Quantum Games

In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of quantum strategies.Comment: 13 pages (LaTeX), 3 figure

Fidelity and Concurrence of conjugated states

We prove some new properties of fidelity (transition probability) and concurrence, the latter defined by straightforward extension of Wootters notation. Choose a conjugation and consider the dependence of fidelity or of concurrence on conjugated pairs of density operators. These functions turn out to be concave or convex roofs. Optimal decompositions are constructed. Some applications to two- and tripartite systems illustrate the general theorem.Comment: 10 pages, RevTex, Correction: Enlarged, reorganized version. More explanation

Trade-off between multiple-copy transformation and entanglement catalysis

We demonstrate that multiple copies of a bipartite entangled pure state may serve as a catalyst for certain entanglement transformations while a single copy cannot. Such a state is termed a "multiple-copy catalyst" for the transformations. A trade-off between the number of copies of source state and that of the catalyst is also observed. These results can be generalized to probabilistic entanglement transformations directly.Comment: Essentially the journal version. 7 pages, no figures. Minor correction

Multiple-copy entanglement transformation and entanglement catalysis

We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A \textbf{65}, 052315 (2002)] can be implemented by a suitable entanglement-assisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. \textbf{83}, 3566 (1999)]. Furthermore, we show that the combination of multiple-copy entanglement transformation and the entanglement-assisted one is still equivalent to the pure entanglement-assisted one. The mathematical structure of multiple-copy entanglement transformations then is carefully investigated. Many interesting properties of multiple-copy entanglement transformations are presented, which exactly coincide with those satisfied by the entanglement-assisted ones. Most interestingly, we show that an arbitrarily large number of copies of state should be considered in multiple-copy entanglement transformations.Comment: 11 pages, RevTex 4. Main results unchanged. Journal versio

Catalysis in non--local quantum operations

We show how entanglement can be used, without being consumed, to accomplish unitary operations that could not be performed with out it. When applied to infinitesimal transformations our method makes equivalent, in the sense of Hamiltonian simulation, a whole class of otherwise inequivalent two-qubit interactions. The new catalysis effect also implies the asymptotic equivalence of all such interactions.Comment: 4 pages, revte

Positive maps, majorization, entropic inequalities, and detection of entanglement

In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map $\Lambda:M_{d}(\mathbb{C})\to M_{d}(\mathbb{C})$ can be written as the difference between two completely positive maps $\Lambda=\Lambda_{1}-\Lambda_{2}$, we propose a possible way to generalize the Nielsen--Kempe majorization criterion. Then we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of the Schur--concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen--Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map $\Lambda$, then there exist entropic inequalities derived from $\Lambda$ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [I\ot\Lambda](\varrho_{AB})\geq 0. Moreover, our inequalities provide a way of constructing multi--copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of recently introduced protocol of state merging and possibility of approximating the mean value of a linear entanglement witness.Comment: the published version, 25 pages in NJP format, 6 figure

Quantifying Graft Detachment after Descemet's Membrane Endothelial Keratoplasty with Deep Convolutional Neural Networks

Purpose: We developed a method to automatically locate and quantify graft detachment after Descemet's Membrane Endothelial Keratoplasty (DMEK) in Anterior Segment Optical Coherence Tomography (AS-OCT) scans. Methods: 1280 AS-OCT B-scans were annotated by a DMEK expert. Using the annotations, a deep learning pipeline was developed to localize scleral spur, center the AS-OCT B-scans and segment the detached graft sections. Detachment segmentation model performance was evaluated per B-scan by comparing (1) length of detachment and (2) horizontal projection of the detached sections with the expert annotations. Horizontal projections were used to construct graft detachment maps. All final evaluations were done on a test set that was set apart during training of the models. A second DMEK expert annotated the test set to determine inter-rater performance. Results: Mean scleral spur localization error was 0.155 mm, whereas the inter-rater difference was 0.090 mm. The estimated graft detachment lengths were in 69% of the cases within a 10-pixel (~150{\mu}m) difference from the ground truth (77% for the second DMEK expert). Dice scores for the horizontal projections of all B-scans with detachments were 0.896 and 0.880 for our model and the second DMEK expert respectively. Conclusion: Our deep learning model can be used to automatically and instantly localize graft detachment in AS-OCT B-scans. Horizontal detachment projections can be determined with the same accuracy as a human DMEK expert, allowing for the construction of accurate graft detachment maps. Translational Relevance: Automated localization and quantification of graft detachment can support DMEK research and standardize clinical decision making.Comment: To be published in Translational Vision Science & Technolog

Body mass index, waist circumference, waist-hip ratio, and glucose intolerance in Chinese and Europid adults in Newcastle, UK.

OBJECTIVE: To compare the prevalence of glucose intolerance (impaired glucose tolerance and diabetes), and its relationship to body mass index (BMI) and waist-hip ratio in Chinese and Europid adults. DESIGN: This was a cross sectional study. SETTING: Newcastle upon Tyne. SUBJECTS: These comprised Chinese and Europid men and women, aged 25-64 years, and resident in Newcastle upon Tyne, UK. MAIN OUTCOME MEASURES: Two hour post load plasma glucose concentration, BMI, waist circumference, and waist-hip ratio. METHODS: Population based samples of Chinese and European adults were recruited. Each subject had a standard WHO oral glucose tolerance test. RESULTS: Complete data were available for 375 Chinese and 610 Europid subjects. The age adjusted prevalences of glucose intolerance in Chinese and Europid men were 13.0% (p = 0.04). Mean BMIs were lower in Chinese men (23.8 v 26.1) and women (23.5 v 26.1) than in the Europids (p values < 0.001), as were waist circumferences (men, 83.3 cm v 90.8, p < 0.001; women, 77.3 cm v 79.2, p < 0.05). Mean waist-hip ratios were lower in Chinese men (0.90 v 0.91, p = 0.02) but higher in Chinese women (0.84 v 0.78, p < 0.001) compared with Europids. In both Chinese and Europid adults, higher BMI, waist circumference, and waist-hip ratio were associated with glucose intolerance. CONCLUSIONS: The prevalence of glucose intolerance in Chinese men and women, despite lower BMIs, is similar to or higher than that in local Europid men and women and intermediate between levels found in China and those in Mauritius. It is suggested that an increase in mean BMI to the levels in the Europid population will be associated with a substantial increase in glucose intolerance in Chinese people

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.Comment: 28 page

Characterizing mixing and measurement in quantum mechanics

What fundamental constraints characterize the relationship between a mixture $\rho = \sum_i p_i \rho_i$ of quantum states, the states $\rho_i$ being mixed, and the probabilities $p_i$? What fundamental constraints characterize the relationship between prior and posterior states in a quantum measurement? In this paper we show that there are many surprisingly strong constraints on these mixing and measurement processes that can be expressed simply in terms of the eigenvalues of the quantum states involved. These constraints capture in a succinct fashion what it means to say that a quantum measurement acquires information about the system being measured, and considerably simplify the proofs of many results about entanglement transformation.Comment: 12 page