Positive Maps From Irreducibly Covariant Operators

In our paper we discuss positive maps arising from (irreducibly) covariant linear operators for finite groups. Family of necessary conditions for positivity for such objects is derived. We present how our conditions work in practice by giving examples of positive irreducibly covariant linear maps for the group $S(4)$ and for the group monomial unitaries. In particular, for two dimensional maps, the necessary and sufficient conditions are given, together with explicit decomposition of such a maps into a sum completely positive and completely co-positive ones. We also present a construction of positive irreducibly covariant linear maps, together with corresponding entanglement witnesses, using the inverse reduction map. As an additional result, a novel interpretation of Fujiwara-Algolet conditions for positivity and complete positivity is presented. Finally, at the end, a new form of irreducible representation of the symmetric group $S(n)$ is constructed, allowing us to calculate in easier way certain Choi-Jamio{\l}kowski images.


I. INTRODUCTION
Symmetry is one of the most prominent properties which physical theory can exhibit. Usually it is induced by some group structure. In this paper we consider linear operators of special importance in quantum information theory, that are irreducibly covariant with respect to some finite group G. We call the map Φ irreducibly covariant (ICLM -irreducibly covariant linear map) if for some unitary representation U of finite group G.
There are two special classes of linear maps that play special role in quantum information theory: positive maps (P ) and completely positive (CP ) maps. Namely, a map Φ, not necessarily covariant, is positive if for every positive semidefinite matrix M ≥ 0, we have Φ(M ) ≥ 0 as well. Secondly, a map Φ is completely positive when Φ ⊗ id, where id stands for identity map, is positive. Completely positive maps model quantum channels, a fundamental blocks for quantum information processing tasks [9]. On the other hand maps that are positive, but not completely positive, which are of the main interest of this manuscript, are useful since they can be used for entanglement detection via so-called entanglement witnesses. The concept of entanglement witness follows from the famous Hanh-Banach Theorem and was introduced firstly in the field of quantum information by Terhal in [21]. This method allows us to detect entanglement without the full knowledge about given quantum state, since one does not have to apply full tomography of the state, which is an expensive process. Instead of that it is enough to check a mean value of an observable representing entanglement witness with a quantum state. Such method is universal since any entangled state posses corresponding entanglement witness.
In general problem of classification or even finding new examples of positive maps, with certain properties, is very hard despite of many attempts and important results in the field [4]. The main reason for that is non-existence of operational criterion for positivity, namely to prove that a given map is positive one has to check for example its block positivity, so expectation value for all product vectors. On the opposite, to check complete positivity property it is enough to compute only all eigenvalues of corresponding Choi-Jamiołkowski image and it can be done effectively at least for finite dimensions. One of the possibility of making progress in the field of positive maps is imposing certain symmetries on the problem and in consequence reducing its complexity. The irreducible covariance with respect to given group G defined as in (1), imposes symmetric patterns from the group on the map which is ICLM with respect to it, that usually help with determining conditions for positivity or complete positivity.
Our paper is a natural continuation of work [17] when authors analysed completely positive irreducibly covariant linear maps. Here, we start from general introduction to the problem by defining rigorously what we mean by irreducibly covariant linear maps and discus their basic properties (Section II). In Section III we present general considerations by presenting necessary conditions on the spectra for ICLM to be positive. Moreover for low dimensional irreducible representations of permutation group S(3) and quaternion group Q we deliver full characterisation of positive ICLM s, together with explicit decomposition into sum of complete positive and complete co-positive maps. We show that in the case of such a ICLM the general necessary condition is also a sufficient condition. To give a flavour of our considerations, in Figure 1 we present two regions for which linear map Φ, which is irreducibly covariant with respect to symmetric group S(3), is positive but not completely positive. One is calculated directly from the definition, and the second one is obtained from inverse reduction map. Both results are provided and discussed in sections III and IV respectively. In Section IV we present a general and effective approach for generating positive ICLM s for irreducible representations of higher dimensions. The main idea is to use properties of the inverse reduction map [10]. Having that we construct families of positive maps for two-dimensional irreducible representation of the permutation group S(3), but our method can be easily applied to an arbitrary finite group, assuming easiness of solving linear inequalities. This fact is illustrated in Section V where we deliver examples of positive maps acting in higher dimensions such as positive ICLM for three-dimensional irreducible representation of S(4). Moreover in the same section, apart from the inverse reduction map, we determine whether a map is positive by checking block positivity of corresponding Choi-Jamiołkowski image. While this method is of a little use in general scenario, it proved efficient in special case of determining positivity of ICLM s with respect to M U (d) for an arbitrary dimension d. The result obtained with this method gives necessary and sufficient conditions for the parameters of ICLM in question, which is significant improvement over sufficient condition, such as e.g. obtained from the inverse reduction.
In Section VI we compare our findings with generalized Choi map [15] in three dimensions by showing that our maps include Choi mapping and generate larger region of positivity. Unfortunately in these new regions we could not find any new indecomposable positive maps.
Finally, in Section VII yet another remarkable property of irreducible covariance is presented. It turns out, that every unital quantum channel can be expresses in the same terms as quantum channel which is irreducibly covariant with respect to quaternion group Q. In the last section we compare the results concerning P and CP obtained in the different approach on the Bloch sphere ref. We show that Fujiwara-Algolet conditions [7] for P and CP for arbitrary unital map obtained in this approach coincide with the conditions for P and CP for quantum channels irreducibly covariant with quaternion group. So in fact the conditions for P and CP for arbitrary unital map are connected with quaternionic invariance of quantum channels.

