Eventually Entanglement Breaking Maps
We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say the map has a finite index of separability. In particular we show that every unital PPT-channel has a finite index of separability and that the class of unital channels that have finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. This analysis is motivated by the PPT-squared conjecture made by M. Christandl that says every PPT channel, when composed with itself, becomes entanglement breaking.
1 Introduction and Motivation
Detecting separability is one of the key aspects of the theory of quantum information. Computationally it is a hard problem to decide whether a state (a positive semi-definite matrix of trace 1) is separable or not. The quantum channel associated to a state is entanglement breaking if the (Choi)state is separable.
An important class of linear maps in this context is the class of PPT maps. The Choi matrices of such maps are positive and have a positive partial transpose. These states turn out to be useful in Quantum Key Distribution (QKD) and dense coding protocols (see ). It was observed that all known PPT maps, when composed twice with themselves, become entanglement breaking. Hence a conjecture was put forward by M. Christandl in 
Conjecture 1.1 (PPT-squared Conjecture).
The composition of two PPT maps is always entanglement breaking.
Our work is closely related to that of Lami and Giovannetti in , who studied quantum channels that are “entanglement saving”, i.e., where no power of the map becomes entanglement breaking. They studied such maps in great detail and also provided conditions on when a channel is “asymptotically entanglement saving”. Although they did not study PPT maps in particular, some of our results concerning PPT maps can be deduced from their work. Our proofs use multiplicative domains and are more direct for our particular results. Very recently in , the authors analyzed the composition of PPT maps and proved an asymptotic version of the above conjecture. They showed that for every PPT map , the sequence of distances goes to zero as , where is the set of entanglement breaking maps. This finding verifies the conjecture asymptotically, that is, is asymptotically entanglement breaking.
In this paper we look at PPT maps as a particular class of maps which become entanglement breaking after finitely many iterations. If a map is such that is entanglement breaking for some , then we say that has finite index of separability. The spectral and multiplicative properties of unital channels are the key ingredients of this investigation. In Section 3, we provide a structure theorem (Theorem 3.2) for a unital PPT map with respect to its action on projections that are mapped to another projection. This reveals that its action can be restricted to independent blocks on the diagonal in a suitable basis. In Section 4, we prove that every unital PPT channel becomes entanglement breaking after finite iterations (see Theorem 4.4). An illustrative example (Example 5.1) is put forward in Section 5 to show that there are maps which are not PPT yet they become eventually entanglement breaking after finite iterations. It is also shown in Theorem 5.2 that the class of such maps which have finite index of separability is dense in the class of unital channels. In Section 6, a large class of channels are shown to be eventually entanglement breaking in asymptotic sense (see Theorem 6.1).
We begin with defining a quantum channel which is a completely positive and trace preserving linear map . Note that every linear map on defines a unique element in which is known as the Choi-matrix of and it is defined as , where are the matrix units in . It holds that is completely positive if and only if is positive semi-definite in . For general introduction of completely positive maps and quantum channels we refer to the monographs , .
One of the main tools we use to investigate iterative properties of a channel is the multiplicative nature of the channel. We note some definitions in this regard.
For a linear map , the following set is known as the multiplicative domain of
Theorem 2.1 (Choi, ).
For a unital completely positive map , the multiplicative domain of is a C-subalgebra of and equals the following set
Note that if the maps are assumed to be unital, trace preserving and completely positive, then the fixed point set of is defined by
This is an algebra and moreover we have . For any unital channel and any , (See ), and hence there is some such that for any , . Following , we denote this algebra and refer to it as the stabilized multiplicative domain of .
Definition 2.2 ().
The multiplicative index of a unital quantum channel is the minimum such that . In other words it is the length of the following decreasing chain of subalgebras
We often denote this number as .
Another useful result is Lemma 2.2 from :
Lemma 2.1 (see ).
If are two unital quantum channels, then
Next, we note down the definition of separability of a positive matrix.
A positive semi-definite matrix is called separable if there exists positive semi-definite matrices and such that .
Separability of states has a close connection to a specific set of channels which we define below.
A linear map is called entanglement breaking if the Choi-matrix of , , is separable in .
The set of separable states is a compact convex set and hence so is the set of entanglement breaking maps. There are many different equivalent criteria for a channel to be entanglement breaking. We note down the following facts about entanglement breaking maps.
Theorem 2.2 ().
For a linear map , the following statements are equivalent:
is entanglement breaking,
has the form , where the are positive matrices with trace 1 in and are all positive semi-definite matrices in .
