# Chapter 7 Quantum channels

Quantum evolution of any

isolatedsystem is unitary but its constituent parts may evolve in a more complicated way. In this chapter we will go beyond unitary evolutions and describe physically realisable transformations of density operators, called quantum channels. Be prepared for some name dropping; you will hear about Karl Kraus, Woody Stinespring, Andrzej Jamiołkowski and Man-Duen Choi. To be sure, knowing names will not give you more insights, but at least you will not be intimidated when you hear about theStinespringand theKraus representations, theJamiołkowski isomorphism, or theChoi matrix.

We discussed how entanglement forced us to describe quantum states of open quantum systems (ones which are part of a larger system) in terms of density operators.
In this chapter we will describe how open systems evolve.
The question we are asking here is: what are the most general physically admissible transformations of density operators?
At the fundamental level — and this should be your quantum mantra^{97} — there is *only one* unitary evolution, and if there is any other evolution then it has to be derived from a unitary evolution.
From this perspective, any non-unitary evolution of an open system is induced by a unitary evolution of a larger system.
But how?
The short answer is: by adding (tensoring) and removing (partial trace) physical systems.
A typical combination of these operations is shown in the following diagram:

First, we prepare our system of interest in an input state ^{98} which is large enough to include everything our system will interact with, and also large enough to be in a pure state *trace out* the auxiliary system, turning the joint state *not* correlated with the input state, in a nice compact way:
**completely positive trace-preserving map**, or, in the parlance of quantum information science, a **quantum channel**.
We will elaborate on the mathematics behind quantum channels shortly, but for now let us only check the essential properties, i.e. that this map preserves both trace and positivity (as its name suggests).

- Trace preserving: since the trace is linear, invariant under cyclic permutations of operators, and we ask that
\sum_i E_i^\dagger E_i=\mathbf{1} , we see that\operatorname{tr}\left(\sum_k E_k\rho E_k^\dagger\right) = \operatorname{tr}\left(\sum_k E^\dagger_k E_k \rho\right) = \operatorname{tr}\rho. - Positivity preserving: since
\rho is a positive^{99}(semi-definite) operator, and so is\sqrt{\rho} , we see that\sum_k E_k\rho E_k^\dagger = \sum_k (E_k\sqrt{\rho})(\sqrt{\rho} E_k^\dagger).

These conditions are certainly *necessary* if we want to map density operators into legal density operators, but we shall see in a moment that they are not *sufficient*: quantum channels are not just positive maps, they are **completely** positive maps.
We will discuss their special properties, describe the most common examples, and, last but not least, specify when the action of quantum quantum channels can be reversed, or corrected, so that we can recover the original input state.
This will set the stage for our subsequent discussion of quantum error correction.

## 7.1 Random evolution

### 7.1.1 Random unitaries

As a first step toward understanding the quantum description of an evolving open system, consider a “two-qubit universe” in which we observe *only one* of the qubits.
Let us revisit the controlled-

- if the input state of the control qubit is
|0\rangle , the target qubit evolves unitarily according to the identity operator\mathbf{1} ; - if the input state of the control qubit is
|1\rangle , the target qubit evolves unitarily according to the bit flip operatorX ; - but for input states of the control that are superpositions of
|0\rangle and|1\rangle the evolution of the target qubit is*not*unitary.

To justify this last point, note that, if the control qubit is in the state ^{100}

This argument works even if the target qubit is initially in a mixed state, since we are dealing with a linear transformation, and any mixed state can be expressed as a statistical ensemble of pure states (via the convex decomposition of a density matrix).
Thus, in general, we can express the evolution of the target qubit^{101} as

Our discussion can easily be extended beyond the qubits to cover any conditional dynamics of the type

The reason we are paying a particular attention to the random unitaries is that each unitary is invertible, and, as such, they offer a sliver of hope for reversing the action of the channel.
If we can learn, post factum, which particular unitary operation *we cannot figure out which particular unitary was applied by inspecting the target system alone*.
In this case the best we can do is to apply the inverse of the most likely unitary, which will then recover the input state, *but only with some probability of success*.
In order to do better than that we have to look at slightly different channels.

First though, a fundamental example of a random unitary evolution: a **single qubit Pauli channel** applies one of the Pauli operators, **quantum errors**: bit-flip

### 7.1.2 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

Now, if ^{102}, then we can reverse the action of the channel:
we can, at least in principle, perform a measurement on

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^{103}

The three qubits, which form the output of the channel, are given to Bob, whose task is to recover the original state ^{104}

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?