II. DEFINITIONS AND NOTATIONS
In this section we provide necessary mathematical tool for our further considerations. In the first part we present basics notions from linear algebra and group representation theory with one of the most important concept for this manuscript which is adjoint and contragradient representation. Second part is devoted to connection of linear maps and their Choi-Jamiołkowski images. We focus on if and only if conditions for positivity, complete positivity and complete copositivity for an arbitrary linear map. Next, we define concept of entanglement witness and its decomposability. As a exemplary case we discuss the general connection between irreducibly covariant linear maps coming from two-dimensional irreducible representations of the permutation group S(3), the quaternion group Q with unital linear maps, manifesting the importance of our studies.

A. Quantum entanglement, entanglement witnesses and decomposability
In this and further sections by H = C d we denote Hilbert space of dimension d. Using this notation we define set of all states on H. Having two Hilbert spaces H, K we say that state ρ ∈ S(H ⊗ K) is separable if it can be written as ρ = i p i σ i ⊗ ω i , where σ i , ω i are states on H, K respectively, and p i are positive numbers satisfying i p i = 1, otherwise the state ρ is entangled. Having above we are in position to define objects called entanglement witnesses [11], [20]: Definition 1. The hermitian operator W ∈ M(d 2 , C) is called entanglement witness when: 3. There exists at least one entangled state ρ, such that tr (ρW ) < 0.
The so-called Choi Jamiołkowski isomorphism gives useful characterization of entanglement witnesses. Let M(n, C) denote the space of n × n complex matrices and let {E ij } n i,j=1 , where E ij ≡ |i j|, denote a basis of M(n, C) and by {|i } d i=1 we denote a standard basis for C d . For a linear map Φ ∈ End[M(d, C)] its Choi-Jamiołkowski image J(Φ) is given by [3,12]: Isomorphism defined in (3) encodes properties of linear maps into properties of corresponding Choi-Jamiołkowski image. Thanks to Choi-Jamiołkowski isomorphism every entanglement witness is connected with positive, but not completely positive linear map Φ ∈ End[M(d, C)] by where P + d is a projector on maximally entangled state |ψ |ii . In the set of all entanglement witnesses we distinguish subset of decomposable ones. Namely any decomposable entanglement witness W admits the following decomposition with A, B being positive operators on the space H ⊗ H, and Γ being a partial transposition with respect to the standard basis. The corresponding map Φ W is decomposable if it can be expressed as where Φ (A) , Φ (B) are completely positive, and T stands for transposition with respect to standrard basis. Additionally, operators W satisfying Definition 1 for states being PPT entangled, do not admit decomposition (5) and they are called indecomposable entanglement witnesses. At this point for more informations about entanglement witnesses and their properties we refer reader to a review paper [4].

