5.1 Introduction

Consider a typical quantum system such as a string of ions in a trap. To “process” the quantum information they carry, we have to perform, in general, many different processing steps such as free time evolution (including unwanted but unavoidable interactions with the environment), controlled time evolution (e.g., the application of a “quantum gate” in a quantum computer), preparations and measurements. This lecture aims at providing a unified framework for describing all these different operations.

5.2 Completely Positive Maps

The basic idea is to interpret each processing step as a channel, which transforms the system's initial state into the output state the system attains, after completion of the processing. Occasionally, we will represent this picture graphically as in Figure 5.1. To get a mathematical description, consider the two Hilbert spaces , (subsequently called the “initial” and the “target” Hilbert space) with (finite) dimensions¹ and and the algebras , respectively, of (bounded) operators on them. Input and output states are described by density operators on and , which we denote by and again. Using this notation, we can regard a channel as a map , which transforms the input state into the output state .

Each physically reasonable operation should obey the mixing of states, that is, if , are transformed into , the mixture , is mapped to . This implies that can be extended to a linear map

5.1

and since maps density matrices to density matrices it must be positive

5.2

and trace preserving

5.3

Hence, each channel can be described by a positive and trace‐preserving linear map .

However, this picture is incomplete because we can apply a channel not only to the overall system but also to subsystems. A typical example arises, if Alice and Bob share a bipartite system in an entangled state and Alice applies a local quantum operation to her subsystem (and Bob does nothing). Again, the crucial point is that the overall system shared by Alice and Bob end up in a valid quantum state . In other words, the combination of “quantum operation performed by Alice” and “doing nothing by Bob” can be interpreted as a valid channel applied to the bipartite system (cf. Figure 5.2). If and are density matrices on or , respectively, the channel applied by Alice can be described by a positive, trace‐preserving linear map , while “doing nothing” on Bob's system is just represented by the identity . Hence, the output state can be written as . Obviously, is linear and trace preserving; but positivity of is not sufficient for positivity of . The most prominent example where this fails is the transposition. Although the transpose of a positive matrix is positive, the partial transpose is in general not. To describe a physically realizable operation, the map has to satisfy therefore in addition to 5.2 and 5.3 the condition

5.4

and because Bob's system can be arbitrary, this should hold for any dimension of . Let us summarize the discussion up to now in the following definition:

A channel is represented in the Schrödinger picture by a trace‐preserving cp‐map. To get the Heisenberg picture representation, we have to introduce the dual of . It is the map uniquely defined by

5.5

It is easy to see that is completely positive, if is, and that is unital if is trace preserving (Problem 5.1). If is an effect, that is, an operator with , representing a yes/no measurement², its image is an effect as well (since is positive and unital). It should be regarded as the effect we get, if we first apply the channel to the system and then measure the effect (cf. Figure 5.3). Some typical examples of channels are given as follows:

• Unitary time evolution. The most simple example is time evolution, described by a unitary operator on . The corresponding channel is described by .
• Expansion. Another elementary example arises if we expand a given quantum system by a second one (described by the Hilbert spaces and , respectively). Hence, initial and target Hilbert spaces are and , and if the ‐system is in the state the channel becomes .
• Restriction. The inverse operation arises, if we discard a subsystem; that is, the initial Hilbert space is now , the target Hilbert space is and is given by , where denotes the partial trace over .
• Noisy time evolution. The composition of channels is again a channel. Hence, we can combine the three examples just given: First, expand the system, then let it evolve unitarily, and finally discard the system added in the first step: and
  5.6

  with a unitary on and a density matrix on . Physically, this type of channel describes the influence of noise caused by interaction with the environment (represented by the ‐system): is the initial state of the environment and represents the joint evolution of the system and the environment; cf. Figure 5.4. We will see in Section 5.4 that each channel can be written this way.

