Multipartite entangling power by von Neumann entropy

TL;DR

This paper introduces measures for multipartite entangling, assisted entangling, and disentangling power, providing analytical formulas and conditions for their maximum, with applications to common multi-qubit gates like Toffoli and Fredkin.

Contribution

It generalizes entangling power to multipartite systems, derives analytical expressions for Schmidt-rank-two unitaries, and analyzes entangling capabilities of key multi-qubit gates.

Findings

Entangling power of Schmidt-rank-two unitaries derived analytically.

Maximum assisted entangling power conditions established.

Entangling power quantified for Toffoli and Fredkin gates.

Abstract

Quantifying the entanglement generation of a multipartite unitary operation is a key problem in quantum information processing. We introduce the definition of multipartite entangling, assisted entangling, and disentangling power, which is a natural generalization of the bipartite ones. We show that they are assumed at a specified quantum state. We analytically derive the entangling power of Schmidt-rank-two multi-qubit unitary operations by the minimal convex sum of modulo-one complex numbers. Besides we show the necessary and sufficient condition that the assisted entangling power of Schmidt-rank-two unitary operations reaches the maximum. We further investigate the widely-used multi-qubit gates, for example, the entangling and assisted entangling power of the $n$-qubit Toffoli gate is one ebit. The entangling power of the three-qubit Fredkin gate is two ebits, and that of the four-qubit Fredkin gate is in two to $\log_25$ ebits.