Hierarchies among genuine multipartite entangling capabilities of quantum gates

Abstract

We classify quantum gates according to their capability to generate genuine multipartite entanglement (GME), using a hierarchy based on multipartite separable states. In particular, when a fixed unitary operator acts on the set of k-separable states, the maximal genuine multipartite entanglement content produced via that particular unitary operator is determined after maximizing over the set of k-separable input states. We identify unitary operators that are beneficial for generating high GME when the input states are entangled in some bipartition, although the picture can also be reversed, where such initial entanglement offers no advantage. We investigate the maximum entangling power of a broad range of unitary operators, encompassing special classes of quantum gates, as well as diagonal, permutation, and Haar-uniformly generated unitaries by computing generalized geometric measure (GGM) as a GME quantifier. Additionally, we observe a notable distinction in entangling power based on the nature of the input states: when maximization is restricted to separable states with real coefficients, the entangling power is lower than when the optimization is carried out over arbitrary separable states with complex coefficients, thereby highlighting the role of complex amplitudes in entanglement creation. Furthermore, we determine which unitary operators, along with their corresponding optimal inputs, yield output states with the highest achievable GGM.

v1: 8 pages, 8 figures; v2: 11 pages, 6 figures, new sections including new results added; v3: 15 pages, 7 figures, several new results included, two new Authors added; v4: 16 pages, 7 figures, typos corrected, results unchanged, close to the published version

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