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

## 7.2 Evolution of open systems

Needless to say, there is more to evolutions of open systems than random isometries, and what follows is the most general scenario we will discuss.
Consider two interacting systems, *not* assume that their interacting dynamics admits a control-target interpretation.
We will view ^{105} the evolution of system

Let us pick an orthonormal basis *not* need to be unitary, but, for the overall transformation *and* on the initial state of the auxiliary system

Without any loss of generality, we may assume that system ^{106} which can be chosen to be one of the basis states

The resulting density operator for

In summary, we can think about a quantum evolution of subsystem

In summary:

- First we pick up a system of interest which, in general, can be in a mixed state
\rho . It may be the case that this system is entangled with some other degrees of freedom or with some other physical systems, but these other entities will remain passive and will not enter any subsequent dynamics. - Then we dilate the system: we add an ancilla which is large enough to include everything our system will interact with, and also large enough to be in a pure state. The expansion ends when the composed system is (for all practical purposes) isolated and follows a unitary evolution
U . - After the unitary evolution takes place, we discard the ancilla and focus on the system alone. In fact we do not have to discard exactly what we added: we can discard only part of the ancilla, or any other part of the dilated system.
- The output system in this scenario does not have to be the original input system, but usually it is.

It is adding (i.e. tensoring) the auxiliary system in a fixed state, and then discarding it (via the partial trace), that is responsible for the *non-unitary* character of this evolution.

## 7.3 Stinespring’s dilation and Kraus’s ambiguity

Once we start playing with adding physical systems and increasing the dimension of the underlying Hilbert space, it is convenient to switch from unitaries to isometries.
This is more for mathematical simplicity than physical insight, but it is always good to declutter our equations a bit if we can.
Note that when we fix the initial state of system ^{107}
The matrix representation of an isometry is a rectangular matrix given by selecting only a few of the columns from a unitary matrix;
here, with

Let us now rephrase our derivation of the evolution of system

**Stinespring**^{108}**dilation.**Any quantum channel\mathcal{E} can be thought of as arising from a*unitary*evolution on a*dilated*system. When we combine tensoring and the unitary evolution into an isometryV , we can express the action of the channel\mathcal{E} as\rho \longmapsto \rho'= \operatorname{tr}_\mathcal{A} V\rho V^\dagger, where we trace out a suitably chosen ancilla\mathcal{A} . In quantum information science we often refer to this approach as*“the Church of the Larger Hilbert Space”*.**Kraus**^{109}**representation**(a.k.a.**operator-sum decomposition**). It is often more convenient to not deal with a larger Hilbert space, but to instead work with operators between the input and output Hilbert spaces\rho \longmapsto \rho'= \sum_i E_i\rho E_i^\dagger. Here we avoid dragging in the ancilla, which can be a good thing, since ancillas typically represent environments that can be very large and complex. This operator–sum decomposition is not unique, since the operatorsE_i (known as the**Kraus operators**or**effects**) depend on the choice of basis in the ancilla. The Kraus operators must satisfy the normalisation condition\sum_i E^\dagger_iE_i=\mathbf{1} , also known as the**completeness relation**.

We can easily switch between these two equivalent representations:

- We have already seen how to go from a unitary evolution
U on a larger system, to an isometryV , and then to a map on density operators represented by a set of Kraus operatorsE_i (as in Figure 7.1). - Conversely, once we have an operator-sum representation of the channel with a set of Kraus operators
E_i , we can introduce an ancilla of dimension equal to the number of Kraus operators, and use the orthonormal basis|i\rangle to form the isometryV=\sum_i|i\rangle\otimes E_i . In terms of matrices, this corresponds to simply “stacking up” the matricesE_i to form the block column (as shown in Figure 7.1), which gives us the matrix representation ofV . If we want to go further, from an isometryV to a unitaryU , then the next step is somewhat arbitrary: we can choose all the remaining block columns ofU however we please,*as long as*we end up with a unitary matrixU .

In summary:

All linear transformations of density operators that can be written in the Stinespring (or, equivalently, Kraus) form represent *physically realisable operations*, and we call them **quantum channels**.^{110}

We note again that the Kraus decomposition is *not unique*: the operators

In summary:

Suppose ^{111} to ensure that * \mathcal{E} and \mathcal{F} describe the same channel if and only if F_j=\sum_i R_{ji} E_i for some unitary R*.

