## 9.12 *Remarks and exercises*

### 9.12.1 Purifications and isometries

All purifications of a density operator are related by an isometry acting on the purifying system.
That is, if

To show this, we start with the spectral decomposition of

This observation leads to a way of relating *all* convex decompositions of a given density operator: let

### 9.12.2 The Markov approximation

Unitary evolutions form a group, but quantum channels form a **semigroup**, since they are not necessarily invertible.
Indeed, 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).

Anyway, composition of quantum channels 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:^{186}

Here the system, initially in state *not* discarded after the first unitary evolution *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 case^{187} 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.

### 9.12.3 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 that a quantum state of a bipartite system **separable** if **entangled**.
If we apply the partial transpose

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*

Positive (but not completely positive) maps, such as the transpose, can be quite deceptive: you have to include other systems in order to detect their unphysical character.

In particular, positive maps appear to be completely positive *on separable states*.

As an example, consider a quantum state ^{188} state

We say that a state is a **PPT state**^{189} if its partial transpose is positive.
An important thing to note is that separable states are PTT, but the converse is generally *not* true: there exist entangled PPT states.
However, in the specific case of *two* qubits, the converse *is* true: the PPT states are exactly the separable states.

### 9.12.4 Partial inner product

Tensor products bring the possibility to do “partial things” beyond just the partial trace.
Given **partial inner product with |x\rangle.**
It is first defined on the product vectors

For example, the partial inner product of

### 9.12.5 The “control” part of controlled-NOT

Consider a single-qubit channel induced by the action of the

This time we are interested in the evolution of the *control* qubit: the control qubit will be our system, and the target qubit, initially in a fixed state

We can calculate the Kraus operators:

The unitary action of the gate when the state of the target qubit is fixed at

The evolution of the control qubit alone can be expressed in the Kraus form as

As we can see, the diagonal elements of **coherences**) disappear.
The two Kraus operators, *measuring the control qubit in the standard basis and then just forgetting the result*.

### 9.12.6 Surprisingly identical channels

Let us now compare two single qubit-quantum channels:

We are familiar with the first channel from the previous example (9.12.5): it performs the measurement in the standard basis, but *doesn’t* reveal the outcome of this measurement.
The second channel chooses randomly, with equal probability, between two options: it will either let the qubit pass undisturbed, or apply the phase-flip

These two apparently very different physical processes correspond to the same quantum channel:

You can also check that the two channels can be implemented by the following two circuits:

### 9.12.7 Independent ancilla

Another way to understand the freedom in the operator-sum representation is to realise that, once the system and the ancilla cease to interact, any operation on the ancilla alone has no effect on the state of the system.

That is, the two unitaries

### 9.12.8 Order matters?

We know that, given a fixed state of the environment, the unitaries

### 9.12.9 Unchanged reduced density operator

Show that^{190}, for any operator

### 9.12.10 Cooling down

We can show that the process of cooling a qubit to its ground state, described the map

### 9.12.11 No pancakes

Consider a single-qubit operation which causes the

Explain why we cannot physically implement such a map.

### 9.12.12 Pauli twirl

Show that randomly applying the Pauli operators **Pauli twirl**) results in the maximally mixed state

### 9.12.13 Depolarising channel

The “most popular” Pauli channel^{191} is the **depolarising channel**

Show, using the Pauli twirl (Exercise 9.12.12) or otherwise, that we can rewrite the depolarising channel as

In particular then, we can say that, for

It is also instructive to see how the depolarising channel acts on the Bloch sphere.
An arbitrary density matrix for a single qubit can be written as

### 9.12.14 Toffoli gate

Consider the **Toffoli gate**

Express

### 9.12.15 Expressing vectors using the maximally mixed state

Show that any vector

### 9.12.16 Complete positivity of a certain map

Let *positive*, and the range for which it is *completely positive*.

### 9.12.17 Duals

We say that **dual** of a linear map

- Show that, if
\mathcal{E} is trace preserving, then\mathcal{E}^\star is**unital**(i.e. that it sends the identity to the identity, or equivalently that its Kraus operatorsF_j satisfy\sum_j F_jF_j^\dagger=\mathbf{1} ). - Show that, if
\sum_i E_i E_i^\dagger is an operator-sum decomposition of\mathcal{E} , then\sum_i E^\dagger_i E_i is an operator-sum decomposition of\mathcal{E}^\star .

### 9.12.18 Trace, transpose, Choi

Let

(For example, if we are interested in the component

### 9.12.19 Entanglement witness

Show that, if *separable* states.

### 9.12.20 Almost Kraus decomposition

Show that^{192} any linear map

### 9.12.21 Tricks with a maximally mixed state

A maximally mixed state of a bipartite system can be written, using the Schmidt decomposition (from Exercise 5.14.13), as

If we take the transpose in the Schmidt basis of

|\Omega\rangle , then\langle\Omega|A\otimes B|\Omega\rangle = \frac{1}{d}\operatorname{tr}(A^T B). Any pure state

|\psi\rangle=\sum_{i,j} c_{ij}|i\rangle|j\rangle of the bipartite system can be written as(C\otimes\mathbf{1})|\Omega\rangle = (\mathbf{1}\otimes C^T)|\Omega\rangle. This implies that(U\otimes U^\star)|\Omega\rangle=|\Omega\rangle (whereU^\star denotes the matrix given by taking the complex conjugate, entry-wise, ofU , i.e.*without*also taking the transpose).The swap operation

\texttt{SWAP}=S\colon|i\rangle|j\rangle\mapsto|j\rangle|i\rangle can be expressed as\begin{aligned} S &= d |\Omega\rangle\langle\Omega|^{T_{\mathcal{A}}} \\&= d \sum_{i,j} \big(|i\rangle\langle j|\big)^T\otimes|i\rangle\langle j| \\&= d \sum_{i,j} |j\rangle\langle i|\otimes|i\rangle\langle j| \end{aligned} where we writeX^{T_{\mathcal{A}}} to mean the partial transpose over\mathcal{A} , i.e.T\otimes\mathbf{1} . This implies that\operatorname{tr}[(A\otimes B)S] = \operatorname{tr}AB and that(A\otimes\mathbf{1})S = S(\mathbf{1}\otimes A).

Here we have reverted to the convention of writing the ancilla/environment

*after*the system of interest instead of*before*.↩︎A

**quantum Markov process**! Andrey Markov (1929–2012) was a Russian mathematician best known for his work on stochastic processes.↩︎Recall that a state is said to be

**maximally mixed**if the outcomes of*any*measurement on that state are completely random.↩︎“PPT” stands for

*positive partial transpose*.↩︎*Hint: show this for separable operators*↩︎\rho=A\otimes B and then extend the result to any operator\rho by linearity.Recall that a single-qubit Pauli channel is a channel that applies one of the Pauli operators,

X ,Y orZ , chosen randomly with some prescribed probabilitiesp_x ,p_y andp_z .↩︎*Hint: use the singular-value decomposition of the Choi matrix.*↩︎