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.
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 \mathcal{H} and \mathcal{H}' be Hilbert spaces such that \dim\mathcal{H}\leqslant\dim\mathcal{H}'.
An isometry is a linear map V\colon\mathcal{H}\to\mathcal{H}' such that V^\dagger V=\mathbf{1}_{\mathcal{H}}
Isometries preserve inner products, and therefore also the norm and the metric induced by the norm.
An isometry V\colon\mathcal{H}\to\mathcal{H}' maps the whole Hilbert space \mathcal{H} onto a subspace of \mathcal{H}'.
As a consequence, the matrix representation of an isometry is a rectangular matrix formed by selecting only a few of the columns from a unitary matrix.
For example, given a unitary U we can construct an isometry V as follows:
The fact that an isometry V preserves the inner products comes from the fact that we require V^\dagger V=\mathbf{1}_{\mathcal{H}};
we do not require VV^\dagger=\mathbf{1}_{\mathcal{H'}}.
Indeed, if we required both of these, then that would be equivalent to asking for V to be unitary.
The operator VV^\dagger is a projector operator acting on \mathcal{H}', which projects onto the image of \mathcal{H} under the isometry V, as we can see by expressing V in Dirac notation:
V = \sum_i |b_i\rangle\langle a_i|,
where the |a_i\rangle form an orthonormal basis in \mathcal{H}, and the |b_i\rangle are just orthonormal (but not necessarily spanning) vectors in \mathcal{H}';
in the special case where V is unitary, the orthonormal vectors |b_i\rangle form an orthonormal basis in \mathcal{H}'.
Writing V in this form, it is clear that V^\dagger V=\sum_i |a_i\rangle\langle a_i|=\mathbf{1}, and that VV^\dagger = \sum_i |b_i\rangle\langle b_i| projects on the subspace spanned by |b_i\rangle.
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 \mathcal{A} in state |\psi\rangle, and bring in another system \mathcal{B} in some fixed state |b\rangle;
applying some unitary U to the combined system \mathcal{A}\mathcal{B} then gives an isometry from \mathcal{H}=\mathcal{H}_\mathcal{A} to \mathcal{H}'=\mathcal{H}_\mathcal{A}\otimes\mathcal{H}_\mathcal{B}, i.e. the result is a linear map V defined by
V\colon
|\psi\rangle
\longmapsto
|\psi\rangle|b\rangle
\longmapsto
U(|\psi\rangle|b\rangle).
Any isometry is a quantum channel, since any quantum state described by the state vector |\psi\rangle (or by a density operator \rho) is transformed as
|\psi\rangle\longmapsto V|\psi\rangle
(or as \rho\mapsto V\rho V^\dagger), and the normalisation condition is exactly the defining property of isometries:
V^\dagger V =\mathbf{1}.
Now suppose we have isometries V_1,\ldots,V_n\colon\mathcal{H}\to\mathcal{H}'.
If \mathcal{H}' is “sufficiently bigger” than \mathcal{H}, and if the images \mathcal{H}'_i\coloneqq V_i(\mathcal{H}) do not “overlap” (that is, if the subspaces \mathcal{H}'_i are mutually orthogonal), 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\oplus\mathcal{H}'_n, 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.
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 V^\dagger_i V_j=\delta_{ij}\mathbf{1}.
We shall return to such channels later on.
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 |\psi\rangle=\alpha_0|0\rangle+\alpha_1|1\rangle, introduce two auxiliary qubits in a fixed state |0\rangle|0\rangle, and apply a unitary operation to the three qubits, namely two controlled-\texttt{NOT} gates:
The result is the isometric embedding of the 2-dimensional Hilbert space of the first qubit (spanned by |0\rangle and |1\rangle) into the 2-dimensional subspace (spanned by |000\rangle and |111\rangle) of the 8-dimensional Hilbert space of the three qubits.
The isometric operator
V = |000\rangle\langle 0| + |111\rangle\langle 1|
acts via
\alpha_0|0\rangle+\alpha_1|1\rangle
\longmapsto \alpha_0|000\rangle+\alpha_1|111\rangle.
This three qubit-encoding can be reversed by the mirror image circuit:
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 |\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?
