The Structure of Qubit Unextendible Product Bases

Unextendible product bases have been shown to have many important uses in quantum information theory, particularly in the qubit case. However, very little is known about their mathematical structure beyond three qubits. We present several new results about qubit unextendible product bases, including a complete characterization of all four-qubit unextendible product bases, which we show there are exactly 1446 of. We also show that there exist p-qubit UPBs of almost all sizes less than $2^p$.Comment: 20 pages, 3 tables, 7 figure

Three-by-three bound entanglement with general unextendible product bases

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 x 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 the 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.Comment: 36 page

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.Comment: 24 pages RevTex, 15 figures; appendix removed, several small corrections, to appear in Comm. Math. Phy

Distinguishability of complete and unextendible product bases

It is not always possible to distinguish multipartite orthogonal states if only local operation and classical communication (LOCC) are allowed. We prove that we cannot distinguish the states of an unextendible product basis (UPB) by LOCC even when infinite resources (infinite-dimensional ancillas, infinite number of operations). Moreover we give a necessary and sufficient condition for the LOCC distinguishability of complete product bases.Comment: added necessary and sufficient condition for complete product bases, example Lagarias-Shor ten-parties complete basi

A Family of Indecomposable Positive Linear Maps based on Entangled Quantum States

We introduce a new family of indecomposable positive linear maps based on entangled quantum states. Central to our construction is the notion of an unextendible product basis. The construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension.Comment: 16 pages LaTex: updated and a derivation of a lower bound on epsilon is added and calculated for one of the examples. Submitted to Lin. Alg. and Its App

Unextendible Mutually Unbiased Bases from Pauli Classes

We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.Comment: 22 pages, v2: minor changes, references added; published versio