Matrix probing and its conditioning

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media

Riesz transforms on generalized Heisenberg groups and Riesz transforms

Let 1 < q < \infty. We prove that the Riesz transforms $R_{k}=X_{k} L^{-\frac{1}{2}}$ on a generalized Heisenberg group $G$ satisfy $\left\|\left(\sum_{k=1}^{K}\left| R_{k}(f)\right| ^{2}\right)^{\frac{1}{2}}\right\| _{L^{q}(G)}\leq C(q,J)\left\| f\right\| _{L^{q}(G)}$ where $K$, $J$ are respectively the dimensions of the first and second layer of the Lie algebra of $G$. We prove similar inequalities on Schatten spaces $S^{q}(H)$, with dimension free constants, for Riesz transforms associated to commuting inner $*$-derivations $D_{k}$ and a suitable substitute of the square function. An example is given by the derivations associated to $n$ commuting pairs of operators $(P_{j},Q_{j})$ on a Hilbert space $H$ satisfying the canonical commutation relations [P$_{j},Q_{j}]=iI_{H}$

Structured Random Matrices

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and Applications" (Springer

Quasiparticle Andreev scattering in the ν=1/3\nu=1/3 fractional quantum Hall regime

The scattering of exotic quasiparticles may follow different rules than electrons. In the fractional quantum Hall regime, a quantum point contact (QPC) provides a source of quasiparticles with field effect selectable charges and statistics, which can be scattered on an 'analyzer' QPC to investigate these rules. Remarkably, for incident quasiparticles dissimilar to those naturally transmitted across the analyzer, electrical conduction conserves neither the nature nor the number of the quasiparticles. In contrast with standard elastic scattering, theory predicts the emergence of a mechanism akin to the Andreev reflection at a normal-superconductor interface. Here, we observe the predicted Andreev-like reflection of an $e/3$ quasiparticle into a $-2e/3$ hole accompanied by the transmission of an $e$ quasielectron. Combining shot noise and cross-correlation measurements, we independently determine the charge of the different particles and ascertain the coincidence of quasielectron and fractional hole. The present work advances our understanding on the unconventional behavior of fractional quasiparticles, with implications toward the generation of novel quasi-particles/holes and non-local entanglements

Observing the universal screening of a Kondo impurity

The Kondo effect, deriving from a local magnetic impurity mediating electron-electron interactions, constitutes a flourishing basis for understanding a large variety of intricate many-body problems. Its experimental implementation in tunable circuits has made possible important advances through well-controlled investigations. However, these have mostly concerned transport properties, whereas thermodynamic observations - notably the fundamental measurement of the spin of the Kondo impurity - remain elusive in test-bed circuits. Here, with a novel combination of a "charge" Kondo circuit with a charge sensor, we directly observe the state of the impurity and its progressive screening. We establish the universal renormalization flow from a single free spin to a screened singlet, the associated reduction in the magnetization, and the relationship between scaling Kondo temperature and microscopic parameters. In our device, a Kondo pseudospin is realized by two degenerate charge states of a metallic island, which we measure with a non-invasive, capacitively coupled charge sensor. Such pseudospin probe of an engineered Kondo system opens the way to the thermodynamic investigation of many exotic quantum states, including the clear observation of Majorana zero modes through their fractional entropy

Observation of the scaling dimension of fractional quantum Hall anyons

Unconventional quasiparticles emerging in the fractional quantum Hall regime present the challenge of observing their exotic properties unambiguously. Although the fractional charge of quasiparticles has been demonstrated since nearly three decades, the first convincing evidence of their anyonic quantum statistics has only recently been obtained and, so far, the so-called scaling dimension that determines the quasiparticles propagation dynamics remains elusive. In particular, while the non-linearity of the tunneling quasiparticle current should reveal their scaling dimension, the measurements fail to match theory, arguably because this observable is not robust to non-universal complications. Here we achieve an unambiguous measurement of the scaling dimension from the thermal to shot noise cross-over, and observe a long-awaited agreement with expectations. Measurements are fitted to the predicted finite temperature expression involving both the quasiparticles scaling dimension and their charge, in contrast to previous charge investigations focusing on the high bias shot noise regime. A systematic analysis, repeated on multiple constrictions and experimental conditions, consistently matches the theoretical scaling dimensions for the fractional quasiparticles emerging at filling factors 1/3, 2/5 and 2/3. This establishes a central property of fractional quantum Hall anyons, and demonstrates a powerful and complementary window into exotic quasiparticles

Signature of anyonic statistics in the integer quantum Hall regime

Anyons are exotic low-dimensional quasiparticles whose unconventional quantum statistics extends the binary particle division into fermions and bosons. The fractional quantum Hall regime provides a natural host, with first convincing anyon signatures recently observed through interferometry and cross-correlations of colliding beams. However, the fractional regime is rife with experimental complications, such as an anomalous tunneling density of states, which impede the manipulation of anyons. Here we show experimentally that the canonical integer quantum Hall regime can provide a robust anyon platform. Exploiting the Coulomb interaction between two co-propagating quantum Hall channels, an electron injected into one channel splits into two fractional charges behaving as abelian anyons. Their unconventional statistics is revealed by negative cross-correlations between dilute quasiparticle beams. Similarly to fractional quantum Hall observations, we show that the negative signal stems from a time-domain braiding process, here involving the incident fractional quasiparticles and spontaneously generated electron-hole pairs. Beyond the dilute limit, a theoretical understanding is achieved via the edge magnetoplasmon description of interacting integer quantum Hall channels. Our findings establish that, counter-intuitively, the integer quantum Hall regime provides a platform of choice for exploring and manipulating quasiparticles with fractional quantum statistics.Comment: 6 pages, 4 figures, 4 Extended Data figures, Methods, Supplemental Informatio

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's inequality has been moved to a separate note; other martingale bounds are described in Caltech ACM Report 2011-0

Systematic stratospheric observations on the Antarctic continent at Dumont d'Urville

Results of different routine measurements performed in Dumont d'Urville (66 deg S, 140 deg E) since 1988 are presented. They include the seasonal variation of total ozone and NO2 as measured by a SAOZ UV-Visible spectrometer, Polar Stratospheric Cloud observations by a backscatter lidar and more recently, vertical ozone profiles by ECC sondes and ozone and aerosols stratospheric profiles by a DIAL lidar. The particular results of 1991 in relation with the volcanic events of Mount Pinatubo and Mount Hudson, and the position of the polar vortex over Dumont d'Urville are discussed