Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure

The Groverian entanglement measure $G\left(\psi \right)$ is applied to characterize pure quantum states $\mid \psi ⟩$ of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state $\mid \psi ⟩$ as an initial state for the search algorithm. A convenient parametrization is presented, which allows us to calculate the Groverian measure analytically for certain states of high symmetry. A numerical procedure is used in order to calculate it for arbitrary pure states of multiple qubits. Using the Groverian measure to evaluate the entanglement produced by quantum algorithms may provide useful insight into the role of entanglement in making quantum algorithms powerful. Here we calculate $G\left(\psi \right)$ for the intermediate states generated during the evolution of Grover’s algorithm for various initial states and for different sets of marked states. It is shown that Grover’s iterations generate highly entangled states in intermediate stages of the quantum search process, even if the initial state and the target state are product states.

Figure 1

The success probability $P_{\mathrm{s}}\left(\psi \right)$ of Grover’s algorithm using $\mid \psi ⟩$ as the initial state (dashed line) and the maximal success probability $P_{\max }\left(\psi \right)$ (solid line) for $\mid \psi ⟩=a_{\eta }\mid \eta ⟩+a_{\text{GHZ}}\mid {\psi }_{\text{GHZ}}⟩$ as a function of $a_{\text{GHZ}}^2$. The two functions coincide up to $a_{\text{GHZ}}^2\simeq 0.65$, while above this point a gap appears. The gap broadens as the GHZ state is approached, demonstrating the effect of the maximization by the local preprocessing.

Figure 2

Analytical results (solid line) and numerical results (open circles) for the Groverian entanglement measure of generalized GHZ states [Eq. (47)] as a function of ${\mid a_0\mid }^2$. The largest value of $G\left(\psi \right)=1∕\sqrt{2}$ is obtained for ${\mid a_0\mid }^2=1∕2$, where the state coincides with the GHZ state.

Figure 3

The Groverian entanglement measure $G\mathbf{\left(}\psi \right(t\left)\mathbf{\right)}$ for states that are generated by Grover’s algorithm as a function of the number of iterations $t$ for $n=12$ and one marked state, where the initial state is $\mid \psi \left(0\right)⟩$. The curves were obtained for the states $\mid \psi \left(0\right)⟩$ given by Eq. (64) with $a_{\text{even}}=1∕\sqrt{2}$—namely, the $\mid \eta ⟩$ state (dashed line), 0.984 (dotted line), 0.994 (dash-dotted line), and 1—namely, the state $H^{\otimes n}\mid {\psi }_{\text{GHZ}}⟩$ (solid line).

Figure 4

The Groverian measure for the states $\mid \psi \left(t\right)⟩$ obtained after $t$ Grover iterations where $\mid \psi \left(0\right)⟩=\mid \eta ⟩$, for 12 qubits and two marked states. In the case that $\mid 0⟩$ and $\mid N-1\mid$ are marked (solid line) the resulting state after $\tau$ iterations is $\mid {\psi }_{\text{GHZ}}⟩$. If, instead, the marked states are $\mid 0⟩$ and $\mid 1⟩$ (dashed line), the register approaches a superposition of these two states, which is not entangled.

Figure 5

The measure $G\mathbf{\left(}\psi \right(t\left)\mathbf{\right)}$ for states that are generated by Grover’s algorithm as a function of the number of iterations $t$ for $n=12$ and 12 marked states, where the initial state is $\mid \eta ⟩$. When the marked states are $\mid 00\cdots 01⟩$, $\mid 00\cdots 10⟩$, $\cdots$, $\mid 10\cdots 0⟩$ (solid line), the state obtained after 14 iterations is the $W$ state (up to a tiny correction due to the discrete nature of the iterations). When the marked states are $\mid i⟩$, $i=0,\dots ,11$ (dashed line), the resulting state is not as strongly entangled.