Spectral Properties of Tensor Products of Channels
We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper *-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recently-introduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.
If we have a linear map acting on a matrix algebra that can be expressed as a tensor product of matrix algebras, and the map itself can be expressed as a tensor product of two other linear maps, there may be few similarities between the constituent maps and the larger linear map they produce. If we restrict ourselves to special classes of linear maps and special domains of matrix algebras, then the tensor product adds no extra complexity. Our goal in this paper is to use the multiplicative domain to characterize some of these properties for trace-preserving, completely positive maps on matrix algebras. These maps are also known as quantum channels, which we refer to as channels.
The multiplicative domain of a linear map is the set of all matrices such that, for all , and . When the linear map is completely positive, this specific set has received much attention in operator theory and operator algebras (, -chapter 4, -section 2.1). In this context, it characterizes certain distinguishability measures. A completely positive linear map acts like a homomorphism on the multiplicative domain, and hence studying this domain can reveal structure and properties of the linear map.
In quantum information theory (, , ), the multiplicative domain contains the unitarily correctable codes and noiseless subsystems. Studying the multiplicative domain of tensor products sheds light on error correction in bipartite systems.
It turns out that we can capture most of the spectral properties of the tensor product of channels simply by investigating the multiplicative behavior. Note that the spectral properties of a channel acting on one copy of a quantum system have been well explored (, , , ) for various purposes, mainly in an effort to understand the dynamics of a system evolving through quantum measurements. In quantum dynamical systems, the ergodicity of a channel  and its decoherence-free subspaces  are important spectral properties . When the underlying domain is a bipartite system, the spectral properties of product channels can be hard to analyze, but we can use the multiplicative domain as a tool to understand them.
As in previous work on quantum error correction (e.g., ), we restrict our focus to unital channels because the multiplicative domain has less structure in non-unital channels. In particular, the multiplicative domain of a unital channel can be described using the Kraus operators. Using this, we can characterize certain channels and derive facts beyond the multiplicative structure.
The paper is organized as follows: firstly, in Section 2, we show that the multiplicative domain of a tensor product of unital channels “splits” nicely with the tensor product. We use this to prove that the peripheral spectra of two unital channels will precisely determine whether the fixed points of their tensor product will also split or not. This analysis provides the necessary and sufficient condition on when the tensor product of two ergodic (or primitive) channels is again ergodic (or primitive). Here we recapture some of the results obtained by ,  in a very different way based on the analysis of multiplicative domain.
Since  showed that repeated applications of a finite-dimensional channel produces a chain in the lattice of unital *-subalgebras of , we characterize such algebras in Section 3.1. This provides an easy way to enumerate the lattice of unital *-subalgebras of , as well as providing a limit on the length of chains in the lattice that is linear in the dimension. This finding can be of independent interest because it provides a finer analysis of the structure of unital *-subalgebras in . In turn, this allows us to use the multiplicative index, introduced in , to show that certain channels cannot be product channels. We give examples of channels with large multiplicative indices in Sections 3.2 and 3.3, thus showing that these cannot be product channels.
Next, in Section 4, we consider channels which are strictly contractive with respect to some distinguishability measures that frequently arise in information theory. Using our results in the previous sections we prove that the tensor product of two strictly contractive channels with respect to certain distinguishability measures is again strictly contractive provided the measures allow recovery maps. We make use of the reversibility and monotonicity properties of these measures under channels, which is a wide topic of current research (, , , ).
As a final application, we show that unitary-correctable quantum codes (UCC) gain nothing through tensor products.
1.1 Background and Notation
Throughout this paper we will use the following notation:
will refer to quantum channels, that is, completely positive, trace-preserving linear operators from to for some finite dimensional Hilbert space . In this paper we identify with , the complex matrices. It is well known that a quantum channel is always represented by a set of (non-unique) Kraus operators in such that for all , we have
where . Here represents the identity matrix in . In the dual picture is realized as a unital completely positive map and denoted by acting on again and satisfying the relation
for every . This linear map is the adjoint of with respect to the Hilbert-Schmidt inner product which is defined as , for all .
An important note is that many papers work in the dual framework, where a quantum channel is necessarily unital but may not be trace-preserving. Hence, these papers refer to these maps as unital channels, or UCP maps. In our work, where all channels are trace-preserving, unitality is an extra condition that limits our results to a particular subset of quantum channels.
Lowercase letters from the end of the Latin alphabets, , will refer to matrices in . The letters will refer to projections in .
Greek letters, will refer to either vectors in or partitions of for . We will use to denote the set .
