## 7.12 *Appendices*

### 7.12.1 Isometries

In many applications, including quantum communication and quantum error correction, it is useful to encode a quantum state of one system into a quantum state of a larger system.
Such operations are described by *isometries*.^{121}
You may think about isometries as a generalisation of unitaries: like unitaries, they preserve inner products; unlike unitaries, they are maps between spaces of *different* dimensions.

Let **isometry** is a linear map

Isometries preserve inner products, and therefore also the norm and the metric induced by the norm.

An isometry *whole* Hilbert space *subspace* of

The fact that an isometry *not* require *unitary*.
The operator

The reason that we care about this more general notion of isometry (instead of specifically unitaries) is that *isometries represent physically admissible operations*: they can be implemented by bringing two systems together (via tensoring) and then applying unitary transformations to the composite system.
That is, take some system

Any isometry is a quantum channel, since any quantum state described by the state vector

An example which we will later return to is that of the **three-qubit code**.
Take a qubit in some pure state

The result is the isometric embedding of the

### 7.12.2 The Markov approximation

Composition of quantum channels^{122} in the Kraus representation is rather straightforward, but do not be deceived by its mathematical simplicity!
We must remember that *quantum channels do not capture all possible quantum evolutions*: the assumption that the system and the environment are *not initially correlated* is crucial, and it does impose some restrictions on the applicability of our formalism.
Compare, for example, the following two scenarios.

Firstly:

Here the system, initially in state *not* discarded after the first unitary evolution *not*: it falls outside the remit of our formalism because the input state of the system and the state of the environment are *not independent*.

Secondly:

Here we have two stages of evolution, as before, but we *discard* the environment after the first unitary, and start the second unitary evolution in a fresh tensor-product state, with a *new* environment;
the two stages involve *independent environments*.
In this case^{123} all three evolutions (

In practice we often deal with complex environments that have internal dynamics that “hides” any entanglement with the system as quickly as it arises.
For example, suppose that our system is an atom, surrounded by the electromagnetic field (which serves as the environment).
Let the field start in the vacuum state.
If the atom emits a photon into the environment, then the photon quickly propagates away, and the immediate vicinity of the atom appears to be empty, i.e. resets to the vacuum state.
In this approximate model, we assume that the environment quickly forgets about the state resulting from any previous evolution.
This is known as the **Markov approximation**;
in a quantum Markov process the environment has essentially no memory.

### 7.12.3 What use are positive maps?

Positive maps that are not completely positive are not completely useless. True, they cannot describe any quantum dynamics, but still they have useful applications — for example, they can help us to determine if a given state is entangled or not.

Recall: a quantum state of a bipartite system **separable** if **entangled**.
If we apply the partial transpose

Positive maps, such as the transpose, can be quite deceptive: you have to include other systems in order to detect their unphysical character.

In separable states, one subsystem does not really know about the existence of the other, and so applying a positive map to one part produces a proper density operator, and thus does *not* reveal the unphysical character of the map.
So, for *any separable state*

As an example, consider a quantum state of two qubits which is a mixture of the maximally entangled state

Note that the implication “if separable then the partial transpose is positive” does not imply the converse: there exist entangled states for which the partial transpose is positive, and they are known as the **entangled PPT states**^{124}
However, for *two* qubits, the PPT states are exactly the separable states.

### 7.12.4 The Choi–Jamiołkowski isomorphism

The correspondence between linear maps **Choi–Jamiołkowski isomorphism** or **channel–state duality**, is another example of a well known correspondence between vectors in

It is slightly confusing at first, but the **Choi isomorphism**, the **Jamiołkowski isomorphism**, and the **Choi–Jamiołkowski** isomorphism are really three distinct things:

- the first is very nice, but non-canonical (i.e. is dependent on the choice of basis);
- the second (for which I have no nice citation, but is basically given by considering
\sum|j\rangle\langle i|\otimes\mathcal{E}(|i\rangle\langle j|) instead of\sum|i\rangle\langle j|\otimes\mathcal{E}(|i\rangle\langle j|) ) is canonical, but doesn’t always map CP maps to positive semidefinite matrices; - the third brings together the two similar, but distinct, results by the respective authors. However, people often say “Choi–Jamiołkowski” to mean any one of the three. Such is life.

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 |i\rangle|j\rangle and use the ket-flipping approach 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 entangled state

### 7.12.5 Block matrices and partial trace

For any matrix *that is written in the tensor product basis*, the partial trace over the first subsystem (here

For example, for any

The same holds for general

The word isometric (like pretty much most of the fancy words you come across in this course) comes from Greek, meaning “of the same measures”:

*isos*means “equal”, and*metron*means “a measure”, and so an “isometry” is a transformation that preserves distances.↩︎Unitary evolutions form a group; quantum channels form a semigroup. Quantum operations are invertible only if they are either unitary operations or simple isometric embeddings (such as the process of bringing in the environment in some fixed state and then

*immediately*discarding it, without any intermediate interaction).↩︎A

**quantum Markov process**! Andrey Markov (1929–2012) was a Russian mathematician best known for his work on stochastic processes.↩︎“PPT” stands for

*positive partial transpose*.↩︎