Papers by Łukasz Skowronek
arXiv (Cornell University), May 10, 2013
Journal of Mathematical Physics, 2016
Linear Algebra and its Applications, 2011
In the finite-dimensional case, we present a new approach to the theory of cones with a mapping c... more In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio lkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.
Journal of Mathematical Physics, 2011
We discuss the subject of Unextendible Product Bases with the orthogonality condition dropped and... more We discuss the subject of Unextendible Product Bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e. states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal Unextendible Product Basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general Unextendible Product Bases, nevertheless they can always be locally transformed to the orthogonal form. This fully confirms surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's Theorem from algebraic geometry.
International Journal of Quantum Information, 2010
We present a survey on mathematical topics relating to separable states and entanglement witnesse... more We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank...
Arxiv preprint arXiv: …, 2009
Abstract: We study operators acting on a composite Hilbert space and investigate their local nume... more Abstract: We study operators acting on a composite Hilbert space and investigate their local numerical range, local spectral radius and local $C$--spectral radius. Concrete bounds for the local numerical range for Hermitian operators are derived. Local numerical range of a non-Hermitian ...
Schedae Informaticae, 2018
We investigate performance of a gradient descent optimization (GR) applied to the traffic signal ... more We investigate performance of a gradient descent optimization (GR) applied to the traffic signal setting problem and compare it to genetic algorithms. We used neural networks as metamodels evaluating quality of signal settings and discovered that both optimization methods produce similar results, e.g., in both cases the accuracy of neural networks close to local optima depends on an activation function (e.g., TANH activation makes optimization process converge to different minima than ReLU activation).
Acta Physica Polonica Series B
We show analytically that Newtonian iterations, when applied to a polynomial equation, have a pos... more We show analytically that Newtonian iterations, when applied to a polynomial equation, have a positive topological entropy. In a specific example of an attempt to ``find'' the real solutions of the equation $x^2+1=0$, we show that the Newton method is chaotic. We analytically find the invariant density and show how this problem relates to that of a piecewise linear map.
Linear Algebra and its Applications, 2011
We study operators acting on a tensor product Hilbert space and investigate their product numeric... more We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.
Linear Algebra and its Applications, 2013
We answer in the affirmative a recently-posed question that asked if there exists an "untypical" ... more We answer in the affirmative a recently-posed question that asked if there exists an "untypical" convex mapping cone-i.e., one that does not arise from the transpose map and the cones of k-positive and k-superpositive maps. We explicitly construct such a cone based on atomic positive maps. Our general technique is to consider the smallest convex mapping cone generated by a single map, and we derive several results on such mapping cones. We use this technique to also present several other examples of untypical mapping cones, including a family of cones generated by spin factors. We also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality.
Journal of Physics A: Mathematical and Theoretical, 2009
We link the study of positive quantum maps, block positive operators, and entanglement witnesses ... more We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Journal of Mathematical Physics, 2010
Numerical range of a Hermitian operator X is defined as the set of all possible expectation value... more Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers for instance the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
Journal of Mathematical Physics, 2009
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert spa... more The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamio lkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies. 1 Let us emphasize here the difference between k-positive maps defined for an integer k and K-positive maps [12, 24], in which K denotes a certain cone of operators.
Journal of Functional Analysis, 2012
We study positive maps of B (K) into B (H) for finite-dimensional Hilbert spaces K and H. Our mai... more We study positive maps of B (K) into B (H) for finite-dimensional Hilbert spaces K and H. Our main emphasis is on how Choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps. Special attention will be given to entanglement properties and k-positive maps, in particular tensor products of 2-positive maps. The latter problem is directly related to the question of n-copy distillability of quantum states, for which we obtain a partial result.
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Papers by Łukasz Skowronek