Stylized letters from the beginning of the Latin alphabet, , will refer to sub-algebras of . For a set , the algebra generated by will be denoted and the *-algebra generated by will be denoted .
For a quantum channel , denotes the multiplicative domain and also denotes the set of fixed points of , that is,
There are a number of useful characterizations of the multiplicative domain we will use extensively.
Theorem 1.1 (See ).
For a unital completely positive map , the multiplicative domain is a C-subalgebra of and moreover, it is equal to the following set:
The following theorem is also useful in describing the multiplicative domain of a unital channel.
The next theorem connects the fixed points set and the Kraus operators of a channel.
Theorem 1.3 (See ).
Let be a unital channel represented as . Then the fixed point set is an algebra and it equals to the commutant of the *-algebra () generated by . That is
where represents the commutant of the algebra .
It follows that is a *-closed subalgebra of containing the fixed points of . As with all finite *-algebras, it is generated by a set of projections. For any projection , is a projection of the same rank, and is also in . We say that is trivial if ; if is non-trivial, then it must contain at least two orthogonal projections.
Definition 1.4 ().
The multiplicative index of a unital quantum channel is the minimum such that .
We denote the multiplicative index of by . Another useful result is Lemma 2.2 from :
If are two unital quantum channels, then
2 The Multiplicative Domain of Product Channels
2.1 Splitting problem for subalgebras in tensor product
The splitting problem for a von Neumann subalgebra (or a C-subalgebra) of a tensor product of algebras has remained one of the most important problems in operator algebra. One of the early results that drew a lot of attention on this problem is due to L. Ge and R. Kadison:
(Ge-Kadison, 1996, ) Let be two von Neumann algebras and assume that is a factor. If is a subalgebra that contains , then
for some von Neumann subalgebra of .
Here we examine the multiplicative domain of the tensor product of two channels acting on and separately. Since the multiplicative domain is a C-subalgebra of , it is natural to ask whether this subalgebra splits into tensor product of two subalgebras. We show that for unital channels, the multiplicative domain is unchanged by the tensor product. To prove this claim we need the following lemma. For our purposes, if and are two algebras and and are two sets, possibly without any algebraic structure themselves, then is defined as .
Let and . If, for every , there is a projection such that , and for every there is a projection such that , then . If there also exist such for for all and all , then .
Note that for any two sets and , and , so , so .
For the reverse inclusion, free products of elements of will span , and free products of elements of will span . The tensor product of spanning sets is a spanning set of the tensor product, so elements of the form , with and , will span . We take an arbitrary element of this form, , and then take such that and such that . Then:
Since , then there are elements in such that . But for any , , so the sum is in . Similarly, . Thus, the product above is also in , and thus all the basis elements of are in , so .
For the *-algebras, a very similar logic holds. Free products of the form , and , will span . Take and arbitrary element of this form and let and be projections defined as before, i.e., and . Suppose is in , so , with . Projections are self-adjoint, so is in . Similarly, if is in , then . Thus the same decomposition can be done as the one above:
And since all of the terms on the right-hand side are in , then . ∎
For any two unital quantum channels ,
Let and be the Kraus decomposition of and respectively. Trace preservation implies .
The Kraus operators of are for any . Define . Similarly, let be the set Kraus operators of . Then the Kraus operators of are , or . Since and , we have the necessary projections to use Lemma 2.2. Hence we have that
Now the finite dimensional *-algebras are von Neumann algebras and by the commutant-tensor product theorem for von Neumann algebras ( , Theorem 11.2.16) we have that
Then by Theorem 1.3 and and , thus
Now invoking Theorem 1.2 and noting that we immediately obtain
Since the multiplicative domain behaves well with the tensor product, it leads to a simple form for the multiplicative index:
Given two unital channels , then (where is the multiplicative index).
If , then:
That is, the multiplicative domain is constant after , so . Then suppose (and, without loss of generality, suppose ). By a similar logic:
Since the multiplicative domain is still strictly decreasing with , then and the result follows. ∎
The above proposition implies the following corollary:
For unital channels we have
2.2 Fixed Points of Product Channels
For a unital channel , the fixed point set is a subalgebra of and unlike the multiplicative domain case, this subalgebra does not split nicely. However, using Theorem 2.3, we can provide an exact description of this algebra and characterize when this subalgebra splits and recapture the result of . Our results are specific cases of  and , but through a vastly different approach. The spectrum of the tensor product of two channels is known to be the set product of the two spectra, but this theorem characterizes the eigen operators as only the obvious choices. In what follows represents the unit circle in the complex plane. Note that (see ) for any quantum channel , all the eigenvalues lie in the closed unit disc of the complex plane. We define the spectrum () of as follows
where is the identity operator on . The set is called the peripheral eigenvalues and the corresponding eigenoperators are called peripheral eigenvectors.
