Relative phase change during quantum operations

Abstract

Quantum operations represented by completely positive maps encompass many 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 a quantum operation. We find that the relative phase shift of a system not only depends on the initial state of the system, but also depends on the initial state of an 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 “in-phase quantum channels.” We will present the relative phase shifts for a qubit undergoing both a depolarizing channel and complete randomization, and discuss their implications.