Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 32 million-dimensional bi-photon probability distribution

Abstract

We demonstrate how to efficiently implement extremely high-dimensional compressive imaging of a bi-photon probability distribution. Our method uses fast-Hadamard-transform Kronecker-based compressive sensing to acquire the joint space distribution. We list, in detail, the operations necessary to enable fast-transform-based matrix-vector operations in the joint space to reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a subspace of that image exists a 3.2 million-dimensional bi-photon probability distribution. In addition, we demonstrate how the marginal distributions can aid in the accuracy of joint space distribution reconstructions.

Quality of spatial entanglement propagation

“…The conjugate nature of these variables underlies imaging and propagation, while their infinite-dimensional Hilbert space holds much potential for quantum computation [6][7][8][9]. Typically, the photon source for continuous-variable entanglement is spontaneous parametric down-conversion (SPDC) [10][11][12][13][14] but, remarkably, there have been few investigations into its amount and distribution upon propagation [15,16].A universal metric to quantify the degree of entanglement is the Schmidt number, which is directly related to the non-separability of the state's (two) subsystems [17][18][19]. While interferometric measurements of the Schmidt number have been proposed [15] and demonstrated [16], such methods do not examine the manifestation of the entanglement, i.e., non-separability of amplitude or phase.…”

“…The conjugate nature of these variables underlies imaging and propagation, while their infinite-dimensional Hilbert space holds much potential for quantum computation [6][7][8][9]. Typically, the photon source for continuous-variable entanglement is spontaneous parametric down-conversion (SPDC) [10][11][12][13][14] but, remarkably, there have been few investigations into its amount and distribution upon propagation [15,16].…”

Compressively Characterizing High-Dimensional Entangled States with Complementary, Random Filtering

“…Single-particle sensing matrices a (k1) , a (k2) , b (x1) , and b (x2) are generated by taking M rows from randomly permuted n × n Hadamard matrices. This allows the repeated calculations of AK and BX performed by the solver to use a Fast Hadamard transform, decreasing computational requirements [48]. Because we only collect transmitted modes from both position and momentum filters, we require 16 separate measurements to collect all coincident combinations of transmission and rejection for the 4 filters (described in supplemental material).…”

“…The full measurement and reconstruction recipe we follow is similar to that described in Ref. [48]. Note that our choice of a single momentum SLM and two position DMDs was due to available equipment.…”

A performance comparison of measurement matrices in compressive sensing

“…The sensing signal is prerandomized by scrambling its sample locations and flipping its sample signs and then fast‐transform the randomized samples and, finally, subsample the resulting transform coefficients to obtain the final sensing measurements. Lum et al used fast‐Hadamard‐transform Kronecker‐based sensing matrices to acquire the joint space distribution for high‐dimensional compressive imaging of bi‐photon probability distribution. The sensing matrix is obtained by randomizing Hadamard matrix and using fast Hadamard transform.…”