Editing Quantum operation (section)

=== 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.