The tensor product of p-adic Hilbert spaces
Paolo Aniello, Lorenzo Guglielmi, Stefano Mancini, Vincenzo Parisi
TL;DR
This work constructs a tensor product for $p$-adic Hilbert spaces by introducing a non-Archimedean projective-type norm on the algebraic tensor product, completing to a $p$-adic Banach space, and equipping it with a compatible inner product to form $\mathcal{H} \widehat{\otimes}_\pi \mathcal{K}$. It then proves a natural isometric isomorphism between this tensor product and the $p$-adic trace/Hilbert-Schmidt class $\mathcal{T}(\mathcal{H},\mathcal{K})$, using an anti-unitary construction, thereby establishing the correct $p$-adic analogue of the complex tensor product. The paper further analyzes the tensor product of subspaces, showing how $c_0(I)\widehat{\otimes}_\pi X$ corresponds to $c_0(I,X)$ and identifying regularity properties that parallel the complex setting while highlighting essential $p$-adic differences. Collectively, these results furnish the mathematical foundation for composite systems and potential entanglement phenomena in $p$-adic quantum information theory, and outline directions for studying duality, reflexivity, and broader implications.
Abstract
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.