  Starting from the meaning of the conjugate of a complex Hilbert space,
including a related application of the theorem of Fr\'{e}chet-Riesz (by which
an analysis of semilinear operators can be reduced to - linear - operator
theory) to a revisit of applications of nuclear and absolutely $p$-summing
operators in algebraic quantum field theory in the sense of Araki, Haag and
Kastler ($p=2$) and more recently in the framework of general probabilistic
spaces ($p=1$), we will outline that Banach operator ideals in the sense of
Pietsch, or equivalently tensor products of Banach spaces in the sense of
Grothendieck are even lurking in the foundations and philosophy of quantum
physics and quantum information theory. In particular, we concentrate on their
importance in algebraic quantum field theory. In doing so, we establish a
canonical isometric isomorphism between the Hilbert spaces $H\otimes_2 (K
\otimes_2 L)$ and $(H \otimes_2 K) \otimes_2 L$ (Theorem 3.8) and revisit the
role of trace class operators.
  A few applications are specified, including the appropriateness of the class
of Hilbert-Schmidt operators and an implied Banach operator ideal
representation of the tensor product of two complex Hilbert spaces $H \otimes_2
K$ (Proposition 3.4) and a purely linear algebraic description of the quantum
teleportation process (Example 3.10).