B. Irreducibly covariant linear maps and quantum channels
Here we summarize all necessary facts and definitions about irreducibly covariant linear maps. For more details we refer reader to [17].
Above definition is equivalent to say, that Φ ∈ Int G (Ad U ) or mat(Φ) ∈ Int G (U ⊗ U c ), where Int G (Ad U ) is the commutant of the adjoint representation given as and , where χ U : G → C is the character of the representation U : G → M(d, C) given as χ U (g) = tr U (g), and |G| denotes cardinality of the group G. In the most of the cases we study properties of Int G (U ⊗ U c ) in the matrix space M(d 2 , C), which is simpler (but equivalent) than studying Int G (Ad U ) in the space End[M(d, C)]. As it was shown in [8] the representation U ⊗ U c : G → M(d 2 , C) is not irreducible and we have where Θ is the set of irreps of the group G that appear in the decomposition U ⊗ U c , ϕ α are unitary irreps of the group G: ∀ g ∈ G, ϕ α (g) = ϕ α ij (g) ∈ M (d α , C), and m α is the multiplicity of the irrep ϕ α of dimension d α ≡ dim ϕ α . The multiplicity m α is given by the following expression [2]: χ α is the character of the irrep ϕ α (g), and χ Ad U (g) = χ U (g) 2 is the character of the adjoint representation Ad U .
The identity irrep ϕ id is always included in the decomposition (10) with the multiplicity one.
Here we restrict ourselves to the case when decomposition (10) is multiplicity free, i.e. all irreps in the set Θ occur with the multiplicity one. In this situation we can write The operators Π α ∈ End[M(d, C)] are of the form and where id End[M(d,C)] is identity map on End[M(d, C)]. Equivalently we can write where Orthogonality properties from (14) are also valid for matrix images Π α . The linear map Φ that is irreducible covariant with respect to unitary irreducible representation U of a finite group G admits decomposition The numbers l α are eigenvalues of a map Φ. Moreover, it was shown in [6], that l α can be chosen to be real if we restrict ourselves to CP T P maps. To ensure that map Φ is trace-preserving (T P ) we put l id = 1, where id denotes trivial representation. The complete positivity (CP ) is obtained from the Choi-Jamiołkowski isomporphism, via theorem below.
Theorem 3. The Choi-Jamiołkowski isomorphism given through (3) has the following properties: 1. A linear map is completely positive if and only if its Choi-Jamiolkowski image J(Φ) is a positive semidefinite matrix, i.e. J(Φ) ≥ 0.

2.
A linear map is positive but not completely positive (P) if and only if its Choi-Jamiolkowski image is blockpositive matrix, but not positive semifefinite. i.e. x| ⊗ y|J(Φ)|x ⊗ |y ≥ 0 for all vectors |x , |y ∈ C d , but there exists |z ∈ C d ⊗ C d for which z|J(Φ)|z < 0.

