Separability for mixed states with operator Schmidt rank two

Gemma De las Cuevas1, Tom Drescher2, and Tim Netzer2

1Institute for Theoretical Physics, Technikerstr. 21a, A-6020 Innsbruck, Austria
2Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria

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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.

The study of entanglement criteria is a “classic” problem in quantum information theory. Here we provide a new proof of the fact that if a mixed bipartite state can be expressed as a sum of two elementary tensor factors, then it is separable. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable. 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", Journal of Mathematical Physics 61 4, 041901 (2020).

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