Extremal positive maps on
and idempotent matrices
A new method of analysing positive bistochastic maps on the algebra of complex matrices has been proposed. By identifying the set of such maps with a convex set of linear operators on , one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been show that all positive bistochastic maps, extremal in the set of all positive maps of , that are not Jordan isomorphisms of are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.
Keywords: positive maps, extremal, idempotent, semigroup.
Positive maps of operator algebras constitute a rich area of research, directly connected to the theory of quantum entanglement. In 1990s, A. Peres and P. M. R. Horo-deckis pointed out at the intrinsic relation between separable states of composite quantum systems and positive maps of algebras of observables [12, 8]. The well established criterion of separability, proposed in the mentioned papers, reveals a one-to-one correspondence between positive maps and entanglement witnesses . This Peres-Horodecki criterion, originally proposed for maps on algebras of square complex matrices of size , which holds true even in the most general setting of injective von Neumann algebras , is computationally feasible as long as the structure of general positive maps on operator algebras representing a composite quantum system in question is known. To this day, the complete characterisation of positive maps have been obtained only for the algebra and the maps between and [14, 16].
To analyse the structure of positive maps in the simplest, still unresolved case of maps on , we propose a continuation of the reasoning conducted in our previous paper . We have established a connection between positive maps that preserve both the trace of matrices and the identity matrix, the so-called bistochastic maps, and their stable subspaces that have the structure of Jordan algebras. Here, we go one step further and explore the relation between those stable subspaces and the idempotent real matrices that represent the conditional expectations projecting onto the spaces. To this end, we employ mostly geometrical techniques that allowed us before to establish the structure theorem for maps on the algebra , as well as the methods from the theory of compact semigroups [13, 4].
The main result of this paper makes possible to outline a program, suggesting where the extremal positive maps on are to be found with respect to their connection to associated idempotents (see Theorem 3 below). We have shown that all positive bistochastic maps, extremal in the set of all positive maps of , that are not Jordan isomorphisms of are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one dimensional orthogonal projection. As a corollary, we specify some norm conditions for matrices representing possible extremal maps. The structure of the paper concentrates on building a mathematical framework necessary to prove the final result. We start with a brief listing of necessary notation and definitions.
Let be grater than 0. Let denote the algebra of complex square matrices of size . The algebra of real matrices will always be denoted explicitly by . For , the norm is understood to be the standard operator norm, i.e. the maximal singular value of the matrix . For the Hilbert-Schmidt norm (HS-norm) of , we reserve the symbol , where tr denotes the trace operation, and is the conjugate-transpose of . The HS-norm of can be computed as the sum of squares of singular values of . The Hilbert-Schmidt inner product, induced by the HS-norm, is defined as , for . The identity matrix of , we denote by , or simply , and the null matrix by .
We say that a matrix is positive-semidefinite, or simply positive, if the inner product , for any vector (i.e. and has a non-negative spectrum). A linear map is said to be positive, indicated: , if for any such that , we have . The operator norm of the linear map is given by
It is true that if is positive, then . Any positive map is Hermitian, i.e. , for all . The identity map of is labelled , or simply . The convex cone of all positive maps of is denoted by . A positive map is extremal, if for any positive map such that , i.e. , we have for some number . It is true that every positive map can be written as a convex combination of extremal ones. If is a positive map such that and , for any , then we call it bistochastic, or doubly stochastic.
