On almost randomizing channels with a short Kraus decomposition

For large d, we study quantum channels on C^d obtained by selecting randomly N independent Kraus operators according to a probability measure mu on the unitary group U(d). When mu is the Haar measure, we show that for N>d/epsilon^2$, such a channel is epsilon-randomizing with high probability, which means that it maps every state within distance epsilon/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general mu, we obtain a epsilon-randomizing channel provided N > d (\log d)^6/epsilon^2$. For d=2^k (k qubits), this includes Kraus operators obtained by tensoring k random Pauli matrices. The proof uses recent results on empirical processes in Banach spaces.Comment: We added some background on geometry of Banach space

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

Learning Arbitrary Statistical Mixtures of Discrete Distributions

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, $\vartheta$, is a probability distribution over probability distributions $p$, where each such $p$ is a probability distribution over $[n] = \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$ directly, but only indirectly and in very noisy fashion, by sampling from $[n]$ repeatedly, independently $K$ times from the distribution $p$. The problem is to infer $\vartheta$ to high accuracy in transportation (earthmover) distance. We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution $\vartheta$. We bound the quality of the solution as a function of the size of the samples $K$ and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.Comment: 23 pages. Preliminary version in the Proceeding of the 47th ACM Symposium on the Theory of Computing (STOC15

Recent Cases

This is a summary of the case law from 1965

Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. {}.Comment: 18 p., princeton/ecel/7-12-9

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

Spectroscopic investigation of the deeply buried Cu In,Ga S,Se 2 Mo interface in thin film solar cells

The Cu In,Ga S,Se 2 Mo interface in thin film solar cells has been investigated by surface sensitive photoelectron spectroscopy, bulk sensitive X ray emission spectroscopy, and atomic force microscopy. It is possible to access this deeply buried interface by using a suitable lift off technique, which allows to investigate the back side of the absorber layer as well as the front side of the Mo back contact. We find a layer of Mo S,Se 2 on the surface of the Mo back contact and a copper poor stoichiometry at the back side of the Cu In,Ga S,Se 2 absorber. Furthermore, we observe that the Na content at the Cu In,Ga S,Se 2 Mo interface as well as at the inner grain boundaries in the back contact region is significantly lower than at the absorber front surfac

Covering convex bodies by cylinders and lattice points by flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d

Improvement of composition of CdTe thin films during heat treatment in the presence of CdCl2

CdCl2 treatment is a crucial step in development of CdS/CdTe solar cells. Although this rocessing step has been used over a period of three decades, full understanding is not yet achieved. This paper reports the experimental evidence for improvement of composition of CdTe layers during CdCl2 treatment. This investigation makes use of four selected analytical techniques; Photo-electro-chemical (PEC) cell, X-ray diffraction (XRD), Raman spectroscopy and Scanning electron microscopy (SEM). CdTe layers used were electroplated using three Cd precursors; CdSO4, Cd(NO3)2 and CdCl2. Results show the improvement of stoichiometry of CdTe layers during CdCl2 treatment through chemical reaction between Cd from CdCl2 and elemental Te that usually precipitate during CdTe growth, due to its natural behaviour. XRD and SEM results show that the low-temperature (~85ºC) electroplated CdTe layers consist of ~(20-60) nm size crystallites, but after CdCl2 treatment, the layers show drastic recrystallisation with grains becoming a few microns in size. These CdCl2 treated layers are then comparable to high temperature grown CdTe layers by the size of grains