Narrow Results

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 in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation to other distances is investigated. As an illustrative application, the proposed metric is evaluated for one-qubit mixed states.

The effect of filtering operation with respect to purification and concentration of entanglement in quantum states are discussed in this paper. It is shown, through examples, that the local action of the filtering operator on a part of the composite quantum state allows for purification of the remaining part of the state. The redistribution of entanglement in the subsystems of a noise affected state is shown to be due to the action of local filtering on the non-decohering part of the system. The varying effects of the filtering parameter, on the entanglement transfer between the subsystems, depending on the choice of the initial quantum state is illustrated.