Contents

# Contents

## Idea

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 a process should be a linear map of vector spaces of matrices

$U \;\colon\; Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})$

(so far this is a general “superoperator”) 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.

Such a map is then called a quantum operation. The notion of a quantum operation is built from the Stinespring factorization theorem.

## Definition

We first give the traditional definition in terms of linear algebra and matrices in

Then we consider the general abstract formulation

### In terms of matrices

Let $k,n \in \mathbb{N}$.

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

A linear map (morphism of vector spaces of matrices)

$\Phi \;\colon\; Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})$

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

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

$\Phi \otimes Id_{Mat(p\times p),\mathbb{C}} \;\colon\; Mat(n \times n , \mathbb{C}) \otimes Mat(p \times p , \mathbb{C}) \longrightarrow Mat(k \times k , \mathbb{C}) \otimes Mat(p \times p , \mathbb{C})$

is positive.

A quantum operation (or quantum channel) is a map that is both completely positive and trace preserving (often abbreviated to CPTP).

### In terms of compact closed categories

… due to (Selinger 05) … see for instance (Coecke-Heunen 11, section 2) for a quick summary …

## Properties

### Characterization of complete positivity

###### Theorem

A map $\Phi$ as above is completely positive precisely if there exists a set $I$ and an $I$-family $\{E_i \in Mat(k \times n, \mathbb{C}| i \in I)\}$ of matrices, such that for all $A \in Mat(n \times n, \mathbb{C})$ we have

$\Phi(A) = \sum_{i \in I} E_i A E_i^\dagger \,.$

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

$\sum_{i \in I} E_i^\dagger E_i = Id_{Mat(n \times n, \mathbb{C})} \,.$

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 $\dagger$-categories is given in (Coecke 07).

The matrices $\{E_i\}$ 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 $QChan \subset Vect$ – a subcategory of Vect${}_{\mathbb{C}}$ – whose

• objects are the vector spaces $Mat(n \times n, \mathbb{C})$ for all $n \in \mathbb{N}$;

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

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

## Universal property

The category whose objects are indexed by natural numbers $n,m, \cdots$ and whose morphisms are quantum operations from $n \times n$ to $m \times m$ matrices is a semicartesian monoidal category with the monoidal structure given by multiplication of numbers. Being semicartesian, the monoidal tensor unit (the number $1$) has a unique morphism to it from any object: this morphism is the trace.

In fact, this category has the universal property of the semicartesian reflection of the monoidal category of isometries. This is the category whose objects are natural numbers, considered as Hilbert spaces, and whose morphisms are isometries between them, where an isometry $m\to n$ is an $m\times n$ complex matrix $V$ such that $VV*=I$.

In detail, the universal property says that for any strict semicartesian monoidal category $\mathcal{D}$ and any monoidal functor $\mathbf{Isometries}\to \mathcal{D}$, there is a unique symmetric monoidal functor making the following diagram commute:

$\array{ \mathbf{Isometries} &\rightarrow& \mathbf{Quantum Channels} \\ &\searrow&\downarrow\\ && \mathcal{D} }$

This fits a physical intuition as follows. Suppose that the isometries are a model of reality, as in the the many worlds interpretation and the Church of the larger Hilbert space. But in practice the observer cannot access the entirety of reality, and so some bits are hidden. The canonical way to model this hiding is to do it freely, which is to form the semicartesian reflection.

## Examples

### Quantum measurement and POVMs

A quantum measurement is formally represented by a quantum operation that is induced by a positive-operator valued probability measure (POVM).

### Systems in a bath

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 $\rho_{env}$ be the state of the environment. The action of a unitary transformation, $U$, on the system is

$T(\rho) = Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}.$

The Kraus-decomposition characterization of completely positive maps is due to

• W. Forrest Stinespring, Positive functions on $C^\ast$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.
• K. Kraus, General state changes in quantum theory, Ann. Physics 64 (1971), no. 2, 311–335.
• M. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications Volume 10, Issue 3, (1975), Pages 285-290

Reviews and surveys include

• Greg Kuperberg, section 1.5 of A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)
• Michael Nielsen, Isaac Chuang, section 8.2 of Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000) (pdf)

• 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

• Peter Selinger, Dagger-compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer

Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 (pdf, ps)

• Bob Coecke, Complete positivity without compactness, 2007 (pdf)

This is further explored in

For the universal property, see

• Mathieu Huot, Sam Staton, Universal properties in quantum theory (QPL 2018) (pdf).