# Noiseless subsystems and the structure of the commutant in quantum error correction

John A. Holbrook, David W. Kribs and Raymond Laflamme Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1.
Institute for Quantum Computing, University of Waterloo, Waterloo, ON, CANADA N2L 3G1.
Perimeter Institute for Theoretical Physics, 35 King St. North, Waterloo, ON, CANADA N2J 2W9. ; ;
###### Abstract

The effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise.

## I Introduction

Quantum mechanics promises to manipulate information for communication, cryptography and computation in a way fundamentally different from its classical counterpartNC . Although it is possible to manipulate small quantum systems in the laboratory, the task to do so for large ones is daunting, especially because in absence of control of noise and imperfection of realistic devices the quantum properties of the state are destroyed. Quantum error correction methods have recently been discovered which protect quantum information against corruption. In particular it was shown that if the error rate is below a certain threshold AB ; Kit ; KLZ ; Preskillplenum then it is possible implement quantum error correction and obtain a fixed probability of success with only a polynomial amount of resources. The work on the accuracy threshold assumes certain error models such as independent or quasi independent errors. The theorem shows that under reasonable assumptions, imperfections and imprecisions of realistic devices do not create fundamental objections to scalable quantum information processing.

The estimated value of the error threshold is small and, today, an extraordinary challenge for experimentalists. From the experimental point of view there is a need to better understand the details of error models of quantum devices, the form of the Kraus noise operators which occur during the evolution of a quantum computation or during the transmission in a quantum channel, the correlation between errors, their strength, etc. From the theoretical side we need to find efficient ways to estimate the error model, determine the threshold for various physically relevant error models and find ways to achieve better threshold. This can be done by optimizing quantum error correction procedures or simply finding better ways to protect quantum information.

Recently a unified view of error correction was given which brought together quantum error correcting codes and noiseless (decoherence-free) subspaces DG ; KLV ; LCW ; Zan ; ZR . This investigation led to the discovery of noiseless subsystems. A quantum bit, called a qubit, is described by a unit vector in a 2-dimensional Hilbert space. Often this 2-dimensional space is directly associated with a two-level physical system such as the ground and excited states of an atom, a nuclear spin pointing up or down, etc. Noiseless subsystems are instead described by a set of operators such as the Pauli matrices, a more general concept than a subspace of a Hilbert spaceKLV . An example where the difference can be appreciated is in cases where the systems at hand are subject to noise with some symmetries and we are interested in a fraction of the operators acting on the whole Hilbert space of the systems. For example, the polarization of photons moving in an optic fiber undergo collective rotation. When a symmetry of the noise exists, it is possible to find conserved quantities and thus invariant, or fixed, operators. These operators form the noise commutant of the noise.

When the channel determined by the transmission is unital (i.e. the unit matrix is preserved), the commutant is a -algebra Kchannel . Such an algebra has a block diagonal matrix form

 A′=⎡⎢ ⎢ ⎢⎣∗0∗⋱0∗⎤⎥ ⎥ ⎥⎦, (1)

where the compression to each block forms a full matrix algebra. But some of these blocks may be ‘linked’ in a sense that we make precise below. Once the block matrix form and the various links between blocks have been found, the explicit form of the algebra is revealed. In particular, there are unique positive integers for such that the algebra is unitarily (spatially) equivalent to an orthogonal direct sum

where is the full matrix algebra on -dimensional Hilbert space and is the identity operator on -dimensional space. A tensor product in this decomposition corresponds to blocks of matrices in (1) which are linked, and is referred to as an ampliation of when . From the representation theory perspective, the integers correspond to the multiplicities of the identity representations of which determine the structure of .

If there are invariant operators which can be decomposed into projectors of rank one, so they are supported on blocks with , then they correspond to invariant subspaces and the usual notion of qubits. Otherwise the invariant operators correspond to noiseless subsystems on higher dimensional spaces. This can be read off the structure of the commutant. For instance, in the above discussion there will be a copy of every operator from contained in the component of the decomposition. Thus the structure of the commutant indicates how quantum information can be encoded to be preserved under the influence of the noise.