5.3 The Choi–Jamiolkowski Isomorphism

The subject of this section is a relation between completely positive maps and states of bipartite systems first discovered by Choi (1) and Jamiolkowski (2), which is very useful in establishing several fundamental properties of cp‐maps.

The idea is based on the setup already discussed in Figure 5.2: Alice and Bob share a bipartite system in a maximally entangled state

5.7

(where denotes an orthonormal basis of ) and Alice applies to her subsystem a channel while Bob does nothing. At the end of the processing, the overall system ends up in a state

5.8

Mathematically, Eq. 5.8 makes sense, if is only linear but not necessarily positive or completely positive (but then isn't positive either). If we denote the space of all linear maps from into by we therefore get a map

5.9

which is easily shown to be linear (i.e., for all and all ). Furthermore, this map is bijective, hence a linear isomorphism.

The proof of this theorem is left as an exercise to the reader (Problem 5.2). From the definition of in Eq. 5.8, it is obvious that is positive, if is completely positive. To see that the converse is also true, is not as trivial, because a transposition (which is not completely positive) is involved in the definition of 5.11. It is therefore useful to rewrite Eq. 5.11 in terms of a purification of . Hence, consider an auxiliary Hilbert space and such that . Note that the existence of such a requires positivity of , but not normalization. If denotes an orthonormal basis, we can define an operator by

5.12

Now we get with Eq. 5.11

5.13

5.14

5.15

Let us summarize this result for later reference in the following lemma.

Note that the definition of in terms of from Eq. 5.12 depends on the choice of the basis but not on the (the are fixed already by the choice of in 5.8). However, this ambiguity does not affect the expression , because all operators arising from different bases in are related by unitary operators on .

From Eq. 5.16, we see immediately that is completely positive, if is positive. Together with Theorem 5.2 this leads to

As an immediate consequence of this theorem, we can simplify the original characterization of complete positivity in Definition 5.1. To this end, let us define for each a map to be ‐positive if is positive (where denotes as in Definition 5.1 the identity on ). Note that this is in general a weaker condition than complete positivity, because is completely positive, if is ‐positive for each . In the finite‐dimensional case, however, it is sufficient to have ‐positivity for sufficiently large .

5.4 The Stinespring Dilation Theorem

At the end of section 5.2, we have claimed that each channel can be written in terms of an ancilla as in Eq. 5.6. We are now prepared to prove this statement. The following theorem, which goes back to Stinespring (3), is the central structure theorem about completely positive maps.

Let us consider now the uniqueness of Stinespring representations. Obviously, we can always enlarge the dilation space by adding extra dimensions (i.e., replacing by and leaving untouched). Hence, Stinespring representations are not unique. But what happens, if we assume that is “as small as possible,” that is, if the dimension of cannot be reduced by discarding “superfluous” components? This situation is characterized by the condition

5.18

Now we have the following theorem:

This lemma shows that we get a minimal Stinespring representation if we define and in terms of 5.19 with a minimal purification of . Its uniqueness follows from the uniqueness (up to unitary equivalence) of the minimal purification.

Let us consider now two alternative representation theorems, which can be derived directly from the Stinespring Theorem. The first is the ancilla form of a channel, which we have encountered already in Eq. 5.6.

Even if the Stinespring representation used in the proof is the minimal one, there is a lot of freedom to define the unitary , because it depends on the choice of and of many matrix elements, which in the end drop out of all results. This is a disadvantage of the ancilla approach in practical computations.

Let us come back now to a general (i.e., not necessarily trace preserving) cp‐map and consider a Stinespring representation of it. If we choose vectors with we can define a family of operators by

5.27

In terms of these operators, Eq. 5.17 can be rewritten as follows (cf. Problem 5.4 and ( 1,4)).

