Relative phase change during quantum operation ∗

Quantum operations represented by completely positive maps encompass many of the physical processes and have been very powerful in describing quantum computation and information processing tasks. We introduce the notion of relative phase change for a quantum system undergoing quantum operation. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The relative phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a ‘memory’ to a quantum channel. Also the notion of relative phase shift helps us to define what we call ‘inphase quantum channels’. We will present the relative phase shift for a qubit undergoing depolarizing channel and complete randomization and discuss their implications. ∗Invited talk presented at First Feynman Festival on Quantum Computation at University of Maryland, College Park from August 23-28th, 2002 under the title “Quantum phase during quantum operation”.