# Chapter 7 Quantum channels

Quantum evolution of any

isolatedsystem is unitary but its constituent parts may evolve in a more complicated way. In this chapter we will go beyond unitary evolutions and describe physically realisable transformations of density operators, called quantum channels. Be prepared for some name dropping; you will hear about Karl Kraus, Woody Stinespring, Andrzej Jamiołkowski and Man-Duen Choi. To be sure, knowing names will not give you more insights, but at least you will not be intimidated when you hear about theStinespringand theKraus representations, theJamiołkowski isomorphism, or theChoi matrix.

We discussed how entanglement forced us to describe quantum states of open quantum systems (ones which are part of a larger system) in terms of density operators.
In this chapter we will describe how open systems evolve.
The question we are asking here is: what are the most general physically admissible transformations of density operators?
At the fundamental level — and this should be your quantum mantra^{97} — there is *only one* unitary evolution, and if there is any other evolution then it has to be derived from a unitary evolution.
From this perspective, any non-unitary evolution of an open system is induced by a unitary evolution of a larger system.
But how?
The short answer is: by adding (tensoring) and removing (partial trace) physical systems.
A typical combination of these operations is shown in the following diagram:

First, we prepare our system of interest in an input state ^{98} which is large enough to include everything our system will interact with, and also large enough to be in a pure state *trace out* the auxiliary system, turning the joint state *not* correlated with the input state, in a nice compact way:
**completely positive trace-preserving map**, or, in the parlance of quantum information science, a **quantum channel**.
We will elaborate on the mathematics behind quantum channels shortly, but for now let us only check the essential properties, i.e. that this map preserves both trace and positivity (as its name suggests).

- Trace preserving: since the trace is linear, invariant under cyclic permutations of operators, and we ask that
\sum_i E_i^\dagger E_i=\mathbf{1} , we see that\operatorname{tr}\left(\sum_k E_k\rho E_k^\dagger\right) = \operatorname{tr}\left(\sum_k E^\dagger_k E_k \rho\right) = \operatorname{tr}\rho. - Positivity preserving: since
\rho is a positive^{99}(semi-definite) operator, and so is\sqrt{\rho} , we see that\sum_k E_k\rho E_k^\dagger = \sum_k (E_k\sqrt{\rho})(\sqrt{\rho} E_k^\dagger).

These conditions are certainly *necessary* if we want to map density operators into legal density operators, but we shall see in a moment that they are not *sufficient*: quantum channels are not just positive maps, they are **completely** positive maps.
We will discuss their special properties, describe the most common examples, and, last but not least, specify when the action of quantum channels can be reversed, or corrected, so that we can recover the original input state.
This will set the stage for our subsequent discussion of quantum error correction.

…there is only one unitary evolution, there is only one unitary evolution, there only one unitary evolution… …and everything else is cheating↩︎

Depending on the context, the auxiliary system is either called the

**ancilla**(usually when we can control it) or the**environment**(usually when we cannot control it).↩︎Recall that an operator is positive if and only if it can be written in the form

XX^\dagger for someX (hereX=E_k\sqrt{\rho} ). Also, the sum of positive operators is again a positive operator.↩︎