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The group symmetries inherent in quantum channels often make them tractable and applicable to various problems in quantum information theory. In this paper, we introduce natural probability distributions for covariant quantum channels. Specifically, this is achieved through the application of ``twirling operations'' on random quantum channels derived from the Stinespring representation that use Haar-distributed random isometries. We explore various types of group symmetries, including unitary and orthogonal covariance, hyperoctahedral covariance, diagonal orthogonal covariance (DOC), and analyze their properties related to quantum entanglement based on the model parameters. In particular, we discuss the threshold phenomenon for positive partial transpose and entanglement breaking properties, comparing thresholds among different classes of random covariant channels. Finally, we contribute to the PPT$^2$ conjecture by showing that the composition between two random DOC channels is generically entanglement breaking., Comment: minor modifications

In this paper, we study $k$-positivity and Schmidt number under standard orthogonal group symmetries. The Schmidt number is a natural quantification of entanglement in quantum information theory. First of all, we exhibit a complete characterization of all orthogonally covariant $k$-positive maps. This generalizes earlier results in [Tom85]. Furthermore, we optimize duality relations between $k$-positivity and Schmidt numbers under compact group symmetries. This new framework enables us to efficiently compute the Schmidt numbers of all orthogonally invariant quantum states.

We introduce a new technique for generating retinal fundus images that have anatomically accurate vascular structures, using diffusion models. We generate artery/vein masks to create the vascular structure, which we then condition to produce retinal fundus images. The proposed method can generate high-quality images with more realistic vascular structures and can create a diverse range of images based on the strengths of the diffusion model. We present quantitative evaluations that demonstrate the performance improvement using our method for data augmentation on vessel segmentation and artery/vein classification. We also present Turing test results by clinical experts, showing that our generated images are difficult to distinguish with real images. We believe that our method can be applied to construct stand-alone datasets that are irrelevant of patient privacy., Comment: 9 pages, 6 figures

One of the most fundamental questions in quantum information theory is PPT-entanglement of quantum states, which is an NP-hard problem in general. In this paper, however, we prove that all PPT $(\overline{\pi}_A\otimes \pi_B)$-invariant quantum states are separable if and only if all extremal unital positive $(\pi_B,\pi_A)$-covariant maps are decomposable where $\pi_A,\pi_B$ are unitary representations of a compact group and $\pi_A$ is irreducible. Moreover, an extremal unital positive $(\pi_B,\pi_A)$-covariant map $\mathcal{L}$ is decomposable if and only if $\mathcal{L}$ is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form $$\Phi(\rho)=a\frac{\text{Tr}(\rho)}{d}\text{Id}_d+ b\rho+c\rho^T+(1-a-b-c)\text{diag}(\rho)$$ are entanglement-breaking, and that all A-BC PPT $(U\otimes \overline{U}\otimes U)$-invariant tripartite quantum states are A-BC separable. The former strengthens some conclusions in [VW01,KMS20], and the latter provides a strong contrast to the fact that there exist PPT-entangled $(U\otimes U\otimes U)$-invariant tripartite Werner states [EW01] and resolves some open questions raised in [COS18].

We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group $G$ with a symplectic structure determined by a 2-cocycle on $G$. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form $G= F\times\hat{F}$ endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of $n$-bosonic modes, $n$-qudit systems with odd $d\ge 3$, and $p$-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem \cite{Gro} to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space $\mathbb{T}^n\times\mathbb{Z}^n$ and fermionic/hard-core bosonic systems with phase space $\mathbb{Z}^{2n}_2$ (which are not 2-regular), and completely characterize their Gaussian states., Comment: Some parts of the article arXiv:2004.13860 have been included and the article arXiv:2004.13860 will not be published anywhere

We use a noncommutative generalization of Fourier analysis to define a broad class of pseudo-probability representations, which includes the known bosonic and discrete Wigner functions. We characterize the groups of quantum unitary operations which correspond to phase-space transformations, generalizing Gaussian and Clifford operations. As examples, we find Wigner representations for fermions, hard-core bosons, and angle-number systems., Comment: It has been specified that we are recovering Mukunda's Wigner function for angle-number system. There have been some typo corrections

In the case of two-dimensional gradient index cavities designed by the conformal transformation optics, we propose a boundary integral equation method for the calculation of resonant mode functions by employing a fictitious space which is reciprocally equivalent to the physical space. Using the Green's function of the interior region of the uniform index cavity in the fictitious space, resonant mode functions and their far-field distributions in the physical space can be obtained. As a verification, resonant modes in lima\c{c}on-shaped transformation cavities were calculated and mode patterns and far-field intensity distributions were compared with those of the same modes obtained from the finite element method., Comment: 13 pages, 6 figures

Fundoscopic images are often investigated by ophthalmologists to spot abnormal lesions to make diagnoses. Recent successes of convolutional neural networks are confined to diagnoses of few diseases without proper localization of lesion. In this paper, we propose an efficient annotation method for localizing lesions and a CNN architecture that can classify an individual finding and localize the lesions at the same time. Also, we introduce a new loss function to guide the network to learn meaningful patterns with the guidance of the regional annotations. In experiments, we demonstrate that our network performed better than the widely used network and the guidance loss helps achieve higher AUROC up to 4.1% and superior localization capability., Comment: 8 pages, Computational Pathology and Ophthalmic Medical Image Analysis, pp.176-184

Retinal images have the highest resolution and clarity among medical images. Thus, vessel analysis in retinal images may facilitate early diagnosis and treatment of many chronic diseases. In this paper, we propose a novel multi-scale residual convolutional neural network structure based on a \emph{scale-space approximation (SSA)} block of layers, comprising subsampling and subsequent upsampling, for multi-scale representation. Through analysis in the frequency domain, we show that this block structure is a close approximation of Gaussian filtering, the operation to achieve scale variations in scale-space theory. Experimental evaluations demonstrate that the proposed network outperforms current state-of-the-art methods. Ablative analysis shows that the SSA is indeed an important factor in performance improvement., Comment: 10 pages, 7 figures

Retinal vessel segmentation is an indispensable step for automatic detection of retinal diseases with fundoscopic images. Though many approaches have been proposed, existing methods tend to miss fine vessels or allow false positives at terminal branches. Let alone under-segmentation, over-segmentation is also problematic when quantitative studies need to measure the precise width of vessels. In this paper, we present a method that generates the precise map of retinal vessels using generative adversarial training. Our methods achieve dice coefficient of 0.829 on DRIVE dataset and 0.834 on STARE dataset which is the state-of-the-art performance on both datasets., Comment: 9 pages, submitted to DLMIA 2017