Editing Quantum operation (section)

===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.