Reduced state uniquely defines the Groverian measure of the original pure state

Abstract

Groverian and geometric entanglement measures of the $n$-party pure state are expressed by the $(n\ensuremath{-}1)$-party reduced state density operator directly. This main theorem derives several important consequences. First, if two pure $n$-qudit states have reduced states of $(n\ensuremath{-}1)$-qudits, which are equivalent under local unitary 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 the 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.