Revisit mixed state geometric phase

Abstract

Quantum computation (QC) offers more efficient ways for certain tasks than classical computers. In recent years, it has witnessed remarkable experimental attention in the various physical two-level systems (e.g., trapped-ion, Cavity QED, nuclear spin, Josef Junction, photons). Unfortunately, processors working according to the rules of quantum mechanics are extremely delicate objects. On the one hand, the unavoidable coupling with the uncontrollable environment, which would bring about the undesired decoherence, spoils the unitary nature of the dynamical evolution and transforms the pure states to mixed states. On the other hand, extreme capabilities in quantum state control are required, since even very small manipulation imperfections will eventually drive the processing system into a "wrong" output state. Therefore for the purpose of robust quantum computation, some strategies (namely, quantum error-correction, error avoiding, and error-suppression techniques) have been developed at the theoretical level. But all of these strategies require extra physical resources in terms of either qubits or additional manipulations. A further, conceptually fascinating, strategy for the stabilization of quantum information is provided by the topological approach. In such QIP schemes, gate operations depend on topological features of geometric phase instead on the trace of the loops that are actually realized, and are therefore largely insensitive to local inaccuracies and fluctuations. It is this built-in fault-tolerant features that lead scientists to investigate various schemes for robust quantumcomputation. Although there is no ambiguity in defining pure state geometric phase and applying it to quantumcomputation, the unavoidable decoherence compels us to consider the case of mixed state. The problem is, there exist different definitions of mixed state geometric phase. In this article, we firstly give a brief introduction on the pure state geometric phase, and then mainly discuss the situation of mixed state under unitary evolution. We concern with the pure components of the mixed state. For each pure component we consider the corresponding parallel transport. For different components we consider the relationship between one and another. Based on the latter consideration wepropose symmetric and anti-symmetric evolution, which respectively lead to above mentioned two different results of mixed state geometric phase existing so far. At the end of this article, we make a brief report on our recent experimental observation of mixed state geometric phase. © 2011 by Nova Science Publishers, Inc. All rights reserved.