A linear map is completely copositive (CoP) iff Choi-Jamiołkowski image of its composition with transposition
Thus, to have complete positivity, the coefficients l α have to satisfy the following set of inequalities: In the above, V β i ∈ M(n, C) denote the normalized eigenvectors of rank-one projectors where n is the dimension of irrep with respect to ICLM is constructed. Considering special case when |x = |i , |y = |j , where |i , |j belong to orthonormal basis of C n we can formulate necessary condition for positivity of a ICLM Φ ∈ End[M(d, C)]. Namely we have the following

A. General Considerations
In this section we will focus on general conditions on positivity of covariant linear maps. First, a general condition for the positivity of a linear map Φ : M(n, C) → M(n, C) can be made through the following general and known: This condition, although general, is not particularly tractable. We shall later see, that it can be successfully applied only to low-dimensional cases. To have more tractable conditions let us formulate first two facts.
Fact 6. Let Π α be projectors given in 17. Then we have, for all X ∈ M(n, C): Fact 7. For all X ∈ M(n, C) In the following section we discuss the case when the dimensions of the corresponding irreducible representation is small (two-dimensional). The general structure of every low-dimensional positive maps is known due to works by Størmer and Woronowicz [19,22]. Namely, we know that every such map can be written as a sum of completely positive and completely co-positive maps. Nevertheless, it does not give us explicit method how to built positive maps for a given finite group. First we focus on the case of two-dimensional irrep λ = (2, 1) of the symmetric group S(3) and then we switch to appropriate irrep of the quaternion group Q. In addition we present direct decomposition of maps into sum of CP and CoP maps and show that components are also irreducibly covariant maps. In these particular considerations we set l id = 1 since it is only a scaling factor and we restrict to normalized maps (i.e. having trace equal to one).

Positive irreducibly covariant linear maps for S(3)
Here we consider positivity ICLM from (17) Using explicit form of ICLM considered here is Φ = Π id + l sgn Π sgn + l λ Π λ (see [17] or Section II B) we have From this we deduce at once that l sgn , l λ ∈ R, otherwise Φ(P ) would not be hermitian Now calculating the eigenvalues of this hermitian matrix we deduce that for any p = p 1 p 2 , |p 1 | 2 + |p 2 | 2 = 1.

Now let us observe that
so we have Therefore we may the main result for this paragraph: Theorem 10. The linear map Φ = Π id + l sgn Π sgn + l λ Π λ is positive if and only if l sgn , l λ ∈ R and lies in the rectangle |l sgn | ≤ 1, |l λ | ≤ 1.
As a corollary of above two propositions we get

C. Direct Approach to Decomposability -Low Dimensions
Spectral parameters l i of ICLM with respect to two-dimensional irrep of quaternion group Q, given by 32, are connected with eigenvalues of Choi-Jamiołkowski image of Φ, denoted by δ i with matrix I, namely Since I is involutive, we can write conversely Since for an ICLM that is positive we have |l i | ≤ l id , then writing all posible combinations δ α + δ β , α = β , i.e.
one can observe that for P but not CP map Φ, exactly one δ α is negative. We want to decompose such Φ into sum of completely positive map and completely co-positive map, both of which are ICLM with respect to quaternion group, namely where T denotes transposition operator. Since Ψ (1) and Ψ (2) are completely positive ICLM s with respect to quaternion group, they depend on spectral parameters (a i ), (b i ), i = id, . . . , 3 respectively, satisfying Combining (38) with (35) and (39) we obtain From the above equations one can see, that δ α + δ β ≥ 0 if α = β. Without the loss of generality one can assume that δ id < 0 (the exactly one negative eigenvalue mentioned earlier), thus δ id ≥ |δ id |. Rewriting (41) we get Since δ i > |δ id | we can write So we obtain For arbitrary set of eigenvalues δ i corresponding to ICLM that is P but not CP the above equation has solution of the form As we know, when we set in (32) l 1 = l 3 we obtain ICLM with respect to two-dimensional irrep of S(3) group. Considering (35) we get that the equality must hold for appropriate Choi eigenvalues as well (i.e. δ 1 = δ 3 ). Thus from (35) and the fact that only one δ i can be negative we conclude, that for no choice of δ i there is a solution for (γ i ) such that Ψ (2) is ICLM with respect to S(3) group. This shows again the special role quaternion of quaterion qroup Q among the low-dimensional ICLM s.

