Entangled bases with fixed Schmidt number

Abstract

An entangled basis with fixed Schmidt number $k$ (EBk) is a set of orthonormal basis states with the same Schmidt number $k$ in a product Hilbert space ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$. It is a generalization of both the product basis and the maximally entangled basis. We show here that, for any $k\le \min \{d,d^{\prime }\}$, EBk exists in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for any $d$ and $d^{\prime }$. Consequently, general methods of constructing SEBk (EBk with the same Schmidt coefficients) and EBk (but not SEBk) are proposed. Moreover, we extend the concept of EBk to multipartite case and find out that the multipartite EBk can be constructed similarly.

7 pages. Minors are corrected

References in corpus (12)

• Quantum entanglement
• Entanglement detection
• Unextendible Product Bases and Bound Entanglement
• The Uniqueness Theorem for Entanglement Measures
• The state space for two qutrits has a phase space structure in its core
• On Multipartite Pure-State Entanglement
• Higher order Schmidt decompositions
• A special simplex in the state space for entangled qudits
• Unextendible maximally entangled bases and mutually unbiased bases
• Unextendible maximally entangled bases in ${\mathbb{C}}^d⨂{\mathbb{C}}^{d^{\prime }}$
• Unextendible maximally entangled bases in dxd
• Unextendible entangled bases with fixed Schmidt number

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• Constructions of Unextendible Maximally Entangled Bases in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\)
• Volume of the set of locally diagonalizable bipartite states
• Special unextendible entangled bases with continuous integer cardinality
• Schmidt rank constraints in Quantum Information Theory
• Constructions of mutually unbiased entangled bases
• Constructions of unextendible entangled bases
• Spin exchange of two spin-1/2 atoms