Editing Quantum operation (section)

== Kraus operators ==
'''Kraus{{'}} theorem''' (named after [[Karl Kraus (physicist)|Karl Kraus]]) characterizes [[Completely positive map|completely positive maps]], which model quantum operations between quantum states. Informally, the theorem ensures that the action of any such quantum operation <math>\Phi</math> on a state <math>\rho</math> can always be written as <math display="inline">\Phi(\rho) = \sum_k B_k\rho B_k^*</math>, for some set of operators <math>\{B_k\}_k</math> satisfying <math display="inline">\sum_k B_k^* B_k \leq \mathbf{1}</math>, where <math>\mathbf{1}</math> is the identity operator.

=== Statement of the theorem ===
'''Theorem'''.<ref>This theorem is proved in {{harvp|Nielsen|Chuang|2010}}, Theorems 8.1 and 8.3.</ref>  Let <math>\mathcal H</math> and <math>\mathcal G</math> be Hilbert spaces of dimension <math>n</math> and <math>m</math> respectively, and <math>\Phi</math> be a quantum operation between <math>\mathcal H</math> and <math>\mathcal G</math>. Then, there are matrices <math display="block">\{ B_i \}_{1 \leq i \leq nm}</math> mapping <math>\mathcal H</math> to <math>\mathcal G</math> such that, for any state <math> \rho </math>, <math display="block"> \Phi(\rho) = \sum_i B_i \rho B_i^*.</math>
Conversely, any map <math> \Phi </math> of this form is a quantum operation provided <math display="inline">\sum_k B_k^* B_k \leq \mathbf{1}</math>.

The matrices <math>\{ B_i \}</math> are called ''Kraus operators''.  (Sometimes they are known as ''noise operators'' or ''error operators'', especially in the context of [[quantum information processing]], where the quantum operation represents the noisy, error-producing effects of the environment.)  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 <math>\{ B_i \}</math> by a sequence of bounded operators.

===Unitary equivalence===
Kraus matrices are not uniquely determined by the quantum operation <math>\Phi</math> in general. For example, different [[Cholesky factorization]]s of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices representing the same quantum operation are related by a unitary transformation:

'''Theorem'''. Let <math>\Phi</math> be a (not necessarily trace-preserving) quantum operation on a finite-dimensional Hilbert space ''H'' with two representing sequences of Kraus matrices <math>\{ B_i \}_{i\leq N}</math> and <math>\{ C_i \}_{i\leq N}</math>.  Then there is a unitary operator matrix <math>(u_{ij})_{ij}</math> such that <math display="block"> C_i = \sum_j u_{ij} B_j. </math>

In the infinite-dimensional case, this generalizes to a relationship between two [[Stinespring factorization theorem|minimal Stinespring representations]].

It is a consequence of Stinespring's theorem that all quantum operations can be implemented by unitary evolution after coupling a suitable [[ancilla (quantum computing)|ancilla]] to the original system.

===Remarks===
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. Among all possible Kraus representations of a given [[quantum channel|channel]], there exists a canonical form distinguished by the orthogonality relation of Kraus operators, <math>\operatorname{Tr} A^\dagger_i A_j \sim \delta_{ij} </math>. 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.