IV. POSITIVE IRREDUCIBLY COVARIANT MAPS FROM INVERSE REDUCTION MAP
In this section we present an efficient method allowing for the construction of irreducibly covariant positive maps using inverse reduction map. We start from the definition of the inverse reduction map. Namely we have Reader can check by direct calculation that indeed map defined above is inverse map with respect to reduction map defined in [10]. Moreover map R −1 is surjective between set P d k of rank k projectors an the set P d 1 of rank one projectors. Later in [18] it has been shown that map R −1 from Definition 14 is the only one with such property. The following theorem proved in [16] holds Theorem 15. Let W ∈ M(d 2 , C) be hermitian and non-positive operator such that W = (1 ⊗ R −1 )W ≥ 0. Then the operator W is an entanglement witness.
As it was mentioned in introduction in this manuscript we restrict ourselves to special class of linear maps which are irreducibly covariant, see Section II B. and where P + d is projector on maximally entangled sate |ψ Proof. First let us prove equation (47). For an arbitrary X ∈ M(d, C) we have Using explicit form of the map Φ from equation (17) we compute where in the last equality we used one of the orthogonality relation for characters given in (B1) of Appendix B. Substituting (50) into (49) we get desired expression. Having explicit equation for composition R −1 • Φ we calculate W in the following way since tr(E ij ) = δ ij .
where numbers β i are eigenvalues of the Choi-Jamiołkowski image J(Φ) of the map Φ from (17). Having above and Theorem 15 from Section II A we can formulate the following Corollary 18. An ICLM Φ ∈ End[M(d, C)] is positive when all inequalities given in (52) of Corollary 17 are satisfied and the map Φ is not completely positive, i.e. when at least one inequality from (18) is not fulfilled.
As an illustration of above statements let us consider symmetric group S(3) and its two-dimensional irrep labelled by λ = (2, 1). In this particular case conditions (52) (left-hand side) and (18) (right-hand side) have a form Taking Choi-Jamiołkowski image J(Φ) and computing overlap with maximally entangled state P + we have tr(P + J(Φ)) = l sgn + 2l λ + l id . We see that point (l id , l sgn , l λ ) = (1, −1, −1) belongs to the region for which map Φ is positive but not completely positive, and J(Φ) is appropriate entanglement witness since it detects at least one entangled state.

V. NEW EXAMPLES OF (IRREDUCIBLY) COVARIANT POSITIVE LINEAR MAPS IN HIGHER DIMENSIONS
In the following two subsections we present irreducibly covariant positive linear maps induced by three-dimensional irreducible representation of the permutation group S(4) and d−dimensional representations of the group of monomial unitaries. In the case of the group S(4) our results have been obtained using construction of novel irreducible representation (Appendix A), allowing for simplification of respective Choi-Jamiołkowski image with respect to previous approach presented in [17].

B. Covariant positive maps induced by group of monomial unitaries
As it was stated in introductory section, although in general the condition for block positivity of Choi-Jamiołkowski image of given map is hard to work with, it proved efficient in a class of d−dimensional maps covariant with respect to monomial unitary group. Before we move to main considerations for this section we start from the following Definition 19. The group of monomial unitary matrices M U (d) is given as the collection of unitaries U ∈ U (d) of the form U = DP , where D, P ∈ U (d) and D is diagonal with respect to orthonormal basis of C d and P is a permutation matrix.
It turns out monomial group M U (d) plays a role in many-body state formalism and some aspects of quantum computations, for the details please see [5] and references within it. Here, due to our further purposes we define subgroup M U (d, n) of M U (d) as Definition 20. We define M U (d, n) to be the subgroup of the monomial unitary matrices of dimension d whose non-zero entries consist only of m−th roots of unity.
Above defined subgroup is also under interest of quantum information community, since it contains so called T −gate [1], defined as T = |0 0| + exp( i π 4 )|1 1|, whenever n ≥ 8. This gate together with Clifford gates forms a universal set for quantum computations [1]. Finally the linear map M covariant with respect to group M U (d, n) from Definition 20 is given by: for every X ∈ M(n, d) and α, β ∈ R. Above map has been defined in [6] and used in the same paper to generalize randomized benchmarking protocol, when gates are representations of finite group, but not necessarily irreducible or 2-design.

