Projective tensor product of protoquantum spaces

Aharon Razon, Lawrence Paul Horwitz

Sergey N. Filippov

Wilhelm Ochs

Let {WK} be a projective system of state operators defined on the finite tensor products of some family {ℋt} of Hilbert spaces. We prove that all projective limits of {WK} on the complete tensor product ⊗t∈T ℋt can be generated from every single projective limit by certain operations. In addition, we provide a necessary and sufficient condition for {WK} to have projective limits on incomplete tensor products of the ℋt’s.

Alexander Ya. Helemskii

Paolo Aniello, Lorenzo Guglielmi, Stefano Mancini +1 more

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.