Let be two unital quantum channels. Then for any :
Let . For the left inclusion, suppose there are two numbers such that and for matrices and . Then
For the right inclusion, let be a matrix such that . By Theorem 2.5 from , we know that the peripheral eigenvectors of a channel are precisely the stabilized multiplicative domain. Thus:
We can then represent as , where and . By the same theorem, we know that is a linear combination of peripheral eigenvectors of . Thus we can further decompose as
where the are linearly independent and with . This gives us:
But by choice of , . By the linear independence of , we have that , i.e., . This holds for all , giving the required inclusion. ∎
Using the above theorem we obtain the following corollary which first appeared in , Corollary 13 in a more general context. However our method of obtaining this result is significantly different from .
For two unital channels and with spectra and respectively, the fixed point algebra splits if and only if the intersection of the peripheral spectra is trivial. That is,
if and only if .
The fixed points are the special case of peripheral eigen-operators where . Using Theorem 2.5, we have that the fixed points are given by
This set will equal if and only if there is no with . Since the spectrum of a quantum channel is closed under conjugation, this means would need to be in both spectra. Thus, the spectrum will split if and only if the intersection of the spectra is trivial. ∎
Theorem 2.5 is particularly helpful to analyze the ergodicity or irreducibility of tensor product of quantum channels. We provide the definition of such channels below:
A channel is called irreducible if there is no non-trivial projection such that , for .
An irreducible channel is called primitive if the set of peripheral eigenvalues contains only 1, that is if .
We note down some properties of irreducible positive linear maps:
(see ) Let be a positive linear map on and let be its spectral radius. Then
There is a non zero positive element such that
If is irreducible and if a positive is an eigenvector of corresponding to some eigenvalue of , then and is a positive scalar multiple of .
If is unital, irreducible and satisfies the Schwarz inequality for positive linear maps then
Every peripheral eigenvalue is simple and the corresponding eigenspace is spanned by a unitary which satisfies , for all .
The set is a cyclic subgroup of the group and the corresponding eigenvectors form a cyclic group which is isomorphic to under the isomorphism .
Often irreducible channels are called ergodic channels. Ergodic/irreducible positive maps have been a great topic of interest (see , , ). The study of such maps enriched the analysis of non-commutative Perron-Frobenius theory. Although ergodicity of a quantum dynamical system (discrete or continuous) has received much attention, the same analysis in the tensor product framework has been talked about less except  and . Here we present necessary and sufficient conditions for a channel to be irreducible and primitive in the tensor product system. By the aid of Theorem 2.5 we recapture Theorem 5.3 in .
Let be an irreducible unital quantum channel with peripheral eigenvalues . Then:
The product is irreducible if and only if is also primitive, in which case is also primitive.
For any primitive unital channel , is irreducible.
If is irreducible with peripheral eigenvalues , then is irreducible if and only if .
(1) For to be irreducible, its fixed points would need to be , meaning the fixed points would have to split. By Corollary 2.6, this would occur if and and only if the peripheral spectrum of is trivial, meaning is primitive. Since the spectrum of a quantum channel is contained in the unit disc, in this case the peripheral spectrum of will still be trivial and thus it will be primitive.
(2) Since is primitive, its only eigenvalue is 1 with eigenvector . Thus the fixed points of will split, and since both fixed point algebras are trivial, the product will also be trivial.
For item (3), if , then the two cyclic groups intersect trivially and hence by Corollary 2.6 we get .
Conversely, if is irreducible, then . From Theorem 2.9 we know that the peripheral spectrum of is a cyclic subgroup of some order . Since , it is evident that this can only happen if the fixed point algebra splits. By Corollary 2.6 again we conclude that ; that is, . ∎
Theorem 2.5 gives structure to the eigenspaces of these eigenvalues. For some intuition on this, a channel acts like an automorphism on its stabilized multiplicative domain, so in some sense it is “normal” on this subalgebra. The eigenspaces of the tensor product of two normal matrices will simply be the products of the original eigenspaces, and here something similar holds for the “normal part” of the channel.
