# Contents

## Idea

Any physical process is supposed to take physical states into physical states. If we use density matrices to describe states in quantum mechanics, then it must be some operation that sends density matrices to density matrices: it should be a linear map of vector spaces

$U:\mathrm{Mat}\left(n×n,ℂ\right)\to \mathrm{Mat}\left(k×k,ℂ\right)$U : Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})

that preserves the subset of density matrices, in that it

• preserves the trace of matrices;

• takes hermitian matrices with non-negative eigenvalues to hermitian matrices with non-negative eigenvalues.

The notion of a quantum operation is built from the Stinespring factorization theorem.

## Definition

Let $k,n\in ℕ$.

A matrix $A\in \mathrm{Mat}\left(n×n,ℂ\right)$ is called positive if it is hermitian – if ${A}^{†}=A$ – and if all its eigenvalues (which then are necessarily real) are non-negative.

A linear map (morphism of vector spaces of matrices)

$\Phi :\mathrm{Mat}\left(n×n,ℂ\right)\to \mathrm{Mat}\left(k×k,ℂ\right)$\Phi : Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})

is called positive if it takes positive matrices to positive matrices.

The map $\Phi$ is called completely positive if for all $p\in ℕ$ the tensor product

$\Phi \otimes {\mathrm{Id}}_{\mathrm{Mat}\left(p×p\right),ℂ}:\mathrm{Mat}\left(n×n,ℂ\right)\otimes \mathrm{Mat}\left(p×p,ℂ\right)\to \mathrm{Mat}\left(k×k,ℂ\right)\otimes \mathrm{Mat}\left(p×p,ℂ\right)$\Phi \otimes Id_{Mat(p\times p),\mathbb{C}} : Mat(n \times n , \mathbb{C})\otimes Mat(p \times p , \mathbb{C}) \to Mat(k \times k , \mathbb{C})\otimes Mat(p \times p , \mathbb{C})

is positive.

###### Theorem

A map $\Phi$ as above is completely positive precisely if there exists a set $I$ and an $I$-family $\left\{{E}_{i}\in \mathrm{Mat}\left(k×n,ℂ\mid i\in I\right)\right\}$ of matrices, such that for all $A\in \mathrm{Mat}\left(n×n,ℂ\right)$ we have

$\Phi \left(A\right)=\sum _{i\in I}{E}_{i}A{E}_{i}^{†}\phantom{\rule{thinmathspace}{0ex}}.$\Phi(A) = \sum_{i \in I} E_i A E_i^\dagger \,.

Moreover, such $\Phi$ preserves the trace of matrices precisely if

$\sum _{i\in I}{E}_{i}^{†}{E}_{i}={\mathrm{Id}}_{\mathrm{Mat}\left(n×n,ℂ\right)}\phantom{\rule{thinmathspace}{0ex}}.$\sum_{i \in I} E_i^\dagger E_i = Id_{Mat(n \times n, \mathbb{C})} \,.

This is originally due to (Choi, theorem 1). A proof in terms of †-categories is given in (Selinger). A characterization of completely positive maps entirely in terms of $†$-categories is given in (Coecke).

The matrices $\left\{{E}_{i}\right\}$ 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 $\mathrm{QChan}\subset \mathrm{Vect}$ – a subcategory of Vect${}_{ℂ}$ – whose

• objects are the vector spaces $\mathrm{Mat}\left(n×n,ℂ\right)$ for all $n\in ℕ$;

• morphism are completely positive and trace-preserving linear maps $\Phi :\mathrm{Mat}\left(n×n,ℂ\right)\to \mathrm{Mat}\left(m×m,ℂ\right)$;

• composition of morphisms is, of course, the composition in Vect, i.e. the ordinary composition of linear maps.

## Example

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 $\rho$ be the state of some quantum system and ${\rho }_{\mathrm{env}}$ be the state of the environment. The action of a unitary transformation, $U$, on the system is

$T\left(\rho \right)={\mathrm{Tr}}_{\mathrm{env}}U\left(\rho \otimes {\rho }_{\mathrm{env}}\right){U}^{†}.$T(\rho) = Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}.

## References

An early influential reference on (completely) positive trace-preserving maps (CPTP) is:

• M. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications Volume 10, Issue 3, (1975), Pages 285-290

A useful review of the notion is provided by

• 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)

• Michael Nielsen, Isaac Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000)

• 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 goes back to

• Peter Selinger, Dagger-compact closed categories and completely positive maps (ps)

This was further developed in

Revised on February 10, 2012 13:48:38 by Urs Schreiber (89.204.139.196)