Fig. 1: Theoretical comparison and schematic representation of tomography methods.

a Schematic representing the DEMESST method. Rather than sampling an entire multimode operator space (3D space), if a state lives in some number of polynomial subspaces (blue 2D plane), we restrict the sampling to each of those instead. The {O} basis operators are of the form \({O}_{\overrightarrow{{{{{{{{\bf{n}}}}}}}}},{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}^{{\prime} }}=\left\vert \overrightarrow{{{{{{{{\bf{n}}}}}}}}}\right\rangle \left\langle {\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}^{{\prime} }\right\vert\) for generic basis states \(\left\vert \overrightarrow{{{{{{{{\bf{n}}}}}}}}}\right\rangle,\left\vert {\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}^{{\prime} }\right\rangle\) (see Supplementary Note 4). Assuming an orthonormal basis, the state ρ is given by \(\rho={\sum }_{{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{1}}}}}}}}},{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{2}}}}}}}}}}\,{{\mbox{Tr}}}\,[\rho {O}_{{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{2}}}}}}}}},{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{1}}}}}}}}}}]{O}_{{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{1}}}}}}}}},{\overrightarrow{{{{{{{{\bf{n}}}}}}}}}}_{{{{{{{{\bf{2}}}}}}}}}}\). This improves the overall efficiency of the sampling, especially for states with support across large numbers of modes. In practice, we use Hermitian {O} that are accessible through experiment. b Number of measurements required for the DEMESST (purple, circles) and OLI (orange, squares) methods to reach a 90% state reconstruction fidelity on W states of up to 7 modes, assuming perfect state preparation. Dashed lines indicate fits to exponential and polynomial functions \(y=\exp (a+bx)\) with a = 0.1, b = 3.6 and \(y=\exp (a+b\log (x))\) with a = 7.9, b = 2.8, respectively. OLI scales exponentially with the number of modes M, while DEMESST scales only polynomially.