Identification of the mode, polarization, wavelength and intensity of light using a one-pixel device on an optical fibre tip

52 6.6 Tensor decomposition of the response signals We extracted ~104 experimentally measured states and reconstructed the complete 5-dimensional response tensor of our device: 𝒳∈ℝ=×?×@×A×+ corresponding to mode (M) × intensity (I) × polarization (θ) × wavelength (λ) × port numbers (N). This tensor encodes the full 4D optical input and the six-port electrical response. CP decomposition analysis CP decomposition factorizes this tensor into a sum of rank-one tensors: 𝑋 ≈𝒳OB = . 𝑎/ B {/-"} ∘𝑏/ ∘𝑐/ ∘𝑑/ ∘𝑒/ where R is the CP rank, ar, br, cr, dr, er are factor vectors along each dimension, and ∘ denotes the outer product. We performed CP decomposition on the full tensor, sweeping the rank from 1 to 6, and computed the relative reconstruction error and explained variance. The relative reconstruction error is defined as 𝐸𝑟𝑟(𝑅) = V𝒳−𝒳OBV ‖𝒳‖ representing the discrepancy between the rank-R approximation and the original tensor. The explained variance is 𝑅# = 1 −𝐸𝑟𝑟(𝑅)# indicating the fraction of total variance captured by the rank-R decomposition. As shown in Supplementary figure 35, a rank-1 model explains approximately 96.7% of the total variance, indicating a dominant global response component. Increasing the rank further reduces the reconstruction error, and a rank-4 model captures ~99.5% of the variance, corresponding to a clear inflection point. Increasing the rank beyond four yields only marginal improvements (<0.2% per additional component), indicating saturation behavior. These results demonstrate that the effective dimensionality of the device response is four, rather than six, and that higher-rank components mainly capture structured redundancy and experimental noise rather than additional independent physical interaction channels. This effective rank-four behavior suggests that the full information content of the multidimensional optical response can, in principle, be accessed using four linearly independent response ports. Using fewer ports would generally fail to capture all independent components, whereas additional ports primarily provide redundancy, enhancing robustness and noise tolerance without introducing new independent information. We further validate this implication experimentally in the following analysis.