On the relation between Schmidt coefficients and entanglement

Abstract

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the `realignment' or `computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance -- with regard to the characterization of entanglement -- of the `symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

References in corpus (6)

• Entanglement monotones
• Some Properties of the Computable Cross Norm Criterion for Separability
• Covariance matrices and the separability problem
• Entanglement detection beyond the cross-norm or realignment criterion
• Bipartite quantum systems: on the realignment criterion and beyond
• A class of inequalities inducing new separability criteria for bipartite quantum systems

Cited by in corpus (8)

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• A note on the realignment criterion
• Classical Tensors and Quantum Entanglement II: Mixed States
• Channel discrimination power of bipartite quantum states
• Locality, Correlations, Information, and non-Hermitian Quantum Systems
• Trade-off between Information Gain and Disturbance in Local Discrimination of Entangled Quantum States