## 7.3 Random 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*.^{123}
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

Although isometries are strictly more general than unitaries, an fundamentally important fact is that *isometries still 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

Now suppose we have isometries

Apart from single unitaries or isometries, it turns out that the only reversible, or **correctable**, channels (i.e. channels in which the input state can be recovered) are exactly the mixtures of mutually orthogonal isometries

### 7.3.1 Three-qubit codes

Here is a simple, but important, example, which we will revisit several times in different disguises: that of the **three-qubit code**.
Take a qubit in some pure state

The result is the isometric embedding of the

This isometry is just one member of a family, as we now explain.

Alice constructs a quantum channel which is a mixture of four isometries.
The input is a single qubit, and the output is a dilated system composed of three qubits.
She prepares the input qubit in a state^{124}

The three qubits, which form the output of the channel, are given to Bob, whose task is to recover the original state

V_1 maps\mathcal{H} to\mathcal{H}'_1 , which is a subspace of\mathcal{H}' spanned by|000\rangle and|111\rangle ;V_2 maps\mathcal{H} to\mathcal{H}'_2 , which is a subspace of\mathcal{H}' spanned by|001\rangle and|110\rangle ;V_3 maps\mathcal{H} to\mathcal{H}'_3 , which is a subspace of\mathcal{H}' spanned by|010\rangle and|101\rangle ;V_4 maps\mathcal{H} to\mathcal{H}'_4 , which is a subspace of\mathcal{H}' spanned by|100\rangle and|011\rangle .

Given that these subspaces are mutually orthogonal, and

Just in case you are curious (as you should be!), below is a diagram of how the four isometries are implemented. How would you reverse these operations?

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.↩︎Our arguments here can be easily extended to any mixed state

\rho , but for simplicity we consider the case of a pure state.↩︎