In this paper we study how to find the structure of the commutant once the noise (Kraus) operators are known. A constructive proof of this structure is given in the case of unital evolution. The paper is structured as follows. In sections 2 and 3 we review background concepts. In particular, we recall basic properties of quantum channels and results from Kchannel on the fixed point set of a unital channel. Section 4 contains the core theoretical component of the paper. Based on operator theory and operator algebra methods, we present a detailed algorithm for computing the noise commutant of a unital channel. We apply this method in Section 5 to find the commutant for a number of simple illustrative examples which are related to the phase damping channel FVHTC ; KBLW ; NC . Section 6 contains a short discussion on how to turn a non-unital channel into a unital one, when the noise operators satisfy certain constraints. In the final three sections we conduct an analysis of special cases of collective noise channels which arise from collective rotation Fi ; KLV ; KBLW ; Lnoiseless ; VFPKLC ; VKL ; Zan . Specifically, section 7 includes an introductory discussion on this class, and in the final two sections we use the algorithm to explicitly compute the commutant for the 3-qubit KBLW ; Lnoiseless ; VFPKLC ; VKL and 4-qubit channels from this class.

## Ii Channel Preliminaries

Let be a finite dimensional Hilbert space and let be the set of all bounded (continuous) operators acting on . A quantum channel (or quantum evolution) on is a linear map which is completely positive and trace preserving. We mention the texts Paulsentext ; Paulsentext2 for basic properties of completely positive maps, and NC for an introduction to channels in quantum information theory.

It is well-known Choi ; Kraus that to every completely positive map there corresponds a (non-unique) family of operators in which determine the map through the equation

 Φ(T)=n∑k=1AkTA†kforT∈B(H). (2)

Trace preservation of is equivalent to these operators satisfying

 n∑k=1A†kAk=1l,

where is the identity operator on . (This is also equivalent to the ‘dual map’ of being unital.) We say is unital if also,

 Φ(1l)=n∑k=1AkA†k=1l.

For a quantum channel , the family of operators determining as in (2) are called the Kraus noise operators for the channel.

The fixed point set for will be denoted by

 Fix(Φ)={T∈B(H):Φ(T)=T}.

We shall also let be the so-called interaction algebra KLV generated by the noise operators,

 A=Alg{A1,…,An}.

In other words, is the set of all polynomials in the typically noncommuting variables . Note that as an application of the Cayley-Hamilton theorem from linear algebra, the degree of any such polynomial may be reduced below some uniform bound. The noise commutant of is the set of all operators which commute with every operator in ; it is the algebra

 A′ = {T∈B(H):TA=ATfor allA∈A} = {T∈B(H):TAk=AkTfork=1,…,n}
###### Remark II.1

The interaction algebra, and hence the noise commutant, are usually defined as -algebras KLV . However, in the case considered here, that of unital noise, the discussion below shows that our ‘nonselfadjoint’ definitions are actually a red herring, the algebras turn out to be selfadjoint.

In this paper, we shall focus our attention on unital quantum channels. These are sometimes called bistochastic channels in the literature. For such channels, it was proved in Kchannel that:

• is a -algebra; in other words, belongs to if and only if belongs to . Furthermore, the algebra is independent of the choice of noise operators which determine the channel as in (2); the same algebra is obtained whatever the choice of operators .

• The commutant is a -algebra which is equal to the fixed point set of ,

 A′=Fix(Φ).

These results are recalled in the next section. Taken together, they may be regarded as a structure theorem for the fixed point set of a unital quantum channel. In particular, the fixed point set (which is just a -closed subspace for general completely positive maps) is itself an algebra, hence closed under multiplication, equal to the noise commutant. But there is a tight decomposition theory for finite dimensional -algebras (see section IV for details). This motivates the following fundamental problem which we consider in this paper:

###### Problem II.2

Given a unital quantum channel , explicitly determine the algebra structure of the noise commutant .