Finally, let us state a third result, which is closely related to the Stinespring theorem. It characterizes all decompositions of a given completely positive map into completely positive summands. It shows in particular that all “Kraus representations” of a given cp‐map (i.e., Eq. 5.28 with appropriate operators ) can be derived as in 5.27. By analogy with results from measure theory, we will call it a Radon–Nikodym theorem (cf. (5))

The properties of completely positive maps we have just discussed are only the most elementary ones. For a much more complete, in‐depth presentation of this subsection, we would like to refer the reader to the book of Paulsen (6).

5.5 Classical Systems as a Special Case

Up to now we have only treated pure quantum systems, for which the possible observables are given by all bounded operators on a Hilbert space. Classical systems can be understood as a special case, with a constraint on what we can measure: namely only those observables, which are diagonal in some fixed basis. Since diagonal matrices commute, this is the same as choosing a commutative subalgebra of observables.

The transition from a quantum system to a classical subdescription is made by a particular channel , which simply kills all off‐diagonal terms, sometimes called “interference terms.” When denotes the particular orthonormal basis in which we want to go classical, we set

5.40

This is also called a complete von Neumann measurement: the term in this sum is the corresponding basis state, multiplied with the probability for obtaining the result . It is easily verified that the formula for the Heisenberg picture of this channel is exactly the same as 5.40.

Clearly, the specification of elements in the classical observable algebra require only rather than real parameters, as in the quantum case. Therefore, channels with one classical input or output can also be described by fewer parameters. For example, a channel with classical input has the property that : its output depends only on the diagonal matrix elements of the input matrix. Hence, it can be written as

5.41

where the are arbitrary states of the final system, which characterize . The input state merely selects the weights in a convex combination of these states. Dually, channels with classical output are of the form

5.42

where the are positive operators adding up to the identity operator. Thus, is an observable, or positive operator‐valued measure.

An important special case is also the channels whose output is the tensor product of a classical and a quantum output. If , is classical basis in , the general form of such a channel is , with

5.43

where each of the is completely positive. Such a channel is called an instrument (7). Since there are two outputs, we get two “marginals,” that is, the channels obtained by ignoring either output: If we do not look at the quantum output, we get an observable in the sense of 5.42 by . On the other hand, if we do not select according to the results , we get the channel .

5.6 Channels with Memory

During a realistic communication process, the same channel is used many times in succession, which raises the question in which way each invocation can depend on the previous ones. A mathematical analysis of this problem leads to the concept of a channel with memory, which is described below. Since this is a very large field, we can only give a very brief overview. A detailed discussion can be found in (8) and (9) and the references therein.

The most simple case is a memoryless channel transmitting ‐level systems. It is described by a trace‐preserving cp‐map with . If Alice uses to send systems in the joint state to Bob, is invoked ‐times independently. The invocation is given by the tensor product

5.44

such that the overall operation becomes the concatenation

5.45

Hence, Bob receives the output systems in the joint state . This is the appropriate model for situations where memory effects are not present or can be ignored.

If in contrast memory effects have to be taken into account, we have to replace by a trace‐preserving cp‐map

5.46

Here, is a (finite‐dimensional) Hilbert space, which describes the memory, and is called a channel with memory. If Alice transmits one system in a state with the memory in the initial state , the output system received by Bob is in general correlated with the memory and the joint output state is . If Bob is not interested in the memory (or cannot access it), we have to trace away such that the real output state becomes . To send an ‐fold system in the state , we have to invoke the channel times in succession. The invocation is again a tensor product, but now the memory has to be taken into account such that we get

5.47

which is a map of the form

5.48

Note that the factor is shifted here from the to the position. This allows us to write the overall operation as in 5.45 as a concatenation

5.49

5.50

If the memory is ignored at the end, Bob receives the ‐fold system as above in the final state . Note that in contrast to we cannot write as a tensor product and even if the input state is a product state the output state in general is not.

The scheme just constructed describes a channel that can act on an arbitrary number of systems (via the concatenations ). Furthermore, it satisfies the natural causality condition that the invocation depends on the previous ones but not on the that will take place in the future. It can be shown that any channel that is causal in this way can be written as a concatenation of a memory channel ; cf. (9).

