Three-Qubit Groverian Measure

TL;DR

This paper analytically computes the Groverian measure for various three-qubit states, expressing results through local-unitary invariants and discussing implications for understanding quantum entanglement.

Contribution

It provides analytical expressions for the Groverian measure in multiple three-qubit state types and explores their invariance properties, advancing the understanding of quantum entanglement measures.

Findings

Analytical expressions derived for several three-qubit state types.

Groverian measure expressed in terms of local-unitary invariants.

Identified invariance of the measure with respect to phase factors in certain states.

Abstract

The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in type 4 should be independent of the phase factor $\phi$, which appear in the three-qubit state $|\psi>$. This fact with geometric interpretation on the Groverian measure may enable us to derive the analytical expressions for general arbitrary three-qubit states in near future.