Measurement-Induced Chaos with Entangled States

The dynamics of an ensemble of identically prepared two-qubit systems is investigated which is subjected to the iteratively applied measurements and conditional selection of a typical entanglement purification protocol. The resulting dynamics exhibits strong sensitivity to initial conditions. For one class of initial states two types of islands characterize the asymptotic limit. They correspond to a separable and a fully entangled two-qubit state, respectively, and their boundaries form fractal-like structures. In the presence of incoherent noise an additional stable asymptotic cycle appears.

Figure 1

Schematic representation of one iteration step of the entanglement purification process. The diamond shaped elements denote acceptance of the pair of qubits only when both measurements yield 0. The depicted procedure is used to prepare from an ensemble $\mathcal{E}$ made up of pairs of qubits all in a state $|\psi ⟩$, an ensemble ${\mathcal{E}}^{\prime }$ made up of pairs of qubits in the state $|{\psi }^{\prime }⟩$. During the procedure, half of the pairs in $\mathcal{E}$ are completely used up, while a portion of the other half is retained depending on the measurement outcome. Here we depicted the special case $U=H\otimes H$, where $H$ is the Hadamard gate.

Figure 2

Convergence of initial states of Eq. (1) to the two limiting cycles as a function of the complex parameter $\zeta$. The dark (blue) color denotes convergence to the maximally entangled state of Eq. (4), and the light (green) color the convergence to the cycle of the two separable states in Eqs. (5, 6).

Figure 3

Convergence of mixed initial states $\rho (\lambda ,\zeta )$ of Eq. (7) to the three limiting cycles as a function of the complex parameter $\zeta$, at $\lambda =0.75$. The darkest (blue) and second-darkest (green) colors denote convergence to the same cycles as in Fig. 2, while the lightest (yellow) color denotes convergence to the separable mixed cycle in Eq. (8). As $\lambda$ is decreased from 1 to 0, the corresponding convergence plots gradually turn from those in Fig. 2 to uniform lightest (yellow) color.