Let us change our point of view now slightly and look at the final state of the memory while the transmitted system is ignored, that is, is given by

5.51

The interesting question is how much information about the initial state is still contained in . The most extreme case arises if for some (and therefore for all , as well) it does not depend on at all. Channels of this type are called forgetful (since after at least invocations the initial state is completely “forgotten”). A simple example for a forgetful channel is the “shift Channel” given by (with )

5.52

which exchanges the input with the memory (note the flipped positions of the Hilbert spaces at the output side). Hence, the memory is completely overridden after only one invocation. In contrast to this, the identity channel (taking again into account the flipped Hilbert spaces) is not forgetful, since the memory is passed unchanged. Forgetful channels play a special role since they can be treated in many respects (in particular if channel capacities are discussed) in the same way as memoryless channels.

5.7 Examples

5.7.2 Depolarizing Channel

At the opposite extreme is a channel that destroys all input information, replacing it by a completely chaotic output state , where . Slightly more generally, we can look at the channel that does this with probability , and otherwise ideally transmits the input:

5.53

Here, we have included the trace factor (which is 1 for input states), so that becomes a linear map. This channel is often used as a noise model, usually with a small depolarization probability . Interestingly, this channel is completely positive even for some . For qubits, this has a quite intuitive interpretation in terms of transformations of the Poicaré sphere: as increases, the set of output states shrinks, until at it coincides with the origin. Increasing further means that the Poincaré sphere becomes inverted. For , we would get a complete inversion, the so‐called Universal‐NOT operation, which sends every pure state to its orthogonal complement. This map is positive, but not completely positive, so it is an impossible operation. Its best approximation by completely positive channels is obtained by taking as large as possible ( for qubits),

The Kraus decompositions of the fully depolarizing channel ( ) are characterized by the equation

5.54

This can be solved for any by operators , which are unitary up to a factor. Such orthogonal sets of unitaries play a central role in teleportation and dense coding schemes.

5.7.3 Entanglement Breaking Channels

Can quantum information be transmitted via classical channels? This would mean to first make a measurement , transmit the results via a classical channel, and to let the receiver try to reconstruct the quantum input state by a repreparation , which depends on the results of the measurement. The form of such a channel is , where is the von Neumann measurement for the intermediate classical channel. When and are given as in 5.41 and 5.42, respectively, and the classical signals transmitted are labeled by , this gives a channel of the form

5.55

It turns out that these channels are characterized by the property that turns every entangled state into a separable state, that is, they destroy all entanglement (Problem 5.6).

5.7.4 Covariant Channels

Many channels of interest have a simple characterization in terms of symmetries. For example, the depolarizing channels 5.53 are the only ones that do not distinguish any basis in Hilbert space, in the sense that a basis change by a unitary operator does not change the action of the channel: . More general characterizations of symmetries involve subgroups of unitary operators, which may differ for initial and target space:

5.56

where is some abstract group and and are unitary representations of this group on the initial and target Hilbert space, respectively. Channels satisfying this condition are called covariant.

Since the minimal Stinespring representation is unique up to unitary equivalence, the covariance of the channel is also reflected at that level, and this often allows us to give concise formulas for all channels satisfying 5.56, given and the representations. Let be the Stinespring isometry. Then, for every , is again a dilation, which means that this dilation must be connected with by a unitary of the form . In other words, we find the condition

5.57

One readily verifies that must be a unitary representation of on . In the language of group representation theory, this relation says that must be an intertwining operator between the representations of , and there is a highly developed formalism to determine such operators. Let us consider two cases:

When the group is , the irreducible representations are labelled by the spin parameter . Let us take both input and output representations to be irreducible with spin and , respectively. This fixes the dimensions to be and . Now it is easy to see that decomposing the representation into irreducibles corresponds to a convex decomposition of . Therefore, to find the extremal covariant channels, we can assume to be irreducible, as well, and hence to be fixed by a spin parameter . Then, the Clebsch–Gordan theory of adding angular momenta tells us that a nonzero intertwiner exists if and only if , and is integer. Moreover, the intertwiner in these cases is a unique isometry, whose matrix elements are the well‐known Clebsch–Gordan coefficients.

