The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations -the Donsker-Varadhan formalism and the optimal fluctuation method -manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, which has previously appeared in other problems, and we identify the reason for this coincidence.
A single photon has many physical degrees of freedom (DOF) that can carry the state of a high-dimensional quantum system. Nevertheless, only a single DOF is usually used in any specific demonstration. Furthermore, when more DOF are being used, they are analyzed and measured one at a time. We introduce a two-qubit information system, realized by two degrees of freedom of a single photon: polarization and time. The photon arrival time is divided into two time-bins representing a qubit, while its polarization state represents a second qubit. The time difference between the two time-bins is created without an interferometer at the picosecond scale, which is much smaller than the detector's response time. The two physically different DOF are analyzed simultaneously by photon bunching between the analyzed photon and an ancilla photon. Full twoqubit states encoded in single photons were reconstructed using quantum state tomography, both when the two DOF were entangled and when they were not, with fidelities higher than 96%.
Low rank matrix recovery problems, including matrix completion and matrix sensing, appear in a broad range of applications. In this work we present GNMR -an extremely simple iterative algorithm for low rank matrix recovery, based on a Gauss-Newton linearization. On the theoretical front, we derive recovery guarantees for GNMR in both the matrix sensing and matrix completion settings. A key property of GNMR is that it implicitly keeps the factor matrices approximately balanced throughout its iterations. On the empirical front, we show that for matrix completion with uniform sampling, GNMR performs better than several popular methods, especially when given very few observations close to the information limit.
We present a new, simple, and computationally efficient iterative method for low rank matrix completion. Our method is inspired by the class of factorization-type iterative algorithms, but substantially differs from them in the way the problem is cast. Precisely, given a target rank r, instead of optimizing on the manifold of rank r matrices, we allow our interim estimated matrix to have a specific overparametrized rank 2r structure. Our algorithm, denoted R2RILS, for rank 2r iterative least squares, thus has low memory requirements, and at each iteration it solves a computationally cheap sparse least squares problem. We motivate our algorithm by its theoretical analysis for the simplified case of a rank 1 matrix. Empirically, R2RILS is able to recover ill-conditioned low rank matrices from very few observations---near the information limit---and it is stable to additive noise.
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