## 7.9 Channel-state duality

Suppose that

After scaling by a factor of **Choi matrix**^{139} 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, the Choi–Jamiołkowski isomorphism gives us a simple necessary and sufficient condition for a linear map to be a quantum channel:

**Channel-state duality.** *if and only if*

One direction of this claim is immediate: we already know that any quantum channel

Now, any vector ^{140}

Using this, we see that

Finally, ^{141}

This establishes the desired isomorphism between *states* and *channels*.
The equations

We can summarise the flow of implications in a diagram:
*if \mathcal{E} is a quantum channel, then its Choi matrix \widetilde{\mathcal{E}} is a density matrix*.
The reverse implication goes as follows.
The density matrix

*if the Choi matrix*\widetilde{\mathcal{E}} is a density matrix, then \mathcal{E} is a quantum channel.

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.↩︎**Exercise.**Prove this!↩︎**Exercise.**Prove this!*Hint: you can decompose the trace into a partial trace followed by a trace, and it’s helpful to do so here —*↩︎\operatorname{tr}\widetilde{\mathcal{E}}=\operatorname{tr}[(\operatorname{tr}\otimes\mathbf{1})\widetilde{\mathcal{E}}] .