There are compelling reasons in quantum information theory for wishing to solve this problem. For instance, in the noiseless subsystem method of quantum error correction DG ; KLV ; Lnoiseless ; LCW ; VFPKLC ; ZR , the structure of the noise commutant can be used to encode quantum information which is immune to the noise of the corresponding channel.

## Iii Structure Theorem

In Kchannel it was discovered that many properties of a channel are married to properties of the noise operators. Recall that an orthogonal projection on reduces an operator in when both subspaces and are invariant for ; that is, and . This is equivalent to the identity being satisfied, or having a block diagonal matrix form with respect to the orthogonal spatial decomposition . Further recall the operator inequality for a pair of self-adjoint operators on means that for all . The key point in the following result is that invariant subspaces for the noise operators of a unital channel are always reducing subspaces.

###### Lemma III.1

Kchannel Let be a unital completely positive map and let be a projection on . Consider the following conditions.

• .

• The subspace reduces ;

 AkP=PAkfor1≤k≤n.
• .

• The subspace is invariant for ;

 AkP=PAkPfor1≤k≤n.
• .

• The subspace is invariant for ;

 PAk=PAkPfor1≤k≤n.

Then the pairs of conditions , , and are each equivalent. Moreover, if is a unital channel then all six conditions are equivalent to each other. Further, given a projection and a unital channel , conditions , and hold for one choice of noise operators for if and only if they hold for all choices.

Let us outline the proof of for a unital channel , since this is the aspect of Lemma III.1 used in the proof of Theorem III.4. Suppose holds for a projection in . Then

As , trace preservation can be seen to imply through an operator theory argument that . More generally, if is a positive contraction operator with for some projection and has trace equal to the trace of , then in fact . Thus since is unital, and for each

 0 ≤ (P⊥A†jP)†(P⊥A†jP) ≤ ∑i(P⊥A†iP)†(P⊥A†iP)=PΦ(P⊥)P=PP⊥P=0.

Thus, for all , and hence is a reducing subspace for .

We mention that this lemma gives a simple characterization of rank one projections fixed by a unital channel.

###### Corollary III.2

Let be a rank one projection in . Then the following are equivalent:

• .

• The range vector for is a joint eigenvector for the noise operators ;

 Akξ=λkξfor someλk∈Candk=1,…,n.

Proof. Condition here is simply a restatement of from the lemma, in the special case of a rank one projection.

We recall some terminology from linear algebra before proceeding.

###### Definition III.3

Let be a normal operator in , in other words . Then a spectral projection for is a projection onto an eigenspace for ; that is, it is a projection onto a subspace of the form for some . The Spectral Theorem from linear algebra states that every normal operator decomposes as a sum , where are the distinct eigenvalues for and the corresponding spectral projections (which necessarily have mutually orthogonal ranges).

Clearly the fixed point set contains the noise commutant ; if belongs to then

 Φ(T)=n∑k=1AkTA†k=T(n∑k=1AkA†k)=TΦ(1l)=T.

The converse inclusion holds for unital channels. We mention that the following theorem was partly motivated by work from DKS on infinite dimensional operator algebras and dilation theory. The proof in Kchannel is somewhat technical, but after submission we discovered a simpler proof of this fact, and hence, for completeness, we shall present it here.

###### Theorem III.4

Kchannel Let be a unital quantum channel determined as in by operators . Then the fixed point set for is a -algebra and is equal to the noise commutant,

Further, the interaction algebra generated by is a -algebra which depends only on via the equation

 A=Fix(Φ)′.

Proof. Notice that the theorem will be proved if it is shown that . Indeed, is an algebra by definition and is -closed since is a positive map. Moreover, if this identity holds, the algebra depends only on , not the choice of noise operators, since . (This is a special case of von Neumann’s double commutant theorem from operator algebras.)

