## Random isometries

There is another invertible operation in quantum theory: an **isometry**, which is a combination of adding another quantum system and then applying a unitary transformation to the resulting composite system.
So let us take a quick look at a simple generalisation of random unitaries, namely random isometries V_i, which give
\rho\longmapsto
\rho' = \sum_{i=1} p_i V_i \rho V^\dagger_i.
An isometry V is similar to a unitary operator except that it maps states in the Hilbert space \mathcal{H} to states in a larger Hilbert space \mathcal{H}' (cf. {the appendix on isometries}(#isometries)).
Here \mathcal{H} is associated with the input and \mathcal{H}'=\mathcal{H}_\mathcal{A}\otimes\mathcal{H} with the dilated system (i.e. the ancilla \mathcal{A} plus our system of interest).
Isometries satisfy V^\dagger V=\mathbf{1}, and they are usually implemented by adding an ancilla in a fixed state and then applying a unitary operation to the resulting composed system.
They can be then reversed by applying the inverse of that unitary and discarding the ancilla.

Now, if \mathcal{H}' is sufficiently larger than \mathcal{H}, and if the images \mathcal{H}'_i of \mathcal{H} in \mathcal{H}' under the different isometries V_i do not overlap, then we can reverse the action of the channel:
we can, at least in principle, perform a measurement on \mathcal{H}', defined by the partition \mathcal{H}'=\mathcal{H}'_1\oplus\mathcal{H}'_2\oplus\ldots, and find out which subspace contains the output state;
once we know which subspace the input was sent to, we know which particular isometry V_k was applied by the channel;
then we simply apply V^\dagger_k.

In order to see this consider the following simple, but important, example, which we will revisit several times in different disguises.

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 |\psi\rangle and then combines it with the two ancillary qubits which are in a fixed state |0\rangle|0\rangle.
Then she applies one of the four, randomly chosen, unitary operations to the three qubits, to generate the following four isometries:
\begin{aligned}
V_1 &= |000\rangle\langle 0| + |111\rangle\langle 1|
\\V_2 &= |001\rangle\langle 0| + |110\rangle\langle 1|
\\V_3 &= |010\rangle\langle 0| + |101\rangle\langle 1|
\\V_4 &= |100\rangle\langle 0| + |011\rangle\langle 1|.
\end{aligned}

The three qubits, which form the output of the channel, are given to Bob, whose task is to recover the original state |\psi\rangle of the input qubit.
In this scenario, Bob, who knows the four isometries, can find out which particular isometry was applied.
He knows that

- 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 \mathcal{H}'=\mathcal{H}'_1\oplus\mathcal{H}'_2\oplus\mathcal{H}'_3\oplus\mathcal{H}'_4, Bob can perform a measurement defined by the projectors on these subspaces.
For example, if Alice randomly picked V_2, then the input state |\psi\rangle=\alpha_0|0\rangle+\alpha_1|1\rangle will be mapped to the output state \alpha_0|001\rangle+\alpha_1|110\rangle in the \mathcal{H}'_2 subspace.
Bob’s measurement will then detect \mathcal{H}'_2 as the subspace where the output state resides, but the measurement (i.e. the corresponding projection) will not affect any state in that subspace.
Bob can now simply apply V_2^\dagger and obtain |\psi\rangle.

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?

Single unitaries or isometries apart, 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 V^\dagger_i V_j=\delta_{ij}\mathbf{1}.
We shall return to these channels later on.