This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan.
[1] The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment.
Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving.
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.
In general it will be in a statistical mix of a sequence of pure states φ1, ..., φk with respective probabilities λ1, ..., λk.
Numerous mathematical formalisms have been established to handle the case of an interacting system.
In particular, the effect of quantum operations stays within the set of density states.
In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving.
The Stinespring factorization theorem extends the above result to arbitrary separable Hilbert spaces H and G. There, S is replaced by a trace class operator and
For example, different Cholesky factorizations of the Choi matrix might give different sets of Kraus operators.
be a (not necessarily trace-preserving) quantum operation on a finite-dimensional Hilbert space H with two representing sequences of Kraus matrices
It is a consequence of Stinespring's theorem that all quantum operations can be implemented by unitary evolution after coupling a suitable ancilla to the original system.
These results can be also derived from Choi's theorem on completely positive maps, characterizing a completely positive finite-dimensional map by a unique Hermitian-positive density operator (Choi matrix) with respect to the trace.
Such canonical set of orthogonal Kraus operators can be obtained by diagonalising the corresponding Choi matrix and reshaping its eigenvectors into square matrices.
There also exists an infinite-dimensional algebraic generalization of Choi's theorem, known as "Belavkin's Radon-Nikodym theorem for completely positive maps", which defines a density operator as a "Radon–Nikodym derivative" of a quantum channel with respect to a dominating completely positive map (reference channel).
It is used for defining the relative fidelities and mutual informations for quantum channels.
For a non-relativistic quantum mechanical system, its time evolution is described by a one-parameter group of automorphisms {αt}t of Q.
This can be narrowed to unitary transformations: under certain weak technical conditions (see the article on quantum logic and the Varadarajan reference), there is a strongly continuous one-parameter group {Ut}t of unitary transformations of the underlying Hilbert space such that the elements E of Q evolve according to the formula The system time evolution can also be regarded dually as time evolution of the statistical state space.
This can be easily generalized: If G is a connected Lie group of symmetries of Q satisfying the same weak continuity conditions, then the action of any element g of G is given by a unitary operator U:
In the general case, measurements can be made using non-orthogonal operators, via the notions of POVM.
The non-orthogonal case is interesting, as it can improve the overall efficiency of the quantum instrument.
This set of questions can be understood to be chosen from an orthocomplemented lattice Q of propositions in quantum logic.
The lattice is equivalent to the space of self-adjoint projections on a separable complex Hilbert space H. Consider a system in some state S, with the goal of determining whether it has some property E, where E is an element of the lattice of quantum yes-no questions.
Measurement, in this context, means submitting the system to some procedure to determine whether the state satisfies the property.
When an observable A has a pure point spectrum, it can be written in terms of an orthonormal basis of eigenvectors.
Repeated measurements, made on a statistical ensemble S of systems, results in a probability distribution over the eigenvalue spectrum of A.
Shaji and Sudarshan argued in a Physical Review Letters paper that, upon close examination, complete positivity is not a requirement for a good representation of open quantum evolution.
Thus, they show that to get a full understanding of quantum evolution, non completely-positive maps should be considered as well.