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arXiv:quant-ph/0211052 · Physical Review A

Schmidt numbers of low-rank bipartite mixed states

Hao Chen

Published Jun 3, 2003• 7 citations
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Abstract

Schmidt numbers of bipartite mixed states [B. M. Torhal and P. Horodecki, Phys. Rev. A 61, R040301 (2001)] characterize the minimum Schmidt ranks of pure states that are needed to construct such mixed states. Schmidt number is the minimum number of entangled degrees of freedom of a bipartite mixed state. We give a lower bound for the Schmidt numbers of low-rank bipartite mixed states and conclude that generic low-rank mixed states have relatively high Schmidt numbers and thus entangled.

DOI
10.1103/physreva.67.062301

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Schmidt numbers of bipartite mixed states [B. M. Torhal and P. Horodecki, Phys. Rev. A 61, R040301 (2001)] characterize the minimum Schmidt ranks of pure states that are needed to construct such mixed states. Schmidt number is the minimum number of entangled degrees of freedom of a bipartite mixed state. We give a lower bound for the Schmidt numbers of low-rank bipartite mixed states and conclude that generic low-rank mixed states have relatively high Schmidt numbers and thus entangled.
The Schmidt number of a mixed state characterizes the minimum Schmidt rank of the pure states needed to construct it. We investigate the Schmidt number of an arbitrary mixed state by studying Schmidt-number witnesses that detect it. We present a canonical form of such witnesses and provide constructive methods for their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in ${\mathcal{C}}^{3}\ensuremath{\bigotimes}{\mathcal{C}}^{3}$ have Schmidt number 2.
We study Schmidt rank for a vector (i.e., a pure state) and Schmidt number for a mixed state which are entanglement measures. We show that if a subspace of a certain bipartite system contains no vector of Schmidt rank ⩽k\leqslant k⩽k, then any state supported on that space has Schmidt number at least k+1k+1k+1. A construction of subspace of Cm⊗Cn\mathbb{C}^m \otimes \mathbb{C}^nCm⊗Cn of maximal dimension, which does not contain any vector of Schmidt rank less than 333, is given here.
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 characterized 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 [Terhal and Horodecki, 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.
A profound comprehension of quantum entanglement is crucial for the progression of quantum technologies. The degree of entanglement can be assessed by enumerating the entangled degrees of freedom, leading to the determination of a parameter known as the Schmidt number. In this paper, we develop an efficient analytical tool for characterizing high Schmidt number witnesses for bipartite quantum states in arbitrary dimensions. Our methods not only offer viable mathematical methods for constructing high-dimensional Schmidt number witnesses in theory but also simplify the quantification of entanglement and dimensionality. Most notably, we develop high-dimensional Schmidt number witnesses within arbitrary-dimensional systems, with our Schmidt witness coefficients relying solely on the operator Schmidt coefficient. Subsequently, we demonstrate our theoretical advancements and computational superiority by constructing Schmidt number witnesses in arbitrary dimensional bipartite quantum systems with Schmidt numbers four and five.