Unextendible product bases and the construction of inseparable states

Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a finite set B of separable orthonormal states |phi(k)> such that the non-empty space B9perp), the set of vectors orthogonal to B, contains no separable projection. Examples of orthogonal UPB sets were first constructed by Bennett et al [1] and other examples appear, for example, in [2] and [3]. If F denotes the set of convex combinations of the projections |phi(k)><phi(k)|, then F is a face in the set S of separable densities. In this note we show how to use F to construct families of positive partial transform states (PPT) which are not separable. We also show how to make an analogous construction when the condition of orthogonality is dropped. The analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable PPT states from S.Comment: to appear in Lin. Alg. App

Entanglement Measures under Symmetry

We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for UU-invariant states, and we find a counterexample to the additivity conjecture for the relative entropy of entanglement.Comment: RevTeX,16 pages,9 figures, reference added, proof of monotonicity corrected, results unchange

Linear functions and duality on the infinite polytorus

We consider the following question: Are there exponents $2<p<q$ such that the Riesz projection is bounded from $L^q$ to $L^p$ on the infinite polytorus? We are unable to answer the question, but our counter-example improves a result of Marzo and Seip by demonstrating that the Riesz projection is unbounded from $L^\infty$ to $L^p$ if $p\geq 3.31138$. A similar result can be extracted for any $q>2$. Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.Comment: This paper has been accepted for publication in Collectanea Mathematic