In particular, this unitary equivalence of the Kraus operators implies that the identity channel

## 7.4 Single qubit channels

The best way to familiarise ourselves with the concept of a quantum channel is to study a few examples, and we will start with the simplest case: **single qubit channels**.
The single qubit case is special since we can visualise the action of the channel by looking at the corresponding deformation of the Bloch sphere.
Recall that an arbitrary density matrix for a single qubit can be written in the form

*A bit-flip with probability*p :\rho \longmapsto (1-p)\rho+pX\rho X. The Kraus operators are\sqrt{1-p}\mathbf{1} and\sqrt{p}X ; the original Bloch sphere shrinks into a prolate spheroid aligned with thex -axis; for the specific case ofp=\frac{1}{2} , the Bloch sphere degenerates to the[-1,1] interval on thex -axis.*A phase-flip with probability*p :\rho \longmapsto (1-p)\rho+pZ\rho Z. The Kraus operators are\sqrt{1-p}\mathbf{1} and\sqrt{p}Z ; the original Bloch sphere shrinks into a prolate spheroid aligned with thez -axis; for the specific case ofp=\frac{1}{2} , the Bloch sphere degenerates to the[-1,1] interval on thez -axis.*The depolarising channel:*\rho\longmapsto (1-p)\rho + \frac{p}{3}\left(X\rho X+Y\rho Y+Z\rho Z\right). Here the qubit remains intact with probability1-p , while a quantum error occurs with probabilityp . The error can be of any one of three types: bit-flipX , phase-flipZ , or and both bit- and phase-flipY ; each type of error is equally likely. Forp<\frac{3}{4} , the original Bloch sphere contracts uniformly under the action of the channel, and the Bloch vector shrinks by the factor1-\frac{4}{3}p ; for the specific case ofp=\frac{3}{4} , the Bloch sphere degenerates to the point at the centre of the sphere; forp>\frac{3}{4} , the Bloch sphere is flipped, and the Bloch vector starts pointing in the opposite direction increasing the magnitude up to\frac{1}{3} (which occurs forp=1 ).

There are two interesting points that must be mentioned here.
The first one is about the interpretation of the action of the channel in terms of Kraus operators: our narrative may change when we switch to a different set of effects.
For example, take the phase-flip channel with *narrative* is different.
The first set of effects tells us that the channel chooses randomly, with the same probability, between the two options: let the qubit pass undisturbed or apply the phase-flip

Describing actions of quantum channels purely in terms of their effects (i.e. Kraus operators) can be ambiguous.

The second interesting point is that not *all* transformations of the Bloch sphere into spheroids are possible.
For example, we cannot deform the Bloch sphere into a pancake-like oblate spheroid.
This is due to *complete* positivity (instead of mere positivity) of quantum channels, which we will explain shortly.

## 7.5 Composition of quantum channels

We mentioned that quantum channels are combinations of

- adding a physical system in a fixed state (via tensoring),
- discarding a physical system (taking a partial trace),
- and unitary transformations.

For consistency let us note that each of these operations admits an operator-sum decomposition. This is obvious for unitary evolution (

**Adding a system.**Any quantum system can be expanded by bringing in an auxiliary system in a fixed state|a\rangle . This transformation takes vectors in the Hilbert space associated with the original system and tensors them with a fixed vector|a\rangle in the Hilbert space associated with the auxiliary system:|\psi\rangle \longmapsto |a\rangle\otimes|\psi\rangle = (|a\rangle\otimes\mathbf{1}) |\psi\rangle. In terms of density operators, we write this “expansion” transformation as\begin{aligned} \rho \longmapsto \rho' &= |a\rangle\langle a|\otimes\rho \\&= (|a\rangle\otimes\mathbf{1})\rho (\langle a|\otimes\mathbf{1}) \\&= V\rho V^\dagger \end{aligned} whereV=\mathbf{1}\otimes|a\rangle . We note thatV^\dagger V = \mathbf{1}\otimes\langle a|a|=\rangle\mathbf{1} is the identity in the Hilbert space associated with the system, and soV is an isometry. Indeed, this transformation is an*isometric embedding*.**Discarding a system.**Conversely, given a composite system in state\rho , we can discard one of its subsystems. The partial trace over an auxiliary system can be written in the Kraus representation as\begin{aligned} \rho \longmapsto \rho' &= \operatorname{tr}_\mathcal{A}\rho \\&= (\operatorname{tr}\otimes\mathbf{1})\rho \\&= \sum_i (\langle i|\otimes\mathbf{1})\rho(|i\rangle\otimes\mathbf{1}) \\&= \sum_i E_i\rho E^\dagger_i \end{aligned} where the vectors|i\rangle form an orthonormal basis in the Hilbert space associated with the auxiliary system. Again, we can check that the Kraus operatorsE_i=\langle i|\otimes\mathbf{1} satisfy the completeness relation\sum_i E^\dagger_i E_i =\mathbf{1}\otimes\mathbf{1} (using the fact that\sum_i|i\rangle\langle i|=\mathbf{1} ).