For example, when , gives the ideal channel. For we can also define the channel

5.58

where denotes the angular momentum operators of the spin representation. This corresponds precisely to , because the angular momenta are the components of a vector operator, transforming with the spin 1 representation. gives the depolarizing channel.

Another interesting group for constructing covariant channels are the phase space translations or, more precisely, the Heisenberg group, consisting of the phase space translations and the multiples of . The phase space displacement by the phase space vector is then given by the Weyl operators , and we assume these to act irreducibly, so that there are no further degrees of freedom. By the canonical commutation relations, the Weyl operators are also characterized as the eigenvectors of the action of phase space translations on operators: that is, for all implies that must be proportional to a Weyl operator , and contains an exponential factor characterizing . Inserting this condition into the covariance equation 5.56, one readily finds that (in the Heisenberg picture) a phase space covariant channel must take Weyl operators to multiples of Weyl operators:

5.59

Moreover, is a channel if and only if is the Fourier transform of a probability measure, and acts by making a random phase space translation, selected according to this measure.

The theory applies also, however, when the Weyl systems on the input and output sides are different, and the displacement parameters are connected by some linear map between input and output phase space. For example, we could take , with some positive factor . This corresponds to the amplification or attenuation of a quantum optical light field (depending on whether or ). In this case, the complete positivity condition for is a bit more difficult to write down. It forces to contain some noise, as is expected from the no‐cloning theorem. The ancilla form of the dilation is particularly instructive: any such channel can be represented by coupling an ancillary system in a specified state to the input, making a symplectic transformation (any interaction, which is quadratic in positions and momenta), and then tracing out a part of the system. In particular, when the initial state of the ancilla is Gaussian, the channel is Gaussian as well, which means that the factor has Gaussian form.

Problems

1. 5.1 Show that the dual of a completely positive map is completely positive and that is unital iff is trace preserving.
2. 5.2 Give a proof of Theorem 5.1.
3. 5.3 Give a proof of Lemma 5.2. Hint: Assume that Equation 5.18 does not hold and consider a vector orthogonal to the span of .
4. 5.4 Derive the Kraus form (Corollary 5.3) from the Stinespring form (Theorem 5.3).
5. 5.5 Find a Kraus decomposition for the depolarizing channel.
6. 5.6 Show that the channels define in Equation 5.55 are entanglement‐breaking, that is, turns every entangled state into a separable state. Hint: Use the Jamiolkowski isomorphism.

References

1. 1 Choi, M.‐D. (1975) Completely positive linear maps on complex matrices. Linear Algebra Appl., 10, 285–290.
2. 2 Jamiolkowski, A. (1972) Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys., 3, 275–278.
3. 3 Stinespring, W.F. (1955) Positive functions on C*‐algebras. Proc. Am. Math. Soc., 6, 211–216.
4. 4 Kraus, K. (1983) States Effects and Operations, Springer‐Verlag, Berlin.
5. 5 Arveson, W. (1969) Subalgebras of C*‐algebras. Acta Math., 123, 141–224.
6. 6 Paulsen, V.I. (2002) Completely Bounded Maps and Dilations, Cambridge University Press, Cambridge.
7. 7 Davies, E.B. (1976) Quantum Theory of Open Systems, Academic Press, London.
8. 8 Caruso, F., Giovannetti, V., Lupo, C., and Mancini, S. (2014) Quantum channels and memory effects. Rev. Mod. Phys., 86, 1203.
9. 9 Kretschmann, D. and Werner, R.F. (2005) Quantum channels with memory. Phys. Rev. A, 72, 062323.