Modal decomposition of light has been known for a long time, applied mostly to pattern recognition. With the commercialization of liquid-crystal devices, digital holography as an enabling tool has become accessible to all, and with it all-digital tools for the decomposition of light have finally come of age. We review recent advances in unravelling the properties of light, from the modal structure of laser beams to decoding the information stored in orbital angular momentum (OAM)-carrying fields. We show application of these tools to fiber lasers, solid-state lasers, and structured light created in the laboratory by holographic laser beam shaping. We show by experimental implementation how digital holograms may be used to infer the intensity, phase, wavefront, Poynting vector, polarization, and OAM density of some unknown optical field. In particular, we outline how virtually all the previous ISO-standard beam diagnostic techniques may be readily replaced with all-digital equivalents, thus paving the way for unravelling of light in real time. Such tools are highly relevant to the in situ analysis of laser systems, to mode division multiplexing as an emerging tool in optical communication, and for quantum information processing with entangled photons.

Creation and detection of optical modes with spatial light modulators

In the context of quantifying entanglement we study those functions of a multipartite state which do not increase under the set of local transformations. A mathematical characterization of these monotone magnitudes is presented. They are then related to optimal strategies of conversion of shared states. More detailed results are presented for pure states of bipartite systems. It is show that more than one measure are required simultaneously in order to quantify completely the non-local resources contained in a bipartite pure state, while examining how this fact does not hold in the so-called asymptotic limit. Finally, monotonicity under local transformations is proposed as the only natural requirement for measures of entanglement.

On the characterization of entanglement

Confirmatory factor analysis (CFA) has been frequently applied to executive function measurement since first used to identify a three-factor model of inhibition, updating, and shifting; however, subsequent CFAs have supported inconsistent models across the life span, ranging from unidimensional to nested-factor models (i.e., bifactor without inhibition). This systematic review summarized CFAs on performance-based tests of executive functions and reanalyzed summary data to identify best-fitting models. Eligible CFAs involved 46 samples (N = 9,756). The most frequently accepted models varied by age (i.e., preschool = one/two-factor; school-age = three-factor; adolescent/adult = three/nested-factor; older adult = two/three-factor), and most often included updating/working memory, inhibition, and shifting factors. A bootstrap reanalysis simulated 5,000 samples from 21 correlation matrices (11 child/adolescent; 10 adult) from studies including the three most common factors, fitting seven competing models. Model results were summarized as the mean percent accepted (i.e., average rate at which models converged and met fit thresholds: CFI ≥ .90/RMSEA ≤ .08) and mean percent selected (i.e., average rate at which a model showed superior fit to other models: ΔCFI ≥ .005/.010/ΔRMSEA ≤ -.010/-.015). No model consistently converged and met fit criteria in all samples. Among adult samples, the nested-factor was accepted (41-42%) and selected (8-30%) most often. Among child/adolescent samples, the unidimensional model was accepted (32-36%) and selected (21-53%) most often, with some support for two-factor models without a differentiated shifting factor. Results show some evidence for greater unidimensionality of executive function among child/adolescent samples and both unity and diversity among adult samples. However, low rates of model acceptance/selection suggest possible bias toward the publication of well-fitting but potentially nonreplicable models with underpowered samples. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

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The unity and diversity of executive functions: A systematic review and re-analysis of latent variable studies.

Entanglement is an important resource for quantum technologies. There are many ways quantum systems can be entangled, ranging from the two-qubit case to entanglement in high dimensions or between many parties. Consequently, many entanglement quantifiers and classifiers exist, corresponding to different operational paradigms and mathematical techniques. However, for most quantum systems, exactly quantifying the amount of entanglement is extremely demanding, if at all possible. Furthermore, it is difficult to experimentally control and measure complex quantum states. Therefore, there are various approaches to experimentally detect and certify entanglement when exact quantification is not an option. The applicability and performance of these methods strongly depend on the assumptions regarding the involved quantum states and measurements, in short, on the available prior information about the quantum system. In this Review, we discuss the most commonly used quantifiers of entanglement and survey the state-of-the-art detection and certification methods, including their respective underlying assumptions, from both a theoretical and an experimental point of view. Entanglement is often considered the defining feature separating classical physics from quantum physics and provides the basis for many quantum technologies. This Review discusses recent progress in the challenging task of conclusively proving that a physical system features entanglement, surveying detection and certification methods.

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Entanglement certification from theory to experiment

We consider reversible work extraction from identical quantum systems. From an ensemble of individually passive states, work can be produced only via global unitary (and thus entangling) operations. However, we show here that there always exists a method to extract all possible work without creating any entanglement, at the price of generically requiring more operations (i.e., additional time). We then study faster methods to extract work and provide a quantitative relation between the amount of generated multipartite entanglement and extractable work. Our results suggest a general relation between entanglement generation and the power of work extraction.

Entanglement generation is not necessary for optimal work extraction.

We investigate correlations among complementary observables. In particular, we show how to take advantage of mutually unbiased bases for the efficient detection of entanglement in arbitrarily high-dimensional, multipartite, and continuous-variable quantum systems. The introduced entanglement criteria are relatively easy to implement experimentally since they require only a few local measurement settings. In addition, we establish a link between the separability problem and the maximum number of mutually unbiased bases---opening an additional avenue in this long-standing open problem.

Entanglement detection via mutually unbiased bases

We introduce an intuitive measure of genuine multipartite entanglement, which is based on the well-known concurrence. We show how lower bounds on this measure can be derived and also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.

Measure of genuine multipartite entanglement with computable lower bounds

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way. We show explicitly how any element of can be composed of matrix exponential functions of generalized anti-symmetric ?-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank k can be formulated. Our construction can also be used to derive an orthonormal basis of any k-dimensional subspaces of with the minimal number of parameters. As an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the m-concurrence).

A composite parameterization of unitary groups, density matrices and subspaces

We adopt the concept of the composite parameterization of the unitary group U(d) to the special unitary group SU(d). Furthermore, we also consider the Haar measure in terms of the introduced parameters. We show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d). With regard to possible applications of our results, we consider the computation of high-order integrals over unitary groups.

Composite parameterization and Haar measure for all unitary and special unitary groups

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group $\mathcal{U}(d)$ of arbitrary dimension $d$ which is constructed in a composite way. We show explicitly how any element of $\mathcal{U}(d)$ can be composed of matrix exponential functions of generalized anti-symmetric $\sigma$-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank $k$ can be formulated. Our construction can also be used to derive an orthonormal basis of any $k$-dimensional subspaces of $\mathbb{C}^d$ with the minimal number of parameters. As an example it will be shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the $m$-concurrence).

Christoph Spengler

Papers

Entanglement detection via mutually unbiased bases

Measure of genuine multipartite entanglement with computable lower bounds

A composite parameterization of unitary groups, density matrices and subspaces

Composite parameterization and Haar measure for all unitary and special unitary groups

A composite parameterization of unitary groups, density matrices and subspaces