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Laser & Photonics Review

Empowering Spatial‐Mode‐Encoded High‐Dimensional State Reconstruction With Parallel Full Bloch Vector Mapping

Dawei Lyu, Qianke Wang, Jun Liu, Jian Wang

Published Oct 3, 2026• 0 citations

Abstract

ABSTRACT Quantum state tomography plays a fundamental role in quantum information science. Spatial modes of photons offer a promising encoding platform for high‐dimensional quantum states, yet efficient reconstruction of such states remains challenging. Recent advances in diffractive neural networks (DNNs) have introduced powerful capabilities for spatial mode manipulation, offering new opportunities for realizing efficient projective measurements. Here, we propose and experimentally demonstrate an efficient high‐dimensional state reconstruction method, enabled by parallel measurement of the full Bloch vectors for high‐dimensional states encoded in spatial modes of photons. By employing a DNN, our method maps mutually unbiased basis states to spatially separated Gaussian spots, thereby performing parallel projective measurements with a single measurement setting. We successfully reconstruct spatial‐mode‐encoded states in dimensions two through four, achieving state fidelities up to 99% and measurement process fidelities up to 97%. The method is further extended to spin‐orbit non‐separable states and mixed states, demonstrating its capability for reconstructing different classes of states within the selected spatial‐mode encoding space. Our method significantly improves the efficiency of tomography for spatial‐mode‐encoded high‐dimensional states and provides a versatile tool for the rapid characterization of high‐dimensional quantum systems.

DOI: 10.1002/lpor.71981

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ABSTRACT Quantum state tomography plays a fundamental role in quantum information science. Spatial modes of photons offer a promising encoding platform for high‐dimensional quantum states, yet efficient reconstruction of such states remains challenging. Recent advances in diffractive neural networks (DNNs) have introduced powerful capabilities for spatial mode manipulation, offering new opportunities for realizing efficient projective measurements. Here, we propose and experimentally demonstrate an efficient high‐dimensional state reconstruction method, enabled by parallel measurement of the full Bloch vectors for high‐dimensional states encoded in spatial modes of photons. By employing a DNN, our method maps mutually unbiased basis states to spatially separated Gaussian spots, thereby performing parallel projective measurements with a single measurement setting. We successfully reconstruct spatial‐mode‐encoded states in dimensions two through four, achieving state fidelities up to 99% and measurement process fidelities up to 97%. The method is further extended to spin‐orbit non‐separable states and mixed states, demonstrating its capability for reconstructing different classes of states within the selected spatial‐mode encoding space. Our method significantly improves the efficiency of tomography for spatial‐mode‐encoded high‐dimensional states and provides a versatile tool for the rapid characterization of high‐dimensional quantum systems.

We present a quantum state tomography method that enables the reconstruction of arbitrary d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using only d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are spatially multiplexed on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.

We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary}

d-

dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only}

d+1

experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of

d=6-

dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from

42

to only

7

.

Photonic quantum technologies utilize various degrees of freedom (DOFs) of light, such as polarization, frequency, and spatial modes, to encode quantum information. In the effort of further improving channel capacity of quantum communication, and for increasing the complexity of available quantum operations, high-dimensional and hyperentangled states are now gaining interest. However, efficiently measuring these high dimensional states is challenging due to the large number of measurements required for reconstructing the full density matrix via quantum state tomography (QST), and the fact that each measurement requires some modification in the experimental setup. Here, we propose a framework for reconstructing the density matrix of a single-photon hyperentangled across multiple DOFs using a single intensity-measurement obtainable from traditional cameras, and discuss extensions for multiphoton hyperentangled states. Our method hinges on the spatial DOF of the photon and uses it to encode the quantum information from the other DOFs. We numerically demonstrate this method for single-photon OAM-spin and OAM-frequency entangled states using an ideal coupler and a multimode fiber, to perform the information mixing and transfer the encoding to spatial information, where it is detected using a simple camera. This technique simplifies the experimental setup and reduces acquisition time compared to traditional QST-based methods. Moreover, it allows the recovery of DOFs that conventional cameras cannot detect, such as polarization, thus eliminating the need for projection measurements.

Abstract High‐dimensional (HD) entanglement of photonic orbital angular momentum (OAM) offers significant potential for enhancing channel capacity and improving noise resistance in quantum information processing. However, the challenge of achieving simple and rapid measurement has limited its practical applications. In this work, a quantum state tomography (QST) framework is demonstrated that utilizes convolutional neural networks to rapidly reconstruct the density matrix of OAM entanglement from only two coincidence measurements. The experimental results for a 5D OAM entangled state yield a fidelity of 0.973 ± 0.005. This method is also applicable to mixed OAM entangled states and scenarios with incomplete tomographic measurements. These findings represent a significant step toward implementing high‐speed QST for applications involving HD spatial mode quantum state, whether in free space or integrated systems.

Frequency-bin encoding provides a scalable platform for photonic quantum information processing by enabling high-dimensional qudit states compatible with the telecommunications infrastructure. However, characterizing these systems remains challenging, as conventional quantum state tomography requires measurement resources that increase rapidly with dimensionality, compounded by the difficulty of implementing controllable measurements across many frequency modes at once. Here, we address these challenges and demonstrate threshold quantum state tomography of entangled frequency-bin qudits using adaptive measurements that exploit the sparsity of highly entangled biphoton frequency combs. Combined with Bayesian inference, we experimentally demonstrate full state estimation of biphoton frequency combs with Hilbert space dimensions up to

d^2=100

---a 56% increase over the previous record---all while performing a mere 1.9% of the measurements required by conventional tomographically complete approaches. This work confirms the promise of measurement-aware approaches for characterizing high-dimensional photonic quantum systems, unlocking new tools for full tomography that are hardware-efficient, state-aware, and free from a priori assumptions about state structure.

While measuring the orbital angular momentum state of bright light beams can be performed using imaging techniques, a full characterization at the single-photon level is challenging. For applications to quantum optics and quantum information science, such characterization is an essential capability. Here, we present a setup to perform the quantum state tomography of photonic qubits encoded in this degree of freedom. The method is based on a projective technique using spatial mode projection via fork holograms and single-mode fibers inserted into an interferometer. The alignment and calibration of the device is detailed as well as the measurement sequence to reconstruct the associated density matrix. Possible extensions to higher-dimensional spaces are discussed.