Any *sequential* composition of two quantum channels ^{112} described by the Kraus operators

You might wonder why we explicitly called the above composition “sequential” — isn’t this how we always compose functions?
In actual fact, since we have access to tensor products, there is another sort of composition, namely **parallel** composition: if we have systems

Now that we know how to compose quantum channels in terms of Kraus operators, we can see that the Stinespring representation is perfectly consistent with the Kraus representation: the three basic operations that we are allowed to use to build channels in the Stinespring representation (i.e. adding a system, unitary evolution, and discarding a system) are all themselves quantum channels, in that they admit a Kraus decomposition.

Before moving on, we make a small (but important) remark:

When we compose quantum channels, each channel needs its own independent ancilla;
*do not share ancillas between different channels*.

For example, say we have three channels,

For more on this, see the appendix on Markov approximation.

## 7.6 Completely positive trace-preserving maps

It is easy to verify that quantum channels preserve positivity and trace, but the converse is not true!
You may find it surprising, but there are linear maps that preserve positivity and the trace, but which are *not* quantum channels.
The matrix transpose operation *cannot* be written in the Stinespring (or the Kraus) form;
it is not induced by a unitary operation on some larger Hilbert space, and it cannot be implemented.
So, we then ask, *what is* the class of physically admissible maps?

Mathematically speaking, a quantum channel ^{113}

Firstly, for such a map

\mathcal{E} to be a channel it must*respect the mixing of states*. Consider an ensemble of systems, with a fractionp_1 of them in the state\rho_1 , and the remainingp_2 of them in the state\rho_2 . The overall ensemble is described by\rho=p_1\rho_1+p_2\rho_2 . If we apply\mathcal{E} to each member of the ensemble individually, then the overall ensemble will be described by the density operator\rho'=\mathcal{E}(\rho) , which is given by\rho'=p_1\mathcal{E}(\rho_1)+p_2\mathcal{E}(\rho_2) . We conclude that\mathcal{E} must be a*linear map*.Next, since

\mathcal{E} must*map density operators to density operators*it must be both*positive*(\mathcal{E}(\rho)\geqslant 0 whenever\rho\geqslant 0 ) and*trace preserving*(\operatorname{tr}\mathcal{E}(\rho)=\operatorname{tr}\rho for all\rho ).Finally comes a subtle point. It turns out that being positive is not good enough; we must further require that the map

\mathcal{E} *remains positive even when extended to act on a part of a larger system*. Suppose that Alice and Bob share a bipartite system\mathcal{AB} in an entangled state\rho_\mathcal{AB} , and, whilst Alice does nothing, Bob applies the local operation\mathcal{E} to his subsystems, and his subsystems only. Then the resulting map on the whole bipartite system is given by\mathbf{1}\otimes\mathcal{E} , and this must give a proper density operator\rho'_\mathcal{AB} of the composed system. It turns out that this is a strictly stronger property than mere positivity; we are asking for something called**complete positivity**. Needless to say, complete positivity of\mathcal{E} implies positivity, but the converse does not hold: there are maps which are positive but not completely positive. The matrix transpose operation\rho\rightarrow \rho'=\rho^T is a classic example of such a map.

In fact, we can study the matrix transpose a bit further.
Consider the transpose operation on a single qubit: *only one* of the two qubits (say, the second one), then the density matrix of the two qubits evolves under the action of the partial transpose *not* preserve positivity, and therefore *not* a completely positive map.
If you prefer to see this more explicitly, then you can use the matrix representation of

So we have seen that, at the very least, we want to be considering *completely positive trace-preserving* maps, but how do we know whether or not there are any restrictions left to impose?
Needless to say, here is where mathematics alone cannot guide us, since we are trying to characterise maps which are *physically admissible*, and mathematics knows nothing about the reality of our universe!
However, one thing that we can do is compare our abstract approach with the derivations of quantum channels defined in terms of the Stinespring (or the Kraus) representation.
As it happens, we can show that a map is completely positive and trace preserving if and only if it can be written in the Stinespring (or the Kraus) form.
In other words:

Quantum channels are exactly the completely positive trace-preserving (CPTP) maps.

