Indecomposable positive maps on positive semideﬁnite matrices from M n to M n +1

In this paper we obtain a theorem for 2 -positive linear maps from M n ( C ) to M n +1 ( C ) , where n = 2 , 3 , 4 . In addition, we answer in the afﬁrmative a question that asked if there exists every 2 -positive linear map from M 3 ( C ) to M 4 ( C ) is indecomposable using a family of positive linear maps with Choi matrices of 2-positive maps on positive semideﬁnite matrices. Further it is shown that 2 -positive linear map from M 4 ( C ) to M 5 ( C ) are indecomposable.

Proof Let c 1 , c 2 , c 3 , µ be real numbers and X be a selfadjoint matrix. Then the linear maps φ (c1,c2,c3,µ) is positive if the matrix  is positive. where First we look at the determinants of the 2 × 2 principal submatrices, Comparing the coefficients of α 1 α 2 the minimum determinant is obtained when This implies that c 1 ∈ [0, 1] and c 2 , c 3 ≥ 0. The 2 × 2 submatrix has a minimum positive determinant when This implies that c 2 ∈ [0, 1] and c 2 , c 3 ≥ 0. On the other hand, the 2 × 2 submatrix This determinant is positive if and only if 0 < µ ≤ 1. For the case of 2 × 2 submatrix; In this case the minimum determinant is positive when c 3 ∈ [0, 1] and c 1 , The determinant is nonnegative when c 1 , c 2 , c 3 ∈ [0, 1] and 0 < µ ≤ 1 The determinant of the matrix 2.3.6 is given by, From the two coefficients of α 1 α 2 α 4 we have, This is nonnegative when µ −2r−1 − c 3 ≥ 0. Thus the linear map φ (µ,c1,c2,c3) is positive when c 1 , c 2 , c 3 ∈ [0, 1] and 0 < µ ≤ 1 for all r ∈ R + . Definition 32 A map φ n is n−copositive if and only if the map τn ⊗ φ : Mn −→ Mn is positive.
Definition 33 Let φ : Mn(C) −→ Mm(C) be a linear map. Let (E ij ) with i, j = 1, ..., n be a complete set of matrix units for Mn(C). Then the Choi matrix for φ is the operator The map φ → C φ is linear, injective and is surjective, and given an operator E ij ⊗ A ij ∈ Mn ⊗ Mm. We therefore observe that the Choi matrix depends on the choice of matrix units (E ij ). This map is often called the Jamiolkowski isomorphism [6]. The Jamiolkowski Isomorphism as associate with every a positive map φ from Mn to Mm is a unique matrix C φ ∈ Mnm Mn(Mm). The Choi result in [2] affirms that a map φ is completely positive if and only if the Choi matrix C φ is positive definite. A linear φ is completely positive if and only if the block matrix is positive , otherwise it is not completely positive. It is more convenient to express n-positivity by using a block matrix notation. Since

Remark 35
The tensor product of positive operators Mn and Mm isomorphic to Mn ⊗ Mm is the block matrix mn × mn-matrix given by Mn(Mm). That is, Mn(Mm) Mn ⊗ Mm. In addition if nm is even, then Mn(Mm) Mn ⊗ Mm M 2 (M k ) for some k such that nm = 2k Proposition 36 Let φ : Mn −→ M n+1 be a positive linear map such that n(n + 1) is an even integer. Then the following are equivalent is a block positive matrix in Mn(M n+1 ).
(ii)⇒ (i). We show that the map (In ⊗ φ) is positive if the block matrix [φ(E ij )] n i,j=1 is 2-positive. By Theorem ?? and Theorem 34 let the block matrix [φ(E ij )] n i,j=1 be positive and the standard basis for C n be given by e i so that the set E ij = e i e j is a basis for Mn.
. . , Vr ∈ M n(n+1) such that n(n + 1) = 2k for some positive integer k; . Thus block matrix [φ(E ij )] is positive. By the definition of Choi matrix, Proposition 37 Let φ : Mn −→ M n+1 be a selfadjoint linear map such that nm is an even integer. Then the following are equivalent is block 2-positive. By Theorem ?? and Theorem 34, block matrix [φ(E ji )] n ij,i=1 be positive and let the standard basis for C n be given by e i so that the set E ji = (e i e j ) T is a basis for Mn.

Remark 45
We note that the decomposition of the map φ (c1,µ) is not unique. This is one of the reason decomposition of positive maps even in low dimensions is such complicated to be expressed with a unique algorithm.
The next proposition compliments the decomposition of the map φ (µ,c1,c2) with both the conditions of complete positivity and complete copositivity maps given.
is a completely positive map with with positive eigenvalues ; 4 , 2, 2. This map is decomposable to a sum of completely positive map and a completely copositive map respectively. That is, Using Propositions 312 and 313 we show that the first and the second matrices are completely positive and completely copositive respectively.// In the case of the Choi matrix of the completely positive map Z is a zero matrix.  The decomposability of positive maps is complex in higher dimensions. The next proposition shows the indecomposability of the map φ (µ,c1,c2,c3) even when the parameters c 1 , c 2 , c 3 are all zeros.
Woronowicz [9], Theorem 3.1.6 ] showed that every positive linear map φ form M 2 (C) to Mm(C) is decomposable if and only if m ≤ 3. Yang, Leung and Tang [10] proved that every 2-positive linear map from M 3 (C) to M 3 (C) is decomposable and enquired whether this is true for linear maps from M 3 (C) to M 4 (C). From the inequality 4.1.9, It is clear that φ (µ,c1,c2) is indecomposable whenever c 1 , c 2 > 0. However, there exist values of coefficients µ, c 1 and c 2 for which a linear map φ (µ,c1,c2) from M 3 (C) to M 4 (C) is 2-positive and decomposable. By Proposition 48, the linear map φ (µ,c1,c2,c3) from M 4 (C) to M 5 (C) is indecomposable.