Positive covariant maps induced by group of monomial unitaries in arbitrary dimension
In arbitrary dimension, the Choi-Jamiołkowski isomorphism of T is given by Since the action of J(T ) on |k ⊗ |l is given by the condition for block-positivity becomes where |x = (x 1 , . . . , x n ), |y = (y 1 , . . . , y n ).
Lemma 21. For the following values of α, β the map T is positive: (66) Proof. substituting the above values for α and β one obtains the following expressions for x| ⊗ y| J(T ) |x ⊗ |y : which are obviously positive for arbitrary |x ⊗ |y .
Theorem 22. For all α, β within the quadrilateral spanned by the points given in Lemma 21 the map T is positive.
Proof. Let us consider the lower sides of the quadrilateral in question.
Theorem 23. The region described in Theorem 22 is not only sufficient, but necessary as well.
Proof. To show most of the necessary conditions for positivity of map T we choose special |x and |y , presented in Table I along with the condition yield. |x |y x| ⊗ y| WT |x ⊗ |y condition  The final condition that makes the necessary and sufficient regions coincide is obtained by taking β < d d−2 α−(d−1). Let us set Thus, condition for block positivity becomes Clearly, this expression can be negative, e.g. let us take |x = −(e 1 + e 2 ) and |y = e 2 − e 1 , then The regions of positivity of map T for several dimensions are presented in Figure 3. Neglecting the regions where T is completely positive we finally arrive at the solutions we are interested in, that are presented in Figure 3 as well.

C. Connection between maps induced by group of monomial unitaries and group S(4)
One can observe, that the solution for positivity of the map M for d = 4 resembles the solution for S(4) (see Figure  2) taken along the plane l λ1 = l λ3 = α, l λ2 = β. What is more, for such choice of parameters, eigenvalues of J(Φ λ1 ) and J(M ) are the same. Since both matrices are hermitian, if they have common spectra they are similar. Moreover, using properties of isomorphism, M and Φ λ1 have to be similar as well. For direct connection of witnesses in question we have where matrix A is given as