3 Restrictions on the Multiplicative Index
3.1 Maximal Unital Proper *-Subalgebras (MUPSAs)
Proposition 2.1 restricts which channels can be product channels, since the multiplicative index must be the same as the multiplicative index of one of the channels in the product. Our goal is thus to restrict the possible values of the multiplicative index. An obvious bound is the dimension of the matrix algebra, , but in fact we can do much better by looking at chains of maximal unital proper *-subalgebras, defined in the obvious way as follows:
An algebra is a maximal unital proper *-subalgebra (for convenience, a “MUPSA”) of a C*-algebra if is unital proper *-subalgebra of (meaning , , and ) such that if is another unital proper *-algebra with , then either or .
While there are many possible forms of a subalgebra of , restricting to MUPSAs allows us to precisely characterize their structure, up to isomorphism. We use the Wedderburn decomposition extensively. For a matrix algebra , one can always decompose it as
This is the Wedderburn decomposition.
If is a MUPSA of , then (up to isomorphism) , where .
Let be a *-subalgebra of . Then let
be the Wedderburn decomposition of . If , then the following subalgebra
will strictly contain , but be strictly contained in , contradicting the maximality of . If , then (for some number dividing ). Then , contradicting maximality of . Thus , and . If , then is a proper subalgebra of
which in turn is a proper subalgebra of , again contradicting maximality. The same argument applies to , and thus
Since is unital, , so we can write and , for . If or , then , so . ∎
Let be a unital *-subalgebra of with Wedderburn decomposition . If is a MUPSA of , then, up to unitary equivalence, has one of the following forms:
for some and some such that , or
for some such that and .
Before the proof, we recall a result of Bratteli’s from  that will be very useful.
Let and as algebraic isomorphisms, with . Then there exist integers for and such that we can identify with
with the convention that, for any two matrix algebras and , .
This is an informal statement of the proposition, but it says that every block in is embedded into zero or more blocks of . Note that the equivalences ignore the tensor factors in the usual Wedderburn decomposition, since these affect only the norms, not the algebraic structure. Hence, to prove Theorem 3.3 we will first use Bratteli’s result for the algebraic structure, then recover the norms.
Let and be matrix algebras such that is a MUPSA of , with
and the embedding of into has the form
Then, up to a permutation of the blocks of , either:
The number of blocks in is , and there is an index , such that:
For all , and .
There is some with such that , , .
The number of blocks in is , and there are indices , , such that:
For all , and .
For all , and .
This lemma states that, with one or two exceptions, every block of maps surjectively into a block of . For the remaining block(s), either there are two blocks of that map into one block of , or there is one block of that maps to two blocks of .
Note that we assume is equal to the structure without tensor products, but we can only assume is isomorphic to such a structure. The decomposition of given in the statement of Lemma 3.4 ignores the dimension, and the embedding into may not be isometric. Indeed, if case 2 holds, then one block of will contain a tensor product with .
For all , define as the th block of the embedding of , i.e.:
With this notation, we have that
where is the image of of the embedding into . Note that must also be a MUPSA of .
For each , may be a proper subalgebra of or not. Suppose there is some where it is a proper subalgbera. Then we can take the subalgebra defined by
and this will be a proper subalgebra of and it will contain . Since is also a MUPSA, . Thus, must have the form of , so can have at most one such that is a proper subalgebra of .
In this case, we can argue that must itself be a MUPSA of , or would not be maximal - we could take a MUPSA as the th block instead. By Lemma 3.2, must have the form for some with . This proves Part (1).
The other possible situation is where for all . This means that in the notation of Equation 1, there can only be one block of in each block of , so for each , there is a unique such that , and for all . This means that the embedding of into looks like
The direct sum on the left is not all of , it is only isomorphic to . A block on the left might appear twice in the embedding if there is some such that . This is how, even though each block is surjectively covered by the embedding, can still be a proper subalgebra of , since has more freedom between blocks.
If , then each block of embeds surjectively into each block of , implying the contradictory statement that . Thus . This means there must be some and such that . That is, some block of maps to two blocks in . We define an algebra with such that
That is, is just all of the blocks of except the th block; to embed it into , we use the identity on all blocks, and send a copy of the th block to the th block of . Since we required that each block , then . Clearly, is a proper subalgebra of , and by this construction, must contain . Hence .
Thus a MUPSA must have the form of for some blocks and , hence and in all other blocks, and are equal. This proves Part (2). ∎
Proof of Theorem 3.3.
we can define a new algebra as
This will be *-isomorphic, but not isometric, to . The natural isomorphism can be defined as
Then we can let . In fact, will be a MUPSA of , since any subalgebra of can map to a subalgebra of .
Then, ignoring tensor products, we can write
and apply Lemma 3.4 and consider the two cases.
In the first case, , and the decomposition of looks like
By dimension counting, this must actually equal , so