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arXiv:2607.09627

A complete ultrametric on von Neumann's incomplete tensor products

Andrew Lesniewski

quant-phmath-phmath.OASubmitted Jul 10, 2026• Announced Jul 13, 2026• Revised Jul 17, 2026 (v3)
v1 · Jul 10, 2026v2 · Jul 15, 2026v3 · Jul 17, 2026
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Abstract

We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set $Γ$ of equivalence classes of $C_0$-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric $d$: the distance between two classes is the convergence exponent of the series $\sum_j|\langle\varphi_j,ψ_j\rangle-1|$ formed from any pair of representatives. We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so $d$ separates points only after passing to the quotient $\widetildeΓ$ of $Γ$ by the relation $d=0$; the pair $(\widetildeΓ,d)$ is then a complete ultrametric space. As an application, we show that a product unitary $\bigotimes_j U$ whose factor $U$ satisfies $\inf_{\|x\|=1}|\langle x,Ux\rangle-1|>0$ (in particular, a unitary on a finite dimensional space with $1\notinσ(U)$) displaces every class to the maximal distance $1$. Guided by the intended application -- a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds -- we also develop a gauge-invariant variant $\tilde d$ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by $\tilde d$, is class dependent and realizes every value in $[0,1]$. We interpret $\tilde d$ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.

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45 pages

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Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>'s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given.
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 $\tup{C}^\adj$-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).
Nowak [Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 22, 393–5 (1974)] has given an example of a consistent (in the sense of Kolmogorov) family of Gleason measures [A. M. Gleason, J. Math. Mech. 6, 885–94 (1957)] {mn} defined over ⊗ni=1Hi which do not extend to a Gleason measure on ⊙∞i=1 ΦHi for a given construction of the infinite tensor product. In this paper we show: (1) In the example of Nowak it is not necessary to assume, as is done, that the Hi are infinite dimensional. (2) That every consistent family developed from pure states, which is the type considered by Nowak, extends over the complete infinite tensor product of von Neumann [Compositio. Math. 6, 1–77 (1938)]. (3) Even if each Hi is two-dimensional and the complete infinite tensor product of von Neumann is used, it is possible to give a simple counterexample to the conjecture that every consistent family of Gleason measures extends by the use of nonpure states.
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.
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We consider bipartite exact entanglement embezzlement with a catalyst state vector ψψψ in a Hilbert space H\mathcal{H}H using unitaries (or more generally, contractions). If M⊆B(H)\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})M⊆B(H) is a von Neumann algebra and U∈Md⊗MU \in M_d \otimes \mathcal{M}U∈Md​⊗M and V∈M′⊗MdV \in \mathcal{M}' \otimes M_dV∈M′⊗Md​ are unitaries (or more generally contractions), then such a protocol is of the form (U⊗Id)(Id⊗V)(e0⊗ψ⊗e0)=∑i=0d−1αiei⊗ψ⊗ei(U \otimes I_d)(I_d \otimes V)(e_0 \otimes ψ\otimes e_0)=\sum_{i=0}^{d-1} α_i e_i \otimes ψ\otimes e_i(U⊗Id​)(Id​⊗V)(e0​⊗ψ⊗e0​)=∑i=0d−1​αi​ei​⊗ψ⊗ei​, where each αi>0α_i>0αi​>0 and ∑i=0d−1αi2=1\sum_{i=0}^{d-1} α_i^2=1∑i=0d−1​αi2​=1. We show that any such protocol must arise from a unique state on the tensor product Od⊗Od\mathcal{O}_d \otimes \mathcal{O}_dOd​⊗Od​ of the Cuntz algebra with itself. As a result, we prove that exact entanglement embezzlement is a self-test for a collection of ddd Cuntz isometries for each party and a unique quasi-free state on the Cuntz algebra Od\mathcal{O}_dOd​ in the sense of \cite{Iz93}. Moreover, we use modular theory to show that the von Neumann algebra generated by the copy of Od\mathcal{O}_dOd​ is the unique separable approximately finite-dimensional Type IIIλ\text{III}_λIIIλ​ factor for some 0<λ≤10<λ\leq 10<λ≤1, where λλλ can be determined by an algebraic condition on the Schmidt coefficients of the state φ=∑i=0d−1αiei⊗ei\varphi=\sum_{i=0}^{d-1} α_i e_i \otimes e_iφ=∑i=0d−1​αi​ei​⊗ei​.