VI. COMPARISON WITH THE GENERALIZED CHOI MAP
Choi map [3] is an example of positive, but not completely positive endomorphisms of M(3, C). Its various generalizations are discussed, either in non-normalized [15] or normalized form [14]. Since the ICLM with respest to monomial unitary subgroup is trace preserving, we shall compare it to the latter one. It is defined by its action on standard operator basis {e ij } 3 i,j=1 for a, b, c ≥ 0 as follows: where Using above we compute Choi-Jamiołkowski image of Λ[a, b, c]: We know that Λ[a, b, c] is non-decomposable and positive [14] if a) To have conditions for positive but not completely positive maps [? ] only we have to use instead of (79) the following constraint: a. Comparison with generalized Choi map -S(4). First of all we see that Choi-Jamiołkowski image of J(Φ) from (55) can be equal to J(Λ[a, b, c]) from (76), only when we put l λ3 = l λ2 = l λ1 = l in equation (57). This implies the following conditions b. Comparison with generalised Choi map-monomial unitaries. Comparing this matrix with Choi-Jamiołkowski image of UM for d = 3 we arrive at the following set of equations We can immediately see that for positive α pictures cannot coincide. Thus at least for α > 0 we obtain a positive and non completely positive map, which is different from generalised Choi map described in the literature. Another immediate observation is that if both maps coincide, then b = c. This simplifies the requirement for positivity and non-CP to Solving equation 82 and considering the condition 83 one can easily check that for every (α, β) in the remaining region (except for one point where α = 0) one can check the following Fact 24. In the region presented in Figure 4 corresponding to α < 0 the ICLM M U (α, β) can be expressed as generalised Choi map Λ(a, b, b) which is positive, but not completely positive.
It is worth noting that the region of overlap for the two maps is not closed. Both regions are presented in Figure  4. c. Decomposability of ICLM s generated by monomial unitary group Since the ICM LM with respect to monomial unitary group coincides with Generalized Choi map given in [14] we can examine the conditions for positivity and non-decomposability of Λ (a, b, c).
Since for region of coincidence of the two maps in question we have α = −1/(a + b + c) and α ≥ −1/2 we can see that in the overlap region our map is decomposable.
If we consider decomposition of the form we immediately arrive at the trivial solution α = α 1 = α 2 = 0. Requiring where Requiring that Φ 2 (α 2 , β 2 ) is completely positive, we get the following condition Setting α 1 = (1 − p)α, β 1 = (1 − q)α, α 2 = pα, β 2 = qα we get total condition for validity of the decomposition (88) The general solution of the condition above for all α, β was not obtained, but we can make some observations. Firstly, when we set p = q = 1 we obtain the region of complete co-positivity of our map in question (which is thus trivially decomposable) On the other hand, when we set e.g. α = 2/3, β = 0 we immediately get that p ≤ 1/2 and a p ≥ 2/3 which shows, that such decomposition cannot hold for the whole region of α, β. Finally, we give the description of all classes of positive ICLM s with respect to monomial unitary subgroup for d = 3 that we discussed in Figure 5.