As observed above, the commutant is contained in because is unital. To see the converse, first observe that if the only fixed points are scalars, then , and the two sets coincide. Otherwise, let be a non-scalar operator in . Without loss of generality we may assume that is self-adjoint; in other words, . Indeed, by positivity of , and hence both the real, , and imaginary, , parts of are fixed by . Since is non-scalar, at least one of these self-adjoint operators is non-scalar. Furthermore, by replacing with , we may assume is a positive operator. Let be the eigenvalues for , and the corresponding spectral projections. Then by the Spectral Theorem, and . Let be the eigenspace for the extremal eigenvalue . If is a non-zero vector in , observe that

 λr⟨ξ||ξ⟩=⟨Tξ||ξ⟩ = n∑k=1⟨AkTA†kξ||ξ⟩ = n∑k=1⟨TA†kξ||A†kξ⟩ ≤ λrn∑k=1⟨A†kξ||A†kξ⟩=λr⟨ξ||ξ⟩.

The only way this can happen is if each of the inequalities is actually an equality; in particular, belongs to for . Thus is an invariant subspace for , and so it is a reducing subspace for by Lemma III.1. Therefore, it follows that belongs to the commutant , and hence to the fixed point set . Thus the self-adjoint operator belongs to , and we may iterate this argument to find that each of belongs to . Hence belongs to , as required.

We have shown that every in also belongs to . But is a self-adjoint subspace, and hence spanned by its self-adjoint part. It follows that the fixed point set is contained in the commutant , and this completes the proof.

## Iv Computing the Commutant

In this section, using operator theory and operator algebra techniques, we present a constructive process for explicitly computing the noise commutant for a unital quantum channel. This will be accomplished by first identifying the -algebra structure of the interaction algebra . We begin with a short discussion on the decomposition theory for -algebras.

A fundamental result from the theory of operator algebras is that every -algebra is -isomorphic (an isomorphism which preserves adjoints) to a unique orthogonal direct sum of full matrix algebras. Specifically, there are positive integers such that is -isomorphic to

 A≃Mn1⊕…⊕Mnd. (3)

We mention the texts Arvinvite ; byeg ; Tak for basics of -algebras, or equivalently, finite dimensional -algebras. Another important result is that every representation of such an algebra decomposes into a direct sum of multiples of the identity representation on each summand. As a consequence, up to unitary equivalence is given by a unique direct sum of ‘ampliated’ full matrix algebras,

 A≃(Mn1⊗1lm1)⊕…⊕(Mnd⊗1lmd). (4)

In other words, there is a unitary operator such that is equal to the decomposition in (4). See also VKL ; ZL ; Zan ; KBLW ; Lnoiseless for discussions of these decompositions in connection with quantum information processing.

While it is important to know the -isomorphism class of as in (3), we really need the structure of spatially as in (4). More to the point, we need to identify the precise spatial locations of the matrix blocks . The integers are the multiplicities of the identity representation appearing in each summand associated with the corresponding representation of the -isomorphism class model in (3). The invariants are part of what is known as the Wedderburn structure theory for such algebras. When some , we say is an ampliation of . The projections in which, under the unitary equivalence (4), correspond to the identity operators on the blocks in the decomposition (4) are called the minimal central projections for . Each of these projections will be a sum of projections onto some, but perhaps not all, of the matrix blocks of the same size in the block diagonal form (1).

The projections onto the individual matrix blocks in (1) will be minimal inside , in other words minimal -reducing, but may not belong to the algebra . This is precisely the subtlety we must deal with in this process; an identification of ‘links’ between these minimal projections. Thus, broadly speaking, we must first identify the maximal (here meaning that the projections sum to the identity operator) family of non-zero minimal -reducing projections with mutually orthogonal ranges (part I below); it is easy to see that such a family of projections is unique. Then we must detect any links between these projections (part II below) in the following sense.

###### Definition IV.1

Let be the (unique) maximal family of non-zero mutually orthogonal minimal -reducing projections. A subset of this family is linked in if the following two conditions hold.

• The projection belongs to .

• If is a proper subset of , then the projection does not belong to .

Let be the linked subsets of projections from . It follows that each algebra forms a subalgebra of . Notice that, in terms of (4), the cardinality of the sets are equal to the ampliation multiplicities .