One direction of this claim is much simpler than the other.
Any channel *and* the extended map **Choi matrix**^{114}, which is yet another way to characterise linear maps between operators.

**!!to-do: tim footnote: here we work only with finite dimensional spaces, and if**X is f.d. then every linear operatorf\colon X\to Y between normed vector spaces is continuous (and thus bounded)!!- talk about unbounded operators

## 7.7 State-channel duality

Suppose that

We call this block matrix **Choi matrix** of

The Choi matrix is essentially another way of representing a linear map

The Choi matrix

The expression above may look baffling to an untrained eye, but this is often the case when we turn something conceptually obvious into a precise and compact mathematical notation.
In order to gain some intuition here, recall that, for matrices

This state-channel duality thus gives us a one-to-one correspondence between linear maps **Choi–Jamiołkowski isomorphism**, and we discuss it further in the appendix of the same name.

Mathematically, it is not too surprising that the matrix elements of an operator on a tensor product can be reorganised and reinterpreted as the matrix elements of an operator between operator spaces.
What is interesting, and perhaps not so obvious, however, is that the positivity conditions for maps correspond exactly to conditions on their Choi matrices under this correspondence.
In order to see this, let us express the Choi matrix as the result of

The Choi matrix

In this form, the Choi–Jamiołkowski isomorphism gives us a simple necessary and sufficient condition for a linear map to be a quantum channel:

**State-channel duality**:

One direction of this claim is immediate: we already know that any quantum channel

Now, any vector ^{115}

Using this, we see that
^{116} *states* and *channels*.

The equations

We summarise the flow of implications in the following diagram:
*if \mathcal{E} is a quantum channel, then its Choi matrix \widetilde{\mathcal{E}} is a density matrix*.
The reverse implication goes as follows.
The density matrix

*if the Choi matrix*\widetilde{\mathcal{E}} is a density matrix, then \mathcal{E} is a quantum channel.

## 7.8 The mathematics of “can” and “cannot”

So what is state-channel duality good for?
To start with, it can be used to asses whether or not a given map

Consider the map

But is this map a quantum channel?
That is, *does it represent a physical process that can be implemented in a lab*?

The expression for *is not physically admissible* — it cannot be implemented.
But does it mean that the map

In fact, the answer depends on the value of *completely positive* to merely *positive*.
In order to find this critical value of * \mathcal{E} is completely positive (and hence physically realisable) if and only if \widetilde{\mathcal{E}}\geqslant 0*;
the latter is true only when

## 7.9 Kraus operators, revisited

One thing that is very important is that *state-channel duality gives us more than just a one-to-one correspondence* between states *also* gives a one-to-one correspondence between vectors in the statistical ensemble

We already know that two mixtures ^{117}

How many Kraus operators do we really need?
State-channel duality tells us that the *minimal* number of Kraus operators needed to express ^{118}) of the corresponding Kraus operators, * \langle\widetilde{v}_k|\widetilde{v}_l\rangle=0 implies that \operatorname{tr}E_k^\dagger E_l=0*.

A linear map

If this is the case, then this decomposition has the following properties:

\mathcal{E} is trace preserving if and only if\sum_k E^\dagger_kE_k=\mathbf{1} .- Two sets of Kraus operators
\{E_k\} and\{F_l\} represent the same map\mathcal{E} if and only if there exists a unitaryR such thatE_k =\sum_l R_{kl}F_l (where the smaller set of the Kraus operators is padded with zeros, if necessary).

Note that, for *any* *at most*

## 7.10 Correctable channels

Another question that we might ask ourselves is if we can *reverse the action of a quantum channel, recovering the input state*.
Let us first make precise what we mean by this.