For any completely positive map , the maps and are entanglement breaking, whenever the composition is meaningful.
A linear map is said to have finite index of separability if there exists a such that is entanglement breaking.
If a linear map is entanglement breaking, then for any other completely positive map , is entanglement breaking by Theorem 2.2. Thus, having a finite index of separability means that, after enough repeated applications, a linear map becomes entanglement breaking and stays entanglement breaking. Notice here that this index of separability for channels has been studied before in . We further analyze this property keeping the PPT-squared conjecture in mind.
3 Structure of PPT Maps
In this section we analyze some essential features of PPT maps which eventually help us proving that all PPT maps have finite index of separability.
A linear map is called PPT if it is completely positive and co-completely positive, that is, is completely positive where is the transpose map .
Note that in the literature of quantum information theory, the name PPT appears with different meanings ,. We will consider the above definition as PPT maps. It turns out that the Choi matrix of a PPT map is ‘positive partial transpose’ in the tensor product (see Theorem 7.2.2 in ).
To decide whether a channel is entanglement breaking or not amounts to deciding whether the Choi matrix is separable, which is computationally quite a difficult task. Note that since every separable state is positive partial transpose , it is clear that the set of entanglement breaking channels is a subset of the PPT channels. However, there are PPT channels that are not entanglement breaking. One more criterion for separability in terms of the multiplicative domain is recorded below.
Theorem 3.1 (Størmer, ).
Let be a unital channel. Then if is entanglement breaking, then the multiplicative domain is an abelian C- algebra. The converse is also true provided is a conditional expectation, that is, .
Next we extend the results of Størmer’s mentioned in Theorem 3.1. Before that we prove a lemma that will be useful in the subsequent discussion. This result was given in  but for the sake of completeness we outline a proof. See also Corollary 3 in .
Let be unital completely positive map such that the is contained in an abelian C, then is entanglement breaking.
Let the (finite dimensional) abelian C-algebra be for some . Hence we can regard for every . Now for each , define the co-ordinate projections , that is, . These are positive linear functionals on . Hence composing with , we get which is a unital positive functional on . Since every such functional is given by for some density matrix , we have . Now choose many positive elements in to associate each coordinate element of to and eventually we get
which is entanglement breaking. ∎
Now we state and prove the main theorem of this section.
Let a unital PPT map. Let be projection and let . Then for every we have
It follows that is an abelian C-algebra contained in the center of the C-algebra generated by the . If moreover, is faithful, then is abelian.
If contains no projection then our conclusion follows trivially. So assume there exists a projection . So is a projection (that is ) and for all we have . For any , define
By 2-positivity of we have by Schwarz inequality
If the entry of a positive is zero, then the entry will be zero. Hence . Since was arbitrary, replacing by , we get
But we already have from the multiplicative domain property. Hence we obtain for every and any projection ,
So it follows that for every , we have two projections () such that
The other terms vanishes as and the above commutation property. It also shows that is abelian and it is contained in the center of the algebra generated by . Now if is faithful, restricting on we get an isomorphism between and . As is abelian, it follows that is abelian. ∎
We get an immediate useful corollary to the above theorem.
Let be a unital PPT map. If are two orthogonal projections in the multiplicative domain and such that , then .
If , then from the previous theorem we get . ∎
This result shows that for every PPT map, there is some basis where its action can be restricted to independent blocks on the diagonal.
Next we extend the results of Størmar given in Theorem 3.1 for PPT maps.
If is a unital PPT channel, then is abelian and the converse is also true if is a conditional expectation.
The first part is essentially Theorem 3.2 as the trace preservation property implies faithfulness.
Conversely, suppose is a channel such that and the multiplicative domain is abelian. Then we have that is contained in . Indeed, for any we have
But by the Schwarz inequality of . Now by the trace preservation of , from the above equation we get , which is the equality in the Schwarz inequality and hence . As is contained in an abelian C-algebra, by Lemma 3.1 we get is entanglement breaking. Since an entanglement breaking map is automatically PPT, we have the result.
The composition of PPT maps will have a similar structure, with independent projections, but the projections may not “line up”, and so we need to analyze the composition of PPT maps in more detail.
4 Composition of PPT maps and Finite Index of Separability
In this section we prove one of the main results of the paper, that is, every unital PPT map becomes entanglement breaking after a finite number of iterations.
Let be a unital PPT channel and let be the set of minimum mutually orthogonal projections . Then there is some natural number such that
where is a unital PPT channel with trivial multiplicative domain.