Once the -algebra structure of has been identified up to unitary equivalence, we can easily deduce the -algebra structure of . For instance, if is a channel and the spatial structure of is given by (4), then

 Fix(Φ)=A′≃∑k⊕(Mnk⊗1lmk)′≃∑k⊕(1lnk⊗Mmk).

Furthermore, the algorithm presented below will allow us to explicitly identify the locations of the matrix blocks, possibly ampliated, which belong to the noise commutant.

We shall work our way through the finer points of the process, then finish by summarizing the main points. For the rest of this section we let be a given unital quantum channel acting on , with noise operators , and interaction algebra . The following assumption gives us the starting point.

 {\it Assumption:}{\it The noise commutant A′=Fix(Φ) has been}{\it computed as a linear span.}

The algorithm will take us from knowing the vector space structure of , to identifying the algebra structure of , and hence of . There are two broad components to the process:

• Part I. Find the family of non-zero projections inside the noise commutant which have mutually orthogonal ranges, sum to the identity operator, and are each minimal inside .

• Part II. Given the family of projections from Part I, find the minimal central projections for by computing all links between the . This will give the spatial -algebra structure of , and hence of .

### iv.1 Part I

As is unital, always contains the scalar operators. If , then the singleton is the only (non-zero) minimal central projection for . Otherwise, let belong to . The spectral projections for provide the first step toward the maximal family.

###### Lemma IV.2

The following are equivalent for an operator in :

• .

• Every spectral projection of satisfies .

• Every spectral projection of belongs to .

Proof. Condition and are equivalent by Theorem III.4. The equivalence of the first two conditions immediately follows from standard operator algebra for -algebras. We provide a proof for the sake of brevity. The implication follows from the Spectral Theorem of linear algebra; decomposes as a sum where are the distinct eigenvalues for , and the corresponding spectral projections. On the other hand, if , then belongs to the commutant by Theorem III.4. But again by the Spectral Theorem, the spectral projections of commute with everything that commutes with . Hence follows since .

Thus, if we let be the (mutually orthogonal) spectral projections of , then belongs to for each . Observe that for all we may define a map by

 Φj(S)=n∑k=1Ak,jSA†k,j,

where

 Ak,j≡AkPj=PjAk=PjAkPj.

It should be kept in mind that this is a slight abuse of notation; we should really write as the restriction . These are unital channels since

A similar argument can be used to verify trace preservation for ; that is, . Proposition IV.4 below uses an analysis of these maps to give a test for deciding if is minimal.

###### Lemma IV.3

Let be a unital quantum channel, with noise operators , and let . The following are equivalent:

• .

• There are no non-trivial reducing subspaces for the algebra .

Proof. This immediately follows from Theorem III.4, since and is a -algebra.

###### Proposition IV.4

Let be a unital channel with noise operators , and let be a projection which reduces the noise operator algebra for . Then the unital channel given by , where , satisfies

 Fix(ΨP)=(CP)′=PC′P.

Furthermore, is a minimal projection in if and only if

 Fix(ΨP)=PC′P=CP.

Proof. Again, the statement is a notational convenience. We really mean that the compressed commutant, , is equal to the commutant of the restriction, , inside . To see this, first let belong to . Then

 (PTP)(CiP) = P(TCi)P=P(CiT)P = (PCiP)(PTP)=(CiP)(PTP),

so that contains . Conversely, if is in , then

 TCi=TPCi=T(CiP)=(CiP)T=CiT,

and belongs to . Thus Theorem III.4 gives us

 Fix(ΨP)={Ci,P}′={CiP}′=(CP)′=PC′P.

The rest of the result follows from the previous lemma.

If we apply this result for , we see that is a minimal projection inside if and only if . Thus, since we are assuming the noise commutant is known as a linear span, we can simply compute for a set of vector space generators for and check if each belongs to . If this is the case, then is minimal and it belongs to the maximal family of projections we seek. On the other hand, if there is a in such that is not in , then is not minimal; however, in this case we may iterate the process outlined above. Indeed, is a unital channel on with non-scalar fixed points, in fact is such an operator. Thus, we may use to obtain a self-adjoint operator in , and continue the process by examining the spectral projections of this operator, which are, of course, each a subprojection of . As is finite dimensional, this process will eventually terminate. We will be left with the maximal family (since they sum to the identity operator) of mutually orthogonal non-zero minimal projections in , and this completes Part I.

