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.
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.
One of the simplest possible quantum circuits, consisting of a cnot gate, a Hadamard gate, and a measurement on one of the outputs is known to lead to chaotic dynamics when applied iteratively on an ensemble of equally prepared qubits. The evolution of pure initial quantum states is characterized by a fractal (in the space of states), formed by the border of different convergence regions. We examine how the ideal evolution is distorted in the presence of both coherent error and incoherent initial noise, which are typical imperfections in current implementations of quantum computers. It is known that under the influence of initial noise only, the fractal is preserved, moreover, its dimension remains constant below a critical noise level. We systematically analyze the effect of coherent Hadamard gate errors by determining fixed points and cycles of the evolution. We combine analytic and numerical methods to explore to what extent the dynamics is altered by coherent errors in the presence of preparation noise as well. We show that the main features of the dynamics, and especially the fractal borders, are robust against the discussed noise, they will only be slightly distorted. We identify a range of error parameters, for which the characteristic properties of the dynamics are not significantly altered. Hence, our results allow to identify reliable regimes of operation of iterative protocols.
We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.
Complex chaos is specified by an iterated mapping on complex numbers. It has recently been found in the dynamics of qubits where each time step is conditioned on a measurement result on part of the system. We analyse the simplest case of one qubit dynamics with one complex parameter in some detail. We point out that two attractive cycles can exist and provide examples how the fractal like Julia set divides the areas of corresponding initial states. We show how to determine the set of parameters for which one, two or no stable fixed cycles exists and provide the numerically calculated images of the sets. The results can be relevant for the quantum state purification protocol based on the similar dynamics of two or more qubits and in general for any protocol based on conditioned nonlinear dynamics where truly chaotic behavior may occur.
We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics, including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.
Entanglement purification is a vital protocol to produce a high-quality entangled state from an ensemble of identical states. Based on the particular scheme of entanglement purification [Phys. Rev. A 87, 052316 (2013)], the effect of phase fluctuation is investigated. The convergence pattern of the initial states can be divided into two kinds of regions, corresponding to the purified outcomes being a maximally entangled state (MES) or a separable state. And there is the fractal-like structure near the boundary between these two regions. It is found that the phase noise plays a positive role in generating an MES for an initial states near some fractal boundary, which can only become the separable one if the noise is absent. It is also found that the minimal iteration steps to achieve the maximally entangled state with the phase noise can be decreased, which can save the resource in the protocol.