Let be the multiplicative index of defined as in 2.2. Then the multiplicative domain of is just . As , by Theorem 3.2, this domain is an abelian subalgebra of , and thus there is some set of minimum mutually orthogonal projections that span . We can apply Theorem 3.2 repeatedly and get that the action of is
Since , then is also a projection in . If we take a projection of minimum rank, , then must also be a projection of minimum rank. Since these projections are all accounted for in , then for some permutation . We can repeat this process for every projection of minimum rank, and then for the projections of second-smallest rank, until we construct a permutation such that for all , . Then, for any , we can use Equation 1:
Then will have some finite order , so repeating the calculations above gives:
Setting and for , and applying Theorem 3.2 to , which has multiplicative domain equal to , gives the required result. ∎
At this juncture we note down some results that we will use to arrive to the main theorem of the section.
(Gurvits-Barnum, ) Let be a normalized density matrix in a bipartite system of total dimension such that , then is separable, where are the identity matrices in and respectively.
Note that is the Choi matrix for the channel given by , for all . Using the Choi-Jamiolkowski identification with matrices in and the linear maps from to , the above theorem ensures that the map has a neighborhood where each quantum channel in the neighborhood is entanglement breaking.
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 peripheral spectrum () of as follows
where is the identity operator on . The set is called the peripheral eigenvalues and any satisfying , with , is called a peripheral eigenvector corresponding . It is a consequence of the Kadison-Schwarz inequality that for a unital channel , if , for some , then . See , Corollary 2.2 for more details.
(See , Theorem 6.7) Let be a unital channel such that it has trivial peripheral spectrum, that is , then for all .
Channels with trivial peripheral spectrum are known as “primitive” channels. These maps are generic in the sense that they are dense in the set of channels. Following , Corollary 3.5, a unital channel is primitive if and only if the stabilized multiplicative domain is trivial, that is, , where is the identity matrix in .
With all the necessary background we are ready to write down the main result of this section:
Every unital PPT channel has finite index of separability.
Consider the channels in Theorem 4.1. Each channel has trivial multiplicative domain, so by , its peripheral spectrum is trivial. Thus, we can apply Theorem 4.3 and conclude that . Then there will be some finite such that is close enough to that their Choi matrices are within a distance of . By Theorem 4.2, this means that the Choi matrix of is separable, that is is entanglement breaking. Letting , then
will be a direct sum of entanglement breaking channels, and thus is entanglement breaking. ∎
5 Beyond PPT maps
PPT maps turn out to be a special case of channels with finite index of separability. This turns out to be, topologically, an abundant class of channels. We need the following result:
Theorem 5.1 (see , corollary 7.5.5).
Let be a positive map such that , then is entanglement breaking.
The set of unital channels on which have finite index of separability is dense (C.B topology) in the set of unital channels.
Let be a unital channel. We will show that given , there is a unital channel approximating within in CB norm and has finite index of separability.
To this end, let us define the map , , where . It is easily verified that is a unital channel and
For simplicity call . As , it follows that for every . So
As as , it follows that for large and hence is entanglement breaking for large by Theorem 5.1. ∎
5.1 Schur Channels
Next we provide some concrete examples of maps which are not PPT, yet they have finite index of separability. We start with a definition of Schur channels. Recall for two matrices and , the Schur product is defined as .
Given a matrix , we define a map on . Such maps are called Schur product maps.
It is well known that is completely positive if and only if the matrix . Moreover, if has all its diagonal entries equal to 1, then is a unital channel.
For a Schur Channel on , the following statements are equivalent:
is entanglement breaking.
, the identity element in .
is well-known. For , a characterization of the multiplicative domain of Schur channels has been put forward in . It is given below
Hence it is clear that the multiplicative domain of contains the algebra of diagonal matrices of . Since the matrix units for all and noting that we get by Corollary 3.1. Thus .
To prove , if , then can be written as . Since these are rank-1 operators, is entanglement breaking by one of the equivalent criteria given in . Hence, is also entanglement breaking. ∎
The above proposition suggests that there are no non-trivial Schur channels that are PPT, a fact which was proved in  using different method.
Example 5.1 (Non- PPT Maps having finite index of Separability).
and set , where , for all . Hence is a unital channel. One computes
Clearly it is a non-abelian subalgebra of and hence can not be PPT by the Theorem 3.2. But we have that
which, by Lemma 2.1, gives , the algebra of diagonal matrices. Further, we observe that , that is the range of is contained inside the abelian C-algebra , hence by Lemma 3.1, must be entanglement breaking. Thus, is not PPT, but is entanglement breaking.
