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 \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 the 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 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.

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 \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 for any vector |\psi\rangle\in\mathcal{H}_\mathcal{A}.

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}.

An example which we will later return to is that of the three-qubit code. Take a qubit in some pure state |\psi\rangle=\alpha|0\rangle+\beta|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\rangle+\beta|1\rangle \longmapsto \alpha|000\rangle+\beta|111\rangle. This three qubit-encoding can be reversed by the mirror image circuit:

7.12.2 The Markov approximation

Composition of quantum channels122 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 \rho, undergoes two stages of evolution, and the environment is not discarded after the first unitary evolution U_A; the environment persists and participates in the second unitary evolution U_B. In this case the evolutions \rho\mapsto\rho' and \rho\mapsto\rho'' are both well defined quantum channels, but the evolution \rho'\mapsto\rho'' is 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 case123 all three evolutions (\rho\mapsto\rho', \rho'\mapsto\rho'', and \rho\mapsto\rho'') are well defined quantum channels, and they compose: if \mathcal{E}_\mathcal{A} describes the evolution from \rho to \rho', and \mathcal{E}_\mathcal{B} from \rho' to \rho'', then the composition \mathcal{E}_\mathcal{B}\circ\mathcal{E}_\mathcal{A} describes the evolution from \rho to \rho''.

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 \mathcal{AB} described by the density matrix \varrho^{\mathcal{AB}} is said to be separable if \varrho^{\mathcal{AB}} is of the form \varrho^{\mathcal{AB}} = \sum_k p_k \rho^{\mathcal{A}}_k \otimes\rho^{\mathcal{B}}_k where p_k \geqslant 0 and \sum_{k=1} p_k=1; otherwise \varrho^{\mathcal{AB}} is said to be entangled. If we apply the partial transpose \mathbf{1}\otimes T to this state, then it remains separable, since, as we have seen, the transpose \rho^B is a legal density matrix.

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 \rho, we have (\mathbf{1}\otimes T)\rho\geqslant 0.

As an example, consider a quantum state of two qubits which is a mixture of the maximally entangled state |\psi\rangle = \frac{1}{\sqrt 2}(|00\rangle + |11\rangle) and the maximally mixed state described by the density matrix \rho_p = p|\psi\rangle\langle\psi| + \frac{(1-p)}{4}\mathbf{1}\otimes\mathbf{1}, where p\in [0,1]. If we apply the partial transpose \mathbf{1}\otimes T to this state, and check for which values of p the resulting matrix is a density matrix, we see that, for all p\in[\frac{1}{3},1], the density operator \rho describes an 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 states124 However, for two qubits, the PPT states are exactly the separable states.

7.12.4 The Choi–Jamiołkowski isomorphism

The correspondence between linear maps \mathscr{B}(\mathcal{H})\to\mathscr{B}(\mathcal{H'}) and operators in \mathscr{B}(\mathcal{H}\otimes\mathcal{H'}), known as the Choi–Jamiołkowski isomorphism or channel–state duality, is another example of a well known correspondence between vectors in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}} and operators \mathscr{B}(\mathcal{H}_{\mathcal{A}}^\star,\mathcal{H}_{\mathcal{B}}) or \mathscr{B}(\mathcal{H}_{\mathcal{B}}^\star,\mathcal{H}_{\mathcal{A}}).

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

  1. the first is very nice, but non-canonical (i.e. is dependent on the choice of basis);
  2. 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;
  3. 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 |a\rangle\otimes|b\rangle\in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}. Then it defines natural maps in \mathscr{B}(\mathcal{H}_{\mathcal{A}}^\star,\mathcal{H}_{\mathcal{B}}) and \mathscr{B}(\mathcal{H}_{\mathcal{B}}^\star,\mathcal{H}_{\mathcal{A}}), via \begin{aligned} \langle x| &\longmapsto \langle x|a\rangle|b\rangle \\\langle y| &\longmapsto |a\rangle\langle y|b\rangle \end{aligned} for any linear forms \langle x|\in\mathcal{H}^\star_A and \langle y|\in\mathcal{H}^\star_B. We then extend this construction (by linearity) to any vector in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}. These isomorphisms are 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 \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}} and operators in \mathscr{B}(\mathcal{H}_{\mathcal{A}},\mathcal{H}_{\mathcal{B}}) or \mathscr{B}(\mathcal{H}_{\mathcal{B}},\mathcal{H}_{\mathcal{A}}). For example, physicists like to “construct” \mathscr{B}(\mathcal{H}_{\mathcal{B}},\mathcal{H}_{\mathcal{A}}) in a deceptively simple way: |a\rangle|b\rangle \longleftrightarrow |a\rangle\langle b|. (where we have simply omitted the tensor product symbol). Flipping |b\rangle and switching from \mathcal{H}_{\mathcal{B}} to \mathcal{H}^\star_B is an 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 |b\rangle=\sum_j\beta_j|j\rangle, the correspondence looks like |i\rangle|b\rangle \longleftrightarrow \sum_j \beta_j |i\rangle\langle j| and not like |i\rangle|b\rangle \longleftrightarrow |i\rangle\langle b| = \sum_j \beta^\star_j |i\rangle\langle j|. This isomorphism is 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 |\Omega\rangle=\sum_i|i\rangle|i\rangle.

7.12.5 Block matrices and partial trace

For any matrix M in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}} that is written in the tensor product basis, the partial trace over the first subsystem (here \mathcal{A}) is the sum of the diagonal block matrices, and the partial trace over the second subsystem (here \mathcal{B}) is a matrix in which the block sub-matrices are replaced by their traces. You can visualise this as in Figure 7.2.

Visualising the two partial traces of a matrix written in the tensor product basis.

Figure 7.2: Visualising the two partial traces of a matrix written in the tensor product basis.

For example, for any M in the tensor product space associated with two qubits, written in the standard basis \{|00\rangle,|01\rangle,|10\rangle,|11\rangle\} in block form as M = \left[ \begin{array}{c|c} P & Q \\\hline R & S \end{array} \right] where P, Q, R, and S are all (2\times 2) sized sub-matrices, we have that \begin{aligned} \operatorname{tr}_{\mathcal{A}} M &= P+S \\\operatorname{tr}_{\mathcal{B}} M &= \left[ \begin{array}{c|c} \operatorname{tr}P & \operatorname{tr}Q \\\hline \operatorname{tr}R & \operatorname{tr}S \end{array} \right] \end{aligned}

The same holds for general M in any \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}} with such a block form (i.e. m\times m blocks of (n\times n) sized sub-matrices, where m=\dim\mathcal{H}_{\mathcal{A}} and n=\dim\mathcal{H}_{\mathcal{B}}).


  1. 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.↩︎

  2. 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).↩︎

  3. A quantum Markov process! Andrey Markov (1929–2012) was a Russian mathematician best known for his work on stochastic processes.↩︎

  4. “PPT” stands for positive partial transpose.↩︎