From now on, let us fix . We choose the set of normalised Gell-Mann matrices, , by taking:
and ; which is an orthonormal basis for with respect to the HS inner product. Any self-adjoint matrix can be written as , where , and . Sometimes, we will use a simplified notation: , where , and . For and , we have . For a bistochastic map , let us define a matrix by , . Then it is easy to see that acts on a self-adjoint matrix by
For a matrix , we denote a linear map on that preserves both the identity and trace, defined by the relation (1.2), by . Let be a set of those real matrices , for which is a bistochastic map: . Therefore, there is a one-to-one correspondence between and the set of bistochastic maps on . The mapping is in fact a semigroup isomorphism, for which , where generally denotes the HS-adjoint map of , given by: , for all . The structure of the set is fairly complicated. First, it is a closed convex set, i.e. for any , , for . Moreover, is is a compact, convex and real topological semigroup (see e.g. [13, 4] for the definitions) with involution being the matrix transposition: . Obviously, . The structure of the set analogous to , but representing maps on the algebra , has been studied using geometrical methods in . Because there is no such geometrical identification for , this time we exploit the semigroup aspect of the set .
Let denote a closed ball in with respect to the operator norm, centred around , with radius . Then
Suppose that and . Let be such that and are orthogonal projections in . It is easy to check that in that case , where denotes the standard Euclidean norm. Because , we have that
Hence is positive, which means that .
Let denote a diagonal matrix ; and so on for other sets of indices: , etc. It is easy to check that all matrices: are idempotent elements belonging to . In addition, there are no idempotent elements in of rank 6 or 7, and the only idempotent of rank 8 is the identity . That fact will become clear in the light of the proof of Theorem 2 below (see Remark 5). Let also denotes the group of those matrices such that is an automorphism: , for a unitary matrix and for any . It is evident that , the special group of orthogonal matrices. The set of those such that is a bistochastic map, extremal in the set of all positive maps on (not necessarily bistochastic), is denoted by , whereas the set of extremal points of the convex set is labelled as . The matrix , if and only if for all and , if , then . It is true that . The next fact follows from Proposition 1.
Let . Suppose that ; then , for .
Let and . Let be the polar decomposition of , , . Suppose that . Let and . Then , so and by Proposition 1, we have that both . Then , and thus it cannot be extremal, a contradiction. Hence, . ∎
By Proposition 1, we see that implies that . Therefore, , , are the only possible elements in , and so in , with the norm .
2 Idempotent and extremal elements of
Let and let be the semigroup generated by : . By , we denote the closure of in . The proof of the following proposition is presented in [13, Lemma 3].
Since is closed, . The set contains a unique idempotent, denoted by .
For the set , we define the following subsets:
the set of idempotents of : ;
the set of nilpotent elements: ;
the group of invertible elements: .
For an idempotent element , we define the following subsets of :
let be the maximal subgroup of containing ;
, where is the unique idempotent element of .
It is a known fact from the semigroup theory that for each idempotent element of a semigroup, there is exactly one maximal subgroup containing it. In particular, in our case of the semigroup , , , obviously, and moreover . Indeed, if the matrix elements of are , then from the fact that , follows ( belongs to the maximal group containing , and the idempotent is the identity for that group). Hence, . Because there is a sequence of natural numbers , , such that , so . But , and thus .
The following proposition, proven in general terms in [13, Theorem 8], will be repeatedly used in the subsequent reasoning.
Let . Then .
Observe that the semigroup fulfils the assumptions of [13, Theorem 8]. ∎
Our task is to analyse the family of sets , and in particular, to show, where among this family the elements of are to be found. To this end, we present the following series of results.
For , , the sets and are disjoint. Moreover, .
If , by Proposition 3, the sets and must be disjoint. For , we have that , and the assertion follows. ∎
It should be noted that for , because (see Proposition 1), and , then . Thus, is an orthogonal matrix. In particular, for any , .
Let and . Then .
It is clear that . Because the mapping is a homeomorphism, so . Let . Then . Hence . This in turn means that , i.e. , which establishes that . The reverse inclusion follows from an analogous reasoning. ∎
Next, we describe the structure of the set of idempotents in . For , let . At first, in the following lemma, we recall a known fact that an idempotent contractive operator on a Hilbert space is an orthogonal projection (see e.g. [1, Problem 5.3.14]).
Let , then , and hence, is an orthogonal projection in .
Because , so , hence . Since , by Proposition 1, . Suppose that and , the range of the operator . Let . Then
i.e. , for any . It means that , for every and , i.e. , which proves that . ∎
Let and . Then , and .
