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Physical Review Applied

Measuring the complete set of spatial Schmidt modes of entangled two-photon fields

Radhika Prasad, Nilakshi Senapati, Abhinandan Bhattacharjee, Suman Karan, Anand K. Jha

Published Sep 23, 2026• 0 citations
Publisher
DOI
10.1103/rwmz-jp7r

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Abstract High-dimensional spatially entangled two-photon states generated by the spontaneous parametric down-conversion process (SPDC) have become a promising resource for several quantum information science applications. In order to harness high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is due to radial coordinates not having a Schmidt-decomposed form for the SPDC photons or a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that azimuthal averaging of SPDC two-photon states yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of the radial Schmidt spectrum of up to 50 radial Schmidt modes with about 98% fidelity.

Journal of Optics•Jul 1, 2026•1 citation
Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98%\%% fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.
High-dimensional spatially entangled two-photon state generated by spontaneous parametric down-conversion process (SPDC) has become a promising resource for several quantum information science applications. For harnessing high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is because for radial coordinates, there is neither a Schmidt-decomposed form for the SPDC photons nor is there a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that the azimuthal averaging of SPDC two-photon state yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of radial Schmidt spectrum of upto 50 radial Schmidt modes with about 98\% fidelity.
Abstract High-dimensional spatially entangled two-photon states generated by the spontaneous parametric down-conversion process (SPDC) have become a promising resource for several quantum information science applications. In order to harness high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is due to radial coordinates not having a Schmidt-decomposed form for the SPDC photons or a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that azimuthal averaging of SPDC two-photon states yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of the radial Schmidt spectrum of up to 50 radial Schmidt modes with about 98% fidelity.