Compressive sensing for imaging spatial entanglement

Entangled photons—specially connected quantum particles that can act in concert over great distances—are the requisite resource for a variety of emerging technologies, including secure quantum communication, quantum computing, and quantum sensing. To date, their implementations have mostly used entanglement in discrete, low-dimensional degrees of freedom that have few possible measurement outcomes, such as the polarization of a photon (two dimensions). However, entanglement may also exist in continuous degrees of freedom, such as the photons' transverse spatial distribution (or image). Because there are no fundamental restrictions on a photon's transverse location, this entanglement can be very high-dimensional. Einstein, Podolsky, and Rosen (EPR) describe exactly such a state in their famous critique of quantum mechanics.¹ High-dimensional entanglement promises significant advantages for quantum technologies, such as increased security and channel capacity for communication, but it remains challenging to use in practice.

To be useful, any entangled system must first be fully characterized. Entanglement is revealed by the appearance of classically impossible correlations in measurements on entangled particles. Traditionally, measuring spatial correlations at the single-photon level requires raster scanning two photon-counting detectors through the detection plane and correlating their outputs. If the detection plane is divided into n pixels, n²measurements are required to correlate every pixel pair. Entangled photon sources are weak and typically provide on the order of 10,000 photon pairs per second across the entire field. Because the number of available photons per measurement decreases by at least 1/n (perfect correlations), raster scanning scales extremely poorly with increasing resolution. A 32×32=1024 pixel space could take up to a year to raster scan for a signal-to-noise ratio of only 10.

To make the measurement practical, we apply the approach of compressive sensing.^{2, 3} Compressive sensing formulates an optimization problem to find an n-dimensional signal from m << n incoherent measurements (random projections). The reconstruction promotes the simplest, or sparsest, signal consistent with the measurements. Sparse signals can be effectively recovered with as few as 2–3% the number of measurements as a raster scan. The highly correlated distribution for spatially entangled photons is extremely sparse: each pixel for one photon should only be correlated with at most a few pixels for its entangled twin. Furthermore, incoherent random projections use on average half the available light regardless of resolution, mitigating the problem of weak sources. Using compressive sensing, we reduce a 32×32 = 1024pixel measurement requiring 310 days to raster scan to about 8 hours. We demonstrate channel capacities up to 8.4 bits, equivalent to 337 entangled modes. By comparing near-field (position) and far-field (momentum) results, we violate an entropic uncertainty bound to demonstrate entanglement.⁴

Figure 1. Schematic of the experimental system for compressive sensing. Spontaneous parametric downconversion generates pairs of spatially entangled photons (f1 and f2). A beamsplitter (BS) splits the photons into signal and idler modes that are directed to digital micromirror devices (DMDs).

Figure 1 shows our experimental configuration. We use spontaneous parametric downconversion to generate pairs of spatially entangled photons. A beamsplitter splits the photons into signal and idler modes that are directed to digital micromirror devices (DMDs). These are arrays of individually addressable, movable mirrors that direct incoming light either to a photon-counting detector or out of the system. Imaging optics place either the near-field (for position correlations) or far-field (for momentum correlations) of the downconversion crystal on the DMDs. Measurements are taken by placing random, binary patterns on the DMDs and counting the number of correlated photons for each pattern. We then use a gradient projection⁵ algorithm to find the sparsest signal consistent with the measurements, a measure of sparsity.

Figure 2. Sample 16×16 reconstructions for position and momentum configurations showing correlations between DMD pixels.

Sample reconstructions for a 16×16 resolution are given in Figure 2 for position and momentum configurations that show correlations between DMD pixels. DMD pixels are listed in column-wise order. The marginal distributions, which give the overall beam shape, are inset. Only 2500 measurements were required to recover a 16⁴ = 65, 536 dimensional signal in about 40 minutes. The average number of photons per measurement was approximately 1500. The sharply defined diagonal and anti-diagonal features show the expected positive and negative correlations for the position and momentum representations, respectively. The state is found to be correlated to an area of less than 1 pixel, indicating higher resolution is possible.

We measured at resolutions of 16×16, 24×24, and 32×32 pixels. We present the results in terms of the classical mutual information for the position and momentum distributions. This gives the channel capacity (bits) for using the system for communication. Table 1 provides results. Additionally, we recently derived an EPR steering inequality based on summing the measured mutual information in conjugate position and momentum bases.⁶ Unlike most entanglement measures, our inequality does not require a complicated reconstruction of the particles' quantum state to use. Instead, it is calculated directly from simple measurements. For position mutual information I_xx and momentum mutual information I_kk, we find all classically correlated distributions must satisfy

where W_xW_k is the bandwidth product for the detection area. Violation of this inequality witnesses entanglement. For our system, this right-hand-side value was 9.8 bits. We exceed this value for all resolutions.

In summary, we used compressive sensing to efficiently measure the spatial correlations between entangled photons produced by spontaneous parametric downconversion. We dramatically reduce the measurement time with respect to raster scanning, providing a practical approach for characterizing high-dimensional, continuous variable entanglement. Our work supports the idea that compressive sensing will be extremely useful for detecting correlated signals in a large dimensional space. Potential applications range from verifying security in energy-time quantum key distribution to correlated imaging through scattering media. We next intend to apply our technique to entanglement in other variables, such as energy and time or angular position and orbital angular momentum.

This work was supported by the Defense Advanced Projects Agency InPho (Information in a Photon) grant W911NF-10-1-0404.