Reduced State Uniquely Defines Groverian Measure of Original Pure State

TL;DR

This paper demonstrates that the Groverian and Geometric entanglement measures of an n-party pure state can be directly derived from its (n-1)-party reduced state, simplifying their computation and revealing key properties.

Contribution

It establishes that these entanglement measures depend solely on the reduced state and derives linear eigenvalue equations for three-qubit systems, simplifying their analysis.

Findings

Measures are invariant under local unitary transformations of reduced states.

Upper bounds for the measures are identified, with saturation only in two-qubit systems.

Linear eigenvalue equations are derived for three-qubit Groverian measures.

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.