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In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement $$F(x)+\epsilon $$ may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union $$\mathbf{A}$$ of closed linear subspaces of a Hilbert space $$\mathbf{H}$$ from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple $$(\mathbf{A}, \mathbf{M}, \mathbf{H})$$ , and show that the minimizer provides a suboptimal approximation to the original sparse signal $$x^0\in \mathbf{A}$$ when the measurement map F has the sparse Riesz property and the almost linear property on $${\mathbf A}$$ . The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property. [ABSTRACT FROM AUTHOR]

Given a frame F = { f} for a separable Hilbert space H, we introduce the linear subspace $H^{p}_{F}$ of H consisting of elements whose frame coefficient sequences belong to the ℓ-space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as $H^{p}_{F}$-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in $H_{F}^{p}$ converges in both the Hilbert space norm and the ||·||-norm which is induced by the ℓ-norm. [ABSTRACT FROM AUTHOR]

We investigate the distillability of bipartite quantum states in terms of positive and completely positive maps. We construct the so-called generalized Choi states and show that it is distillable when it has negative partial transpose. We convert the distillability problem of 2-copy n × n Werner states into the determination of the positivity of an Hermitian matrix. We obtain several sufficient conditions by which the positivity holds. Further, we investigate the case n = 3 by the classification of 2 × 3 × 3 pure states. [ABSTRACT FROM AUTHOR]