### iv.2 Part II

We emphasize the importance of distinguishing the various links between projections in the maximal family from Part I by presenting the following simple example.

###### Example IV.5

The point is that we must distinguish between bonafide orthogonal direct sums on the one hand and ampliation algebras on the other by finding linked projections. Let

be a scalar ampliation algebra and let be the unlinked orthogonal direct sum given by

 D={(a00b):a,b∈C}≃C1⊕C1,

where both of these -algebras are diagonal with respect to the orthonormal basis for .

The family of rank one projections is the maximal family from Part I for both algebras. However, the commutant of the scalar ampliation algebra is unitarily equivalent to , whereas the commutant of the orthogonal direct sum is also , the direct sum of two copies of the one dimensional scalar algebra. Notice that the projections and are linked in , since contains and neither of these subprojections belongs to . On the other hand, they are unlinked in because both projections already belong to .

Before continuing, let us recall a helpful identity from operator algebras. As a -algebra, is equal to its own double commutant, . This is a special case of von Neumann’s double commutant theorem Arvinvite ; byeg ; Tak . Now, let be the family of projections for obtained in Part I. It follows from the discussion at the start of this section that links can only occur between of the same rank. Hence for , let be the projections amongst with . Further let be the index set for this set of projections. Suppose with in .

Since , we may check if projections of the form , where , belong to simply by computing the commutators

 [PS,Bi]=PSBi−BiPS% fori=1,…,r.

The projections are linked in precisely when for each , and, there is no proper subset of such that for each .As is finite dimensional, there are just finitely many computations required here. Thus, we will eventually exhaust all possibilities and discover the links between the projections .

Testing for links between the rank one projections in is easier than the general case, hence we present it separately. Recall a rank one projection belongs to if and only if is a joint eigenvector for .

###### Lemma IV.6

Let be the rank one projections from the set . Let and suppose are scalars with

 Aiξk=λikξkfori=1,…,n.

Define a function by

 f(Qk)=(λ1k,…,λnk).

If in is such that is nonempty, then the projections in are linked in .

Proof. Let be a nonempty subset of . Observe that if and are range vectors for projections and in , then they are eigenvectors for and we have

 ⟨Aξk||ξk⟩=⟨Aξl||ξl⟩for allA∈A. (5)

Indeed, and hence these inner products agree on a set of generators for . Suppose is a projection in which is the sum of some, but not all, of the projections in . Then there is a pair of vectors and , each a range vector for a projection in , such that and . In particular, the identity (5) does not hold for , and consequently does not belong to .

For a nonempty set , let be the sum of all projections inside . It remains to show that each belongs to . From the structure theory for -algebras discussed at the start of this section, we know that for each the projection belongs to , where is the index set for all projections in of rank . Let be the projection obtained for . Then it is easy to see that the compressions , for , are mutually commuting operators inside which are each diagonal with respect to the basis for the range of . Hence it follows that is a subalgebra of which, up to unitary equivalence, has a block matrix decomposition of the form

Further, under this unitary equivalence, each projection onto a block corresponds to a projection which is the sum of projections from a particular nonempty . Since complete freedom is allowed for the scalars , we see that every belongs to .

###### Remark IV.7

As a prelude to some of the examples below, we note that the case of an abelian algebra corresponds to the case that the family of minimal -reducing projections are all rank one.

### iv.3 Summary of the Algorithm

We present a number of applications in the rest of the paper, but let us finish this section with a brief technical summary of the algorithm. Recall we have assumed that a set of vector space generators is known for .

Part I.

• If , then choose inside .

• Compute the spectral projections for .

• For each , compute . If every belongs to , then and is a (non-zero) minimal reducing projection for , and it belongs to the maximal family.

