Norm Conditions for Separability in \({\mathbb M}_m\otimes {\mathbb M}_n\)
An element \(\mathbf{S}\) of the tensor product \({\mathbb M}_m\otimes {\mathbb M}_n\) is said to be separable if it admits a (separable) decomposition $$ \mathbf{S}\ =\ \sum _pX_p\otimes Y_p \quad \exists \ \ 0 \le X_p \in {\mathbb M}_m,\ \exists \ 0 \le
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Abstract An element S of the tensor product Mm ⊗ Mn is said to be separable if it admits a (separable) decomposition S =
X p ⊗ Y p ∃ 0 ≤ X p ∈ Mm , ∃ 0 ≤ Y p ∈ Mn .
p
This decomposition is not unique. We present some conditions on suitable norms of S which guarantee its separability. Even when separability of S is guaranteed by some method, its separable decomposition itself is difficult to construct. We present a general condition which makes it possible to find a way of an explicit separable decomposition.
1 Introduction Let Mn denote the space of n × n (complex) matrices for each n = 1, 2, . . .. Each element of Mn is considered as a linear map from Cn to itself. Here an element x of Cn is understood as a column n-vector, and correspondingly x ∗ is a row n-vector. Then given x, y ∈ Cn , according to the rule of matrix multiplication, x ∗ y is the inner product of x and y, that is, x ∗ y = x|y while yx ∗ is a matrix of rank ≤ 1 in Mn . Notice here that the inner product √ x|y is linear in y and anti-linear in x. With this inner product and norm x = x|x, Cn becomes a Hilbert space. Correspondingly the space Mn becomes a Hilbert space with the inner product T |S := Tr(T ∗ S) ∀ S, T ∈ Mn
T. Ando (B) Hokkaido University (Emeritus), Hokkaido, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_2
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T. Ando
where Tr(S) is the trace of S, and the Hilbert–Schmidt (or Frobenius) norm S 2 :=
S|S ∀ S ∈ Mn .
For self-adjoint S, T ∈ Mn , the order relation S ≥ T or equivalently T ≤ S is defined as S − T is positive semi-definite. Therefore S ≥ 0 or 0 ≤ S means simply that S is positive semi-definite. The operator (or spectral) norm S of S ∈ Mn is defined by S := sup{ Sx ; x = 1}.
(1)
When S is selfadjoint, the norm S is calculated simply as S = S∗
=⇒
S = sup{|x|Sx|; x = 1}.
(2)
As an immediate consequence of (2), S = S ∗ , I − S ≤ 1
=⇒
S ≥ 0.
(3)
In fact, the assumption guarantees that x 2 − x|Sx ≤ x 2 ∀ x. Next let us turn to the tensor product Mm ⊗ Mn of two matrix spaces Mm and Mn . There are canonical identifications Mm ⊗ Mn ∼ Mm (Mn ) ∼ Mmn . Here Mm (Mn ) denotes the space of m × m block matrices with entries in Mn . The first identification is understood in the following way: X ⊗ Y ∼ [ξ j,k Y ] j,k ∀ X = [ξ j,k ] j,k ∈ Mm , ∀ Y ∈ Mn .
(4)
The second identification is understood by cutting each mn × mn matrix into m × m block matrices in a natural way. Therefore by (4)
E j,k ⊗ S j,k ∼ S = [S j,k ] j,k
j,k
where E j,k ( j, k = 1, 2, . . . , m) is the matrix unit in Mm , that is, E j,k = e j ek∗ ( j, k = 1, 2, . . . , m) with the canonical orthonormal basis e j ( j = 1, 2, . . . , m) of Cm .
Norm Conditions for Separability in Mm ⊗ Mn
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In the following we abuse the notation: S =
E j,k ⊗ S j,k ∀ S = [S j,k ] j,k ∈ Mm (Mn ).
(5)
j,k
According to the general rule, the space Mm (Mn )
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