We say that a quantum channel **correctable** if there exists a **recovery channel**

The action of any unitary operation

If all we know is that an isometry

Consider a channel

Note that, for a measurement on *not* in the image of the channel: we will never see the result of the measurement corresponding to the projection onto that subspace.
However, we *need* to add it in order to obtain a complete decomposition of *the operator-sum decomposition is not unique*, and so the same channel can be described by another, unitarily related, set of Kraus operators,

Conversely, if the channel ^{119} *and* sufficient in order for the channel

Let

\mathcal{E} is correctable;E_i^\dagger E_j = \sigma_{ij}\mathbf{1} for some density matrix\sigma_{ij} ;- there exists a set of orthogonal isometries
\{V_i\} and a probability distribution\{p_i\} such that\mathcal{E} (\rho) = \sum_i p_i V_i\rho V^\dagger_i for every state\rho .

## 7.11 *Appendices*

### 7.11.1 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*.^{120}
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

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

Any isometry is a quantum channel, since any quantum state described by the state vector

An example which we will later return to is that of the **three-qubit code**.
Take a qubit in some pure state

The result is the isometric embedding of the

### 7.11.2 The Markov approximation

Composition of quantum channels^{121} 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 *not* discarded after the first unitary evolution *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^{122} 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.

### 7.11.3 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 **separable** if **entangled**.
If we apply the partial transpose

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*

As an example, consider a quantum state of two qubits which is a mixture of the maximally 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 states**^{123}
However, for *two* qubits, the PPT states are exactly the separable states.

### 7.11.4 The Choi–Jamiołkowski isomorphism

The correspondence between linear maps **Choi–Jamiołkowski isomorphism** or **channel–state duality**, is another example of a well known correspondence between vectors in

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

- the first is very nice, but non-canonical (i.e. is dependent on the choice of basis);
- 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; - 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 **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 *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

*not*like

**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

### 7.11.5 Block matrices and partial trace

For any matrix *that is written in the tensor product basis*, the partial trace over the first subsystem (here

For example, for any

The same holds for general

## 7.12 *Remarks and exercises*

### 7.12.1 Partial inner product

The tensor product structure brings with it 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

### 7.12.2 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*.

### 7.12.3 Surprisingly identical channels

Let us now compare two single qubit quantum channels:

We are familiar with the first channel from the previous example: it performs the measurement in the standard basis, but doesn’t reveal the outcome of the 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:

### 7.12.4 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

### 7.12.5 Cooling down

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

### 7.12.6 Unchanged reduced density operator

Show that, for any operator

*(Hint: show this for operators \rho which are tensor products \rho=A\otimes B and then extend the result to any operator \rho.)*

### 7.12.7 Order matters?

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

### 7.12.8 Pauli twirl

Show that randomly applying the Pauli operators

### 7.12.9 Depolarising channel

The most “popular” Pauli channel is the **depolarising channel**

(For

### 7.12.10 Depolarising channel and the Bloch sphere

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

### 7.12.11 Complete positivity of a certain map

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

### 7.12.12 Toffoli gate

Consider the Toffoli gate

Express

### 7.12.13 Duals

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

- Show that, if
\mathcal{E} is trace preserving, then\mathcal{E}^\star is unital. - Show that, if
\mathcal{E}=\sum_i E_i\cdot E_i^\dagger , then\sum_i E^\dagger_i\cdot E_i is an operator-sum decomposition of\mathcal{E}^\star .

### 7.12.14 Trace, transpose, Choi

Let

(For example, if we are interested in the component

### 7.12.15 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

### 7.12.16 Tricks with a maximally entangled state

A maximally entangled state of a bipartite system can be written, using the Schmidt decomposition, 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 of the bipartite system

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

S\colon|i\rangle|j\rangle\mapsto|j\rangle|i\rangle , can be expressed as^{124}\begin{aligned} S &= d |\Omega\rangle\langle\Omega|^{T_{\mathcal{A}}} \\&= d \sum_{ij} \big(|i\rangle\langle j|\big)^T\otimes|i\rangle\langle j| \\&= d \sum_{ij} |j\rangle\langle i|\otimes|i\rangle\langle j|. \end{aligned} This implies that\operatorname{tr}[(A\otimes B)S] = \operatorname{tr}AB and that(A\otimes\mathbf{1})S = S(\mathbf{1}\otimes A).

### 7.12.17 Trace preserving and partial trace

Show that

*(Hint: show that \operatorname{tr}\mathcal{E}(|i\rangle\langle j|=\delta_{ij}.)*

### 7.12.18 Rotating Kraus operators

Mathematically speaking, Kraus operators

### 7.12.19 No pancakes

Consider a single qubit operation which causes the

Explain why we cannot physically implement such a map.

