Uniquely defined geometric phase of an open system

Various types of unravelings of Lindblad master equation have been used to define the geometric phase for an open quantum system. Approaches of this type were criticized for lacking in unitary symmetry of the Lindblad equation [ A. Bassi and E. Ippoliti Phys. Rev. A 73 062104 (2006)]. We utilize quantum state diffusion (QSD) approach to demonstrate that a geometric phase invariant on the symmetries of the Lindblad equation can be defined. It is then shown that such a definition of the geometric phase could be either invariant on the decomposition of the initial mixed state or gauge invariant, but not both. This alternative is inherent to the definitions based on quantum trajectories. The QSD geometric phase is computed for a qubit in different types of environments.