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arXiv:2605.16025

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

Frank Oertel

quant-phmath.OASubmitted May 15, 2026• Announced May 18, 2026• 0 citations
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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 Fréchet-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).

Comments
This paper should be viewed as a strongly enhanced (and slightly corrected) version of the paper arXiv:2308.04627 which I withdrew from the arXiv server on my own initiative

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Starting from a thorough analysis of the conjugate H‾\overline{H}H of a complex Hilbert space HHH, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fréchet-Riesz over to a revisit of applications of nuclear and absolutely ppp-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler (p=2p=2p=2) and more recently in the framework of general probabilistic spaces (p=1p=1p=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 \tupC\adj\tup{C}^\adj\tupC\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).
Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fréchet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely ppp-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler (p=2p=2p=2) and more recently in the framework of general probabilistic spaces (p=1p=1p=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⊗2(K⊗2L)H\otimes_2 (K \otimes_2 L)H⊗2​(K⊗2​L) and (H⊗2K)⊗2L(H \otimes_2 K) \otimes_2 L(H⊗2​K)⊗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⊗2KH \otimes_2 KH⊗2​K (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert–Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X\mathbb{X}X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X\mathbb{X}X under the semi-inner product, and then the state space S(X)\mathcal{S} (\mathbb{X})S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X\mathbb{X}X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X)ω\in \mathcal{S}(\mathbb{X})ω∈S(X) to be a projection PPP (bounded operator such that P2=PP^2 =PP2=P) in X\mathbb{X}X satisfying the positivity condition 0≤ω(P)≤1,0 \le ω(P) \le 1,0≤ω(P)≤1, and a physical quantity at a quantum state ωωω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω.ω.ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
Mathematical theory of infinite dimensional Hilbert spaces and the theory of operator algebras acting in such spaces (or C algebras in a more abstract approach) provide a standard setting for the formulation of modern quantum mechanics. On the other hand, experimental and theoretical progress achieved in the field of quantum information theory in the last two decades has indicated the practical and technological importance of low-dimensional quantum systems, where only a few basic modes play a significant role. Such modes can often be effectively decoupled from the rest of the system and controlled separately, providing physical realizations of qubits, qutrits and other basic information carriers. Regardless of concrete physical realization, be it photon polarization, electron or nuclear spin, charge in Josephson junctions to name just a few, the mathematical description of such systems requires only finite-dimensional Hilbert space language and finite-dimensional matrix algebras. Such structures are in principle computationally manageable in sharp contrast to the infinite dimensional ones.