integrated quantum photonic circuit: Scaling coherent arrays of identical photon sources and precisely controlling high-dimensional states remains a significant experimental challenge.The approach generalizes source-based qudit entanglement from d coherently pumped sources, but requires stable, scalable integration.
- Large-scale integrated quantum photonic circuit: 120 interference experiments achieved visibility above 0.90 in every case, with more than 80% exceeding 0.98.The approximately 2 kHz photon-pair detection rate was measured under typical conditions.
- Large-scale integrated quantum photonic circuit: At d = 16, projector fidelities reached 97% in the computational basis and 85% in the Fourier basis.At d = 8, the corresponding fidelities were 98% and 97%; residual imperfections were mainly attributed to thermal cross-talk.
- Quantum state tomographies: Quantum compressed sensing reconstructed entangled states with fidelities of 96%, 87%, and 81% for dimensions d = 4, 8, and 12.The method reduces the measurement and computational cost of reconstructing large density matrices compared with standard tomography.
- Certification of system dimensionality: Generalized Bell correlations violated classical bounds for dimensions 2–8, while multidimensional steering certified entanglement up to d = 15.The Bell results closely approached the Tsirelson bound for dimensions 2–4.
- Certification of system dimensionality: CGLMP measurements at d = 4 gave S4 = 2.867 ± 0.014, exceeding the classical bound by 61.9 σ and the two-dimensional quantum maximum by 2.8 σ.The comparison indicates stronger observed Bell nonlocality for the higher-dimensional system in this experiment.
- Multidimensional randomness expansion: The experiments demonstrated multidimensional randomness expansion and self-testing, including 1SDI efficiency Hmin/n > 1 for 4 ≤d ≤14 and DI efficiency Hmin/n > 1 for d = 3 and 4.The average self-tested fidelity for tunable qutrit states was 77%, while arbitrary-dimensional SATWAP self-testing remains open.
1 Device and experimental setup details
The device combines silicon photonic components, programmable interferometers, photon-pair sources, and projectors to generate, manipulate, and measure path-encoded qudits on-chip. Characterization covers source interference, phase-shifter calibration, projector fidelities, and multidimensional operation.
- Device fabrication and components: The chip is fabricated on a commercial silicon-on-insulator platform and integrates grating couplers, waveguide crossers, and tunable MZIs.Figure S1 identifies each tunable MZI as two 2 × 2 MMI beamsplitters with a controllable thermo-optical phase-shifter.
- Qudit operations: Universal path-encoded qudit operations are implemented with triangular MZI networks using integrated beamsplitters and phase-shifters.Arbitrary projections are mapped to a computational-basis state before photon detection.
- Calibration and analysis: Phase-shifters are calibrated by fitting heater I-V curves and mapping dissipated electrical power to optical phase through sinusoidal fringe measurements.Projectors are then characterized using input-output probability distributions in computational Z and Fourier F bases.
- Calibration and analysis: Projector fidelities reach 0.998, 0.990, 0.979, and 0.971 for d = 2, 4, 8, and 16 in the Z-basis.The corresponding F-basis fidelities are 0.990, 0.965, 0.970, and 0.844; residual thermal cross-talk is the main reported imperfection.
- Sources: The circuit uses 16 identical 1.5 cm SFWM photon-pair sources, with pump distributions controlled by input MZIs.Each source produces approximately 2 kHz detected photon-pairs under typical conditions.
- Source characterization: Two-photon interference is measured for all 120 pairs among the 16 sources to assess indistinguishability and source-brightness uniformity.The measurements include reversed-HOM interference fringes and photon-statistics error bars.
2 Compressed sensing tomographies
Compressed-sensing tomography reduces the measurement burden of reconstructing large entangled-qudit density matrices. The paper applies this approach to dimensions up to 12 and visualizes reconstructed states using density-matrix representations.
- Motivation: Complete quantum state tomography becomes costly in measurements and computation, motivating compressed sensing for larger entangled-qudit states.The supplied passages describe complete tomography as having been achieved only up to 8-dimensional systems.
- Method: Compressed-sensing tomography reconstructs density matrices from projective measurements in randomly sampled operator eigenbases.Measured outcome statistics are related to a linear measurement operator and used in a semidefinite program.
- Method: The reconstruction uses convex trace-norm minimization over semipositive-definite matrices, followed by renormalization of the estimated density matrix.The noise parameter ϵ controls the constraint applied to the measurement data.
- Results: The experiment reconstructs bipartite entangled states with local dimension up to d = 12 using compressed-sensing quantum state tomography.The reconstructed density matrices for dimensions 4, 8, and 12 are reported in the main-text figure, while dimensions 2 and 3 are shown separately.
- Results: The method uses 50, 122, and 228 measured operators for dimensions 4, 8, and 12, respectively.The corresponding semidefinite-program runtimes are a few seconds, approximately five minutes, and approximately one hour on a standard laptop.
