Existence of product vectors and their partial conjugates in a pair of spaces

Abstract

Let $D$ and $E$ be subspaces of the tensor product of the $m$ and $n$ dimensional complex spaces, with codimensions $k$ and \ell$,respectively.Weshowthatif$k+\ell<m+n-2$thentheremustexistaproductvectorin$D$whosepartialconjugateliesin$E$.If$k+\ell >m+n-2$thentheremaynotexistsuchaproductvector.If$k+\ell=m+n-2$thenbothcasesmayoccurdependingon$k$and$\ell$.

15 pages, to correct a technical problem in V2

References in corpus (10)

• Generic local distinguishability and completely entangled subspaces
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• Unextendible product bases and extremal density matrices with positive partial transpose
• Description of rank four PPT entangled states of two qutrits
• Separability, entanglement and full families of commuting normal matrices
• Low rank positive partial transpose states and their relation to product vectors
• Bipartite states of low rank are almost surely entangled
• The probability of entanglement

Cited by in corpus (13)

• Facial structures for various notions of positivity and applications to the theory of entanglement
• Separable states with unique decompositions
• Diagonal unitary and orthogonal symmetries in quantum theory
• Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
• Properties and construction of extreme bipartite states having positive partial transpose
• Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
• Faces for two qubit separable states and the convex hulls of trigonometric moment curves
• Product vectors in the ranges of multi-partite states with positive partial transposes and permanents of matrices
• Random covariant quantum channels
• Entangled edge states of corank one with positive partial transposes
• Global geometric difference between separable and Positive partial transpose states
• The number of product vectors and their partial conjugates in a pair of spaces