Quantum Operation - Mathematical Development

Mathematical Development

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.

Kraus' theorem characterizes maps that model quantum operations between density operators of quantum state:

Theorem. Let H and G be Hilbert spaces of dimension n and m respectively, and Φ be a quantum operation taking the density matrices acting on H to those acting on G. Then there are matrices

acting on G such that

Conversely, any map Φ of this form is a quantum operation provided

The matrices are called Kraus operators. (Sometimes they are known as noise operators or error operators, especially in the context 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 by a sequence of bounded operators.

Kraus matrices are not uniquely determined by the quantum operation Φ in general. For example, different Cholesky factorizations of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices which represent the same quantum operation are related by a unitary transformation:

Theorem. Let Φ be a (not necessarily trace preserving) quantum operation on a finite dimensional Hilbert space H with two representing sequences of Kraus matrices {B_i}_{i≤ N} and {C_i}_{i≤ N} . Then there is a unitary operator matrix such that

In the infinite dimensional case, this generalizes to a relationship between two minimal Stinespring representations.

It is a consequence of Stinespring's theorem that all quantum operations can be implemented via 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. Among all possible Kraus representations of a given channel there exists a canonical form distinguished by the orthogonality relation of Kraus operators, . Such a canonical set of orthogonal Kraus operators can be obtained by diagonalising the corresponding Choi matrix and reshaping its eigenvectors into square matrices.

There exists also an infinite dimensional algebraic generalization of Choi's theorem 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.

In the context of quantum information, quantum operations as defined above, i.e. completely positive maps that do not increase the trace, are also called quantum channels or stochastic maps. In the above discussion, we have confined ourselves to channels between quantum states. In other words, both the input and output spaces consist of quantum states. This formulation can be extended to include classical states as well, therefore allowing us to handle quantum and classical information simultaneously.