Separability for mixed states with operator Schmidt rank two

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. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.

12 pages. v2: We thank A. Mueller-Hermes, M. Studinski and N. Johnston for pointing out to us that the first main result of this paper (our Theorem 1) had been first proven by D. Cariello using other techniques. We have updated our manuscript accordingly, have changed the title, and slightly modified the abstract. v3: minor changes. Accepted for publication in Quantum. v4: modified license

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Cited by in corpus (3)

• Analyzing quantum entanglement with the Schmidt decomposition in operator space
• Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices
• Schmidt rank constraints in Quantum Information Theory