Extending Schmidt vector from pure to mixed states for characterizing entanglement

• URL: http://arxiv.org/abs/2407.17199v1
• Date: Wed, 24 Jul 2024 12:00:09 GMT
• Title: Extending Schmidt vector from pure to mixed states for characterizing entanglement
• Authors: F. Meroi, M. Losada, G. M. Bosyk,
• Abstract summary: We extend the concept of Schmidt vector from pure to mixed bipartite states. We show that the Schmidt vector fully characterizes separable and maximally entangled states. We introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this study, we enhance the understanding of entanglement transformations and their quantification by extending the concept of Schmidt vector from pure to mixed bipartite states, exploiting the lattice structure of majorization. The Schmidt vector of a bipartite mixed state is defined using two distinct methods: as a concave roof extension of Schmidt vectors of pure states, or equivalently, from the set of pure states that can be transformed into the mixed state through local operations and classical communication (LOCC). We demonstrate that the Schmidt vector fully characterizes separable and maximally entangled states. Furthermore, we prove that the Schmidt vector is monotonic and strongly monotonic under LOCC, giving necessary conditions for conversions between mixed states. Additionally, we extend the definition of the Schmidt rank from pure states to mixed states as the cardinality of the support of the Schmidt vector and show that it is equal to the Schmidt number introduced in previous work [Phys. Rev. A 61, 040301 (R), 2000]. Finally, we introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.

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