Quantum Noise and Decoherence

No practical system is realistically closed. A system is naturally subjected to its interactions with the surrounding system, which is commonly called the environment. There is also a more fundamental reason for the notion of an open quantum system in quantum mechanics. The theory of quantum mechanics is intrinsically probabilistic, meaning that the verification of any quantum principle should be tested statistically by taking repeated measurements and incorporating the resulting data. The measurement process inevitably requires coupling the system to a measuring device. Moreover, for quantum computation and more generally for quantum information processing, we desire preparation, manipulation, and measurement of quantum states. All those procedures require the system to be coupled to external equipment.

In principle, one can regard the combined system enclosing both the system and the environment as a closed system, and thus apply the quantum mechanical principles to the total system. However, the environment is typically large—and since perfect isolation is impossible, the total system is eventually the whole universe—with a huge number of degrees of freedom. A complete microscopic description incorporating the environmental degrees of freedom is not only impractical but also of little use. First of all, such a description is tremendously complicated and hard to solve. A solution, if any, would lead to an intractable amount of information, the vast majority of which would be irrelevant to the physical effects exhibited by the system itself.

A more reasonable and practical approach is thus to seek an effective description of open quantum systems in terms of only the system’s degrees of freedom. An effective theory is achieved in two stages: First, ignorance of the environmental degrees of freedom brings about a statistical mixture of pure states for the system. The state of the system is no longer a

pure state

. It is a

mixed state

and described by a density operator. Second, the influence of the environment should be reflected on the (effective) dynamical evolution of the density operator in a way that does not depend on the details of the environment and of the system-environment coupling. A powerful mathematical tool is provided by the formalism of quantum operations.

In this collection of tutorial documents, we first take toy models to examine the decoherence process on the elementary and qualitative level. We then introduce quantum operations formalism. The two common and complementary representations of quantum operations are discussed together with simple examples. Quantum operations are used not only for the dynamical processes of open quantum systems but also for the quantum theory of generalized measurement. Next, we turn to the quantum master equation approach to open quantum systems. This is an approximate approach for quantum operations formalism under the Markovian assumption. While quantum operations formalism provides the most general mathematical tool, it is not always possible to find explicit quantum operations for given specific systems. It is far simpler and more insightful to construct the quantum master equation and thus examine the solution to understand the behavior of the open quantum systems in question. In the remaining part, we introduce distance measures between quantum states.