1Institute for Theoretical Physics, Technikerstr. 21a, A-6020 Innsbruck, Austria
2Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria
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Abstract
The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.
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Cited by
[1] Gemma De las Cuevas and Tim Netzer, “Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices”, arXiv:1907.03664.
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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Source: https://quantum-journal.org/papers/q-2019-12-02-203/
