Via state purification The key idea in Uhlmann's approach to the mixed state geometric phase is to lift the system' s density operator acting on the hilbert space to an extended hilbert space by attaching an ancilla p=7r()(q) P=入的|+入的 =)81a+2)82)a Then the geometric phase depends on the dynamics of the ancilla Mixed State Geometric Phases, Entangled Systems, and Local Unitary Transformations Marie Ericsson, Arun K. Pati, Erik Sjoqvist, Johan Brannlund, and Daniel K. L. Oi Phys.Rev.et91,090405(2003)15 Via state purification • The key idea in Uhlmann’s approach to the mixed state geometric phase is to lift the system’s density operator acting on the Hilbert space to an extended Hilbert space by attaching an ancilla. Then the geometric phase depends on the dynamics of the ancilla.    =  Tr(| |) Mixed State Geometric Phases, Entangled Systems, and Local Unitary Transformations Marie Ericsson, Arun K. Pati, Erik Sjöqvist, Johan Brännlund, and Daniel K. L. Oi Phys. Rev. Lett. 91, 090405 (2003) 1 1 1 1 1 2 | | | |, | | |1 | | 2 a a         =  +   =   +  