Quantum Operation - Background

Background

The Schrödinger picture provides a satisfactory account of time evolution of state for a quantum mechanical system under certain assumptions. These assumptions include

• The system is non-relativistic
• The system is isolated.

The Schrödinger picture for time evolution has several mathematically equivalent formulations. One such formulation expresses the time rate of change of the state via the Schrödinger equation. A more suitable formulation for this exposition is expressed as follows:

The effect of the passage of t units of time on the state of an isolated system S is given by a unitary operator U_t on the Hilbert space H associated to S.

This means that if the system is in a state corresponding to v ∈ H at an instant of time s, then the state after t units of time will be U_t v. For relativistic systems, there is no universal time parameter, but we can still formulate the effect of certain reversible transformations on the quantum mechanical system. For instance, state transformations relating observers in different frames of reference are given by unitary transformations. In any case, these state transformations carry pure states into pure states; this is often formulated by saying that in this idealized framework, there is no decoherence.

For interacting (or open) systems, such as those undergoing measurement, the situation is entirely different. To begin with, the state changes experienced by such systems cannot be accounted for exclusively by a transformation on the set of pure states (that is, those associated to vectors of norm 1 in H). After such an interaction, a system in pure state φ may no longer be in the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ₁,..., φ_k with respective probabilities λ₁,..., λ_k. The transition from a pure state to a mixed state is known as decoherence.

Numerous mathematical formalisms have been established to handle the case of an interacting system. The quantum operation formalism emerged around 1983 from work of K. Kraus, who relied on the earlier mathematical work of M. D. Choi. It has the advantage that it expresses operations such as measurement as a mapping from density states to density states. In particular, the effect of quantum operations stays within the set of density states.