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Laser Physics

Geometric phase for mixed states

L. C. Kwek, D. M. Tong, J. L. Chen, Jiangfeng Du, Keng Wah Choo, Ravishankar Ramanathan, Dagomir Kaszlikowski, C. H. Oh

Published Feb 1, 2006• 8 citations
Publisher

Abstract

We present a kinematic approach for obtaining the geometric phase of a mixed state under nonunitary evolution. This phase is gauge-invariant and measurable and leads to well-known results for unitary evolution.
DOI
10.1134/s1054660x06020344

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Comment on Geometric phases for mixed states during cyclic evolutions

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It is shown that a recently suggested concept of mixed state geometric phase in cyclic evolutions (Fu L-B and Chen J-L 2004 J. Phys. A: Math. Gen. 37 3699 ) is gauge dependent.

arXiv:quant-ph/0404174•Journal of Physics A Mathematical and General•Jul 8, 2004•4 citations
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
We consider arbitrary mixed state in unitary evolution and provide a comprehensive description of corresponding geometric phase in which two different points of view prevailing currently can be unified. Introducing an ancillary system and considering the purification of given mixed state, we find that different results of mixed state geometric phase correspond to different choice of the representation of Hilbert space of the ancilla. Moreover we demonstrate that in order to obtain Uhlmann's geometric phase it is not necessary to resort to the unitary evolution of ancilla.