Reduced State Uniquely Defines Groverian Measure of Original Pure State

Abstract

Groverian and Geometric entanglement measures of the n-party pure state are expressed by the (n-1)-party reduced state density operator directly. This main theorem derives several important consequences. First, if two pure n-qudit states have reduced states of (n-1)-qudits, which are equivalent under local unitary(LU) transformations, then they have equal Groverian and Geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two qubit systems. Third, it converts effectively the nonlinear eigenvalue problem for three qubit Groverian measure into linear eigenvalue equations. Some typical solutions of these linear equations are written explicitly and the features of the general solution are discussed in detail.

arXiv commenttitle is changed, proofs are generalized to qudits and simplified, new analytic and numeric results are presented, new references are added, PRA version

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Cited by in corpus (19)

• Multiqubit symmetric states with high geometric entanglement
• The maximally entangled symmetric state in terms of the geometric measure
• Geometric measure of entanglement for pure multipartite states
• Three-Tangle for Rank-3 Mixed States: mixture of Greenberger-Horne-Zeilinger, W and flipped W states
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• Additivity and non-additivity of multipartite entanglement measures
• Classification of Entanglement in Symmetric States
• A comparison of old and new definitions of the geometric measure of entanglement
• Reconstructing quantum states from single-party information
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• Exploring Non-Multiplicativity in the Geometric Measure of Entanglement
• Local Purity Distillation in Quantum Systems: Exploring the Complementarity Between Purity and Entanglement
• Role of Three-Qubit Mixed-States Entanglement in Teleportation Scheme