…there is only one unitary evolution, there is only one unitary evolution, there only one unitary evolution… …and everything else is cheating↩︎

Depending on the context, the auxiliary system is either called the

**ancilla**(usually when we can control it) or the**environment**(usually when we cannot control it).↩︎Recall that an operator is positive if and only if it can be written in the form

XX^\dagger for someX (hereX=E_k\sqrt{\rho} ). Also, the sum of positive operators is again a positive operator.↩︎Recall that, for the basis states,

\operatorname{tr}|i\rangle\langle j|=\langle i|j\rangle=\delta_{ij} .↩︎We can also focus on the evolution of the control qubit: see the examples and exercises section. In fact, we can choose any subset of qubits for our inputs and outputs. For example, our input could be the control qubit, and the output could be

*both*the control*and*the target qubits.↩︎That is, if the subspaces

\mathcal{H}'_i are mutually orthogonal.↩︎Our arguments here can be easily extended to any mixed state

\rho , but for simplicity we consider the case of a pure state.↩︎**!!!to-do!!! picture**↩︎For now, when we write tensor products, we will place the ancilla first and the system of interest second:

\mathcal{H}_\mathcal{A}\otimes\mathcal{H}_\mathcal{B} . We do this to begin with simply because block matrices on tensor products are easier to interpret when written in this particular order. Later on we will revert to the more common convention in which the system of interest is placed first.↩︎If

\mathcal{A} were initially in a mixed state, we could always regard\mathcal{A} as a subsystem of some larger\widetilde{\mathcal{A}} that is in an entangled pure state.↩︎Recall that a map

V is an isometry ifV^\dagger V=\mathbf{1} . For example, adding a system in state|k\rangle gives an isometryV\colon|\psi\rangle\mapsto|k\rangle\otimes|\psi\rangle , and the combination of adding a system in a fixed state followed by a unitary evolution of the combined system is also an isometry. Isometries preserve inner products, and therefore also preserve both the norm and the metric based upon the norm.↩︎William Forrest “Woody” Stinespring (1929–2012) was an American mathematician specialising in operator theory.↩︎

Karl Kraus (1938–1988) was a German physicist known for his contributions to the mathematical foundations of quantum theory. His book

*States, effects, and operations*(Lecture Notes in Physics, Vol.**190**, Springer-Verlag, Berlin 1983) is an early account of the notion of complete positivity in physics.↩︎Quantum channels are also known as

**superoperators**— this way physicists remind themselves that these transformations take operators to operators.↩︎If you wish, instead of appending zeros, you may view

R_{ji} as an isometry from the the smaller to the larger set of Kraus operators.↩︎Here we have tacitly assumed that the dimensions agree, i.e. that the output of

\mathcal{E} and the input of\mathcal{F} are of the same dimension, so that the composition makes sense.↩︎Recall that, given a pair of Hilbert spaces

\mathcal{H} and\mathcal{H}' , we denote the set of (bounded) linear operators from\mathcal{H} to\mathcal{H}' by\mathcal{B}(\mathcal{H},\mathcal{H}') . We write\mathcal{B}(\mathcal{H}) as shorthand for\mathcal{B(H,H)} ↩︎Man-Duen Choi was brought up in Hong Kong. He received his Ph.D. degree under the guidance of Chandler Davis at Toronto. He taught at the University of California, Berkeley, from 1973 to 1976, and has worked since then at the University of Toronto. His research has been mainly in operator algebras, operator theory, and polynomial rings. He is particularly interested in examples/counterexamples and two-by-two matrix manipulations.↩︎

**Exercise.**Prove this!↩︎**Exercise.**Prove this!↩︎The number of vectors contributing to each mixture (and hence the number of corresponding Kraus operators) may be different, so we simply extend the smaller set to the required size by adding zero operators.↩︎

Recall that the Hilbert–Schmidt product

(A|B) of two operatorsA andB is defined by(A|B)=\frac12\operatorname{tr}A^\dagger B .↩︎This is the normalisation condition for the Kraus operators

R_lE_j ↩︎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.↩︎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).↩︎A

**quantum Markov process**! Andrey Markov (1929–2012) was a Russian mathematician best known for his work on stochastic processes.↩︎“PPT” stands for

*positive partial transpose*.↩︎We write

X^{T_{\mathcal{A}}} to mean the partial transpose over\mathcal{A} , i.e.T\otimes\mathcal{I} .↩︎