Types of quantum field thories
Any physical process is supposed to take physical states into physical states (Schrödinger picture). If density matrices are used to describe quantum states in quantum mechanics, then it must be some operation that sends density matrices to density matrices. So for finite-dimensional state spaces then an process it should be a linear map of vector spaces of matrices
it preserves the trace of matrices;
takes hermitian matrices with non-negative eigenvalues to hermitian matrices with non-negative eigenvalues.
Such a map is then called a quantum operation. The notion of a quantum operation is built from the Stinespring factorization theorem.
Then we consider the general abstract formulation
A matrix is called positive if it is hermitian – if – and if all its eigenvalues (which then are necessarily real) are non-negative.
A linear map (morphism of vector spaces of matrices)
is called positive if it takes positive matrices to positive matrices.
The map is called completely positive if for all the tensor product
A map as above is completely positive precisely if there exists a set and an -family of matrices, such that for all we have
Moreover, such preserves the trace of matrices precisely if
This is originally due to (Stinespring 55). The decomposition in the theorem is called Kraus decomposition after (Kraus 71). See also (Choi 76, theorem 1). A brief review is for instance in (Kuperberg 05, theorem 1.5.1). A general abstract proof in terms of †-categories is given in (Selinger 05). A characterization of completely positive maps entirely in terms of -categories is given in (Coecke 07).
The matrices that are associated to a completely positive and trace-preserving map by the above theorem are called Kraus operators.
In the physics literature the above theorem is then phrased as: Every quantum channel can be represented using Kraus operators .
Notice that the identity map is clearly completely positive and trace preserving, and that the composite of two maps that preserve positivity and trace clearly still preserves positivity and trace. Therefore we obtain a category – a subcategory of Vect – whose
objects are the vector spaces for all ;
morphism are completely positive and trace-preserving linear maps ;
composition of morphisms is, of course, the composition in Vect, i.e. the ordinary composition of linear maps.
A very common example of this formalism comes from its use in open quantum systems, that is systems that are coupled to an environment. Let be the state of some quantum system and be the state of the environment. The action of a unitary transformation, , on the system is
The Kraus-decomposition characterization of completely positive maps is due to
Reviews and surveys include
Caleb J. O’Loan (2009), Topics in Estimation of Quantum Channels , PhD thesis, University of St. Andrews, (arXiv)
Christian B. Mendl, Michael M. Wolf, Unital Quantum Channels - Convex Structure and Revivals of Birkhoff’s Theorem , Commun. Math. Phys. 289, 1057-1096 (2009) (arXiv:0806.2820)
Smolin, John A., Verstraete, Frank, and Winter, Andreas Entanglement of assistance and multipartite state distillation , Phys. Rev. A, vol. 72, 052317, 2005 (arXiv:quant-ph/0505038)
John Watrous, Mixing doubly stochastic quantum channels with the completely depolarizing channel (2008) (arXiv)
The description of completely positive maps in terms of dagger-categories (see at finite quantum mechanics in terms of dagger-compact categories) goes back to
This is further explored in