Some Remarks on the Entanglement Number

Abstract

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

The Existence of Barren Plateaus in The Detection of Quantum Entanglement

“…(2) It can be shown that the function µ given by Equation ( 1) is the largest convex function defined on the set of all states of H, which is less than or equal to the function µ on the set of pure states of H. While at first glance it's not clear that the infimum in Equation ( 1) is attained, it has indeed been proven [8,9] that the infimum is attained as long as the measure µ is norm continuous on the set of unit vectors of H, (i.e. on the set of pure states of H).…”