Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices

We study six natural decompositions of mixed states in one spatial dimension: the Matrix Product Density Operator (MPDO) form, the local purification form, the separable decomposition (for separable states), and their three translational invariant (t.i.) analogues. For bipartite states diagonal in t...

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Bibliographic Details
Published in:arXiv.org 2020-04
Main Authors: De las Cuevas, Gemma, Netzer, Tim
Format: Article
Language:English
Subjects:
Decomposition
Economic models
Factorization
Matrices (mathematics)
Operators (mathematics)
Online Access:Get full text
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Summary:We study six natural decompositions of mixed states in one spatial dimension: the Matrix Product Density Operator (MPDO) form, the local purification form, the separable decomposition (for separable states), and their three translational invariant (t.i.) analogues. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorisations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorisation, the nonnegative factorisation, and the positive semidefinite factorisation. We also show that a symmetric version of these decompositions corresponds to the symmetric factorisation, the completely positive factorisation, and the completely positive semidefinite transposed factorisation, respectively. We leverage this correspondence to characterise the six decompositions of mixed states.
ISSN:2331-8422
DOI:10.48550/arxiv.1907.03664