## 9.8 Completely positive trace-preserving maps

A while back we upgraded from working with state vectors ^{180} Hermitian operators *not* quantum channels, and thus which are not “physical operations”.

The matrix transpose operation *cannot* be written in the Stinespring (or the Kraus) form;
it is not induced by a unitary operation on some larger Hilbert space, and it cannot be physically implemented.
So, we then ask, *what is* the class of physically admissible maps?
That is, how can we classify which maps are quantum channels and which are not?

First, some notation.
We say that a linear operator **bounded** if there exists some real number

One reason to care so much about *bounded* operators is the following fact: *a linear operator between normed vector spaces is bounded if and only if it is continuous*.

Another important fact is that the set **involution** given by the **adjoint**.
Formally, **C*-algebra**.

Now here is another example of where working only with finite-dimensional spaces greatly simplifies the mathematics: if *every linear map f\colon X\to Y is bounded (or, equivalently, continuous)*.

In the infinite-dimensional setting, it is important to know whether or not a given operator is bounded, but it turns out that certain unbounded operators are still very useful. There are some technical details, but such operators are used to model observables in the Hilbert-space formalism of quantum mechanics.

Then, mathematically speaking, a quantum channel *any* such maps, of course — the statistical interpretation of quantum theory imposes certain properties on the subset of maps in which we are interested.

Firstly, for such a map *respect the mixing of states*.
Consider an ensemble of systems, with a fraction *linear map*.

Next, since *map density operators to density operators*, it has to be both *positive* (*trace preserving* (

Finally comes a subtle point.
It turns out that being positive is not good enough;
we must further require that the map *remains positive even when extended to act on a part of a larger system*.
Suppose that Alice and Bob share a bipartite system *also* give a density operator *strictly stronger* property than mere positivity;
we are asking for something called **complete positivity**.
Needless to say, complete positivity of

Let’s study this matrix transpose example a bit more.
Consider the transpose operation on a single qubit: *only one* of the two qubits (say, the second one), then the density matrix of the two qubits evolves under the action of the **partial transpose** *not* preserve positivity (since *not* positive), and therefore *not* a completely positive map.

If you prefer to see this more explicitly, then you can use the matrix representation of *not* a density matrix (since it is not positive).

We have seen that, at the very least, we want to be considering *completely positive trace-preserving* maps, but how do we know whether or not there are any further restrictions left to impose?
Needless to say, here is where mathematics alone cannot guide us, since we are trying to characterise maps which are *physically admissible*, and mathematics knows nothing about the reality of our universe!
However, one thing that we can do is compare our abstract approach with the derivations of quantum channels defined in terms of the Stinespring (or Kraus) representation.
As it happens, we can (and will!) show that a map is completely positive and trace preserving *if and only if* it can be written in the Stinespring (or Kraus) form.
In other words:

Quantum channels are *exactly* the completely positive trace-preserving (CPTP) maps.

One direction of this claim is much simpler than the other.
Any quantum channel *and* the extended map

Conversely, showing that CPTP maps are quantum channels is less simple.
In order to prove this, we will now introduce a very convenient tool called the **Choi matrix**, which gives yet another way to characterise linear maps between operators.

It’s a small abuse of notation, but we often simply say “positive” to mean “positive semi-definite” or “non-negative”. We write

\rho\geqslant 0 to mean that\rho is positive.↩︎