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Optics Letters

Parallelized projective measurements for spatial photonic qudits estimation

Quimey Pears Stefano, Lorena Rebón, Claudio Iemmi

Published Aug 28, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1364/ol.570564

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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−d-d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only} d+1d+1d+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−d=6-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 424242 to only 777.
d+1d+1d+1 Measurement Bases are Sufficient for Determining ddd-Dimensional Quantum States: Theory and Experiment
A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only d+1d+1d+1 projective measurement bases to fully reconstruct an arbitrary ddd-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for d=6d=6d=6, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.
We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension ddd, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only 4d4d4d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of d2{d}^{2}d2. By using this technique, we have experimentally reconstructed a large number of states ranging from d=2d=2d=2 up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with ddd slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the ddd slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d\ensuremath{-}3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.
We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension.In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime.Fidelities of reconstruction within the range of 0.81-0.96are obtained for both pure and mixed states.These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.