aqora

The super operator system structures and their applications in quantum entanglement theory

Blerina Xhabli

arXiv:1107.2366Journal of Functional AnalysisNov 16, 201121 citations

Operator system structures on ordered spaces

Vern I. Paulsen, Ivan G. Todorov, Mark Tomforde

Given an Archimedean order unit space (V, V+, e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system 𝒮 is completely boundedly isomorphic to either OMIN(𝒮) or to OMAX(𝒮). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.Given an Archimedean order unit space (V, V+, e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system 𝒮 is completely boundedly isomorphic to either OMIN(𝒮) or to OMAX(𝒮). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.

arXiv:0904.3783Proceedings of the London Mathematical SocietyMay 28, 201092 citations

Rank in Matrix Analysis: On the Preservers of Maximally Entangled States and Fractional Minimal Rank

Benjamin Großmann

For Hilbert spaces X , Y, the set of maximally entangled states, MES X ,Y , is a set of rank-1 positive semidefinite operators over the space X ⊗ Y.In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X , Y.After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES X ,Y , which is a set of low-rank positive semidefinite operators over X ⊗ Y.We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices.We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the "minimal rank" and "triangular minimal ranks" differ in general.For Hilbert spaces X , Y, the set of maximally entangled states, MES X ,Y , is a set of rank-1 positive semidefinite operators over the space X ⊗ Y.In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X , Y.After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES X ,Y , which is a set of low-rank positive semidefinite operators over X ⊗ Y.We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices.We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the "minimal rank" and "triangular minimal ranks" differ in general.

Jan 1, 20190 citations

Maximal vectors in Hilbert space and quantum entanglement

William Arveson

arXiv:0804.1140Journal of Functional AnalysisSep 4, 200818 citations

Rank in matrix analysis

Benjamin Großmann, Hugo J. Woerdeman

For Hilbert spaces X,Y, the set of maximally entangled states, MES_{X,Y}, is a set of rank-1 positive semidefinite operators over the space X [o times] Y. In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X,Y. After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES_{X,Y}, which is a set of low-rank positive semidefinite operators over X [o times] Y. We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices. We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the ''minimal rank'' and ''triangular minimal ranks'' differ in general.For Hilbert spaces X,Y, the set of maximally entangled states, MES_{X,Y}, is a set of rank-1 positive semidefinite operators over the space X [o times] Y. In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X,Y. After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES_{X,Y}, which is a set of low-rank positive semidefinite operators over X [o times] Y. We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices. We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the ''minimal rank'' and ''triangular minimal ranks'' differ in general.

Jun 1, 20190 citations

Geometry of quantum state space and entanglement

Prasenjit Deb, Pratapaditya Bej

arXiv:1805.11292Quantum Information ProcessingJan 28, 20193 citations

Minimal and maximal operator spaces and operator systems in entanglement theory

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