Let us recall that for a bistochastic map , by we denote the stable subspace of defined by (see [10, Eq. (3.3)]):
where is the adjoint map of . The fact that can be generalised to the following result.
Let and be the stable subspace of the map . Then is a Jordan algebra and , the group of Jordan automorphisms of .
Because is idempotent, by [10, Corollary 3], the space is a Jordan algebra. The map is in fact the conditional expectation onto . Let . By Proposition 6, for any , , and the same for . Thus, the stable algebra (compare , eq. (3.3) and below). Again, by [10, Corollary 3], is a Jordan automorphism of the Jordan algebra .
On the other hand, if is an arbitrary Jordan automorphism of , then it could be extended to a bistochastic map on , for some , by . Then , because acts as the identity map on . Hence, . In addition, since is invertible on , by extending to another bistochastic map , , we show that and , which proves that . ∎
From the above theorem, since , we infer that the group consists of those matrices that represent Jordan isomorphism on , i.e. for any , there is a unitary matrix such that either or for all .
The set is a sum of seven disjoint subsets:
If , then as above, is a conditional expectation map onto the Jordan algebra . By Theorems 5.3.8 and 6.2.3 of , all Jordan subalgebras of are isomorphic (unitary equivalent) to one of the following: , , , , , , and itself; where is the Jordan algebra of symmetric matrices of size : ; , are matrix units with 1 at the ith diagonal entry and 0 elsewhere, and . Hence, there is such that , and is the orthogonal projection that represents the projection map onto precisely one of the algebras mentioned above. It is straightforward to check that then , and is equal to the dimension of the respective Jordan algebra associated to it. Hence, . ∎
In the light of the proof of the theorem above, since the are no Jordan subalgebras of of dimension or , nor there are idempotent elements of that have rank or .
For , since is a projection, .
We prove a useful decomposition of elements of in the following lemma.
Let . A matrix belongs to , if and only if, , where , and . This decomposition is unique.
Suppose that , , , and . Because , there is a sequence for , such that . Then , i.e. .
On the other hand, let us suppose that . We have , where . By Proposition 4, both . Hence, , which means that , and also . Let , and . We have that and . Then , for , which implies that . Since , , again by Proposition 4. For , we have , i.e. . There is a sequence of natural numbers such that . Hence, . It follows that , which is enough to say that . Lastly, suppose that , where , and has the property: . For some sequence of natural numbers , we have . Then , and also . ∎
For , since the decomposition described above is unique, let us denote by and the matrices such that , , , and . The above lemma justifies the following definition.
Let . We define: . For , let be a set consisting of those , for which the largest singular value of is equal to 1 with multiplicity .
It is obvious that , and these sets are disjoint (possibly empty).
Let . If , then . If , then for .
If , then , and since by Remark 3, , obviously . Suppose that and . Then , by Proposition 6; , and because for any , , we have that , where is a orthogonal projection onto the space spanned by eigenvectors of with eigenvalue 1. Of course, , so the matrix . It must be that , otherwise , and hence , a contradiction. If , then , because by Theorem 2, there are no idempotent elements of with rank or . Hence and . By the same argument, for , it is impossible that , so , for , because by definition, . ∎
For the sake of convenience, let us introduce the following sequence of elements of : , , , , .
Let be integers such that , and . If , then there exist and such that .
Let and , as above. Since this decomposition is unique, we can write as in the proof of Lemma 3: , and , where . Because , then , and thus , by Proposition 6. Let be the singular value decomposition of , i.e. are orthogonal matrices, and is diagonal with the only possible non-zero entries , such that , and . Then for , , and because is closed: . Also, since , we have that . It follows that is an idempotent and . By Theorem 2, there is such that . A similar argument, applied this time to , shows that there is such that the idempotent could be written as . Let . What remains to show is that . It is evident that . One can easily check that , and hence , where , and ,