We can extend this example to higher dimensions: Set (where is the matrix of all ones), and set to be the permutation corresponding to in the symmetric group on . Then setting , we see that is not PPT, and neither is for , but will be entanglement breaking.
Note that above example suggests that the converse of the PPT conjecture is false, that is, if is channel such that is entanglement breaking, then need not be PPT.
6 Asymptotically Entanglement Breaking
In this section we show the existence of maps that don’t have finite index of separability, but asymptotically they are entanglement breaking.
We call a linear map ‘asymptotically entanglement breaking’ if there is a sequence such that is entanglement breaking, where the limit is in the bounded weak (BW) topology.
In a very recent article , it was shown that PPT maps are asymptotically entanglement breaking. Indeed, they showed that a limit point of iterates of a PPT map becomes entanglement breaking. This could also be deduced from the work of Lami and Giovannetti . In this section we show that there are plenty of channels that are asymptotically entanglement breaking.
Let be a unital channel. Then is asymptotically entanglement breaking map if and only if the stabilized multiplicative domain is abelian.
Note that Kuperberg in  proved that for a unital completely positive map , there is subsequence such that converges to a unique conditional expectation such that is completely positive and also
that is, the range is the span of all the peripheral eigen operators of . From the Theorem 2.5 in , we get is the algebra generated by the set . Hence by the hypothesis, is contained in an abelian C-algebra. Hence by the Lemma 3.1, we have is entanglement breaking. Thus is asymptotically entanglement breaking.
Conversely, let there exist a subsequence of such that
then it follows that the idempotent and commutes for every . It is evident that
Now choosing particularly the subsequence for which we get the conditional expectation onto the subalgebra , we get
Now passing to any subsequence of the sequence we may conclude that
As is entanglement breaking and composing it with any completely positive map yields another entanglement breaking map, it is immediate that is entanglement breaking since the set of entanglement breaking channel is a closed set. Hence by Lemma 3.1 we get is abelian.
If any of the limit points of the set are entanglement breaking, then the above proof technique can be used to show that any other limit point of the set will be a limit of entanglement breaking maps, which is again entanglement breaking. This fact was first proved in , Proposition 20. Note that Lami-Giovannetti in  proved Theorem 24 which is essentially equivalent to Theorem 6.1. Indeed, they prove that a channel is asymptotically entanglement saving if and only if the stabilized multiplicative domain (of ) is non-abelian.
We next demonstrate a large class of maps that have the property mentioned in the above theorem.
A positive linear map is called irreducible if holds for a projection and , then , that is, must be a trivial projection.
Irreducible maps are generic in the sense that these maps are dense in the set of all positive linear maps acting on .
Every unital irreducible channel is asymptotically entanglement breaking.
Note that by the aid of Perron-Frobenius theory of irreducible positive maps, we know that (see ) the peripheral eigen operators of an irreducible channel are generated by a single unitary. It is proved in the Lemma 3.4 in  that for a unital irreducible channel , we have is an abelian C-algebra. Hence by the Theorem 6.1 we have the result. ∎
We end this section by providing an example of a channel which is asymptotically entanglement breaking but does not have finite index of separability, ensuring that being abelian is not sufficient for to have finite index of separability.
Let be defined as a Schur product channel , where , with . It follows that the stabilized multiplicative domain , which is clearly abelian. However following Proposition 5.1, it is evident that is not entanglement breaking for any as , where .
Note that since the irreducible channels are dense in the set of all quantum channels, Corollary 6.1 ensures that the set of unital channels that are asymptotically entanglement breaking is also a dense subset of the unital channels. This result, combined with Theorem 5.2, demonstrates the richness of the class of eventually entanglement breaking maps.
The requirement of unitality of PPT channels can be relaxed in some cases if some properties of the adjoint map are exploited. Note that our method guarantees the existence of a number for a unital PPT channel on such that s entanglement breaking. However, a uniform bound could not be found. A uniform upper bound for the multiplicative index may provide an upper bound for this number. In a recent article , Theorem 3.8, such a bound for multiplicative index of a channel was proposed.
We would like to thank the referee for pointing out a simplified proof of Theorem 5.2. M.R is supported by a Post Doctoral Fellowship of the department of Pure Mathematics, University of Waterloo and Institute of Quantum Computing, S.J is supported by a Graduate Research Fellowship of the department of Combinatorics and Optimization and Institute of Quantum Computing and V.P is supported by NSERC Grant Number 03784.
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