VII. IRREDUCIBLY COVARIANT LINEAR MAPS AND UNITAL QUANTUM CHANNELS
As an example how above methods from previous subsections work in practice, and to illustrate their importance for general studies on linear maps, below we investigate a novel connection between all qubit unital quantum channels and linear maps generated by irreducible representations of the quaternion group Q. In order to understand this connection we have to consider two methods of description of a qubit quantum state transformations. A qubit state is a matrix where σ i are Pauli matrices. Here due to the properties of the Pauli matrices we have a one-to-one correspondence between the states ρ ∈ M(2, C) and the vectors r = (r i ) ∈ R 3 , so we may describe the state transformations as the transformations T ∈ End[R 3 ] ≡ M(3, R). From the paper of Kübler and Braun [13] we know that the most general form of the unital qubit quantum channel Φ T [ρ(r)] is and is fully characterized by T ∈ End[R 3 ] given by The most important quantum properties of the map T like positivity and complete positivity depend on the matrix D(η) only, not on matrices R i ∈ SO(3), so the unital quantum channel is completely characterized by the numbers (η 1 , η 2 , η 3 ), called signed singular values (SSV) in [13]. From [13] Φ T is completely positive if and only if On the other hand, we have non-abelian quaternion group, described in Section II B (see also [17]). Taking in (18) The respective matrix representation mat(Φ) of the trace-preserving ICLM Φ from (17) is of the form The conditions for Φ to be CP map are They are exactly the same conditions as for CP of unital qubit quantum channel Φ T (94) defined through T (η, R 1 , R 2 ) in (95). In particular, if R 1 = R 2 = 1 the operators Φ T (η, 1, 1) and Φ coincide.
In this section we continue analysis of the connection between ICLM generated by irreducible representations of Q and S(3) described briefly in Section II B. Firstly we present what types of linear transformations on End[R 3 ] are induced by ICLM generated by groups Q and S(3). We can summarize our considerations in the following b) The ICLM Φ S(3) (k) = Π id + k sgn Π sgn + k λ Π λ induces the following map in M(3, R) Both maps induce the diagonal transformation of the Bloch vector but the map Φ Q (l) is more general than Φ S(3) (k). What is more, we have Corollary 27. The following relation between maps Φ Q (l), Φ S(3) (k) and the general form of the unital qubit quantum channel T (η, R 1 , R 2 ) holds The spectral parameters l = (l i ) and k = (k sgn , k λ ) of the ICLM Φ Q (l) and Φ S(3) (k) are precisely the SSV of the corresponding maps T (η Q , id SO(3) , id SO (3) ) and T (η S(3) , id SO (3) , id SO (3) ).
Under the identification of parameters (l i ) and (η Q i ) in Corollary 27 we may formulate the following Proposition 28. Any unital map can be decomposed as follows where M Q (l) is the matrix induced by some quaternion irreducible covariant map Φ Q (l). In other words, the SSV of any unital map T ∈ End[R 3 ] are spectral parameters of some quaternion irreducible covariant map Φ Q (l). We know from [13] that all important quantum properties of the unital maps depend only on their SSV . In fact investigated properties depend on the spectral parameters of the quaternion irreducibly covariant map Φ Q (l). Namely, the Fujiwara-Algoet conditions for CP of the unital map T (η, R 1 , R 2 ) [13] 1 + η 3 ≥ |η 1 + η 2 |, 1 − η 3 ≥ |η 1 − η 2 | (104) are exactly the same as our conditions on spectral parameters l = (l i ), derived in [17] (or see (98) in this paper). Moreover the conditions for P of the unital map T (η, R 1 , R 2 ), given in [13] |η i | ≤ 1, i = 1, 2, 3 are also the same as our conditions for P of the irreducibly covariant quaternion map Φ(l) presented in Theorem 13. It may be checked that the Fujiwara-Algoet conditions [7] for CP presented in (104)  1 + k sgn ≥ 2|k λ |, 1 ≥ |k sgn |.
We can summarize the above as follows Proposition 29. The unital map T (l, R 1 , R 2 ) given in Proposition 28 is CP (respectively P ) if and only if the irreducibly covariant quaternion map Φ(l) is CP (respectively P ). Although all unital maps are more complicated objects, they depend on the elements of the group SO(3), their specific properties (CP, P ) are determined by ICLM by the quaternion group Q, which are simpler object in the general description. Moreover, similar forms of the matrices M Q (l) = diag(l 3 , l 1 , l 2 ) and M S(3) (k) = diag(k λ , k λ , k s ) presented in Proposition 26 suggest that in the particular case of the ICLM Φ Q (l) of the form Φ Q (k λ , k λ , k sgn ) is closely related to ICLM Φ S(3) (k λ , k sgn ).

VIII. CONCLUSIONS AND DISCUSSION
The results presented in the paper show how symmetric patterns imposed on linear map via irreducible covariance make examination of its certain properties, like positivity simpler than in general case. A variety of techniques was employed which led to different (both general and partial) results concerning multiple linear maps. In particular the spectral decomposition of ICLM allows to formulate the conditions for the positivity as the conditions on the mentioned spectrum of ICLM , leading to some geometrical interpretations in the spaces of ICLM s spectra. Basing on this idea, the full description of positivity and complete positivity for low dimensions of ICLM is given. The interesting connection of Fujiwara-Algolet conditions for an arbitrary qubit unital map with quaternion symmetry is shown. Such result suggests possibility of similar relation for unital maps and some, possibly different symmetries in higher dimensions. For higher dimensions we present methods allowing us to find necessary conditions for positivity exploiting inverse reduction map, as well as direct approach, working remarkable effective for monomial unitary group.
Still many questions remain open. For example, the ICLM arising from irreducible representation of symmetric group S(4) was examined via inverse reduction map and thus the obtained region of positivity is only sufficient. The general solution despite attempts could not be obtained. Moreover, the (non-)decomposability of ICLM with respect to unitary monomial subgroup was not resolved in general and thus there may be some region from which new, non-decomposable entanglement witnesses may arise.