Geometric Aspects of Entanglement Generating Hamiltonian Evolutions

Abstract

We examine the pertinent geometric characteristics of entanglement that arise from stationary Hamiltonian evolutions transitioning from separable to maximally entangled two-qubit quantum states. From a geometric perspective, each evolution is characterized by means of geodesic efficiency, speed efficiency, and curvature coefficient. Conversely, from the standpoint of entanglement, these evolutions are quantified using various metrics, such as concurrence, entanglement power, and entangling capability. Overall, our findings indicate that time-optimal evolution trajectories are marked by high geodesic efficiency, with no energy resource wastage, no curvature (i.e., zero bending), and an average path entanglement that is less than that observed in time-suboptimal evolutions. Additionally, when analyzing separable-to-maximally entangled evolutions between nonorthogonal states, time-optimal evolutions demonstrate a greater short-time degree of nonlocality compared to time-suboptimal evolutions between the same initial and final states. Interestingly, the reverse is generally true for separable-to-maximally entangled evolutions involving orthogonal states. Our investigation suggests that this phenomenon arises because suboptimal trajectories between orthogonal states are characterized by longer path lengths with smaller curvature, which are traversed with a higher energy resource wastage compared to suboptimal trajectories between nonorthogonal states. Consequently, a higher initial degree of nonlocality in the unitary time propagators appears to be essential for achieving the maximally entangled state from a separable state. Furthermore, when assessing optimal and suboptimal evolutions...

Figures (2)

• Figure 1: In (a), there is a plot of the temporal behavior of the entanglement of the path that connects nonorthogonal initial (separable) and final (maximally entangled) states $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 00\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\left[ \left\vert 00\right\rangle +\left\vert 11\right\rangle \right] /\sqrt{2}$, respectively, when the evolution is time optimal. Entanglement is specified by the concurrence C$_{\mathrm{opt}}^{\mathrm{nonortho}}\left( t\right)$. Moreover, the unitary evolution occurs with energy dispersion $\Delta E=E/\sqrt{2}$ and $0\leq t\leq\left( \pi\hslash\right) /(2\sqrt{2}E)$. In (b), there is a plot of the temporal behavior of the entanglement of the path that connects $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 00\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\left[ \left\vert 00\right\rangle +\left\vert 11\right\rangle \right] /\sqrt{2}$ when the evolution is time suboptimal. Entanglement is specified by the concurrence C$_{\mathrm{subopt}}^{\mathrm{nonortho}}\left( t\right)$. Moreover, the unitary evolution occurs with energy dispersion $\Delta E=E/\sqrt{2}$ and $0\leq t\leq\left( \pi\hslash\right) /(2E)$. In (c), there is a plot of the temporal behavior of the entanglement of the path that connects orthogonal initial (separable) and final (maximally entangled) states $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 00\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\left[ \left\vert 01\right\rangle +\left\vert 10\right\rangle \right] /\sqrt{2}$, respectively, when the evolution is time optimal. Entanglement is specified by the concurrence C$_{\mathrm{opt}}^{\mathrm{ortho}}\left( t\right)$. Moreover, the unitary evolution occurs with energy dispersion $\Delta E=(\sqrt{5/2})E$ and $0\leq t\leq\left( \pi\hslash\right) /(\sqrt{10}E)$. In (d), there is a plot of the temporal behavior of the entanglement of the path that connects $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 00\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\left[ \left\vert 01\right\rangle +\left\vert 10\right\rangle \right] /\sqrt{2}$ when the evolution is time suboptimal. Entanglement is specified by the concurrence C$_{\mathrm{subopt}}^{\mathrm{ortho}}\left( t\right)$. Moreover, the unitary evolution occurs with energy dispersion $\Delta E=(\sqrt{5/2})E$ and $0\leq t\leq\left( \pi\hslash\right) /E$. In all plots, we set $E=1$ and, finally, physical units are chosen using $\hslash=1$.
• Figure 2: Entanglement-based comparative analysis of two distinct optimal time evolutions (Example 2 and Example 3) from an initial (separable) state to a final (maximally entangled) state $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 01\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\frac{1+i}{2}\left\vert 01\right\rangle$$+\frac{1-i}{2}\left\vert 10\right\rangle$, respectively, with $\left\vert \left\langle A\left\vert B\right. \right\rangle \right\vert \neq0$. In (a), there is a plot of the temporal behavior of the entanglement of the paths that connect $\left\vert A\right\rangle$ and $\left\vert B\right\rangle$. State entanglement is specified by the concurrence C$\left( t\right)$ which exhibits an identical time behavior in both cases. Both unitary time evolutions occur with energy dispersion $\Delta E=2E$ and $0\leq t\leq\left( \pi\hslash\right) /(8E)$. In (b), the nonlocal character of the unitary time propagators is captured by Yukalov's entanglement production $\varepsilon_{\mathrm{EP}}^{\mathrm{Yukalov}}\left( t\right)$ versus time $t$ for Example 2 (thin solid line) and Example 3 (thick solid line). In (c), the entanglement capability of the unitary time propagators is characterized by Zanardi's entanglement power $\varepsilon_{\mathrm{EP}}^{\mathrm{Zanardi}}\left( t\right)$ versus time $t$ for Example 2 (thin solid line) and Example 3 (thick solid line). Observe that $\left[ \varepsilon_{\mathrm{EP}}^{\mathrm{Zanardi}}\left( \left( \pi\hslash\right) /(8E)\right) \right] _{\text{Example-}2}=\left[ \varepsilon_{\mathrm{EP}}^{\mathrm{Zanardi}}\left( \left( \pi\hslash\right) /(8E)\right) \right] _{\text{Example-}3}=1/6$. In all plots, we set $E=1$ and, finally, physical units are chosen using $\hslash=1$.