Daniel J. Lum

Photon counting compressive depth mapping

We demonstrate a compressed sensing, photon counting lidar system based on the single-pixel camera. Our technique recovers both depth and intensity maps from a single under-sampled set of incoherent, linear projections of a scene of interest at ultra-low light levels around 0.5 picowatts. Only two-dimensional reconstructions are required to image a three-dimensional scene. We demonstrate intensity imaging and depth mapping at 256 × 256 pixel transverse resolution with acquisition times as short as 3 seconds. We also show novelty filtering, reconstructing only the difference between two instances of a scene. Finally, we acquire 32 × 32 pixel real-time video for three-dimensional object tracking at 14 frames-per-second.

Simultaneous Measurement of Complementary Observables with Compressive Sensing

The more information a measurement provides about a quantum systems position statistics, the less information a subsequent measurement can provide about the systems momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainly relations correspond to resolution limits; increasing a detectors position sensitivity decreases its momentum sensitivity and vice versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.

Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution

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.

Compressive wavefront sensing with weak values

We demonstrate a wavefront sensor that unites weak measurement and the compressive-sensing, single-pixel camera. Using a high-resolution spatial light modulator (SLM) as a variable waveplate, we weakly couple an optical fields transverse-position and polarization degrees of freedom. By placing random, binary patterns on the SLM, polarization serves as a meter for directly measuring random projections of the wavefronts real and imaginary components. Compressive-sensing optimization techniques can then recover the wavefront. We acquire high quality, 256 × 256 pixel images of the wavefront from only 10,000 projections. Photon-counting detectors give sub-picowatt sensitivity.

Quantum enigma machine: Experimentally demonstrating quantum data locking

Shannon proved in 1949 that information-theoretic-secure encryption is possible if the encryption key is used only once, is random, and is at least as long as the message itself. Notwithstanding, when information is encoded in a quantum system, the phenomenon of quantum data locking allows one to encrypt a message with a shorter key and still provide information-theoretic security. We present one of the first feasible experimental demonstrations of quantum data locking for direct communication and propose a scheme for a quantum enigma machine that encrypts 6 bits per photon (containing messages, new encryption keys, and forward error correction bits) with less than 6 bits per photon of encryption key while remaining information-theoretically secure.

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

The resources needed to conventionally characterize a quantum system are overwhelmingly large for high- dimensional systems. This obstacle may be overcome by abandoning traditional cornerstones of quantum measurement, such as general quantum states, strong projective measurement, and assumption-free characterization. Following this reasoning, we demonstrate an efficient technique for characterizing high-dimensional, spatial entanglement with one set of measurements. We recover sharp distributions with local, random filtering of the same ensemble in momentum followed by position---something the uncertainty principle forbids for projective measurements. Exploiting the expectation that entangled signals are highly correlated, we use fewer than 5,000 measurements to characterize a 65, 536-dimensional state. Finally, we use entropic inequalities to witness entanglement without a density matrix. Our method represents the sea change unfolding in quantum measurement where methods influenced by the information theory and signal-processing communities replace unscalable, brute-force techniques---a progression previously followed by classical sensing.

Short-wave infrared compressive imaging of single photons

We show megapixel-resolution short-wave infrared single-photon imaging with a timing resolution of 150 ps utilizing one superconducting nanowire single-photon detector and compressive imaging.

Position-momentum Bell nonlocality with entangled photon pairs

Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Although Bell nearly showed a violation using the Clauser-HorneShimony-Holt (CHSH) inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he does not give a quantitative estimation of the maximum achievable violation for the wave function he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of 2 √ 2. Although Bell’s wave function does not produce a large violation of the CHSH inequality, other states may yet do so. DOI: 10.1103/PhysRevA.93.012105

Using double compressive sensing in simultaneous imaging of spatial entanglement

We use compressive sensing in the image and Fourier planes of a spontaneous parametric downconversion source to simultaneously gather the joint position and momentum distributions. We witness entanglement by violating a continuous variable steering inequality.

Complementary imaging with compressive sensing

Measurements on quantum systems are always constrained by uncertainty relations. For traditional, projective measurements, uncertainty relations correspond to resolution limitations; a detectors position resolution is increased at the cost of its momentum resolution and vice-versa. However, many experiments in quantum measurement are now exploring non- or partially-projective measurements. For these techniques, measurement disturbance need not manifest as a blurring in the complementary domain. Here, we describe a technique for complementary imaging | obtaining sharp position and momentum distributions of a transverse optical field with a single set of measurements. Our technique consists of random, partially-projective filtering in position followed by projective measurements in momentum. The partial-projections extract information about position at the cost of injecting a small amount of noise into the momentum distribution, which can still be directly imaged. The position distribution is recovered via compressive sensing.