Hierarchies among genuine multipartite entangling capabilities of quantum gates

TL;DR

This work introduces a hierarchical framework to classify unitary quantum gates by their genuine multipartite entangling power (GME) when acting on $k$-separable inputs, quantified via the Generalized Geometric Measure (GGM). By refining the input state sets to mutually disjoint $\mathbb{S}^k$ and optimizing over these inputs, the authors reveal nontrivial hierarchies and invariances under local unitaries, showing that complex amplitudes can enhance entanglement generation. They systematically analyze diagonal, permutation, and Haar-random unitaries across three to five qubits, identifying cases where maximal GME ($G_{\max}=0.5$) is achievable and specifying the optimal input states and parameter constraints that realize it; they also demonstrate that initial bipartite entanglement does not always aid GME production. The statistical study shows that, on average, biseparable inputs often outperform fully separable ones for certain gate classes, that complex inputs generally boost entangling power, and that Haar-random gates tend to generate higher GME as system size grows, highlighting practical routes for gate design in multipartite quantum information tasks.

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.

Figures (7)

• Figure 1: (Color online.) A schematic diagram of the notion of hierarchies among entangling powers of a fixed unitary operator, $U$, based on the inputs chosen. Each column represents a different set of separable states. From left to right, set of inputs are chosen to be $N$-separable (i.e., fully separable) to $k$-separable. The parties entangled are marked with blue circles while the brown circles are for the rest. After the action of $U$ on $k$-separable inputs, the maximum genuine multipartite entanglement of the resulting states, $\mathcal{E}^k_{\max}$ in Eq. (\ref{['eq:maindef']}), is computed, which is represented by the vertical green color. Here, the superscript represents the class of separable states considered for maximization. Deeper green indicates more expected GME created in the outputs.
• Figure 2: (Color online.) Enhancement in entanglement generation with initial entanglement. Biseparable inputs are better than the fully separable ones when a special diagonal unitary matrix, $U_D= \hbox{diag}(1,1,1,1,1,1, 1, e^{i\phi})$ with $\phi \in (0, 2 \pi)$Fenner acts on three-party input states. The maximum GGM produced (ordinate) against $\phi$ (abscissa). The optimization is performed over the set of biseparable (circles) and fully separable (stars) states, denoted as $\mathbb{S}^3$ and $\mathbb{S}^2$ respectively, to obtain $G_{\max}$. Both the axes are dimensionless.
• Figure 3: (Color online.) Normalized frequency distribution, $f_{G_{\max}}$ (vertical axis) as calculated via Eq. (\ref{['eq:Nfreq']}) against $G_{\max}$ (horizontal axis) where the unitaries are chosen to be diagonal, $U_D^{gen}$ by generating $\phi_i \in [0, 2\pi]$ randomly from uniform distribution. Stars correspond to $f_{G_{\max}}$, obtained by optimizing over the set of fully separable states, $\mathbb{S}^2$, circles represent the distribution when optimizing over all biseparable states in the $1:23$ bipartition, $\mathbb{S}^3$ and the squares depict the same when the maximization is restricted to the set of biseparable states in $1:23$ bipartition with only real coefficients (having vanishing $\xi'_i)$s), denoted as $\mathbb{S}^2_R$. The patterns of $f_{G_{\max}}$ do not alter if bipartitions of the biseparable states get changed and if the coefficients of the fully separable states are real. Both the axes are dimensionless.
• Figure 4: (Color online.) Contour plots in slices of multipartite entangling power, $G_{\max}^3$ of $U_{sp}^1$, given in Eq. (\ref{['eq:spUFarook']}) with respect to the parameters $J_x$ ($x$-axis), $J_y$ ($x$-axis) and $J_z$ ($z$-axis). The optimization is performed over the set of fully separable states. Both the axes are dimensionless.
• Figure 5: (Color online.) Biseparable vs fully separable states as inputs. When a special unitary operator, given in Eq. (\ref{['eq:spU']}) acts on fully separable (triangles) or biseparable states (circles), the maximum GGM, $G_{\max}$ (ordinate) produced is plotted with respect to $J_x$ (abscissa) for fixed $J_y = J_z =0.1$. This is an example of an unitary operator for which higher GGM can be obtained from fully separable inputs compared to the biseparable ones for high values of $J_x \sim \pi/4$ provided the coefficients of the biseparable states are chosen to be real. Both the axes are dimensionless.