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Jan Lang

Publications and source records attributed to Jan Lang.

27 records · Page 2Linked to original sources
arXiv2022

Non-local temporal interferometry for highly resilient free-space quantum communication

Entanglement distribution via photons over long distances enables many applications, including quantum key distribution (QKD), which provides unprecedented privacy. The inevitable degradation of entanglement through noise accumulated over long distances remains one of the key challenges in this area. Exploiting the potential of higher-dimensional entangled photons promises to address this challenge, but poses extreme demands on the experimental implementation. Here, we present an interstate free-space quantum link, distributing hyper-entanglement over $10.2\,$km with flexible dimensionality of encoding by deploying a phase-stable non-local Franson interferometer. With this distribution of multidimensional energy-time entangled photons, we analyse the achievable key rate in a dimensionally-adaptive QKD protocol that can be optimized with respect to any environmental noise conditions. Our approach enables and emphasises the power of high-dimensional entanglement for quantum communication, yielding a positive asymptotic key rate well into the dawn of the day.

quant-ph↗
arXiv2022

Embeddings between Lorenz sequence spaces are strictly singular

Given $0<p,q, r<\infty $ and $ q<r\le \infty$ we consider the natural embedding $\ell_{p,q}\hookrightarrow \ell_{p,r}$ between Lorenz sequence spaces. We prove that this non-compact embedding is always strictly singular but not finitely strictly singular.

math.FA↗
arXiv2022

Embedding between Lebesgue and weak Lebesgue sequence spaces is strictly singular

Given $1< p < q < \infty$ it is well know that the natural embedding of Lebesgue sequence spaces $\ell_p \hookrightarrow \ell_q$ is strictly singular. In this paper we extend this classical results and show that even the natural non-compact embedding between Lebesgue and weak Lebesgue sequence spaces $\ell_{p}\hookrightarrow \ell_{p,\infty}$ is strictly singular.

math.FA↗
arXiv2020

Compactness of Sobolev embeddings and decay of norms

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed of its decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.

math.FA↗
arXiv2020

Maximal non-compactness of Sobolev embeddings

It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces $L^{p,\infty}$ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as $p<\infty$. Finally, we show that if the target space is $L^{\infty}$ (which formally is also a weak Lebesgue space with $p=\infty$), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.

math.FA↗
arXiv2020

Measure of noncompactness of Sobolev embeddings on strip-like domains

We compute the precise value of the measure of noncompactness of Sobolev embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on strip-like domains $D$ of the form $\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict $s$-numbers of the embeddings in question coincide with their norms. We also prove that the maximal noncompactness of Sobolev embeddings on strip-like domains remains valid even when Sobolev-type spaces built upon general rearrangement-invariant spaces are considered. As a by-product we obtain the explicit form for the first eigenfunction of the pseudo-$p$-Laplacian on an $n$-dimensional rectangle.

math.FA↗
arXiv2017

Strict s-numbers of non-compact Sobolev embeddings into continuous functions

For limiting non-compact Sobolev embeddings into continuous functions we study behavior of Approximation, Gelfand, Kolmogorov, Bernstein and Isomorphism s-numbers. In the one dimensional case the exact values of the above-mentioned strict s-numbers were obtained and in the higher dimensions sharp estimates for asymptotic behavior of strict s-numbers were established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption then our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting case when some of them (Approximation, Gelfand and Kolmogorov) have positive lower bound and others (Bernstein and Isomorphism) are decreasing to zero. From our results also follows that such limiting non-compact Sobolev embeddings are finitely strictly singular maps.

math.FA↗
arXiv2017

Strict s-numbers of the Volterra operator

For Volterra operator $V\colon L^1(0,1) \to C[0,1]$ and summation operator $σ\colon \ell^1 \to c$, we obtain exact values of Approximation, Gelfand, Kolmogorov, Mityagin and Isomorphism numbers.

math.FA↗