Beyond trace-class and Hilbert-Schmidt -- Interaction between operator ideals and von Neumann algebras in quantum physics

TL;DR

This paper explores the deep connections between operator ideals, von Neumann algebras, and quantum physics, highlighting their foundational role in quantum theory and information, with applications to quantum teleportation.

Contribution

It revisits the role of nuclear and absolutely p-summing operators in quantum field theory and quantum information, linking operator ideals to the foundations of quantum physics.

Findings

Banach operator ideals are fundamental in quantum physics and quantum information.

Constructs the enveloping C*-algebra for arbitrary operator ideals.

Provides a linear algebraic description of quantum teleportation.

Abstract

Starting from a thorough analysis of the conjugate $\overline{H}$ of a complex Hilbert space $H$, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fr\'{e}chet-Riesz over to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory (AQFT) 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 AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).