• If some is such that does not belong to , then is not minimal -reducing and the compression channel has non-scalar fixed points; for instance. Then proceed through steps , and , perhaps a number of times, to find minimal -reducing mutually orthogonal projections supported on .

Part II.

• Given the family from Part I, group together those projections with the same rank.

• For the projections of rank , test for links between the by evaluating the commutators where with . Then are linked in if and only if for each , and there is no proper subset of with this property. The projections of rank may be tested using Lemma IV.6. The projections defined by linked sets of projections will form the family of minimal central projections for .

• The various links will give the -algebra structure of up to unitary equivalence, and hence of , along with the precise locations of matrix blocks inside .

## V Phase Damping and Related Examples

In specific cases the computations involved with the method described above can be less cumbersome than what is suggested. We begin with a number of simple illustrative applications of the process. In each of these cases the commutant is already well-known, or at least easy to find, but we shall demonstrate how to compute it using the algorithm. See FVHTC ; KBLW for related examples.

Recall the Pauli matrices are given by

 1l2=(1001),X=(0110),Y=(0−ii0),Z=(100−1).

We regard these as matrix representations of operators acting on with respect to an orthonormal basis corresponding to the base states of a given two-level system. In the case of higher dimensions, for instance with basis , we use notation such as and to denote, respectively, the tensor products of and acting on .

###### Example V.1

Let be a positive real number, . Let be operators on defined on the standard basis by:

 A1=(√1−p)1l2=√1−p(1001)

and

 A2=(√p)Z=√p(100−1).

Clearly , so are noise operators for a unital channel acting on . The quantum operation corresponding to this channel is equivalent to the phase flip or phase damping operation on single qubits NC . Its effect is to flip the phase of relative to that of , and vice-versa, with probability .

A simple computation shows that

 Fix(Φ)=A′ = {A1,A2}′ = {(a00b):a,b∈C}≃C1⊕C1 = Alg{Z}=A.

Let us see how this structure arises through the algorithm. First choose a non-scalar operator in ; in this case will suffice. The rank one projections and are the spectral projections for , hence they are minimal inside since the vectors and are joint eigenvectors for . Thus the family forms the maximal family () of non-zero mutually orthogonal minimal projections inside the commutant .

Further observe that

 {A1|0⟩=√1−p|0⟩A2|0⟩=√p|0⟩and{A1|1⟩=√1−p|1⟩A2|1⟩=−√p|1⟩

The 2-tuples and are distinct, so the projections and are not linked by Lemma IV.6. Hence we have , with and both subalgebras of each unitarily equivalent to the one dimensional scalar algebra . Thus, the commutant is also unitarily equivalent to , and the form of given above is evident.

###### Example V.2

Let be a positive real number, . Let be operators on defined on the standard orthonormal basis by:

and

Once again, it is clear that these noise operators determine a unital channel on . We may compute

 Fix(Φ)=A′ = {A1,A2}′ = Alg{Z1,Z2,X1X2+Y1Y2} = ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜⎝a1100a140a22a2300a32a330a4100a44⎞⎟ ⎟ ⎟⎠:aij∈C⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭.

By observation, we can see that is unitarily equivalent to the orthogonal direct sum . To see this through the algorithm, first choose a non-scalar self-adjoint operator in ;

 Z1=Z⊗1l2=⎛⎜ ⎜ ⎜⎝1000010000−10000−1⎞⎟ ⎟ ⎟⎠=Z†1

is such an operator. The spectral projections for , corresponding to eigenvalues and , are given by

 P1=|00⟩⟨00|+|01⟩⟨01|andP−1=|10⟩⟨10|+|11⟩⟨11|.

Let for . The projections and belong to , but in this case they are not minimal inside . Indeed, observe that

 P1A′P1 = (AP1)′={A1P1,A2P1}′ = ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜⎝a110000a220000000000⎞⎟ ⎟ ⎟⎠:a11,a22∈C⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭≠CP1,

and similarly for . Following the algorithm, we let

 Z1Z2P1=P1Z1Z2=⎛⎜ ⎜ ⎜⎝10000−10000000
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