3 Dimension witness
The experiment uses device-independent dimension witnesses based on correlations from selected measurement settings and quantum-game strategies. These witnesses certify local dimensions across several generated entangled qudit states, including dimensions below 12.
- Witness construction: Magic Square correlations are realized with maximally entangled states of local dimension 4, while Magic Pentagram correlations use local dimension 8.The corresponding measurements are implemented through orthonormal bases associated with commuting observables.
- Witness construction: The local-dimension bound d ≥⌈D(p)⌉ is computed from observed correlation probabilities without prior assumptions about the experimental apparatus.The approach assumes shared randomness is not a free resource.
- Experimental certification: 16-dimensional correlations are obtained by playing two copies of the Magic Square game in parallel, with the resulting witness satisfying D(p) ≥16.The strategy uses a maximally entangled state of local dimension 16.
- Measurement scenarios: Scenario I estimates dimension from partial correlations using one Alice setting and two Bob settings selected from Magic Square or Magic Pentagram strategies.The experiment prepares maximally entangled states and measures only the settings needed for the selected witness.
- Experimental certification: The experimentally observed correlations certify the true dimension for all d < 12.The certification uses experimentally measured correlations rather than ideal correlations.
4 Multidimensional Bell inequality
The paper formulates multidimensional Bell inequalities through joint probabilities and generalized complex correlators, covering maximally and partially entangled two-qudit states. The resulting violations support state self-testing, although maximality for one generalized family remains conjectural.
- The Bell scenario uses joint probabilities p(ab|xy) from Alice’s and Bob’s measurements, expressed through the Born rule.
- Generalized complex correlators are obtained by a two-dimensional Fourier transform of joint distributions, with quantum observables having d-th-root-of-unity eigenvalues.
- Qd = 2(d −1) is the maximal quantum violation of the multidimensional inequality, achieved by maximally entangled two-qudit states with specified observables.
- The inequality family parametrized by ξ connects CGLMP and maximally-entangled-state inequalities while targeting partially entangled states with different γ values.
- The claimed maximal quantum violation for the generalized partially entangled-state inequality is conjectured from numerical studies and tested with the NPA hierarchy rather than proven analytically.
- Self-testing estimates minimum fidelity to a reference state under observed correlations; maximal violation yields fidelity 1, while lower violations reduce fidelity.
5 Multidimensional steering
The steering analysis treats Alice’s and Bob’s devices and shared state as untrusted and uses steering inequalities to certify multidimensional entanglement. Experimental violations are reported up to local dimension 15.
- In the one-sided device-independent scenario, Alice’s measuring device and the shared state are uncharacterized, while the observed data include p(a|x) and Bob’s steered states.
- A local-hidden-state model explains Alice’s outcomes and Bob’s states through a shared hidden variable and corresponding response and hidden-state distributions.
- Violating an EPR steering inequality certifies entanglement and steering in a one-sided device-independent manner.
- Experimental steering values βd are reported up to local dimension 15 against the classical LHS bound βlhs = 1 + 1/ d.
6 Device-independent and one-sided device-indepedent randomness generation
The paper generates private randomness from Bell or steering violations without fully characterizing the devices. It bounds an eavesdropper’s predictability using optimization programs and extracts randomness from the resulting entropy bounds.
- Bell and EPR steering violations provide private randomness in device-independent and one-sided device-independent scenarios, respectively.
- The fully device-independent protocol optimizes Eve’s guessing probability over distributions compatible with observed SATWAP correlations and relaxed quantum constraints from the NPA hierarchy.
- In the one-sided scenario, optimization uses p(a,e|x) and states prepared for Bob to bound predictability of Alice’s output string.
- The resulting guessing-probability bounds are converted into lower bounds on min-entropy, quantifying extractable randomness per output symbol.
- The reported experimental randomness bounds include maximally and partially entangled states in both one-sided and fully device-independent settings.
7 On-chip multidimensional quantum key distribution
The chip implements multidimensional QKD using entangled photons measured in computational and Fourier bases, with security assessed in device-dependent and device-independent settings. Increasing dimension improves measured key-rate and information-capacity performance within the stated assumptions.
- The experiment uses a generalized entanglement-based BB84 protocol for multidimensional QKD.
- Device-dependent security analysis reports measured fidelity, QBER, and secure key rates across dimensions up to 14.
- Higher dimensionality generally corresponds to higher noise tolerance, photon information efficiency, and secure key rates under the stated reconciliation and multi-pair assumptions.
- The implementation places Alice, Bob, and the entanglement source on the same chip at millimetre-scale distance, while chip-to-chip distribution is identified as an adoptable extension.
- The integrated circuit prepares and measures multidimensional path-encoded states using programmable interferometers and phase controls.