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This description is formulated in terms of the density operator description of a quantum mechanical system.
The Schrödinger picture provides a satisfactory account of time evolution of state for a quantum mechanical system under certain assumptions. These assumptions include
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:
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 Ut 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 systems 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 the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ1,..., φk with respective probabiliies λ1,..., λk. This state of affairs is sometimes expressed by saying that the system experiences 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.
In the following remarks, we will refer to the logical and statistical structure of quantum theory, in particular to the orthocomplemented lattice Q of propositions (or yes no questions); this is the space of self-adjoint projections on a separable complex Hilbert space H. Recall that a density operator is a non-negative operator on H of trace 1.
Mathematically, a quantum operation is a linear map γ on the space of trace class operators on H to itself such that
and which is non-negative, then
is also non-negative.
Note that by the first condition quantum operations may not preserve the normalization property of statistical ensembles. In probabilistic terms, quantum operations may be sub-markovian .
Theorem. Let γ be a quantum operation on the trace class operators of H. Then there is a sequence of bounded linear operators {Bi}i ∈ N on H such that
Conversely, any map γ of this form is a quantum operation provided
This theorem is a variant of the Stinespring factorization theorem and follows easily from a result of M. Choi. This is also proved in the Nielsen and Chuang reference, Theorem 8.1.
In case H has finite dimension n, the sequence can be assumed to have only n2 non-zero entries.
The operators Bi are referred to by physicists as Kraus matrices (or more accurately as Kraus operators). Kraus matrices are not uniquely determined by the quantum operation γ, although all systems of Kraus matrices which represent the same quantum operation are related by a unitary transformation:
Theorem. Let γ be a quantum operation on the trace class operators of H with two representing sequences of Kraus matrices {Bi}i∈N and {Ci}i∈N. Then there is an infinite scalar unitary matrix ui j such that
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