## 9.9 Channel-state duality

Suppose that

After scaling by a factor of **Choi matrix**^{181} of

The Choi matrix is essentially another way of representing a linear map

The Choi matrix

The expression above may look baffling at first glance, but this is often the case when we turn something conceptually obvious into more compact mathematical notation.
In order to gain some intuition here, recall that, for matrices

This gives us a one-to-one correspondence between linear maps **Choi–Jamiołkowski isomorphism**

The correspondence between linear maps **Choi–Jamiołkowski isomorphism** (or **channel-state duality** in the specific setting of quantum information), is another example of a well known correspondence between vectors in

Take a tensor product vector in **canonical**: they do not depend on the choice of any bases in the vectors spaces involved.

However, some care must be taken when we want to define correspondence between vectors in *anti-linear* operation (since it involves complex conjugation).
This is fine *when we stick to a specific basis* *only for the basis vectors*.
This means that, for *not* like
**non-canonical**: it depends on the choice of the basis.
But it is still a pretty useful isomorphism!
The Choi–Jamiołkowski isomorphism is of this kind (i.e. non-canonical) — it works in the basis in which you express a maximally mixed state

Mathematically, it is not too surprising that the matrix elements of an operator on a tensor product can be reorganised and reinterpreted as the matrix elements of an operator between operator spaces.
What is interesting, and perhaps not so obvious, however, is that the positivity conditions for maps correspond exactly to conditions on their Choi matrices under this correspondence.
That is, this one-to-one correspondence between linear maps

The Choi matrix

Pictorially, we might represent this by something like

In this form, we can see right away that, if

Let

\mathcal{E} is completely positive if and only if\widetilde{\mathcal{E}} is positive semi-definite.\mathcal{E} is trace preserving if and only if(\mathbf{1}\otimes\operatorname{tr})\widetilde{\mathcal{E}}=\frac{1}{d}\mathbf{1} .\mathcal{E} sends the identity operator to the identity operator if and only if(\operatorname{tr}\otimes\mathbf{1})\widetilde{\mathcal{E}}=\frac{1}{d}\mathbf{1} .\mathcal{E} sends Hermitian operators to Hermitian operators and only if\widetilde{\mathcal{E}} is Hermitian.

We shall prove the first two of these correspondences here, and leave the last two as an exercise.

Let’s start with complete positivity, since one direction is much easier: if

For the trace-preserving correspondence, note first of all that, if

In particular then, completely positive trace-preserving maps (quantum channels) have Choi matrices that are positive semi-definite and such that their partial trace gives the maximally mixed state

**Channel–state duality.**
The following three things are all equivalent to one another:

- quantum channels (i.e. linear maps that can be written in Stinespring or Kraus form)
- completely positive trace-preserving (CPTP) maps
- linear maps
\mathcal{E} whose Choi matrix\widetilde{\mathcal{E}} is positive semi-definite and such that(\mathbf{1}\otimes\operatorname{tr})\widetilde{\mathcal{E}}=\frac{1}{d}\mathbf{1} .

Furthermore, all completely positive maps admit a Kraus decomposition *also* want the map to be trace preserving, then we must additional require that the Kraus operators satisfy

Man-Duen Choi was brought up in Hong Kong. He received his Ph.D. degree under the guidance of Chandler Davis at Toronto. He taught at the University of California, Berkeley, from 1973 to 1976, and has worked since then at the University of Toronto. His research has been mainly in operator algebras, operator theory, and polynomial rings. He is particularly interested in examples/counterexamples and

2\times2 matrix manipulations.↩︎